Subgroups — C₇×#61465 #13238681

Order 1792.

2156 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3483	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3391	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×Q₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#3362	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#3358	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3354	C₂	1	3	yes	maximal, complemented
8	C₇×#3332	C₂	1	3	yes	maximal, complemented
9	C₇×#3325	C₂	1	3	yes	maximal, complemented
10	C₇×#3274	C₂	1	3	yes	maximal, complemented
11	C₇×#3184	C₂	1	3	yes	maximal, complemented
12	C₇×#3115	C₂	1	3	yes	maximal, complemented
13	C₇×#3109	C₂	1	3	yes	maximal, complemented
14	C₇×#2971	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₇×AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
16	C₂²×C₇×#41	—	2	1	no	—
17	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	3	yes	—
18	C₇×Q₈²	C₂²	1	1	yes	characteristic, complemented
19	C₇×#555	—	2	3	no	—
20	C₇×#555	C₂²	1	3	yes	complemented
21	C₇×#550	—	2	1	no	—
22	C₇×#550	C₂²	1	1	yes	characteristic
23	C₇×#550	C₂²	1	3	yes	complemented
24	C₇×#550	C₂²	1	1	yes	characteristic, complemented
25	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
26	C₇×#548	—	2	1	no	—
27	C₇×#547	C₂²	1	3	yes	—
28	C₇×#546	—	2	3	no	—
29	C₇×D₄²	C₂²	1	3	yes	—
30	C₇×#532	—	2	3	no	—
31	C₇×#532	C₂²	1	3	yes	—
32	C₇×#532	C₂²	1	3	yes	complemented
33	C₇×#525	—	2	3	no	—
34	C₄×C₇×#41	—	2	1	no	—
35	C₄×C₇×#41	C₂²	1	1	yes	characteristic
36	C₄×C₇×#41	C₂²	1	3	yes	complemented
37	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
38	C₇×#498	C₂²	1	3	yes	complemented
39	C₇×#493	C₂²	1	3	yes	complemented
40	C₇×#486	C₂²	1	3	yes	complemented
41	C₇×#486	C₂²	1	3	yes	complemented
42	C₇×#482	C₂²	1	6	yes	complemented
43	C₇×#481	C₂²	1	6	yes	complemented
44	C₇×#476	C₂²	1	3	yes	—
45	C₇×#476	C₂²	1	3	yes	complemented
46	C₇×#475	C₂²	1	3	yes	complemented
47	C₇×#474	C₂²	1	3	yes	complemented
48	C₇×#469	C₂²	1	6	yes	complemented
49	C₇×#469	C₂²	1	6	yes	complemented
50	C₇×#460	C₂²	1	3	yes	complemented
51	C₇×#459	C₂²	1	3	yes	—
52	C₇×#459	C₂²	1	3	yes	complemented
53	C₇×#459	C₂²	1	3	yes	complemented
54	C₇×#449	C₂²	1	6	yes	complemented
55	C₇×#447	C₂²	1	6	yes	—
56	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
57	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
58	C₇×#442	C₂²	1	3	yes	—
59	C₇×#442	C₂²	1	3	yes	complemented
60	C₇×#442	C₂²	1	6	yes	complemented
61	C₇×#440	C₂²	1	1	yes	characteristic
62	C₇×#440	C₂²	1	3	yes	complemented
63	C₄×C₇×SD₁₆	C₂²	1	3	yes	—
64	C₄×C₇×SD₁₆	C₂²	1	3	yes	complemented
65	C₄×C₇×D₈	C₂²	1	3	yes	—
66	C₄×C₇×D₈	C₂²	1	3	yes	complemented
67	C₇×#436	C₂²	1	3	yes	complemented
68	C₇×#436	C₂²	1	3	yes	complemented
69	C₇×#432	C₂²	1	3	yes	complemented
70	C₇×#428	C₂²	1	3	yes	complemented
71	C₇×#418	C₂²	1	3	yes	—
72	C₇×#417	C₂²	1	3	yes	complemented
73	C₄×C₇×M₁₆	C₂²	1	3	yes	complemented
74	#61465	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂×C₇×#41	—	2	1	no	—
