Subgroups — C₇×#60705 #13237921

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3455	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#541	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3178	C₂	1	8	yes	maximal, complemented
4	C₇×#3177	C₂	1	4	yes	maximal, complemented
5	C₇×#3174	C₂	1	4	yes	maximal, complemented
6	C₇×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3142	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3056	C₂	1	4	yes	maximal, complemented
11	C₄×C₇×#137	C₂	1	4	yes	maximal, complemented
12	C₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₇×#137	C₂²	1	4	yes	—
14	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
15	C₇×#542	C₂²	1	4	yes	—
16	C₇×#541	—	2	4	no	—
17	C₇×#541	C₂²	1	4	yes	—
18	C₇×#541	C₂²	1	2	yes	—
19	C₇×#541	C₂²	1	4	yes	—
20	C₇×#540	C₂²	1	2	yes	—
21	C₇×#539	—	2	2	no	—
22	C₇×#538	—	2	2	no	—
23	C₇×#533	C₂²	1	1	yes	characteristic
24	C₇×#532	C₂²	1	1	yes	characteristic
25	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₇×#529	—	2	4	no	—
27	C₂×C₇×#126	—	2	2	no	—
28	C₂×C₇×#125	—	2	2	no	—
29	C₂×C₇×#125	C₂²	1	1	yes	characteristic
30	C₇×#525	C₂²	1	2	yes	—
31	C₂×C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₄×C₇×#41	C₂²	1	2	yes	—
33	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
34	C₂×C₇×#117	C₂²	1	1	yes	characteristic
35	C₇×#501	C₂²	1	4	yes	complemented
36	C₇×#499	C₂²	1	4	yes	complemented
37	C₇×#497	C₂²	1	4	yes	complemented
38	C₇×#494	C₂²	1	2	yes	complemented
39	C₇×#492	C₂²	1	2	yes	complemented
40	C₇×#491	C₂²	1	4	yes	complemented
41	C₇×#491	C₂²	1	2	yes	complemented
42	C₇×#490	C₂²	1	2	yes	complemented
43	C₇×#488	C₂²	1	4	yes	complemented
44	C₇×#487	C₂²	1	4	yes	complemented
45	C₇×#470	C₂²	1	8	yes	—
46	C₇×#468	C₂²	1	8	yes	—
47	C₇×#467	C₂²	1	8	yes	—
48	C₇×#465	C₂²	1	8	yes	—
49	C₇×#441	C₂²	1	8	yes	complemented
50	C₇×#440	C₂²	1	8	yes	complemented
51	C₄×C₇×Q₁₆	C₂²	1	8	yes	complemented
52	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
53	C₇×#428	C₂²	1	4	yes	—
54	C₇×#419	C₂²	1	4	yes	—
55	C₇×#417	C₂²	1	4	yes	—
56	C₇×#405	C₂²	1	4	yes	—
57	C₇×#405	C₂²	1	4	yes	—
58	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
59	C₄×C₇×M₁₆	C₂²	1	2	yes	—
60	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
61	#60705	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₇×#41	C₂³	1	2	yes	—
63	C₂²×C₇×Q₈	C₂³	1	2	yes	—
64	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
65	C₇×#137	—	2	16	no	—
66	C₂×C₇×Q₁₆	C₂³	1	8	yes	—
67	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
68	C₂×C₇×M₁₆	—	2	8	no	—
69	C₂×C₇×M₁₆	C₂³	1	4	yes	—
70	C₂×C₇×M₁₆	C₂³	1	2	yes	—
71	C₂×C₇×M₁₆	C₂³	1	2	yes	—
72	C₂²×C₇×C₈	C₂³	1	4	yes	—
73	C₇×C₄⋊Q₈	—	2	2	no	—
74	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₇×#127	C₂³	1	1	yes	characteristic
79	C₇×#126	—	2	4	no	—
