Subgroups — C₇×#61374 #13238590

Order 1792.

1778 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3503	C₂	1	2	yes	maximal, complemented
2	C₇×#3387	C₂	1	4	yes	maximal, complemented
3	C₇×#3384	C₂	1	4	yes	maximal, complemented
4	C₇×#3381	C₂	1	4	yes	maximal, complemented
5	C₇×Q₈×SD₁₆	C₂	1	4	yes	maximal, complemented
6	C₇×#3163	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3084	C₂	1	4	yes	maximal, complemented
9	C₇×#3079	C₂	1	2	yes	maximal, complemented
10	C₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#2952	C₂	1	2	yes	maximal
12	C₄×C₇×#135	C₂	1	1	yes	maximal, characteristic
13	C₂×C₇×#135	C₂²	1	1	yes	characteristic
14	C₇×Q₈²	C₂²	1	4	yes	—
15	C₇×#557	C₂²	1	4	yes	—
16	C₇×#554	—	2	4	no	—
17	C₇×#554	C₂²	1	4	yes	—
18	C₇×D₄×Q₈	—	2	4	no	—
19	C₇×D₄×Q₈	C₂²	1	4	yes	—
20	C₇×#544	—	2	4	no	—
21	C₇×#544	C₂²	1	4	yes	—
22	C₇×#533	—	2	2	no	—
23	C₇×#533	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₄⋊Q₈	—	2	2	no	—
25	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₇×#520	C₂²	1	4	yes	—
27	C₄×C₇×#41	C₂²	1	2	yes	—
28	C₇×#501	C₂²	1	8	yes	—
29	C₇×#499	C₂²	1	4	yes	—
30	C₇×#498	C₂²	1	4	yes	—
31	C₇×#498	C₂²	1	4	yes	—
32	C₇×#498	C₂²	1	4	yes	—
33	C₇×#479	C₂²	1	8	yes	—
34	C₇×#478	C₂²	1	8	yes	—
35	C₇×#477	C₂²	1	4	yes	—
36	C₇×#476	C₂²	1	4	yes	—
37	C₇×#476	C₂²	1	8	yes	—
38	C₇×#475	C₂²	1	4	yes	—
39	C₇×#475	C₂²	1	8	yes	—
40	C₇×#474	C₂²	1	4	yes	—
41	C₇×#446	C₂²	1	4	yes	—
42	C₇×C₈×Q₈	C₂²	1	4	yes	—
43	C₇×#442	C₂²	1	4	yes	—
44	C₇×#441	C₂²	1	4	yes	—
45	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
46	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
47	C₄×C₇×SD₁₆	C₂²	1	2	yes	—
48	C₄×C₇×SD₁₆	C₂²	1	8	yes	—
49	C₄×C₇×D₈	C₂²	1	2	yes	—
50	C₇×#428	C₂²	1	4	yes	—
51	C₇×#427	C₂²	1	1	yes	characteristic
52	C₂×C₇×#106	C₂²	1	2	yes	—
53	C₂×C₇×#106	C₂²	1	2	yes	—
54	C₇×#424	C₂²	1	2	yes	—
55	C₇×#423	C₂²	1	1	yes	characteristic
56	C₇×#417	C₂²	1	4	yes	—
57	C₇×#416	C₂²	1	2	yes	—
58	C₇×#405	C₂²	1	2	yes	—
59	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
60	#61374	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₇×#41	C₂³	1	2	yes	—
62	C₂²×C₇×Q₈	—	2	4	no	—
63	C₂²×C₇×Q₈	—	2	2	no	—
64	C₇×#135	Q₈	1	8	yes	complemented
65	C₂×C₇×Q₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₇×SD₁₆	C₂³	1	2	yes	—
67	C₂×C₇×D₈	C₂³	1	1	yes	characteristic
68	C₂×C₇×M₁₆	C₂³	1	2	yes	—
69	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
70	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
71	C₇×C₄⋊Q₈	—	2	4	no	—
72	C₇×C₄⋊Q₈	—	2	4	no	—
73	C₇×C₄⋊Q₈	—	2	4	no	—
74	C₇×C₄⋊Q₈	—	2	8	no	—