76	C₂×C₇×#41	—	2	1	no	—
77	C₂×C₇×#41	—	2	3	no	—
78	C₂×C₇×#41	—	2	3	no	—
79	C₂×C₇×#41	—	2	3	no	—
80	C₂×C₇×#41	—	4	8	no	—
81	C₂×C₇×#41	D₄	1	1	yes	characteristic
82	C₂×C₇×#41	C₂³	1	3	yes	—
83	C₂²×C₇×Q₈	—	2	1	no	—
84	C₂²×C₇×D₄	—	2	3	no	—
85	C₂²×C₇×D₄	C₂³	1	3	yes	—
86	C₂³×C₄×C₇	—	2	3	no	—
87	C₇×AGL(1,ℤ/8)	—	2	12	no	—
88	C₂×C₇×SD₁₆	C₂³	1	3	yes	—
89	C₂×C₇×SD₁₆	C₂³	1	1	yes	characteristic
90	C₂×C₇×D₈	C₂³	1	3	yes	—
91	C₂×C₇×D₈	C₂³	1	3	yes	—
92	C₂×C₇×D₈	C₂³	1	6	yes	—
93	C₂×C₇×M₁₆	C₂³	1	3	yes	—
94	C₂×C₇×M₁₆	C₂³	1	3	yes	—
95	C₇×C₄⋊Q₈	—	2	3	no	—
96	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
97	C₇×C₄⋊Q₈	C₂³	1	3	yes	—
98	C₇×#127	—	2	3	no	—
99	C₇×#127	C₂³	1	3	yes	—
100	C₇×#127	C₂³	1	3	yes	—
101	C₇×#124	—	2	6	no	—
102	C₇×#124	—	2	3	no	—
103	C₇×#124	—	2	3	no	—
104	C₇×#123	—	2	6	no	—
105	C₇×C₂²⋊Q₈	—	2	3	no	—
106	C₇×C₂²⋊Q₈	—	2	3	no	—
107	C₇×#121	—	2	3	no	—
108	C₇×#121	—	2	3	no	—
109	C₇×#121	—	2	3	no	—
110	C₇×#121	—	2	3	no	—
111	C₇×#121	—	2	3	no	—
112	C₇×#121	—	2	3	no	—
113	C₇×#121	—	2	6	no	—
114	C₇×#121	C₂³	1	6	yes	—
115	C₇×#121	C₂³	1	6	yes	—
116	C₇×#120	—	2	6	no	—
117	C₄×C₇×Q₈	—	2	1	no	—
118	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
119	C₄×C₇×Q₈	C₂³	1	3	yes	—
120	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
121	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
122	C₄×C₇×D₄	—	2	3	no	—
123	C₄×C₇×D₄	—	2	3	no	—
124	C₄×C₇×D₄	—	2	3	no	—
125	C₄×C₇×D₄	—	2	3	no	—
126	C₄×C₇×D₄	—	2	3	no	—
127	C₄×C₇×D₄	—	2	3	no	—
128	C₄×C₇×D₄	—	2	3	no	—
129	C₄×C₇×D₄	D₄	1	3	yes	—
130	C₄×C₇×D₄	C₂³	1	3	yes	—
131	C₄×C₇×D₄	C₂³	1	3	yes	—
132	C₄×C₇×D₄	C₂³	1	3	yes	—
133	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
134	C₄×C₇×D₄	C₂³	1	3	yes	—
135	C₄×C₇×D₄	C₂³	1	3	yes	—
136	C₄×C₇×D₄	C₂³	1	3	yes	—
137	C₇×#117	—	2	3	no	—
138	C₇×#117	—	2	3	no	—
139	C₇×#117	D₄	1	3	yes	—
140	C₇×#117	C₂³	1	3	yes	—
141	C₂×C₇×C₄⋊C₄	C₂³	1	3	yes	—
142	C₂×C₇×C₂²⋊C₄	—	2	3	no	—
143	C₂×C₄²×C₇	—	2	3	no	—
144	C₂×C₄²×C₇	C₂³	1	3	yes	—
145	C₇×#107	C₂³	1	3	yes	—
146	C₇×#107	C₂³	1	1	yes	characteristic
147	C₇×#106	C₂³	1	3	yes	—
148	C₇×#106	C₂³	1	6	yes	—
149	C₇×#106	C₂³	1	3	yes	—
150	C₇×C₄⋊C₈	C₂³	1	3	yes	—
151	C₇×C₄⋊C₈	C₂³	1	3	yes	—
152	C₇×#103	C₂³	1	3	yes	—
153	C₇×#103	C₂³	1	1	yes	characteristic
154	C₇×#102	C₂³	1	3	yes	—
155	C₇×#102	C₂³	1	3	yes	—
156	C₇×#102	C₂³	1	6	yes	—
157	C₇×#102	C₂³	1	6	yes	—
158	C₇×#102	C₂³	1	3	yes	—
159	C₇×#102	C₂³	1	3	yes	—
160	C₇×#102	C₂³	1	3	yes	—
161	C₇×#102	C₂³	1	1	yes	characteristic
162	C₇×C₂²⋊C₈	C₂³	1	6	yes	—
163	C₇×#97	C₂³	1	1	yes	characteristic
164	C₇×#97	C₂³	1	3	yes	—
165	C₄×C₇×C₈	C₂³	1	3	yes	—
166	C₄×C₇×C₈	C₂³	1	3	yes	—
167	#3483	C₂×C₇	1	1	yes	characteristic, complemented
168	D₄×#41	C₂×C₇	1	1	yes	characteristic, complemented
169	#3391	C₂×C₇	1	1	yes	characteristic, complemented
170	Q₈×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
171	#3362	C₂×C₇	1	1	yes	characteristic, complemented
172	#3358	C₂×C₇	1	1	yes	characteristic, complemented
