Subgroups — C₇×#60713 #13237929

Order 1792.

1878 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3510	C₂	1	2	yes	maximal, complemented
2	C₇×#3183	C₂	1	4	yes	maximal, complemented
3	C₇×#3175	C₂	1	4	yes	maximal, complemented
4	C₇×#3173	C₂	1	4	yes	maximal, complemented
5	C₇×#3168	C₂	1	4	yes	maximal, complemented
6	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#3140	C₂	1	2	yes	maximal, complemented
9	C₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#3058	C₂	1	1	yes	maximal, characteristic
11	C₇×#3057	C₂	1	2	yes	maximal, complemented
12	C₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#2949	C₂	1	2	yes	maximal, complemented
14	C₇×#2948	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇×SD₁₆	C₂²	1	2	yes	—
18	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×#131	C₂²	1	1	yes	characteristic
20	C₂×C₇×#131	C₂²	1	1	yes	characteristic
21	C₇×#559	—	2	4	no	—
22	C₇×#556	—	2	4	no	—
23	C₇×#551	—	2	8	no	—
24	C₇×#542	C₂²	1	8	yes	complemented
25	C₇×#536	C₂²	1	2	yes	—
26	C₇×#535	C₂²	1	2	yes	—
27	C₇×#533	C₂²	1	2	yes	—
28	C₇×#529	C₂²	1	8	yes	complemented
29	C₂×C₇×#124	C₂²	1	2	yes	—
30	C₇×#519	C₂²	1	2	yes	—
31	C₇×#518	C₂²	1	2	yes	—
32	C₄×C₇×#41	C₂²	1	1	yes	characteristic
33	C₇×#500	C₂²	1	2	yes	complemented
34	C₇×#499	C₂²	1	2	yes	complemented
35	C₇×#499	C₂²	1	2	yes	complemented
36	C₇×#498	C₂²	1	2	yes	complemented
37	C₇×#495	C₂²	1	2	yes	—
38	C₇×#495	C₂²	1	2	yes	—
39	C₇×#495	C₂²	1	2	yes	—
40	C₇×#495	C₂²	1	2	yes	—
41	C₇×#489	C₂²	1	8	yes	complemented
42	C₇×#488	C₂²	1	8	yes	complemented
43	C₇×#470	C₂²	1	4	yes	complemented
44	C₇×#469	C₂²	1	4	yes	—
45	C₇×#468	C₂²	1	8	yes	complemented
46	C₇×#467	C₂²	1	4	yes	complemented
47	C₇×#466	C₂²	1	4	yes	—
48	C₇×#465	C₂²	1	8	yes	complemented
49	C₇×#442	C₂²	1	4	yes	—
50	C₇×#441	C₂²	1	4	yes	complemented
51	C₇×#440	C₂²	1	8	yes	complemented
52	C₄×C₇×Q₁₆	C₂²	1	4	yes	complemented
53	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
54	C₄×C₇×D₈	C₂²	1	4	yes	—
55	C₇×#427	C₂²	1	1	yes	characteristic
56	C₇×#427	C₂²	1	2	yes	—
57	C₇×#427	C₂²	1	1	yes	characteristic
58	C₇×#419	C₂²	1	2	yes	—
59	C₇×#419	C₂²	1	2	yes	—
60	C₇×#418	C₂²	1	2	yes	—
61	C₇×#418	C₂²	1	2	yes	—
62	C₇×#405	C₂²	1	1	yes	characteristic
63	C₇×#405	C₂²	1	1	yes	characteristic
64	C₇×#405	C₂²	1	2	yes	complemented
65	C₇×#405	C₂²	1	2	yes	complemented
66	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
67	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
68	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
69	C₂×C₄×C₇×C₈	C₂²	1	2	yes	complemented
70	#60713	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×C₇×#41	C₂³	1	1	yes	characteristic
72	C₂²×C₇×Q₈	C₂³	1	2	yes	—
73	C₂²×C₇×D₄	C₂³	1	2	yes	—
74	C₂×C₇×Q₁₆	—	2	4	no	—
75	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
76	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
77	C₂×C₇×SD₁₆	—	2	8	no	—