80	C₇×#126	—	2	8	no	—
81	C₇×#126	—	2	8	no	—
82	C₇×#126	—	2	4	no	—
83	C₇×#125	—	2	8	no	—
84	C₇×#125	—	2	8	no	—
85	C₇×#125	—	2	4	no	—
86	C₇×#125	—	2	2	no	—
87	C₇×#125	C₂³	1	2	yes	—
88	C₇×#125	C₂³	1	2	yes	—
89	C₇×#125	C₂³	1	2	yes	—
90	C₇×#124	—	2	4	no	—
91	C₇×#124	C₂³	1	2	yes	—
92	C₇×#123	—	2	2	no	—
93	C₇×#123	—	2	4	no	—
94	C₇×#123	—	2	4	no	—
95	C₇×C₂²⋊Q₈	—	2	4	no	—
96	C₇×C₂²⋊Q₈	—	2	2	no	—
97	C₇×C₂²⋊Q₈	—	2	4	no	—
98	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
99	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
100	C₇×#121	—	2	2	no	—
101	C₇×#121	—	2	4	no	—
102	C₇×#121	C₂³	1	4	yes	—
103	C₇×#120	—	2	2	no	—
104	C₄×C₇×Q₈	—	2	4	no	—
105	C₄×C₇×Q₈	—	2	2	no	—
106	C₄×C₇×Q₈	C₂³	1	8	yes	—
107	C₄×C₇×Q₈	C₂³	1	4	yes	—
108	C₄×C₇×D₄	—	2	4	no	—
109	C₄×C₇×D₄	—	2	4	no	—
110	C₄×C₇×D₄	—	2	2	no	—
111	C₄×C₇×D₄	C₂³	1	4	yes	—
112	C₇×#117	—	2	2	no	—
113	C₇×#117	D₄	1	4	yes	—
114	C₇×#117	C₂³	1	2	yes	—
115	C₇×#117	C₂³	1	2	yes	—
116	C₇×#117	C₂³	1	2	yes	—
117	C₂×C₇×C₄⋊C₄	—	2	2	no	—
118	C₂×C₇×C₄⋊C₄	—	2	2	no	—
119	C₂×C₇×C₄⋊C₄	—	2	2	no	—
120	C₂×C₇×C₄⋊C₄	—	2	2	no	—
121	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
122	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
123	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
124	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
125	C₂×C₄²×C₇	—	2	2	no	—
126	C₂×C₄²×C₇	—	2	2	no	—
127	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
129	C₇×#107	C₂³	1	8	yes	—
130	C₇×#106	C₂³	1	8	yes	—
131	C₇×#103	C₂³	1	8	yes	—
132	C₇×#103	C₂³	1	8	yes	—
133	C₇×#103	C₂³	1	8	yes	—
134	C₇×#102	C₂³	1	8	yes	—
135	C₇×#97	C₂³	1	2	yes	—
136	C₇×#97	C₂³	1	2	yes	—
137	C₇×#97	C₂³	1	4	yes	—
138	C₄×C₇×C₈	C₂³	1	2	yes	—
139	C₄×C₇×C₈	C₂³	1	2	yes	—
140	C₄×C₇×C₈	C₂³	1	4	yes	—
141	#3455	C₂×C₇	1	1	yes	characteristic
142	C₂×#541	C₂×C₇	1	1	yes	characteristic, complemented
143	#3178	C₂×C₇	1	8	yes	complemented
144	#3177	C₂×C₇	1	4	yes	complemented
145	#3174	C₂×C₇	1	4	yes	complemented
146	#3166	C₂×C₇	1	1	yes	characteristic, complemented
147	#3152	C₂×C₇	1	1	yes	characteristic, complemented
148	#3142	C₂×C₇	1	1	yes	characteristic, complemented
149	#3139	C₂×C₇	1	1	yes	characteristic, complemented
150	#3056	C₂×C₇	1	4	yes	complemented
151	C₄×#137	C₂×C₇	1	4	yes	complemented
152	#2873	C₂×C₇	1	1	yes	characteristic, complemented
153	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
154	C₇×#41	—	2	8	no	—
155	C₂×C₇×Q₈	—	2	2	no	—
156	C₂×C₇×Q₈	—	2	2	no	—
157	C₂×C₇×Q₈	—	2	8	no	—
158	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
159	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
160	C₂×C₇×D₄	—	2	2	no	—
161	C₂×C₇×D₄	—	2	4	no	—
162	C₂×C₇×D₄	—	2	2	no	—
163	C₂×C₇×D₄	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₇	—	2	4	no	—
165	C₂²×C₄×C₇	—	2	2	no	—
166	C₂²×C₄×C₇	—	2	2	no	—
167	C₂²×C₄×C₇	—	2	2	no	—