173	#3354	C₂×C₇	1	3	yes	complemented
174	#3332	C₂×C₇	1	3	yes	complemented
175	#3325	C₂×C₇	1	3	yes	complemented
176	#3274	C₂×C₇	1	3	yes	complemented
177	#3184	C₂×C₇	1	3	yes	complemented
178	#3115	C₂×C₇	1	3	yes	complemented
179	#3109	C₂×C₇	1	3	yes	complemented
180	#2971	C₂×C₇	1	1	yes	characteristic, complemented
181	C₄×AGL(1,ℤ/8)	C₂×C₇	1	3	yes	complemented
182	C₂⁴×C₇	—	2	3	no	—
183	C₇×#41	—	2	4	no	—
184	C₇×#41	—	2	6	no	—
185	C₇×#41	—	4	3	no	—
186	C₇×#41	—	4	12	no	—
187	C₇×#41	—	4	12	no	—
188	C₇×#41	—	4	4	no	—
189	C₇×#41	—	4	4	no	—
190	C₇×#41	—	8	6	no	—
191	C₂×C₇×Q₈	—	2	3	no	—
192	C₂×C₇×Q₈	—	2	3	no	—
193	C₂×C₇×Q₈	—	4	4	no	—
194	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₇×Q₈	C₂⁴	1	1	yes	characteristic
196	C₂×C₇×D₄	—	2	6	no	—
197	C₂×C₇×D₄	—	2	3	no	—
198	C₂×C₇×D₄	—	2	6	no	—
199	C₂×C₇×D₄	—	2	6	no	—
200	C₂×C₇×D₄	—	2	3	no	—
201	C₂×C₇×D₄	—	2	3	no	—
202	C₂×C₇×D₄	—	2	3	no	—
203	C₂×C₇×D₄	—	2	3	no	—
204	C₂×C₇×D₄	—	2	3	no	—
205	C₂×C₇×D₄	—	4	3	no	—
206	C₂×C₇×D₄	—	4	12	no	—
207	C₂×C₇×D₄	C₂×D₄	1	3	yes	—
208	C₂×C₇×D₄	C₂⁴	1	3	yes	—
209	C₂×C₇×D₄	C₂⁴	1	3	yes	—
210	C₂×C₇×D₄	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₇	—	2	3	no	—
212	C₂²×C₄×C₇	—	2	3	no	—
213	C₂²×C₄×C₇	—	2	3	no	—
214	C₂²×C₄×C₇	—	2	3	no	—
215	C₂²×C₄×C₇	—	2	1	no	—
216	C₂²×C₄×C₇	—	2	3	no	—
217	C₂²×C₄×C₇	—	2	3	no	—
218	C₂²×C₄×C₇	—	2	3	no	—
219	C₂²×C₄×C₇	—	4	12	no	—
220	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
221	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
222	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
223	C₇×SD₁₆	—	2	6	no	—
224	C₇×SD₁₆	#41	1	4	yes	complemented
225	C₇×D₈	—	2	6	no	—
226	C₇×D₈	—	2	6	no	—
227	C₇×D₈	—	4	6	no	—
228	C₇×M₁₆	—	2	6	no	—
229	C₇×M₁₆	—	4	3	no	—
230	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
231	C₂×C₇×C₈	C₂⁴	1	3	yes	—
232	C₂×C₇×C₈	C₂⁴	1	3	yes	—
233	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
234	C₇×C₄⋊C₄	—	2	3	no	—
235	C₇×C₄⋊C₄	—	2	1	no	—
236	C₇×C₄⋊C₄	—	2	3	no	—
237	C₇×C₄⋊C₄	—	2	3	no	—
238	C₇×C₄⋊C₄	—	2	3	no	—
239	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
240	C₇×C₄⋊C₄	C₂×D₄	1	3	yes	—
241	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
242	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
243	C₇×C₄⋊C₄	C₂⁴	1	1	yes	characteristic
244	C₇×C₄⋊C₄	C₂⁴	1	3	yes	—
245	C₇×C₂²⋊C₄	—	2	3	no	—
246	C₇×C₂²⋊C₄	—	2	1	no	—
247	C₇×C₂²⋊C₄	—	2	3	no	—
248	C₇×C₂²⋊C₄	—	2	3	no	—
249	C₇×C₂²⋊C₄	—	2	3	no	—
250	C₇×C₂²⋊C₄	—	2	6	no	—
251	C₇×C₂²⋊C₄	—	2	3	no	—
252	C₇×C₂²⋊C₄	—	2	3	no	—
253	C₇×C₂²⋊C₄	—	2	6	no	—
254	C₇×C₂²⋊C₄	C₂×D₄	1	6	yes	—
255	C₄²×C₇	—	2	1	no	—
256	C₄²×C₇	—	2	3	no	—
257	C₄²×C₇	—	2	3	no	—
258	C₄²×C₇	C₂×D₄	1	3	yes	—
259	C₄²×C₇	C₂×D₄	1	3	yes	—
260	C₄²×C₇	C₂⁴	1	3	yes	—
261	C₄²×C₇	C₂⁴	1	1	yes	characteristic
262	C₂²×#41	—	2	1	no	—
263	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	3	yes	—
264	Q₈²	C₂²×C₇	1	1	yes	characteristic, complemented
265	#555	—	2	3	no	—
266	#555	C₂²×C₇	1	3	yes	complemented
267	#550	—	2	1	no	—