78	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
79	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
80	C₂×C₇×D₈	—	2	4	no	—
81	C₂×C₇×D₈	C₂³	1	2	yes	—
82	C₂×C₇×D₈	C₂³	1	2	yes	—
83	C₇×#131	—	2	4	no	—
84	C₇×#131	—	2	4	no	—
85	C₂×C₇×M₁₆	C₂³	1	2	yes	—
86	C₂×C₇×M₁₆	C₂³	1	2	yes	—
87	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
89	C₂²×C₇×C₈	C₂³	1	2	yes	—
90	C₂²×C₇×C₈	C₂³	1	2	yes	—
91	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
92	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
93	C₇×C₄⋊Q₈	—	2	2	no	—
94	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
95	C₇×#127	—	2	2	no	—
96	C₇×#126	—	2	8	no	—
97	C₇×#126	—	2	8	no	—
98	C₇×#125	C₂³	1	4	yes	—
99	C₇×#124	—	2	2	no	—
100	C₇×#124	—	2	2	no	—
101	C₇×#124	—	2	8	no	—
102	C₇×#124	—	2	4	no	—
103	C₇×#124	C₂³	1	4	yes	—
104	C₇×#124	C₂³	1	4	yes	—
105	C₇×#123	—	2	4	no	—
106	C₇×#123	—	2	8	no	—
107	C₇×#123	—	2	8	no	—
108	C₇×C₂²⋊Q₈	—	2	4	no	—
109	C₇×C₂²⋊Q₈	—	2	8	no	—
110	C₇×C₂²⋊Q₈	C₂³	1	8	yes	—
111	C₇×#121	C₂³	1	8	yes	—
112	C₇×#120	—	2	4	no	—
113	C₄×C₇×Q₈	—	2	4	no	—
114	C₄×C₇×Q₈	—	2	4	no	—
115	C₄×C₇×Q₈	D₄	1	2	yes	—
116	C₄×C₇×Q₈	C₂³	1	8	yes	—
117	C₄×C₇×D₄	—	2	8	no	—
118	C₄×C₇×D₄	—	2	4	no	—
119	C₄×C₇×D₄	D₄	1	4	yes	—
120	C₄×C₇×D₄	D₄	1	2	yes	—
121	C₄×C₇×D₄	C₂³	1	8	yes	—
122	C₇×#117	—	2	4	no	—
123	C₇×#117	—	2	8	no	—
124	C₇×#117	C₂³	1	2	yes	—
125	C₇×#117	C₂³	1	2	yes	—
126	C₇×#117	C₂³	1	1	yes	characteristic
127	C₇×#117	C₂³	1	2	yes	—
128	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
129	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
130	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₇	C₂³	1	2	yes	—
132	C₇×#107	C₂³	1	2	yes	—
133	C₇×#107	C₂³	1	2	yes	—
134	C₇×#107	C₂³	1	4	yes	—
135	C₇×#106	C₂³	1	2	yes	—
136	C₇×#106	C₂³	1	2	yes	—
137	C₇×#106	C₂³	1	4	yes	—
138	C₇×#103	C₂³	1	8	yes	—
139	C₇×#103	C₂³	1	8	yes	—
140	C₇×#102	C₂³	1	8	yes	—
141	C₇×#102	C₂³	1	8	yes	—
142	C₇×#97	C₂³	1	2	yes	—
143	C₇×#97	C₂³	1	2	yes	—
144	C₇×#97	C₂³	1	2	yes	—
145	C₄×C₇×C₈	C₂³	1	2	yes	—
146	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
147	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
148	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
149	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
150	C₄×C₇×C₈	C₂³	1	2	yes	—
151	C₄×C₇×C₈	C₂³	1	2	yes	—
152	#3510	C₂×C₇	1	2	yes	complemented
153	#3183	C₂×C₇	1	4	yes	complemented
154	#3175	C₂×C₇	1	4	yes	complemented
155	#3173	C₂×C₇	1	4	yes	complemented
156	#3168	C₂×C₇	1	4	yes	complemented
157	#3160	C₂×C₇	1	1	yes	characteristic, complemented
158	C₂×#495	C₂×C₇	1	1	yes	characteristic, complemented
159	#3140	C₂×C₇	1	2	yes	complemented
160	#3059	C₂×C₇	1	1	yes	characteristic, complemented
161	#3058	C₂×C₇	1	1	yes	characteristic
162	#3057	C₂×C₇	1	2	yes	complemented
163	#2950	C₂×C₇	1	1	yes	characteristic, complemented
164	#2949	C₂×C₇	1	2	yes	complemented