168	C₂²×C₄×C₇	—	2	4	no	—
169	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
175	C₇×Q₁₆	—	2	16	no	—
176	C₇×SD₁₆	—	2	16	no	—
177	C₇×M₁₆	—	2	4	no	—
178	C₇×M₁₆	—	2	8	no	—
179	C₇×M₁₆	#41	1	8	yes	—
180	C₂×C₇×C₈	—	2	8	no	—
181	C₂×C₇×C₈	C₂⁴	1	4	yes	—
182	C₂×C₇×C₈	C₂⁴	1	4	yes	—
183	C₇×C₄⋊C₄	—	2	8	no	—
184	C₇×C₄⋊C₄	—	2	8	no	—
185	C₇×C₄⋊C₄	—	2	4	no	—
186	C₇×C₄⋊C₄	—	2	4	no	—
187	C₇×C₄⋊C₄	—	2	4	no	—
188	C₇×C₄⋊C₄	—	2	4	no	—
189	C₇×C₄⋊C₄	—	2	2	no	—
190	C₇×C₄⋊C₄	—	2	4	no	—
191	C₇×C₄⋊C₄	—	2	2	no	—
192	C₇×C₄⋊C₄	—	2	4	no	—
193	C₇×C₄⋊C₄	—	2	4	no	—
194	C₇×C₄⋊C₄	—	2	2	no	—
195	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
196	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
197	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
198	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
199	C₇×C₂²⋊C₄	—	2	8	no	—
200	C₇×C₂²⋊C₄	—	2	2	no	—
201	C₇×C₂²⋊C₄	—	2	4	no	—
202	C₇×C₂²⋊C₄	—	2	4	no	—
203	C₇×C₂²⋊C₄	—	2	4	no	—
204	C₇×C₂²⋊C₄	—	2	4	no	—
205	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
206	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
207	C₄²×C₇	—	2	2	no	—
208	C₄²×C₇	—	2	4	no	—
209	C₄²×C₇	—	2	8	no	—
210	C₄²×C₇	C₂×D₄	1	2	yes	—
211	C₄²×C₇	C₂×D₄	1	2	yes	—
212	C₄²×C₇	C₂⁴	1	2	yes	—
213	C₄²×C₇	C₂⁴	1	1	yes	characteristic
214	C₄²×C₇	C₂⁴	1	1	yes	characteristic
215	C₂×#137	C₂²×C₇	1	4	yes	—
216	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
217	#542	C₂²×C₇	1	4	yes	—
218	#541	—	2	4	no	—
219	#541	C₂²×C₇	1	4	yes	—
220	#541	C₂²×C₇	1	2	yes	—
221	#541	C₂²×C₇	1	4	yes	—
222	#540	C₂²×C₇	1	2	yes	—
223	#539	—	2	2	no	—
224	#538	—	2	2	no	—
225	#533	C₂²×C₇	1	1	yes	characteristic
226	#532	C₂²×C₇	1	1	yes	characteristic
227	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
228	#529	—	2	4	no	—
229	C₂×#126	—	2	2	no	—
230	C₂×#125	—	2	2	no	—
231	C₂×#125	C₂²×C₇	1	1	yes	characteristic
232	#525	C₂²×C₇	1	2	yes	—
233	C₂×C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
234	C₄×#41	C₂²×C₇	1	2	yes	—
235	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
236	C₂×#117	C₂²×C₇	1	1	yes	characteristic
237	#501	C₂²×C₇	1	4	yes	complemented
238	#499	C₂²×C₇	1	4	yes	complemented
239	#497	C₂²×C₇	1	4	yes	complemented
240	#494	C₂²×C₇	1	2	yes	complemented
241	#492	C₂²×C₇	1	2	yes	complemented
242	#491	C₂²×C₇	1	4	yes	complemented
243	#491	C₂²×C₇	1	2	yes	complemented
244	#490	C₂²×C₇	1	2	yes	complemented
245	#488	C₂²×C₇	1	4	yes	complemented
246	#487	C₂²×C₇	1	4	yes	complemented
247	#470	C₂²×C₇	1	8	yes	—
248	#468	C₂²×C₇	1	8	yes	—
249	#467	C₂²×C₇	1	8	yes	—
250	#465	C₂²×C₇	1	8	yes	—
251	#441	C₂²×C₇	1	8	yes	complemented
252	#440	C₂²×C₇	1	8	yes	complemented
253	C₄×Q₁₆	C₂²×C₇	1	8	yes	complemented
254	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
255	#428	C₂²×C₇	1	4	yes	—
256	#419	C₂²×C₇	1	4	yes	—
257	#417	C₂²×C₇	1	4	yes	—