75	C₇×C₄⋊Q₈	—	2	2	no	—
76	C₇×C₄⋊Q₈	—	2	2	no	—
77	C₇×C₄⋊Q₈	D₄	1	2	yes	—
78	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₇×C₄⋊Q₈	C₂³	1	8	yes	—
80	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
81	C₇×#125	—	2	4	no	—
82	C₇×#125	—	2	8	no	—
83	C₇×#125	D₄	1	2	yes	—
84	C₇×#125	C₂³	1	8	yes	—
85	C₇×C₂²⋊Q₈	—	2	4	no	—
86	C₇×C₂²⋊Q₈	—	2	4	no	—
87	C₇×C₂²⋊Q₈	—	2	8	no	—
88	C₇×C₂²⋊Q₈	—	2	8	no	—
89	C₇×C₂²⋊Q₈	—	2	8	no	—
90	C₇×C₂²⋊Q₈	D₄	1	4	yes	—
91	C₄×C₇×Q₈	—	2	4	no	—
92	C₄×C₇×Q₈	—	2	4	no	—
93	C₄×C₇×Q₈	C₂³	1	2	yes	—
94	C₄×C₇×Q₈	C₂³	1	8	yes	—
95	C₄×C₇×Q₈	C₂³	1	2	yes	—
96	C₄×C₇×Q₈	C₂³	1	4	yes	—
97	C₄×C₇×D₄	—	2	2	no	—
98	C₄×C₇×D₄	—	2	2	no	—
99	C₄×C₇×D₄	—	2	8	no	—
100	C₄×C₇×D₄	C₂³	1	2	yes	—
101	C₄×C₇×D₄	C₂³	1	2	yes	—
102	C₄×C₇×D₄	C₂³	1	8	yes	—
103	C₇×#117	—	2	4	no	—
104	C₇×#117	—	2	2	no	—
105	C₇×#117	C₂³	1	2	yes	—
106	C₇×#117	C₂³	1	2	yes	—
107	C₂×C₇×C₄⋊C₄	—	2	2	no	—
108	C₂×C₇×C₄⋊C₄	—	2	4	no	—
109	C₂×C₇×C₄⋊C₄	—	2	4	no	—
110	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
111	C₂×C₄²×C₇	—	2	2	no	—
112	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
113	C₇×#107	C₂³	1	8	yes	—
114	C₇×#107	C₂³	1	2	yes	—
115	C₇×#106	C₂³	1	8	yes	—
116	C₇×#106	C₂³	1	8	yes	—
117	C₇×#106	C₂³	1	1	yes	characteristic
118	C₇×#106	C₂³	1	4	yes	—
119	C₇×#106	C₂³	1	4	yes	—
120	C₇×#106	C₂³	1	1	yes	characteristic
121	C₇×C₄⋊C₈	C₂³	1	4	yes	—
122	C₇×C₄⋊C₈	C₂³	1	8	yes	—
123	C₇×#103	C₂³	1	2	yes	—
124	C₇×#103	C₂³	1	8	yes	—
125	C₇×#103	C₂³	1	2	yes	—
126	C₇×#102	C₂³	1	8	yes	—
127	C₇×#102	C₂³	1	2	yes	—
128	C₇×#102	C₂³	1	2	yes	—
129	C₇×#97	C₂³	1	4	yes	—
130	C₄×C₇×C₈	C₂³	1	2	yes	—
131	C₄×C₇×C₈	C₂³	1	2	yes	—
132	C₄×C₇×C₈	C₂³	1	4	yes	—
133	#3503	C₂×C₇	1	2	yes	complemented
134	#3387	C₂×C₇	1	4	yes	complemented
135	#3384	C₂×C₇	1	4	yes	complemented
136	#3381	C₂×C₇	1	4	yes	complemented
137	Q₈×SD₁₆	C₂×C₇	1	4	yes	complemented
138	#3163	C₂×C₇	1	2	yes	complemented
139	C₂×#498	C₂×C₇	1	1	yes	characteristic, complemented
140	#3084	C₂×C₇	1	4	yes	complemented
141	#3079	C₂×C₇	1	2	yes	complemented
142	#2960	C₂×C₇	1	1	yes	characteristic, complemented
143	#2952	C₂×C₇	1	2	yes	—
144	C₄×#135	C₂×C₇	1	1	yes	characteristic
145	C₇×#41	—	2	4	no	—
146	C₇×#41	C₂×Q₈	1	8	yes	complemented
147	C₂×C₇×Q₈	—	2	4	no	—
148	C₂×C₇×Q₈	—	2	8	no	—
149	C₂×C₇×Q₈	—	2	4	no	—
150	C₂×C₇×Q₈	—	8	4	no	—
151	C₂×C₇×Q₈	—	8	8	no	—
152	C₂×C₇×Q₈	C₂×D₄	1	4	yes	—
153	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
154	C₂×C₇×D₄	—	2	2	no	—
155	C₂×C₇×D₄	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₇	—	2	2	no	—
157	C₂²×C₄×C₇	—	2	4	no	—