268	#550	C₂²×C₇	1	1	yes	characteristic
269	#550	C₂²×C₇	1	3	yes	complemented
270	#550	C₂²×C₇	1	1	yes	characteristic, complemented
271	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
272	#548	—	2	1	no	—
273	#547	C₂²×C₇	1	3	yes	—
274	#546	—	2	3	no	—
275	D₄²	C₂²×C₇	1	3	yes	—
276	#532	—	2	3	no	—
277	#532	C₂²×C₇	1	3	yes	—
278	#532	C₂²×C₇	1	3	yes	complemented
279	#525	—	2	3	no	—
280	C₄×#41	—	2	1	no	—
281	C₄×#41	C₂²×C₇	1	1	yes	characteristic
282	C₄×#41	C₂²×C₇	1	3	yes	complemented
283	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
284	#498	C₂²×C₇	1	3	yes	complemented
285	#493	C₂²×C₇	1	3	yes	complemented
286	#486	C₂²×C₇	1	3	yes	complemented
287	#486	C₂²×C₇	1	3	yes	complemented
288	#482	C₂²×C₇	1	6	yes	complemented
289	#481	C₂²×C₇	1	6	yes	complemented
290	#476	C₂²×C₇	1	3	yes	—
291	#476	C₂²×C₇	1	3	yes	complemented
292	#475	C₂²×C₇	1	3	yes	complemented
293	#474	C₂²×C₇	1	3	yes	complemented
294	#469	C₂²×C₇	1	6	yes	complemented
295	#469	C₂²×C₇	1	6	yes	complemented
296	#460	C₂²×C₇	1	3	yes	complemented
297	#459	C₂²×C₇	1	3	yes	—
298	#459	C₂²×C₇	1	3	yes	complemented
299	#459	C₂²×C₇	1	3	yes	complemented
300	#449	C₂²×C₇	1	6	yes	complemented
301	#447	C₂²×C₇	1	6	yes	—
302	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
303	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
304	#442	C₂²×C₇	1	3	yes	—
305	#442	C₂²×C₇	1	6	yes	complemented
306	#442	C₂²×C₇	1	3	yes	complemented
307	#440	C₂²×C₇	1	1	yes	characteristic
308	#440	C₂²×C₇	1	3	yes	complemented
309	C₄×SD₁₆	C₂²×C₇	1	3	yes	—
310	C₄×SD₁₆	C₂²×C₇	1	3	yes	complemented
311	C₄×D₈	C₂²×C₇	1	3	yes	—
312	C₄×D₈	C₂²×C₇	1	3	yes	complemented
313	#436	C₂²×C₇	1	3	yes	complemented
314	#436	C₂²×C₇	1	3	yes	complemented
315	#432	C₂²×C₇	1	3	yes	complemented
316	#428	C₂²×C₇	1	3	yes	complemented
317	#418	C₂²×C₇	1	3	yes	—
318	#417	C₂²×C₇	1	3	yes	complemented
319	C₄×M₁₆	C₂²×C₇	1	3	yes	complemented
320	C₂³×C₇	—	2	1	no	—
321	C₂³×C₇	—	2	3	no	—
322	C₂³×C₇	—	2	3	no	—
323	C₂³×C₇	—	4	6	no	—
324	C₂³×C₇	—	4	6	no	—
325	C₂³×C₇	—	8	3	no	—
326	C₂³×C₇	C₂²×D₄	1	3	yes	—
327	C₇×Q₈	—	2	1	no	—
328	C₇×Q₈	—	4	6	no	—
329	C₇×Q₈	—	8	3	no	—
330	C₇×Q₈	AGL(1,ℤ/8)	1	4	yes	complemented
331	C₇×Q₈	C₂×#41	1	2	yes	complemented
332	C₇×D₄	—	2	6	no	—
333	C₇×D₄	—	2	3	no	—
334	C₇×D₄	—	2	3	no	—
335	C₇×D₄	—	2	1	no	—
336	C₇×D₄	—	2	3	no	—
337	C₇×D₄	—	2	3	no	—
338	C₇×D₄	—	4	6	no	—
339	C₇×D₄	—	4	6	no	—
340	C₇×D₄	—	4	12	no	—
341	C₇×D₄	—	4	6	no	—
342	C₇×D₄	—	4	6	no	—
343	C₇×D₄	—	8	3	no	—
344	C₇×D₄	—	8	3	no	—
345	C₇×D₄	—	8	6	no	—
346	C₇×D₄	C₂×#41	1	2	yes	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	2	6	no	—
349	C₂×C₄×C₇	—	2	3	no	—
350	C₂×C₄×C₇	—	2	3	no	—
351	C₂×C₄×C₇	—	2	1	no	—
352	C₂×C₄×C₇	—	2	3	no	—
353	C₂×C₄×C₇	—	2	1	no	—
354	C₂×C₄×C₇	—	2	3	no	—
355	C₂×C₄×C₇	—	4	3	no	—
356	C₂×C₄×C₇	—	4	6	no	—
357	C₂×C₄×C₇	—	4	2	no	—
358	C₂×C₄×C₇	—	4	6	no	—
359	C₂×C₄×C₇	—	4	6	no	—
360	C₂×C₄×C₇	—	4	6	no	—
361	C₂×C₄×C₇	—	4	6	no	—
362	C₂×C₄×C₇	—	4	6	no	—
363	C₂×C₄×C₇	—	4	6	no	—