258	#405	C₂²×C₇	1	4	yes	—
259	#405	C₂²×C₇	1	4	yes	—
260	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
261	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
262	C₄×M₁₆	C₂²×C₇	1	2	yes	—
263	C₂³×C₇	—	2	2	no	—
264	C₂³×C₇	—	2	2	no	—
265	C₂³×C₇	—	4	2	no	—
266	C₂³×C₇	C₂²×D₄	1	2	yes	—
267	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
268	C₇×Q₈	—	2	8	no	—
269	C₇×Q₈	—	2	4	no	—
270	C₇×Q₈	—	4	4	no	—
271	C₇×Q₈	—	8	2	no	—
272	C₇×D₄	—	2	4	no	—
273	C₇×D₄	—	4	4	no	—
274	C₇×D₄	—	8	2	no	—
275	C₇×D₄	—	8	4	no	—
276	C₂×C₄×C₇	—	2	2	no	—
277	C₂×C₄×C₇	—	2	4	no	—
278	C₂×C₄×C₇	—	2	2	no	—
279	C₂×C₄×C₇	—	2	2	no	—
280	C₂×C₄×C₇	—	2	2	no	—
281	C₂×C₄×C₇	—	2	4	no	—
282	C₂×C₄×C₇	—	2	4	no	—
283	C₂×C₄×C₇	—	2	4	no	—
284	C₂×C₄×C₇	—	4	2	no	—
285	C₂×C₄×C₇	—	4	2	no	—
286	C₂×C₄×C₇	—	4	4	no	—
287	C₂×C₄×C₇	—	4	4	no	—
288	C₂×C₄×C₇	—	4	4	no	—
289	C₂×C₄×C₇	—	8	4	no	—
290	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
291	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
293	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₇	C₂×#41	1	4	yes	—
296	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
297	C₇×C₈	—	2	4	no	—
298	C₇×C₈	C₂×#41	1	8	yes	—
299	C₂×#41	C₂³×C₇	1	2	yes	—
300	C₂²×Q₈	C₂³×C₇	1	2	yes	—
301	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
302	#137	—	2	16	no	—
303	C₂×Q₁₆	C₂³×C₇	1	8	yes	—
304	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
305	C₂×M₁₆	—	2	8	no	—
306	C₂×M₁₆	C₂³×C₇	1	2	yes	—
307	C₂×M₁₆	C₂³×C₇	1	4	yes	—
308	C₂×M₁₆	C₂³×C₇	1	2	yes	—
309	C₂²×C₈	C₂³×C₇	1	4	yes	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
312	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
313	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
314	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
315	#127	C₂³×C₇	1	1	yes	characteristic
316	#126	—	2	4	no	—
317	#126	—	2	8	no	—
318	#126	—	2	8	no	—
319	#126	—	2	4	no	—
320	#125	—	2	8	no	—
321	#125	—	2	8	no	—
322	#125	—	2	4	no	—
323	#125	—	2	2	no	—
324	#125	C₂³×C₇	1	2	yes	—
325	#125	C₂³×C₇	1	2	yes	—
326	#125	C₂³×C₇	1	2	yes	—
327	#124	—	2	4	no	—
328	#124	C₂³×C₇	1	2	yes	—
329	#123	—	2	2	no	—
330	#123	—	2	4	no	—
331	#123	—	2	4	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	—	2	2	no	—
334	C₂²⋊Q₈	—	2	4	no	—
335	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
336	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
337	#121	—	2	2	no	—
338	#121	—	2	4	no	—
339	#121	C₂³×C₇	1	4	yes	—
340	#120	—	2	2	no	—
341	C₄×Q₈	—	2	4	no	—
342	C₄×Q₈	—	2	2	no	—
343	C₄×Q₈	C₂³×C₇	1	8	yes	—
344	C₄×Q₈	C₂³×C₇	1	4	yes	—
345	C₄×D₄	—	2	4	no	—
346	C₄×D₄	—	2	4	no	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	C₂³×C₇	1	4	yes	—
349	#117	—	2	2	no	—
350	#117	C₇×D₄	1	4	yes	—
351	#117	C₂³×C₇	1	2	yes	—
352	#117	C₂³×C₇	1	2	yes	—
353	#117	C₂³×C₇	1	2	yes	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	2	no	—