158	C₂²×C₄×C₇	—	2	2	no	—
159	C₂²×C₄×C₇	—	2	2	no	—
160	C₂²×C₄×C₇	—	2	8	no	—
161	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
163	C₇×Q₁₆	C₂×Q₈	1	4	yes	complemented
164	C₇×SD₁₆	C₂×Q₈	1	8	yes	complemented
165	C₇×D₈	C₂×Q₈	1	4	yes	complemented
166	C₇×M₁₆	—	2	4	no	—
167	C₂×C₇×C₈	—	2	2	no	—
168	C₂×C₇×C₈	C₂×Q₈	1	4	yes	complemented
169	C₂×C₇×C₈	C₂⁴	1	2	yes	—
170	C₂×C₇×C₈	C₂⁴	1	4	yes	—
171	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
172	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
173	C₇×C₄⋊C₄	—	2	4	no	—
174	C₇×C₄⋊C₄	—	2	4	no	—
175	C₇×C₄⋊C₄	—	2	8	no	—
176	C₇×C₄⋊C₄	—	2	4	no	—
177	C₇×C₄⋊C₄	—	2	8	no	—
178	C₇×C₄⋊C₄	—	2	2	no	—
179	C₇×C₄⋊C₄	—	2	2	no	—
180	C₇×C₄⋊C₄	—	2	8	no	—
181	C₇×C₄⋊C₄	—	2	8	no	—
182	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
183	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
184	C₇×C₄⋊C₄	C₂×D₄	1	8	yes	—
185	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
186	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
187	C₇×C₄⋊C₄	C₂⁴	1	2	yes	—
188	C₇×C₂²⋊C₄	—	2	2	no	—
189	C₇×C₂²⋊C₄	—	2	2	no	—
190	C₇×C₂²⋊C₄	—	2	2	no	—
191	C₇×C₂²⋊C₄	—	2	8	no	—
192	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
193	C₄²×C₇	—	2	2	no	—
194	C₄²×C₇	—	2	2	no	—
195	C₄²×C₇	—	2	2	no	—
196	C₄²×C₇	—	2	8	no	—
197	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₇	C₂⁴	1	2	yes	—
200	C₄²×C₇	C₂⁴	1	4	yes	—
201	C₂×#135	C₂²×C₇	1	1	yes	characteristic
202	Q₈²	C₂²×C₇	1	4	yes	—
203	#557	C₂²×C₇	1	4	yes	—
204	#554	—	2	4	no	—
205	#554	C₂²×C₇	1	4	yes	—
206	D₄×Q₈	—	2	4	no	—
207	D₄×Q₈	C₂²×C₇	1	4	yes	—
208	#544	—	2	4	no	—
209	#544	C₂²×C₇	1	4	yes	—
210	#533	—	2	2	no	—
211	#533	C₂²×C₇	1	1	yes	characteristic
212	C₂×C₄⋊Q₈	—	2	2	no	—
213	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
214	#520	C₂²×C₇	1	4	yes	—
215	C₄×#41	C₂²×C₇	1	2	yes	—
216	#501	C₂²×C₇	1	8	yes	—
217	#499	C₂²×C₇	1	4	yes	—
218	#498	C₂²×C₇	1	4	yes	—
219	#498	C₂²×C₇	1	4	yes	—
220	#498	C₂²×C₇	1	4	yes	—
221	#479	C₂²×C₇	1	8	yes	—
222	#478	C₂²×C₇	1	8	yes	—
223	#477	C₂²×C₇	1	4	yes	—
224	#476	C₂²×C₇	1	4	yes	—
225	#476	C₂²×C₇	1	8	yes	—
226	#475	C₂²×C₇	1	4	yes	—
227	#475	C₂²×C₇	1	8	yes	—
228	#474	C₂²×C₇	1	4	yes	—
229	#446	C₂²×C₇	1	4	yes	—
230	C₈×Q₈	C₂²×C₇	1	4	yes	—
231	#442	C₂²×C₇	1	4	yes	—
232	#441	C₂²×C₇	1	4	yes	—
233	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
234	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
235	C₄×SD₁₆	C₂²×C₇	1	8	yes	—
236	C₄×SD₁₆	C₂²×C₇	1	2	yes	—
237	C₄×D₈	C₂²×C₇	1	2	yes	—
238	#428	C₂²×C₇	1	4	yes	—
239	#427	C₂²×C₇	1	1	yes	characteristic
