Subgroups — C₇×#12787 #13190003

Order 1792.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×#131	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#350	C₂	1	2	yes	maximal
5	C₇×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#131	C₄	1	1	yes	characteristic
7	C₂×C₇×#131	C₂²	1	1	yes	characteristic
8	C₄×C₇×#41	C₄	1	1	yes	characteristic
9	C₂×C₇×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×#405	—	2	1	no	—
12	C₇×#405	—	2	1	no	—
13	C₇×#405	C₄	1	1	yes	characteristic
14	C₇×#405	C₂²	1	1	yes	characteristic
15	C₄×C₇×M₁₆	—	2	1	no	—
16	C₄×C₇×M₁₆	C₄	1	1	yes	characteristic
17	C₂×C₄×C₇×C₈	—	2	1	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₇×C₄⋊C₁₆	—	2	1	no	—
20	C₇×C₄⋊C₁₆	—	2	1	no	—
21	C₇×C₄⋊C₁₆	—	2	1	no	—
22	C₇×C₄⋊C₁₆	—	2	1	no	—
23	C₇×#350	—	2	4	no	—
24	#12787	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×#41	C₂×C₄	1	1	yes	characteristic
26	C₇×#131	—	2	2	no	—
27	C₇×#131	—	2	2	no	—
28	C₇×#131	—	2	1	no	—
29	C₇×#131	—	2	1	no	—
30	C₇×#131	D₄	1	2	yes	—
31	C₇×#131	Q₈	1	2	yes	—
32	C₂×C₇×M₁₆	—	2	1	no	—
33	C₂×C₇×M₁₆	—	2	1	no	—
34	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	—	2	1	no	—
37	C₂²×C₇×C₈	—	2	1	no	—
38	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
41	C₄×C₇×Q₈	C₈	1	1	yes	characteristic
42	C₄×C₇×D₄	—	2	1	no	—
43	C₄×C₇×D₄	—	2	1	no	—
44	C₄×C₇×D₄	C₈	1	1	yes	characteristic
45	C₄×C₇×D₄	C₈	1	1	yes	characteristic
46	C₇×#117	—	2	1	no	—
47	C₇×#117	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇	—	2	1	no	—
49	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
50	C₇×M₃₂	—	2	2	no	—
51	C₇×M₃₂	—	2	2	no	—
52	C₂×C₇×C₁₆	—	2	2	no	—
53	C₂×C₇×C₁₆	—	2	2	no	—
54	C₂×C₇×C₁₆	—	2	2	no	—
55	C₂×C₇×C₁₆	—	2	2	no	—
56	C₇×#97	—	2	1	no	—
57	C₇×#97	—	2	1	no	—
58	C₇×#97	—	2	1	no	—
59	C₄×C₇×C₈	—	2	1	no	—
60	C₄×C₇×C₈	—	2	1	no	—
61	C₄×C₇×C₈	—	2	1	no	—
62	C₄×C₇×C₈	D₄	1	1	yes	characteristic
63	C₄×C₇×C₈	D₄	1	1	yes	characteristic
64	C₄×C₇×C₈	D₄	1	1	yes	characteristic
65	C₄×C₇×C₈	D₄	1	1	yes	characteristic
66	C₄×#131	C₂×C₇	1	1	yes	characteristic
67	#2150	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂×#350	C₂×C₇	1	2	yes	—
70	#1379	C₂×C₇	1	2	yes	complemented
71	C₇×#41	—	2	1	no	—
72	C₇×#41	—	2	1	no	—
73	C₇×#41	C₄⋊C₄	1	2	yes	—
74	C₇×#41	M₁₆	1	2	yes	—
75	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₂×C₇×D₄	—	2	1	no	—
77	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₇	—	2	1	no	—
79	C₂²×C₄×C₇	—	2	1	no	—
80	C₂²×C₄×C₇	—	2	1	no	—
81	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₇×M₁₆	—	2	4	no	—
85	C₇×M₁₆	—	2	1	no	—
86	C₇×M₁₆	—	2	1	no	—
87	C₇×M₁₆	—	2	2	no	—
88	C₇×M₁₆	—	2	2	no	—
89	C₇×M₁₆	C₄⋊C₄	1	2	yes	—
90	C₇×M₁₆	M₁₆	1	2	yes	—
91	C₂×C₇×C₈	—	2	4	no	—
92	C₂×C₇×C₈	—	2	1	no	—