357	C₂×C₄⋊C₄	—	2	2	no	—
358	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
362	C₂×C₄²	—	2	2	no	—
363	C₂×C₄²	—	2	2	no	—
364	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
366	#107	C₂³×C₇	1	8	yes	—
367	#106	C₂³×C₇	1	8	yes	—
368	#103	C₂³×C₇	1	8	yes	—
369	#103	C₂³×C₇	1	8	yes	—
370	#103	C₂³×C₇	1	8	yes	—
371	#102	C₂³×C₇	1	8	yes	—
372	#97	C₂³×C₇	1	4	yes	—
373	#97	C₂³×C₇	1	2	yes	—
374	#97	C₂³×C₇	1	2	yes	—
375	C₄×C₈	C₂³×C₇	1	4	yes	—
376	C₄×C₈	C₂³×C₇	1	2	yes	—
377	C₄×C₈	C₂³×C₇	1	2	yes	—
378	C₂²×C₇	—	2	2	no	—
379	C₂²×C₇	—	2	4	no	—
380	C₂²×C₇	—	2	2	no	—
381	C₂²×C₇	—	4	4	no	—
382	C₂²×C₇	—	4	2	no	—
383	C₂²×C₇	—	8	2	no	—
384	C₂²×C₇	#532	1	2	yes	—
385	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
386	C₄×C₇	—	2	4	no	—
387	C₄×C₇	—	2	2	no	—
388	C₄×C₇	—	2	2	no	—
389	C₄×C₇	—	2	2	no	—
390	C₄×C₇	—	4	4	no	—
391	C₄×C₇	—	4	4	no	—
392	C₄×C₇	—	4	2	no	—
393	C₄×C₇	—	4	4	no	—
394	C₄×C₇	#532	1	2	yes	—
395	C₄×C₇	C₂²×#41	1	2	yes	—
396	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
397	#41	—	2	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	—	2	8	no	—
401	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
402	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	C₂⁴×C₇	1	2	yes	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
413	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
414	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
415	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
418	Q₁₆	—	2	16	no	—
419	SD₁₆	—	2	16	no	—
420	M₁₆	—	2	4	no	—
421	M₁₆	—	2	8	no	—
422	M₁₆	C₇×#41	1	8	yes	—
423	C₂×C₈	—	2	8	no	—
424	C₂×C₈	C₂⁴×C₇	1	4	yes	—
425	C₂×C₈	C₂⁴×C₇	1	4	yes	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	8	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
439	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
440	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
441	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
449	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
450	C₄²	—	2	2	no	—
451	C₄²	—	2	4	no	—
452	C₄²	—	2	8	no	—
453	C₄²	C₂×C₇×D₄	1	2	yes	—
454	C₄²	C₂×C₇×D₄	1	2	yes	—
455	C₄²	C₂⁴×C₇	1	2	yes	—
456	C₄²	C₂⁴×C₇	1	1	yes	characteristic
457	C₄²	C₂⁴×C₇	1	1	yes	characteristic
458	C₂×C₇	—	2	2	no	—
459	C₂×C₇	—	4	2	no	—
460	C₂×C₇	—	8	2	no	—
461	C₂×C₇	C₂×#532	1	1	yes	characteristic
462	C₂×C₇	#3586	1	2	yes	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	4	2	no	—
466	C₂³	C₂²×C₇×D₄	1	2	yes	—
467	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
468	Q₈	—	2	8	no	—
469	Q₈	—	2	4	no	—
470	Q₈	—	4	4	no	—
471	Q₈	—	8	2	no	—
472	D₄	—	2	4	no	—
473	D₄	—	4	4	no	—
474	D₄	—	8	2	no	—
475	D₄	—	8	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	4	no	—
487	C₂×C₄	—	4	4	no	—
488	C₂×C₄	—	4	4	no	—
489	C₂×C₄	—	8	4	no	—