240	C₂×#106	C₂²×C₇	1	2	yes	—
241	C₂×#106	C₂²×C₇	1	2	yes	—
242	#424	C₂²×C₇	1	2	yes	—
243	#423	C₂²×C₇	1	1	yes	characteristic
244	#417	C₂²×C₇	1	4	yes	—
245	#416	C₂²×C₇	1	2	yes	—
246	#405	C₂²×C₇	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
248	C₂³×C₇	—	2	2	no	—
249	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
250	C₇×Q₈	—	2	8	no	—
251	C₇×Q₈	—	2	2	no	—
252	C₇×Q₈	—	4	8	no	—
253	C₇×Q₈	—	8	4	no	—
254	C₇×Q₈	—	8	8	no	—
255	C₇×Q₈	C₂²×Q₈	1	4	yes	—
256	C₇×D₄	—	2	8	no	—
257	C₇×D₄	—	2	2	no	—
258	C₇×D₄	C₂²×Q₈	1	4	yes	—
259	C₂×C₄×C₇	—	2	2	no	—
260	C₂×C₄×C₇	—	2	8	no	—
261	C₂×C₄×C₇	—	2	4	no	—
262	C₂×C₄×C₇	—	2	2	no	—
263	C₂×C₄×C₇	—	2	2	no	—
264	C₂×C₄×C₇	—	2	8	no	—
265	C₂×C₄×C₇	—	4	4	no	—
266	C₂×C₄×C₇	—	4	4	no	—
267	C₂×C₄×C₇	—	8	8	no	—
268	C₂×C₄×C₇	—	8	4	no	—
269	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
270	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
274	C₂×C₄×C₇	2₋¹⁺⁴	1	2	yes	—
275	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
276	C₇×C₈	—	2	2	no	—
277	C₇×C₈	—	2	4	no	—
278	C₇×C₈	C₂²×Q₈	1	2	yes	—
279	C₇×C₈	C₂²×Q₈	1	2	yes	—
280	C₂×#41	C₂³×C₇	1	2	yes	—
281	C₂²×Q₈	—	2	4	no	—
282	C₂²×Q₈	—	2	2	no	—
283	#135	C₇×Q₈	1	8	yes	complemented
284	C₂×Q₁₆	C₂³×C₇	1	1	yes	characteristic
285	C₂×SD₁₆	C₂³×C₇	1	2	yes	—
286	C₂×D₈	C₂³×C₇	1	1	yes	characteristic
287	C₂×M₁₆	C₂³×C₇	1	2	yes	—
288	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
290	C₄⋊Q₈	—	2	4	no	—
291	C₄⋊Q₈	—	2	4	no	—
292	C₄⋊Q₈	—	2	4	no	—
293	C₄⋊Q₈	—	2	8	no	—
294	C₄⋊Q₈	—	2	2	no	—
295	C₄⋊Q₈	—	2	2	no	—
296	C₄⋊Q₈	C₇×D₄	1	2	yes	—
297	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
298	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
299	C₄⋊Q₈	C₂³×C₇	1	8	yes	—
300	#125	—	2	4	no	—
301	#125	—	2	8	no	—
302	#125	C₇×D₄	1	2	yes	—
303	#125	C₂³×C₇	1	8	yes	—
304	C₂²⋊Q₈	—	2	4	no	—
305	C₂²⋊Q₈	—	2	4	no	—
306	C₂²⋊Q₈	—	2	8	no	—
307	C₂²⋊Q₈	—	2	8	no	—
308	C₂²⋊Q₈	—	2	8	no	—
309	C₂²⋊Q₈	C₇×D₄	1	4	yes	—
310	C₄×Q₈	—	2	4	no	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	C₂³×C₇	1	2	yes	—
313	C₄×Q₈	C₂³×C₇	1	2	yes	—
314	C₄×Q₈	C₂³×C₇	1	8	yes	—
315	C₄×Q₈	C₂³×C₇	1	4	yes	—
316	C₄×D₄	—	2	2	no	—
317	C₄×D₄	—	2	2	no	—
318	C₄×D₄	—	2	8	no	—
319	C₄×D₄	C₂³×C₇	1	2	yes	—
320	C₄×D₄	C₂³×C₇	1	2	yes	—
321	C₄×D₄	C₂³×C₇	1	8	yes	—
322	#117	—	2	4	no	—
323	#117	—	2	2	no	—
324	#117	C₂³×C₇	1	2	yes	—
325	#117	C₂³×C₇	1	2	yes	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	4	no	—
328	C₂×C₄⋊C₄	—	2	4	no	—
329	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
332	#107	C₂³×C₇	1	2	yes	—
333	#107	C₂³×C₇	1	8	yes	—