364	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
365	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
366	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
367	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
368	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
369	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
370	C₇×C₈	—	2	3	no	—
371	C₇×C₈	—	2	1	no	—
372	C₇×C₈	—	2	3	no	—
373	C₇×C₈	C₂×#41	1	2	yes	complemented
374	C₂×#41	—	2	1	no	—
375	C₂×#41	—	2	1	no	—
376	C₂×#41	—	2	3	no	—
377	C₂×#41	—	2	3	no	—
378	C₂×#41	—	2	3	no	—
379	C₂×#41	—	4	8	no	—
380	C₂×#41	C₇×D₄	1	1	yes	characteristic
381	C₂×#41	C₂³×C₇	1	3	yes	—
382	C₂²×Q₈	—	2	1	no	—
383	C₂²×D₄	—	2	3	no	—
384	C₂²×D₄	C₂³×C₇	1	3	yes	—
385	C₂³×C₄	—	2	3	no	—
386	AGL(1,ℤ/8)	—	2	12	no	—
387	C₂×SD₁₆	C₂³×C₇	1	3	yes	—
388	C₂×SD₁₆	C₂³×C₇	1	1	yes	characteristic
389	C₂×D₈	C₂³×C₇	1	3	yes	—
390	C₂×D₈	C₂³×C₇	1	3	yes	—
391	C₂×D₈	C₂³×C₇	1	6	yes	—
392	C₂×M₁₆	C₂³×C₇	1	3	yes	—
393	C₂×M₁₆	C₂³×C₇	1	3	yes	—
394	C₄⋊Q₈	—	2	3	no	—
395	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
396	C₄⋊Q₈	C₂³×C₇	1	3	yes	—
397	#127	—	2	3	no	—
398	#127	C₂³×C₇	1	3	yes	—
399	#127	C₂³×C₇	1	3	yes	—
400	#124	—	2	6	no	—
401	#124	—	2	3	no	—
402	#124	—	2	3	no	—
403	#123	—	2	6	no	—
404	C₂²⋊Q₈	—	2	3	no	—
405	C₂²⋊Q₈	—	2	3	no	—
406	#121	—	2	3	no	—
407	#121	—	2	3	no	—
408	#121	—	2	3	no	—
409	#121	—	2	3	no	—
410	#121	—	2	3	no	—
411	#121	—	2	3	no	—
412	#121	—	2	6	no	—
413	#121	C₂³×C₇	1	6	yes	—
414	#121	C₂³×C₇	1	6	yes	—
415	#120	—	2	6	no	—
416	C₄×Q₈	—	2	1	no	—
417	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
418	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
419	C₄×Q₈	C₂³×C₇	1	3	yes	—
420	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
421	C₄×D₄	—	2	3	no	—
422	C₄×D₄	—	2	3	no	—
423	C₄×D₄	—	2	3	no	—
424	C₄×D₄	—	2	3	no	—
425	C₄×D₄	—	2	3	no	—
426	C₄×D₄	—	2	3	no	—
427	C₄×D₄	—	2	3	no	—
428	C₄×D₄	C₇×D₄	1	3	yes	—
429	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
430	C₄×D₄	C₂³×C₇	1	3	yes	—
431	C₄×D₄	C₂³×C₇	1	3	yes	—
432	C₄×D₄	C₂³×C₇	1	3	yes	—
433	C₄×D₄	C₂³×C₇	1	3	yes	—
434	C₄×D₄	C₂³×C₇	1	3	yes	—
435	C₄×D₄	C₂³×C₇	1	3	yes	—
436	#117	—	2	3	no	—
437	#117	—	2	3	no	—
438	#117	C₇×D₄	1	3	yes	—
439	#117	C₂³×C₇	1	3	yes	—
440	C₂×C₄⋊C₄	C₂³×C₇	1	3	yes	—
441	C₂×C₂²⋊C₄	—	2	3	no	—
442	C₂×C₄²	—	2	3	no	—
443	C₂×C₄²	C₂³×C₇	1	3	yes	—
444	#107	C₂³×C₇	1	1	yes	characteristic
445	#107	C₂³×C₇	1	3	yes	—
446	#106	C₂³×C₇	1	3	yes	—
447	#106	C₂³×C₇	1	3	yes	—
448	#106	C₂³×C₇	1	6	yes	—
449	C₄⋊C₈	C₂³×C₇	1	3	yes	—
450	C₄⋊C₈	C₂³×C₇	1	3	yes	—
451	#103	C₂³×C₇	1	1	yes	characteristic
452	#103	C₂³×C₇	1	3	yes	—
453	#102	C₂³×C₇	1	3	yes	—
454	#102	C₂³×C₇	1	3	yes	—
455	#102	C₂³×C₇	1	1	yes	characteristic
456	#102	C₂³×C₇	1	3	yes	—
457	#102	C₂³×C₇	1	3	yes	—
458	#102	C₂³×C₇	1	3	yes	—
459	#102	C₂³×C₇	1	6	yes	—
460	#102	C₂³×C₇	1	6	yes	—
461	C₂²⋊C₈	C₂³×C₇	1	6	yes	—
462	#97	C₂³×C₇	1	1	yes	characteristic
463	#97	C₂³×C₇	1	3	yes	—
464	C₄×C₈	C₂³×C₇	1	3	yes	—
465	C₄×C₈	C₂³×C₇	1	3	yes	—