490	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
491	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
492	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
493	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
494	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
495	C₂×C₄	C₂×C₇×#41	1	4	yes	—
496	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
497	C₈	—	2	4	no	—
498	C₈	C₂×C₇×#41	1	8	yes	—
499	C₇	#60705	1	1	yes	characteristic, direct product factor, Sylow
500	C₂²	—	2	2	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	2	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	C₇×#532	1	2	yes	—
507	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
508	C₄	—	2	4	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	2	no	—
515	C₄	—	4	4	no	—
516	C₄	C₇×#532	1	2	yes	—
517	C₄	C₂²×C₇×#41	1	2	yes	—
518	C₂	—	2	2	no	—
519	C₂	—	4	2	no	—
520	C₂	—	8	2	no	—
521	C₂	C₂×C₇×#532	1	1	yes	characteristic
522	C₂	C₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3455	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#541	C₂	semidirect
3. C₇×#3178	C₂	semidirect
4. C₇×#3177	C₂	semidirect
5. C₇×#3174	C₂	semidirect
6. C₇×#3166	C₂	semidirect
7. C₇×#3152	C₂	semidirect
8. C₇×#3142	C₂	semidirect
9. C₇×#3139	C₂	semidirect
10. C₇×#3056	C₂	semidirect
11. C₄×C₇×#137	C₂	semidirect
12. C₇×#2873	C₂	semidirect
35. C₇×#501	C₂²	semidirect
36. C₇×#499	C₂²	semidirect
37. C₇×#497	C₂²	semidirect
38. C₇×#494	C₂²	semidirect
39. C₇×#492	C₂²	semidirect
40. C₇×#491	C₂²	semidirect
41. C₇×#491	C₂²	semidirect
42. C₇×#490	C₂²	semidirect
43. C₇×#488	C₂²	semidirect
44. C₇×#487	C₂²	semidirect
49. C₇×#441	C₂²	semidirect
50. C₇×#440	C₂²	semidirect
51. C₄×C₇×Q₁₆	C₂²	semidirect
52. C₄×C₇×SD₁₆	C₂²	semidirect
61. #60705	C₇	direct (N × H)
142. C₂×#541	C₂×C₇	semidirect
143. #3178	C₂×C₇	semidirect
144. #3177	C₂×C₇	semidirect
145. #3174	C₂×C₇	semidirect
146. #3166	C₂×C₇	semidirect
147. #3152	C₂×C₇	semidirect
148. #3142	C₂×C₇	semidirect
149. #3139	C₂×C₇	semidirect
150. #3056	C₂×C₇	semidirect
151. C₄×#137	C₂×C₇	semidirect
152. #2873	C₂×C₇	semidirect
237. #501	C₂²×C₇	semidirect
238. #499	C₂²×C₇	semidirect
239. #497	C₂²×C₇	semidirect
240. #494	C₂²×C₇	semidirect
241. #492	C₂²×C₇	semidirect
242. #491	C₂²×C₇	semidirect
243. #491	C₂²×C₇	semidirect
244. #490	C₂²×C₇	semidirect
245. #488	C₂²×C₇	semidirect
246. #487	C₂²×C₇	semidirect
251. #441	C₂²×C₇	semidirect
252. #440	C₂²×C₇	semidirect
253. C₄×Q₁₆	C₂²×C₇	semidirect
254. C₄×SD₁₆	C₂²×C₇	semidirect
499. C₇	#60705	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×#3455	2
C₂×C₇×#541	2
C₇×#3178	2
C₇×#3177	2
C₇×#3174	2
C₇×#3166	2
C₇×#3152	2
C₇×#3142	2
C₇×#3139	2
C₇×#3056	2
C₄×C₇×#137	2
C₇×#2873	2
#60705	7

Subgroups — C7×#60705 #13237921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60705 #13237921