334	#106	C₂³×C₇	1	4	yes	—
335	#106	C₂³×C₇	1	4	yes	—
336	#106	C₂³×C₇	1	8	yes	—
337	#106	C₂³×C₇	1	1	yes	characteristic
338	#106	C₂³×C₇	1	1	yes	characteristic
339	#106	C₂³×C₇	1	8	yes	—
340	C₄⋊C₈	C₂³×C₇	1	4	yes	—
341	C₄⋊C₈	C₂³×C₇	1	8	yes	—
342	#103	C₂³×C₇	1	2	yes	—
343	#103	C₂³×C₇	1	2	yes	—
344	#103	C₂³×C₇	1	8	yes	—
345	#102	C₂³×C₇	1	2	yes	—
346	#102	C₂³×C₇	1	2	yes	—
347	#102	C₂³×C₇	1	8	yes	—
348	#97	C₂³×C₇	1	4	yes	—
349	C₄×C₈	C₂³×C₇	1	2	yes	—
350	C₄×C₈	C₂³×C₇	1	2	yes	—
351	C₄×C₈	C₂³×C₇	1	4	yes	—
352	C₂²×C₇	—	2	2	no	—
353	C₂²×C₇	—	2	4	no	—
354	C₂²×C₇	—	4	2	no	—
355	C₂²×C₇	—	4	2	no	—
356	C₂²×C₇	D₄×Q₈	1	2	yes	—
357	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
358	C₄×C₇	—	2	4	no	—
359	C₄×C₇	—	2	2	no	—
360	C₄×C₇	—	2	8	no	—
361	C₄×C₇	—	4	2	no	—
362	C₄×C₇	—	4	2	no	—
363	C₄×C₇	—	4	8	no	—
364	C₄×C₇	D₄×Q₈	1	2	yes	—
365	C₄×C₇	#577	1	4	yes	—
366	C₄×C₇	C₂³×Q₈	1	1	yes	characteristic
367	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
368	#41	—	2	4	no	—
369	#41	C₂×C₇×Q₈	1	8	yes	complemented
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	8	4	no	—
374	C₂×Q₈	—	8	8	no	—
375	C₂×Q₈	C₂×C₇×D₄	1	4	yes	—
376	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	C₂⁴×C₇	1	2	yes	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
386	Q₁₆	C₂×C₇×Q₈	1	4	yes	complemented
387	SD₁₆	C₂×C₇×Q₈	1	8	yes	complemented
388	D₈	C₂×C₇×Q₈	1	4	yes	complemented
389	M₁₆	—	2	4	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	C₂×C₇×Q₈	1	4	yes	complemented
392	C₂×C₈	C₂⁴×C₇	1	2	yes	—
393	C₂×C₈	C₂⁴×C₇	1	4	yes	—
394	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
395	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	8	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
406	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
407	C₄⋊C₄	C₂×C₇×D₄	1	8	yes	—
408	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
409	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
410	C₄⋊C₄	C₂⁴×C₇	1	2	yes	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	2	2	no	—
419	C₄²	—	2	8	no	—
420	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
421	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
422	C₄²	C₂⁴×C₇	1	2	yes	—
423	C₄²	C₂⁴×C₇	1	4	yes	—
424	C₂×C₇	—	2	2	no	—
425	C₂×C₇	—	4	4	no	—
426	C₂×C₇	C₂×D₄×Q₈	1	1	yes	characteristic
427	C₂×C₇	C₂×#577	1	2	yes	—
428	C₂³	—	2	2	no	—
429	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
430	Q₈	—	2	2	no	—
431	Q₈	—	2	8	no	—
432	Q₈	—	4	8	no	—
433	Q₈	—	8	4	no	—
434	Q₈	—	8	8	no	—
435	Q₈	C₂²×C₇×Q₈	1	4	yes	—
436	D₄	—	2	8	no	—
437	D₄	—	2	2	no	—