165	#2948	C₂×C₇	1	1	yes	characteristic, complemented
166	C₄×#131	C₂×C₇	1	1	yes	characteristic, complemented
167	C₂⁴×C₇	—	2	2	no	—
168	C₇×#41	—	2	2	no	—
169	C₇×#41	—	2	2	no	—
170	C₂×C₇×Q₈	—	2	2	no	—
171	C₂×C₇×Q₈	—	2	4	no	—
172	C₂×C₇×Q₈	—	2	4	no	—
173	C₂×C₇×Q₈	—	2	4	no	—
174	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
175	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
176	C₂×C₇×D₄	—	2	8	no	—
177	C₂×C₇×D₄	—	2	2	no	—
178	C₂×C₇×D₄	—	2	4	no	—
179	C₂×C₇×D₄	—	2	4	no	—
180	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
181	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
182	C₂×C₇×D₄	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₇	—	2	2	no	—
184	C₂²×C₄×C₇	—	2	8	no	—
185	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
189	C₇×Q₁₆	—	2	4	no	—
190	C₇×Q₁₆	—	2	4	no	—
191	C₇×SD₁₆	—	2	8	no	—
192	C₇×SD₁₆	—	2	8	no	—
193	C₇×D₈	—	2	4	no	—
194	C₇×D₈	—	2	4	no	—
195	C₇×M₁₆	—	2	2	no	—
196	C₇×M₁₆	—	2	2	no	—
197	C₇×M₁₆	—	2	4	no	—
198	C₇×M₁₆	—	2	4	no	—
199	C₂×C₇×C₈	—	2	4	no	—
200	C₂×C₇×C₈	—	2	4	no	—
201	C₂×C₇×C₈	#41	1	4	yes	—
202	C₂×C₇×C₈	#41	1	4	yes	—
203	C₂×C₇×C₈	C₂⁴	1	2	yes	—
204	C₂×C₇×C₈	C₂⁴	1	2	yes	—
205	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
206	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
207	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
208	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
209	C₇×C₄⋊C₄	—	2	4	no	—
210	C₇×C₄⋊C₄	—	2	8	no	—
211	C₇×C₄⋊C₄	—	2	8	no	—
212	C₇×C₄⋊C₄	—	2	4	no	—
213	C₇×C₄⋊C₄	—	2	4	no	—
214	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
215	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
216	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
217	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
218	C₇×C₂²⋊C₄	—	2	2	no	—
219	C₇×C₂²⋊C₄	—	2	2	no	—
220	C₇×C₂²⋊C₄	—	2	8	no	—
221	C₇×C₂²⋊C₄	—	2	8	no	—
222	C₇×C₂²⋊C₄	—	2	8	no	—
223	C₇×C₂²⋊C₄	—	2	8	no	—
224	C₇×C₂²⋊C₄	—	2	8	no	—
225	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
226	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
227	C₄²×C₇	—	2	8	no	—
228	C₄²×C₇	—	2	2	no	—
229	C₄²×C₇	—	2	2	no	—
230	C₄²×C₇	C₂×D₄	1	4	yes	—
231	C₄²×C₇	C₂×D₄	1	2	yes	—
232	C₄²×C₇	C₂⁴	1	1	yes	characteristic
233	C₄²×C₇	C₂⁴	1	2	yes	—
234	C₄²×C₇	C₂⁴	1	1	yes	characteristic
235	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
236	C₂²×SD₁₆	C₂²×C₇	1	2	yes	—
237	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
238	C₂×#131	C₂²×C₇	1	1	yes	characteristic
239	C₂×#131	C₂²×C₇	1	1	yes	characteristic
240	#559	—	2	4	no	—
241	#556	—	2	4	no	—
242	#551	—	2	8	no	—
243	#542	C₂²×C₇	1	8	yes	complemented
244	#536	C₂²×C₇	1	2	yes	—
245	#535	C₂²×C₇	1	2	yes	—
246	#533	C₂²×C₇	1	2	yes	—
247	#529	C₂²×C₇	1	8	yes	complemented
248	C₂×#124	C₂²×C₇	1	2	yes	—
249	#519	C₂²×C₇	1	2	yes	—
250	#518	C₂²×C₇	1	2	yes	—
251	C₄×#41	C₂²×C₇	1	1	yes	characteristic
