Subgroups — C₇×#7885 #13185101

Order 1792.

666 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2920	C₂	1	1	yes	maximal, characteristic
2	C₇×#2920	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#1278	C₂	1	4	yes	maximal, complemented
5	C₂³×C₇×C₈	—	2	1	no	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₇×#432	—	2	1	no	—
8	C₇×#432	—	2	1	no	—
9	C₇×#432	—	2	1	no	—
10	C₇×#432	C₄	1	2	yes	—
11	C₇×#431	—	2	1	no	—
12	C₇×#431	—	2	1	no	—
13	C₇×#431	—	2	1	no	—
14	C₇×#431	C₂²	1	2	yes	—
15	C₇×#423	C₄	1	2	yes	—
16	C₇×#423	C₄	1	2	yes	—
17	C₇×#407	—	2	1	no	—
18	C₇×#407	C₄	1	2	yes	—
19	C₄×C₇×M₁₆	—	2	1	no	—
20	C₄×C₇×M₁₆	C₂²	1	2	yes	—
21	C₂×C₄×C₇×C₈	—	2	1	no	—
22	C₂×C₄×C₇×C₈	—	2	1	no	—
23	C₂×C₄×C₇×C₈	—	2	1	no	—
24	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
25	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
26	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
27	#7885	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
31	C₂×C₇×M₁₆	—	2	2	no	—
32	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₇×C₈	—	2	8	no	—
34	C₂²×C₇×C₈	—	2	2	no	—
35	C₂²×C₇×C₈	—	2	2	no	—
36	C₂²×C₇×C₈	—	2	2	no	—
37	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×C₄⋊C₈	—	2	2	no	—
50	C₇×C₄⋊C₈	—	2	2	no	—
51	C₇×C₄⋊C₈	—	2	2	no	—
52	C₇×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₇×C₂²⋊C₈	—	2	2	no	—
54	C₇×C₂²⋊C₈	—	2	2	no	—
55	C₇×C₂²⋊C₈	—	2	2	no	—
56	C₇×C₂²⋊C₈	—	2	2	no	—
57	C₇×#97	—	2	2	no	—
58	C₇×#97	—	2	2	no	—
59	C₄×C₇×C₈	—	2	2	no	—
60	C₄×C₇×C₈	—	2	2	no	—
61	C₄×C₇×C₈	—	2	2	no	—
62	C₄×C₇×C₈	—	2	2	no	—
63	C₄×C₇×C₈	—	2	2	no	—
64	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
65	#2920	C₂×C₇	1	1	yes	characteristic
66	#2920	C₂×C₇	1	1	yes	characteristic
67	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
68	#1278	C₂×C₇	1	4	yes	complemented
69	C₂⁴×C₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇	—	2	4	no	—
71	C₂²×C₄×C₇	—	2	4	no	—
72	C₂²×C₄×C₇	—	2	4	no	—
73	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
91	C₇×M₁₆	—	2	8	no	—
92	C₇×M₁₆	—	4	4	no	—
93	C₂×C₇×C₈	—	2	8	no	—
94	C₂×C₇×C₈	—	2	8	no	—
95	C₂×C₇×C₈	—	2	8	no	—
96	C₂×C₇×C₈	—	2	4	no	—
97	C₂×C₇×C₈	—	2	4	no	—
98	C₂×C₇×C₈	—	2	4	no	—
99	C₄²×C₇	C₄²	1	1	yes	characteristic
100	C₄²×C₇	C₄²	1	1	yes	characteristic
101	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
106	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
107	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₇	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
115	C₂³×C₈	—	2	1	no	—
116	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
117	#432	—	2	1	no	—
118	#432	—	2	1	no	—
119	#432	—	2	1	no	—
120	#432	C₄×C₇	1	2	yes	—
121	#431	—	2	1	no	—
122	#431	—	2	1	no	—
123	#431	—	2	1	no	—
124	#431	C₂²×C₇	1	2	yes	—
125	#423	C₄×C₇	1	2	yes	—
126	#423	C₄×C₇	1	2	yes	—
127	#407	—	2	1	no	—
128	#407	C₄×C₇	1	2	yes	—
129	C₄×M₁₆	—	2	1	no	—
130	C₄×M₁₆	C₂²×C₇	1	2	yes	—
131	C₂×C₄×C₈	—	2	1	no	—
132	C₂×C₄×C₈	—	2	1	no	—
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
135	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
137	C₂³×C₇	—	2	2	no	—
138	C₂³×C₇	—	2	2	no	—
139	C₂³×C₇	—	2	2	no	—
140	C₂³×C₇	#95	1	1	yes	characteristic
141	C₂³×C₇	#95	1	1	yes	characteristic
142	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	2	2	no	—
149	C₂×C₄×C₇	—	2	2	no	—
150	C₂×C₄×C₇	—	2	2	no	—
151	C₂×C₄×C₇	—	2	2	no	—
152	C₂×C₄×C₇	—	2	2	no	—
153	C₂×C₄×C₇	—	2	2	no	—
154	C₂×C₄×C₇	—	2	2	no	—
155	C₂×C₄×C₇	—	2	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	2	2	no	—
158	C₂×C₄×C₇	#95	1	1	yes	characteristic
159	C₂×C₄×C₇	#95	1	1	yes	characteristic
160	C₂×C₄×C₇	#95	1	1	yes	characteristic
161	C₂×C₄×C₇	#95	1	1	yes	characteristic
162	C₂×C₄×C₇	#95	1	1	yes	characteristic
163	C₂×C₄×C₇	#95	1	1	yes	characteristic
164	C₂×C₄×C₇	#104	1	4	yes	—
165	C₂×C₄×C₇	#108	1	4	yes	—
166	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₇	#131	1	4	yes	—
173	C₇×C₈	—	2	8	no	—
174	C₇×C₈	—	4	4	no	—
175	C₇×C₈	—	4	4	no	—
176	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
177	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
178	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
181	C₂²×C₈	—	2	8	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
186	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