466	C₂²×C₇	—	2	3	no	—
467	C₂²×C₇	—	2	3	no	—
468	C₂²×C₇	—	2	2	no	—
469	C₂²×C₇	—	2	3	no	—
470	C₂²×C₇	—	4	1	no	—
471	C₂²×C₇	—	4	1	no	—
472	C₂²×C₇	—	4	3	no	—
473	C₂²×C₇	—	4	3	no	—
474	C₂²×C₇	—	4	3	no	—
475	C₂²×C₇	—	4	3	no	—
476	C₂²×C₇	—	4	3	no	—
477	C₂²×C₇	—	4	3	no	—
478	C₂²×C₇	—	8	12	no	—
479	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
480	C₄×C₇	—	2	1	no	—
481	C₄×C₇	—	2	3	no	—
482	C₄×C₇	—	2	3	no	—
483	C₄×C₇	—	2	2	no	—
484	C₄×C₇	—	2	1	no	—
485	C₄×C₇	—	4	3	no	—
486	C₄×C₇	—	4	1	no	—
487	C₄×C₇	—	4	1	no	—
488	C₄×C₇	—	4	3	no	—
489	C₄×C₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
490	C₄×C₇	C₂²×#41	1	1	yes	characteristic
491	C₄×C₇	#585	1	1	yes	characteristic
492	C₂⁴	—	2	3	no	—
493	#41	—	2	4	no	—
494	#41	—	2	6	no	—
495	#41	—	4	3	no	—
496	#41	—	4	12	no	—
497	#41	—	4	12	no	—
498	#41	—	4	4	no	—
499	#41	—	4	4	no	—
500	#41	—	8	6	no	—
501	C₂×Q₈	—	2	3	no	—
502	C₂×Q₈	—	2	3	no	—
503	C₂×Q₈	—	4	4	no	—
504	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
505	C₂×Q₈	C₂⁴×C₇	1	1	yes	characteristic
506	C₂×D₄	—	2	6	no	—
507	C₂×D₄	—	2	3	no	—
508	C₂×D₄	—	2	6	no	—
509	C₂×D₄	—	2	6	no	—
510	C₂×D₄	—	2	3	no	—
511	C₂×D₄	—	2	3	no	—
512	C₂×D₄	—	2	3	no	—
513	C₂×D₄	—	2	3	no	—
514	C₂×D₄	—	2	3	no	—
515	C₂×D₄	—	4	3	no	—
516	C₂×D₄	—	4	12	no	—
517	C₂×D₄	C₂×C₇×D₄	1	3	yes	—
518	C₂×D₄	C₂⁴×C₇	1	3	yes	—
519	C₂×D₄	C₂⁴×C₇	1	3	yes	—
520	C₂×D₄	C₂⁴×C₇	1	1	yes	characteristic
521	C₂²×C₄	—	2	3	no	—
522	C₂²×C₄	—	2	3	no	—
523	C₂²×C₄	—	2	3	no	—
524	C₂²×C₄	—	2	3	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	3	no	—
527	C₂²×C₄	—	2	3	no	—
528	C₂²×C₄	—	2	3	no	—
529	C₂²×C₄	—	4	12	no	—
530	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
531	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
532	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
533	SD₁₆	—	2	6	no	—
534	SD₁₆	C₇×#41	1	4	yes	complemented
535	D₈	—	2	6	no	—
536	D₈	—	2	6	no	—
537	D₈	—	4	6	no	—
538	M₁₆	—	2	6	no	—
539	M₁₆	—	4	3	no	—
540	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
541	C₂×C₈	C₂⁴×C₇	1	3	yes	—
542	C₂×C₈	C₂⁴×C₇	1	3	yes	—
543	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
544	C₄⋊C₄	—	2	3	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	3	no	—
547	C₄⋊C₄	—	2	3	no	—
548	C₄⋊C₄	—	2	3	no	—
549	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
550	C₄⋊C₄	C₂×C₇×D₄	1	3	yes	—
551	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
552	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
553	C₄⋊C₄	C₂⁴×C₇	1	1	yes	characteristic
554	C₄⋊C₄	C₂⁴×C₇	1	3	yes	—
555	C₂²⋊C₄	—	2	3	no	—
556	C₂²⋊C₄	—	2	1	no	—
557	C₂²⋊C₄	—	2	3	no	—
558	C₂²⋊C₄	—	2	3	no	—
559	C₂²⋊C₄	—	2	3	no	—
560	C₂²⋊C₄	—	2	6	no	—
561	C₂²⋊C₄	—	2	3	no	—
562	C₂²⋊C₄	—	2	3	no	—
563	C₂²⋊C₄	—	2	6	no	—
564	C₂²⋊C₄	C₂×C₇×D₄	1	6	yes	—
565	C₄²	—	2	1	no	—
566	C₄²	—	2	3	no	—
567	C₄²	—	2	3	no	—
568	C₄²	C₂×C₇×D₄	1	3	yes	—
569	C₄²	C₂×C₇×D₄	1	3	yes	—
570	C₄²	C₂⁴×C₇	1	3	yes	—
571	C₄²	C₂⁴×C₇	1	1	yes	characteristic
572	C₂×C₇	—	4	3	no	—
573	C₂×C₇	—	4	2	no	—
574	C₂×C₇	—	8	3	no	—