252	#500	C₂²×C₇	1	2	yes	complemented
253	#499	C₂²×C₇	1	2	yes	complemented
254	#499	C₂²×C₇	1	2	yes	complemented
255	#498	C₂²×C₇	1	2	yes	complemented
256	#495	C₂²×C₇	1	2	yes	—
257	#495	C₂²×C₇	1	2	yes	—
258	#495	C₂²×C₇	1	2	yes	—
259	#495	C₂²×C₇	1	2	yes	—
260	#489	C₂²×C₇	1	8	yes	complemented
261	#488	C₂²×C₇	1	8	yes	complemented
262	#470	C₂²×C₇	1	4	yes	complemented
263	#469	C₂²×C₇	1	4	yes	—
264	#468	C₂²×C₇	1	8	yes	complemented
265	#467	C₂²×C₇	1	4	yes	complemented
266	#466	C₂²×C₇	1	4	yes	—
267	#465	C₂²×C₇	1	8	yes	complemented
268	#442	C₂²×C₇	1	4	yes	—
269	#441	C₂²×C₇	1	4	yes	complemented
270	#440	C₂²×C₇	1	8	yes	complemented
271	C₄×Q₁₆	C₂²×C₇	1	4	yes	complemented
272	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
273	C₄×D₈	C₂²×C₇	1	4	yes	—
274	#427	C₂²×C₇	1	1	yes	characteristic
275	#427	C₂²×C₇	1	2	yes	—
276	#427	C₂²×C₇	1	1	yes	characteristic
277	#419	C₂²×C₇	1	2	yes	—
278	#419	C₂²×C₇	1	2	yes	—
279	#418	C₂²×C₇	1	2	yes	—
280	#418	C₂²×C₇	1	2	yes	—
281	#405	C₂²×C₇	1	1	yes	characteristic
282	#405	C₂²×C₇	1	1	yes	characteristic
283	#405	C₂²×C₇	1	2	yes	complemented
284	#405	C₂²×C₇	1	2	yes	complemented
285	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
286	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
287	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
288	C₂×C₄×C₈	C₂²×C₇	1	2	yes	complemented
289	C₂³×C₇	—	2	4	no	—
290	C₂³×C₇	—	4	4	no	—
291	C₂³×C₇	—	8	4	no	—
292	C₂³×C₇	C₂²×D₄	1	2	yes	—
293	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
294	C₇×Q₈	—	2	2	no	—
295	C₇×Q₈	—	2	4	no	—
296	C₇×Q₈	—	2	4	no	—
297	C₇×Q₈	—	4	4	no	—
298	C₇×Q₈	—	8	4	no	—
299	C₇×D₄	—	2	4	no	—
300	C₇×D₄	—	2	2	no	—
301	C₇×D₄	—	2	4	no	—
302	C₇×D₄	—	2	4	no	—
303	C₇×D₄	—	4	4	no	—
304	C₇×D₄	—	8	8	no	—
305	C₂×C₄×C₇	—	2	2	no	—
306	C₂×C₄×C₇	—	2	2	no	—
307	C₂×C₄×C₇	—	2	4	no	—
308	C₂×C₄×C₇	—	2	4	no	—
309	C₂×C₄×C₇	—	2	4	no	—
310	C₂×C₄×C₇	—	4	4	no	—
311	C₂×C₄×C₇	—	4	2	no	—
312	C₂×C₄×C₇	—	4	4	no	—
313	C₂×C₄×C₇	—	8	8	no	—
314	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
315	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
316	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
317	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
318	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₇	C₂×#41	1	2	yes	—
320	C₂×C₄×C₇	C₂×#41	1	2	yes	—
321	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
322	C₇×C₈	—	2	2	no	—
323	C₇×C₈	—	2	2	no	—
324	C₇×C₈	C₂×#41	1	2	yes	—
325	C₇×C₈	C₂×#41	1	2	yes	—
326	C₇×C₈	C₂×#41	1	2	yes	—
327	C₇×C₈	C₂×#41	1	2	yes	—
328	C₂×#41	C₂³×C₇	1	1	yes	characteristic
329	C₂²×Q₈	C₂³×C₇	1	2	yes	—
330	C₂²×D₄	C₂³×C₇	1	2	yes	—
331	C₂×Q₁₆	—	2	4	no	—
332	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
333	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
334	C₂×SD₁₆	—	2	8	no	—
335	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