190	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
191	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
192	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
193	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
194	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
195	C₂×C₄²	C₇×Q₈	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	C₂×C₄×C₇	1	4	yes	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₂²⋊C₈	—	2	2	no	—
205	#97	—	2	2	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	2	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
213	C₂²×C₇	—	2	8	no	—
214	C₂²×C₇	—	2	1	no	—
215	C₂²×C₇	—	2	1	no	—
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	1	no	—
219	C₂²×C₇	—	2	1	no	—
220	C₂²×C₇	#339	1	2	yes	—
221	C₂²×C₇	#341	1	2	yes	—
222	C₂²×C₇	C₂×#95	1	1	yes	characteristic
223	C₂²×C₇	#424	1	2	yes	—
224	C₄×C₇	—	2	1	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	#339	1	2	yes	—
231	C₄×C₇	#341	1	2	yes	—
232	C₄×C₇	#405	1	2	yes	—
233	C₄×C₇	#408	1	2	yes	—
234	C₄×C₇	C₂×#104	1	2	yes	—
235	C₄×C₇	C₂×#108	1	2	yes	—
236	C₂⁴	C₇×C₄⋊C₄	1	1	yes	characteristic
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
241	C₂²×C₄	C₄²×C₇	1	1	yes	characteristic
242	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	C₇×C₄⋊C₄	1	1	yes	characteristic
255	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
258	M₁₆	—	2	8	no	—
259	M₁₆	—	4	4	no	—
260	C₂×C₈	—	2	8	no	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	—	2	8	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	4	no	—
266	C₄²	C₄²×C₇	1	1	yes	characteristic
267	C₄²	C₄²×C₇	1	1	yes	characteristic
268	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
269	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
270	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
271	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
272	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
273	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
274	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
275	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
276	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
277	C₄²	C₇×C₄⋊C₄	1	1	yes	characteristic
278	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
279	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
280	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
281	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
282	C₂×C₇	—	2	2	no	—
283	C₂×C₇	—	2	2	no	—
284	C₂×C₇	—	2	2	no	—
285	C₂×C₇	#1729	1	1	yes	characteristic
286	C₂×C₇	C₂×#339	1	1	yes	characteristic
287	C₂×C₇	C₂×#341	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	C₇×#95	1	1	yes	characteristic
292	C₂³	C₇×#95	1	1	yes	characteristic
293	C₂³	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	C₇×#95	1	1	yes	characteristic
310	C₂×C₄	C₇×#95	1	1	yes	characteristic
311	C₂×C₄	C₇×#95	1	1	yes	characteristic
312	C₂×C₄	C₇×#95	1	1	yes	characteristic
313	C₂×C₄	C₇×#95	1	1	yes	characteristic
314	C₂×C₄	C₇×#95	1	1	yes	characteristic
315	C₂×C₄	C₇×#104	1	4	yes	—
316	C₂×C₄	C₇×#108	1	4	yes	—
317	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
318	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₇×#131	1	4	yes	—
324	C₈	—	2	8	no	—
325	C₈	—	4	4	no	—
326	C₈	—	4	4	no	—
327	C₇	#7885	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²	—	2	8	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	C₇×#339	1	2	yes	—
336	C₂²	C₇×#341	1	2	yes	—
337	C₂²	C₂×C₇×#95	1	1	yes	characteristic
338	C₂²	C₇×#424	1	2	yes	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	C₇×#339	1	2	yes	—
346	C₄	C₇×#341	1	2	yes	—
347	C₄	C₇×#405	1	2	yes	—
348	C₄	C₇×#408	1	2	yes	—
349	C₄	C₂×C₇×#104	1	2	yes	—
350	C₄	C₂×C₇×#108	1	2	yes	—
351	C₂	—	2	2	no	—
352	C₂	—	2	2	no	—
353	C₂	—	2	2	no	—
354	C₂	C₇×#1729	1	1	yes	characteristic
355	C₂	C₂×C₇×#339	1	1	yes	characteristic
356	C₂	C₂×C₇×#341	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	C₂×#95
prime radical	C₂⁴×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1278	C₂	semidirect
27. #7885	C₇	direct (N × H)
68. #1278	C₂×C₇	semidirect
327. C₇	#7885	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×#2920	2
C₂²×C₄×C₇×C₈	2
C₇×#1278	2
#7885	7

Subgroups — C7×#7885 #13185101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#7885 #13185101