575	C₂×C₇	—	8	3	no	—
576	C₂×C₇	D₄×#41	1	1	yes	characteristic
577	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
578	C₂×C₇	#3585	1	1	yes	characteristic
579	C₂³	—	2	1	no	—
580	C₂³	—	2	3	no	—
581	C₂³	—	2	3	no	—
582	C₂³	—	4	6	no	—
583	C₂³	—	4	6	no	—
584	C₂³	—	8	3	no	—
585	C₂³	C₂²×C₇×D₄	1	3	yes	—
586	Q₈	—	2	1	no	—
587	Q₈	—	4	6	no	—
588	Q₈	—	8	3	no	—
589	Q₈	C₇×AGL(1,ℤ/8)	1	4	yes	complemented
590	Q₈	C₂×C₇×#41	1	2	yes	complemented
591	D₄	—	2	3	no	—
592	D₄	—	2	3	no	—
593	D₄	—	2	1	no	—
594	D₄	—	2	3	no	—
595	D₄	—	2	3	no	—
596	D₄	—	2	6	no	—
597	D₄	—	4	6	no	—
598	D₄	—	4	6	no	—
599	D₄	—	4	12	no	—
600	D₄	—	4	6	no	—
601	D₄	—	4	6	no	—
602	D₄	—	8	3	no	—
603	D₄	—	8	3	no	—
604	D₄	—	8	6	no	—
605	D₄	C₂×C₇×#41	1	2	yes	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	3	no	—
608	C₂×C₄	—	2	3	no	—
609	C₂×C₄	—	2	6	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	3	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	3	no	—
614	C₂×C₄	—	4	3	no	—
615	C₂×C₄	—	4	6	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	6	no	—
618	C₂×C₄	—	4	6	no	—
619	C₂×C₄	—	4	6	no	—
620	C₂×C₄	—	4	6	no	—
621	C₂×C₄	—	4	6	no	—
622	C₂×C₄	—	4	6	no	—
623	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
624	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
625	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
626	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
627	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
628	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
629	C₈	—	2	3	no	—
630	C₈	—	2	1	no	—
631	C₈	—	2	3	no	—
632	C₈	C₂×C₇×#41	1	2	yes	complemented
633	C₇	#61465	1	1	yes	characteristic, direct product factor, Sylow
634	C₂²	—	2	3	no	—
635	C₂²	—	2	3	no	—
636	C₂²	—	2	3	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	3	no	—
641	C₂²	—	4	3	no	—
642	C₂²	—	4	3	no	—
643	C₂²	—	4	3	no	—
644	C₂²	—	4	3	no	—
645	C₂²	—	4	3	no	—
646	C₂²	—	8	12	no	—
647	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
648	C₄	—	2	1	no	—
649	C₄	—	2	1	no	—
650	C₄	—	2	3	no	—
651	C₄	—	2	3	no	—
652	C₄	—	2	2	no	—
653	C₄	—	4	1	no	—
654	C₄	—	4	3	no	—
655	C₄	—	4	1	no	—
656	C₄	—	4	3	no	—
657	C₄	C₂×C₇×AGL(1,ℤ/8)	1	6	yes	—
658	C₄	C₂²×C₇×#41	1	1	yes	characteristic
659	C₄	C₇×#585	1	1	yes	characteristic
660	C₂	—	4	3	no	—
661	C₂	—	4	2	no	—
662	C₂	—	8	3	no	—
663	C₂	—	8	3	no	—
664	C₂	C₇×D₄×#41	1	1	yes	characteristic
665	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
666	C₂	C₇×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3483	C₂	semidirect
2. C₇×D₄×#41	C₂	semidirect
3. C₇×#3391	C₂	semidirect
4. C₇×Q₈×SD₁₆	C₂	semidirect
5. C₇×#3362	C₂	semidirect
6. C₇×#3358	C₂	semidirect
7. C₇×#3354	C₂	semidirect
8. C₇×#3332	C₂	semidirect
9. C₇×#3325	C₂	semidirect
10. C₇×#3274	C₂	semidirect
11. C₇×#3184	C₂	semidirect
12. C₇×#3115	C₂	semidirect
13. C₇×#3109	C₂	semidirect
14. C₇×#2971	C₂	semidirect
15. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
18. C₇×Q₈²	C₂²	semidirect
20. C₇×#555	C₂²	semidirect