336	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
337	C₂×D₈	—	2	4	no	—
338	C₂×D₈	C₂³×C₇	1	2	yes	—
339	C₂×D₈	C₂³×C₇	1	2	yes	—
340	#131	—	2	4	no	—
341	#131	—	2	4	no	—
342	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
343	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
344	C₂×M₁₆	C₂³×C₇	1	2	yes	—
345	C₂×M₁₆	C₂³×C₇	1	2	yes	—
346	C₂²×C₈	C₂³×C₇	1	2	yes	—
347	C₂²×C₈	C₂³×C₇	1	2	yes	—
348	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
349	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
350	C₄⋊Q₈	—	2	2	no	—
351	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
352	#127	—	2	2	no	—
353	#126	—	2	8	no	—
354	#126	—	2	8	no	—
355	#125	C₂³×C₇	1	4	yes	—
356	#124	—	2	2	no	—
357	#124	—	2	2	no	—
358	#124	—	2	8	no	—
359	#124	—	2	4	no	—
360	#124	C₂³×C₇	1	4	yes	—
361	#124	C₂³×C₇	1	4	yes	—
362	#123	—	2	4	no	—
363	#123	—	2	8	no	—
364	#123	—	2	8	no	—
365	C₂²⋊Q₈	—	2	4	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	C₂³×C₇	1	8	yes	—
368	#121	C₂³×C₇	1	8	yes	—
369	#120	—	2	4	no	—
370	C₄×Q₈	—	2	4	no	—
371	C₄×Q₈	—	2	4	no	—
372	C₄×Q₈	C₇×D₄	1	2	yes	—
373	C₄×Q₈	C₂³×C₇	1	8	yes	—
374	C₄×D₄	—	2	8	no	—
375	C₄×D₄	—	2	4	no	—
376	C₄×D₄	C₇×D₄	1	4	yes	—
377	C₄×D₄	C₇×D₄	1	2	yes	—
378	C₄×D₄	C₂³×C₇	1	8	yes	—
379	#117	—	2	4	no	—
380	#117	—	2	8	no	—
381	#117	C₂³×C₇	1	2	yes	—
382	#117	C₂³×C₇	1	2	yes	—
383	#117	C₂³×C₇	1	1	yes	characteristic
384	#117	C₂³×C₇	1	2	yes	—
385	C₂×C₂²⋊C₄	—	2	4	no	—
386	C₂×C₂²⋊C₄	—	2	4	no	—
387	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₇	1	2	yes	—
389	#107	C₂³×C₇	1	2	yes	—
390	#107	C₂³×C₇	1	2	yes	—
391	#107	C₂³×C₇	1	4	yes	—
392	#106	C₂³×C₇	1	4	yes	—
393	#106	C₂³×C₇	1	2	yes	—
394	#106	C₂³×C₇	1	2	yes	—
395	#103	C₂³×C₇	1	8	yes	—
396	#103	C₂³×C₇	1	8	yes	—
397	#102	C₂³×C₇	1	8	yes	—
398	#102	C₂³×C₇	1	8	yes	—
399	#97	C₂³×C₇	1	2	yes	—
400	#97	C₂³×C₇	1	2	yes	—
401	#97	C₂³×C₇	1	2	yes	—
402	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
404	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
405	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
406	C₄×C₈	C₂³×C₇	1	2	yes	—
407	C₄×C₈	C₂³×C₇	1	2	yes	—
408	C₄×C₈	C₂³×C₇	1	2	yes	—
409	C₂²×C₇	—	2	2	no	—
410	C₂²×C₇	—	2	4	no	—
411	C₂²×C₇	—	2	2	no	—
412	C₂²×C₇	—	4	8	no	—
413	C₂²×C₇	—	4	4	no	—
414	C₂²×C₇	—	16	4	no	—
415	C₂²×C₇	#532	1	2	yes	—
416	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
417	C₄×C₇	—	2	2	no	—
418	C₄×C₇	—	2	4	no	—
419	C₄×C₇	—	2	2	no	—
420	C₄×C₇	—	2	2	no	—
421	C₄×C₇	—	4	4	no	—
422	C₄×C₇	—	4	4	no	—
423	C₄×C₇	—	4	4	no	—
424	C₄×C₇	#532	1	2	yes	—
425	C₄×C₇	C₂²×#41	1	1	yes	characteristic
426	C₄×C₇	C₂²×#41	1	1	yes	characteristic
427	C₂⁴	—	2	2	no	—
428	#41	—	2	2	no	—
429	#41	—	2	2	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	4	no	—
433	C₂×Q₈	—	2	4	no	—
434	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
435	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	4	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