438	D₄	C₂²×C₇×Q₈	1	4	yes	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	8	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	—	8	8	no	—
448	C₂×C₄	—	8	4	no	—
449	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
450	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
451	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
452	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
453	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
454	C₂×C₄	C₇×2₋¹⁺⁴	1	2	yes	—
455	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
456	C₈	—	2	2	no	—
457	C₈	—	2	4	no	—
458	C₈	C₂²×C₇×Q₈	1	2	yes	—
459	C₈	C₂²×C₇×Q₈	1	2	yes	—
460	C₇	#61374	1	1	yes	characteristic, direct product factor, Sylow
461	C₂²	—	2	2	no	—
462	C₂²	—	2	4	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	C₇×D₄×Q₈	1	2	yes	—
466	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
467	C₄	—	2	2	no	—
468	C₄	—	2	8	no	—
469	C₄	—	2	4	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	8	no	—
473	C₄	C₇×D₄×Q₈	1	2	yes	—
474	C₄	C₇×#577	1	4	yes	—
475	C₄	C₂³×C₇×Q₈	1	1	yes	characteristic
476	C₄	C₂×C₇×2₋¹⁺⁴	1	1	yes	characteristic
477	C₂	—	2	2	no	—
478	C₂	—	4	4	no	—
479	C₂	C₂×C₇×D₄×Q₈	1	1	yes	characteristic
480	C₂	C₂×C₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3503	C₂	semidirect
2. C₇×#3387	C₂	semidirect
3. C₇×#3384	C₂	semidirect
4. C₇×#3381	C₂	semidirect
5. C₇×Q₈×SD₁₆	C₂	semidirect
6. C₇×#3163	C₂	semidirect
7. C₂×C₇×#498	C₂	semidirect
8. C₇×#3084	C₂	semidirect
9. C₇×#3079	C₂	semidirect
10. C₇×#2960	C₂	semidirect
60. #61374	C₇	direct (N × H)
64. C₇×#135	Q₈	semidirect
133. #3503	C₂×C₇	semidirect
134. #3387	C₂×C₇	semidirect
135. #3384	C₂×C₇	semidirect
136. #3381	C₂×C₇	semidirect
137. Q₈×SD₁₆	C₂×C₇	semidirect
138. #3163	C₂×C₇	semidirect
139. C₂×#498	C₂×C₇	semidirect
140. #3084	C₂×C₇	semidirect
141. #3079	C₂×C₇	semidirect
142. #2960	C₂×C₇	semidirect
146. C₇×#41	C₂×Q₈	semidirect
163. C₇×Q₁₆	C₂×Q₈	semidirect
164. C₇×SD₁₆	C₂×Q₈	semidirect
165. C₇×D₈	C₂×Q₈	semidirect
168. C₂×C₇×C₈	C₂×Q₈	semidirect
283. #135	C₇×Q₈	semidirect
369. #41	C₂×C₇×Q₈	semidirect
386. Q₁₆	C₂×C₇×Q₈	semidirect
387. SD₁₆	C₂×C₇×Q₈	semidirect
388. D₈	C₂×C₇×Q₈	semidirect
391. C₂×C₈	C₂×C₇×Q₈	semidirect
460. C₇	#61374	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×#3503	2
C₇×#3387	2
C₇×#3384	2
C₇×#3381	2
C₇×Q₈×SD₁₆	2
C₇×#3163	2
C₂×C₇×#498	2
C₇×#3084	2
C₇×#3079	2
C₇×#2960	2
C₇×#2952	2
C₄×C₇×#135	2
#61374	7

Subgroups — C7×#61374 #13238590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61374 #13238590