23. C₇×#550	C₂²	semidirect
24. C₇×#550	C₂²	semidirect
32. C₇×#532	C₂²	semidirect
36. C₄×C₇×#41	C₂²	semidirect
38. C₇×#498	C₂²	semidirect
39. C₇×#493	C₂²	semidirect
40. C₇×#486	C₂²	semidirect
41. C₇×#486	C₂²	semidirect
42. C₇×#482	C₂²	semidirect
43. C₇×#481	C₂²	semidirect
45. C₇×#476	C₂²	semidirect
46. C₇×#475	C₂²	semidirect
47. C₇×#474	C₂²	semidirect
48. C₇×#469	C₂²	semidirect
49. C₇×#469	C₂²	semidirect
50. C₇×#460	C₂²	semidirect
52. C₇×#459	C₂²	semidirect
53. C₇×#459	C₂²	semidirect
54. C₇×#449	C₂²	semidirect
56. C₇×C₈×Q₈	C₂²	semidirect
57. C₇×C₈×Q₈	C₂²	semidirect
59. C₇×#442	C₂²	semidirect
60. C₇×#442	C₂²	semidirect
62. C₇×#440	C₂²	semidirect
64. C₄×C₇×SD₁₆	C₂²	semidirect
66. C₄×C₇×D₈	C₂²	semidirect
67. C₇×#436	C₂²	semidirect
68. C₇×#436	C₂²	semidirect
69. C₇×#432	C₂²	semidirect
70. C₇×#428	C₂²	semidirect
72. C₇×#417	C₂²	semidirect
73. C₄×C₇×M₁₆	C₂²	semidirect
74. #61465	C₇	direct (N × H)
167. #3483	C₂×C₇	semidirect
168. D₄×#41	C₂×C₇	semidirect
169. #3391	C₂×C₇	semidirect
170. Q₈×SD₁₆	C₂×C₇	semidirect
171. #3362	C₂×C₇	semidirect
172. #3358	C₂×C₇	semidirect
173. #3354	C₂×C₇	semidirect
174. #3332	C₂×C₇	semidirect
175. #3325	C₂×C₇	semidirect
176. #3274	C₂×C₇	semidirect
177. #3184	C₂×C₇	semidirect
178. #3115	C₂×C₇	semidirect
179. #3109	C₂×C₇	semidirect
180. #2971	C₂×C₇	semidirect
181. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
224. C₇×SD₁₆	#41	semidirect
264. Q₈²	C₂²×C₇	semidirect
266. #555	C₂²×C₇	semidirect
269. #550	C₂²×C₇	semidirect
270. #550	C₂²×C₇	semidirect
278. #532	C₂²×C₇	semidirect
282. C₄×#41	C₂²×C₇	semidirect
284. #498	C₂²×C₇	semidirect
285. #493	C₂²×C₇	semidirect
286. #486	C₂²×C₇	semidirect
287. #486	C₂²×C₇	semidirect
288. #482	C₂²×C₇	semidirect
289. #481	C₂²×C₇	semidirect
291. #476	C₂²×C₇	semidirect
292. #475	C₂²×C₇	semidirect
293. #474	C₂²×C₇	semidirect
294. #469	C₂²×C₇	semidirect
295. #469	C₂²×C₇	semidirect
296. #460	C₂²×C₇	semidirect
298. #459	C₂²×C₇	semidirect
299. #459	C₂²×C₇	semidirect
300. #449	C₂²×C₇	semidirect
302. C₈×Q₈	C₂²×C₇	semidirect
303. C₈×Q₈	C₂²×C₇	semidirect
305. #442	C₂²×C₇	semidirect
306. #442	C₂²×C₇	semidirect
308. #440	C₂²×C₇	semidirect
310. C₄×SD₁₆	C₂²×C₇	semidirect
312. C₄×D₈	C₂²×C₇	semidirect
313. #436	C₂²×C₇	semidirect
314. #436	C₂²×C₇	semidirect
315. #432	C₂²×C₇	semidirect
316. #428	C₂²×C₇	semidirect
318. #417	C₂²×C₇	semidirect
319. C₄×M₁₆	C₂²×C₇	semidirect
330. C₇×Q₈	AGL(1,ℤ/8)	semidirect
331. C₇×Q₈	C₂×#41	semidirect
373. C₇×C₈	C₂×#41	semidirect
534. SD₁₆	C₇×#41	semidirect
589. Q₈	C₇×AGL(1,ℤ/8)	semidirect
590. Q₈	C₂×C₇×#41	semidirect
632. C₈	C₂×C₇×#41	semidirect
633. C₇	#61465	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3483	2
C₇×D₄×#41	2
C₇×#3391	2
C₇×Q₈×SD₁₆	2
C₇×#3362	2
C₇×#3358	2
C₇×#3354	2
C₇×#3332	2
C₇×#3325	2
C₇×#3274	2
C₇×#3184	2
C₇×#3115	2
C₇×#3109	2
C₇×#2971	2
C₄×C₇×AGL(1,ℤ/8)	2
#61465	7

Subgroups — C7×#61465 #13238681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61465 #13238681