441	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
442	C₂×D₄	C₂⁴×C₇	1	4	yes	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	8	no	—
445	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
446	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
448	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
449	Q₁₆	—	2	4	no	—
450	Q₁₆	—	2	4	no	—
451	SD₁₆	—	2	8	no	—
452	SD₁₆	—	2	8	no	—
453	D₈	—	2	4	no	—
454	D₈	—	2	4	no	—
455	M₁₆	—	2	2	no	—
456	M₁₆	—	2	2	no	—
457	M₁₆	—	2	4	no	—
458	M₁₆	—	2	4	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	2	4	no	—
461	C₂×C₈	C₇×#41	1	4	yes	—
462	C₂×C₈	C₇×#41	1	4	yes	—
463	C₂×C₈	C₂⁴×C₇	1	2	yes	—
464	C₂×C₈	C₂⁴×C₇	1	2	yes	—
465	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
466	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
467	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
468	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	8	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
475	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
476	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
477	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
478	C₂²⋊C₄	—	2	2	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
486	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
487	C₄²	—	2	8	no	—
488	C₄²	—	2	2	no	—
489	C₄²	—	2	2	no	—
490	C₄²	C₂×C₇×D₄	1	4	yes	—
491	C₄²	C₂×C₇×D₄	1	2	yes	—
492	C₄²	C₂⁴×C₇	1	1	yes	characteristic
493	C₄²	C₂⁴×C₇	1	2	yes	—
494	C₄²	C₂⁴×C₇	1	1	yes	characteristic
495	C₂×C₇	—	2	2	no	—
496	C₂×C₇	—	4	2	no	—
497	C₂×C₇	—	8	4	no	—
498	C₂×C₇	C₂×#532	1	1	yes	characteristic
499	C₂×C₇	#3584	1	2	yes	—
500	C₂³	—	2	4	no	—
501	C₂³	—	4	4	no	—
502	C₂³	—	8	4	no	—
503	C₂³	C₂²×C₇×D₄	1	2	yes	—
504	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
505	Q₈	—	2	2	no	—
506	Q₈	—	2	4	no	—
507	Q₈	—	2	4	no	—
508	Q₈	—	4	4	no	—
509	Q₈	—	8	4	no	—
510	D₄	—	2	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	4	no	—
513	D₄	—	2	2	no	—
514	D₄	—	4	4	no	—
515	D₄	—	8	8	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	8	8	no	—
525	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
526	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
527	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
528	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
529	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
530	C₂×C₄	C₂×C₇×#41	1	2	yes	—
531	C₂×C₄	C₂×C₇×#41	1	2	yes	—
532	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
533	C₈	—	2	2	no	—
534	C₈	—	2	2	no	—
535	C₈	C₂×C₇×#41	1	2	yes	—
536	C₈	C₂×C₇×#41	1	2	yes	—
537	C₈	C₂×C₇×#41	1	2	yes	—
538	C₈	C₂×C₇×#41	1	2	yes	—
539	C₇	#60713	1	1	yes	characteristic, direct product factor, Sylow
540	C₂²	—	2	2	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	4	8	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	16	4	no	—
546	C₂²	C₇×#532	1	2	yes	—
547	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
548	C₄	—	2	2	no	—
549	C₄	—	2	4	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	4	no	—
554	C₄	—	4	4	no	—
555	C₄	C₇×#532	1	2	yes	—
556	C₄	C₂²×C₇×#41	1	1	yes	characteristic
557	C₄	C₂²×C₇×#41	1	1	yes	characteristic
558	C₂	—	2	2	no	—
559	C₂	—	4	2	no	—
560	C₂	—	8	4	no	—
561	C₂	C₂×C₇×#532	1	1	yes	characteristic
562	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3058	C₂
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×#3510	C₂	semidirect
2. C₇×#3183	C₂	semidirect
3. C₇×#3175	C₂	semidirect
4. C₇×#3173	C₂	semidirect
5. C₇×#3168	C₂	semidirect
6. C₇×#3160	C₂	semidirect
7. C₂×C₇×#495	C₂	semidirect
8. C₇×#3140	C₂	semidirect
9. C₇×#3059	C₂	semidirect
11. C₇×#3057	C₂	semidirect
12. C₇×#2950	C₂	semidirect
13. C₇×#2949	C₂	semidirect
14. C₇×#2948	C₂	semidirect
15. C₄×C₇×#131	C₂	semidirect
24. C₇×#542	C₂²	semidirect
28. C₇×#529	C₂²	semidirect
33. C₇×#500	C₂²	semidirect
34. C₇×#499	C₂²	semidirect
35. C₇×#499	C₂²	semidirect
36. C₇×#498	C₂²	semidirect
41. C₇×#489	C₂²	semidirect
42. C₇×#488	C₂²	semidirect
43. C₇×#470	C₂²	semidirect
45. C₇×#468	C₂²	semidirect
46. C₇×#467	C₂²	semidirect
48. C₇×#465	C₂²	semidirect
50. C₇×#441	C₂²	semidirect
51. C₇×#440	C₂²	semidirect
52. C₄×C₇×Q₁₆	C₂²	semidirect
53. C₄×C₇×SD₁₆	C₂²	semidirect
64. C₇×#405	C₂²	semidirect
65. C₇×#405	C₂²	semidirect
67. C₄×C₇×M₁₆	C₂²	semidirect
69. C₂×C₄×C₇×C₈	C₂²	semidirect
70. #60713	C₇	direct (N × H)
152. #3510	C₂×C₇	semidirect
153. #3183	C₂×C₇	semidirect
154. #3175	C₂×C₇	semidirect
155. #3173	C₂×C₇	semidirect
156. #3168	C₂×C₇	semidirect
157. #3160	C₂×C₇	semidirect
158. C₂×#495	C₂×C₇	semidirect
159. #3140	C₂×C₇	semidirect
160. #3059	C₂×C₇	semidirect
162. #3057	C₂×C₇	semidirect
163. #2950	C₂×C₇	semidirect
164. #2949	C₂×C₇	semidirect
165. #2948	C₂×C₇	semidirect
166. C₄×#131	C₂×C₇	semidirect
243. #542	C₂²×C₇	semidirect
247. #529	C₂²×C₇	semidirect
252. #500	C₂²×C₇	semidirect
253. #499	C₂²×C₇	semidirect
254. #499	C₂²×C₇	semidirect
255. #498	C₂²×C₇	semidirect
260. #489	C₂²×C₇	semidirect
261. #488	C₂²×C₇	semidirect
262. #470	C₂²×C₇	semidirect
264. #468	C₂²×C₇	semidirect
265. #467	C₂²×C₇	semidirect
267. #465	C₂²×C₇	semidirect
269. #441	C₂²×C₇	semidirect
270. #440	C₂²×C₇	semidirect
271. C₄×Q₁₆	C₂²×C₇	semidirect
272. C₄×SD₁₆	C₂²×C₇	semidirect
283. #405	C₂²×C₇	semidirect
284. #405	C₂²×C₇	semidirect
286. C₄×M₁₆	C₂²×C₇	semidirect
288. C₂×C₄×C₈	C₂²×C₇	semidirect
539. C₇	#60713	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×#3510	2
C₇×#3183	2
C₇×#3175	2
C₇×#3173	2
C₇×#3168	2
C₇×#3160	2
C₂×C₇×#495	2
C₇×#3140	2
C₇×#3059	2
C₇×#3058	2
C₇×#3057	2
C₇×#2950	2
C₇×#2949	2
C₇×#2948	2
C₄×C₇×#131	2
#60713	7

Subgroups — C7×#60713 #13237929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60713 #13237929