93	C₂×C₇×C₈	—	2	1	no	—
94	C₂×C₇×C₈	—	2	1	no	—
95	C₂×C₇×C₈	—	2	1	no	—
96	C₂×C₇×C₈	—	2	2	no	—
97	C₂×C₇×C₈	—	2	2	no	—
98	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
104	C₂×C₇×C₈	#41	1	1	yes	characteristic
105	C₂×C₇×C₈	#41	1	1	yes	characteristic
106	C₇×C₄⋊C₄	—	2	1	no	—
107	C₇×C₄⋊C₄	—	2	1	no	—
108	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
109	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
110	C₇×C₂²⋊C₄	—	2	1	no	—
111	C₇×C₂²⋊C₄	—	2	1	no	—
112	C₇×C₂²⋊C₄	—	2	1	no	—
113	C₄²×C₇	—	2	1	no	—
114	C₄²×C₇	—	2	1	no	—
115	C₄²×C₇	—	2	1	no	—
116	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
117	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
118	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
119	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
120	C₇×C₁₆	—	4	2	no	—
121	C₇×C₁₆	—	4	2	no	—
122	C₂×#131	C₄×C₇	1	1	yes	characteristic
123	C₂×#131	C₂²×C₇	1	1	yes	characteristic
124	C₄×#41	C₄×C₇	1	1	yes	characteristic
125	C₂×M₃₂	C₂²×C₇	1	2	yes	—
126	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
127	#405	—	2	1	no	—
128	#405	—	2	1	no	—
129	#405	C₄×C₇	1	1	yes	characteristic
130	#405	C₂²×C₇	1	1	yes	characteristic
131	C₄×M₁₆	—	2	1	no	—
132	C₄×M₁₆	C₄×C₇	1	1	yes	characteristic
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
135	C₄⋊C₁₆	—	2	1	no	—
136	C₄⋊C₁₆	—	2	1	no	—
137	C₄⋊C₁₆	—	2	1	no	—
138	C₄⋊C₁₆	—	2	1	no	—
139	#350	—	2	4	no	—
140	C₂³×C₇	—	2	1	no	—
141	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
142	C₇×Q₈	—	2	1	no	—
143	C₇×Q₈	C₄⋊C₈	1	2	yes	—
144	C₇×D₄	—	2	2	no	—
145	C₇×D₄	—	2	2	no	—
146	C₇×D₄	—	2	1	no	—
147	C₇×D₄	C₄⋊C₈	1	2	yes	—
148	C₂×C₄×C₇	—	2	1	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	1	no	—
153	C₂×C₄×C₇	—	2	1	no	—
154	C₂×C₄×C₇	—	4	1	no	—
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂²⋊C₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₇	#117	1	1	yes	characteristic
164	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
167	C₇×C₈	—	2	2	no	—
168	C₇×C₈	—	2	1	no	—
169	C₇×C₈	—	2	2	no	—
170	C₇×C₈	—	2	1	no	—
171	C₇×C₈	#121	1	1	yes	characteristic
172	C₇×C₈	#121	1	1	yes	characteristic
173	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
174	C₇×C₈	C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×#41	C₂×C₄×C₇	1	1	yes	characteristic
176	#131	—	2	2	no	—
177	#131	—	2	2	no	—
178	#131	—	2	1	no	—
179	#131	—	2	1	no	—
180	#131	C₇×D₄	1	2	yes	—
181	#131	C₇×Q₈	1	2	yes	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
185	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
190	C₄×Q₈	C₇×C₈	1	1	yes	characteristic
191	C₄×Q₈	C₇×C₈	1	1	yes	characteristic
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₇×C₈	1	1	yes	characteristic
195	C₄×D₄	C₇×C₈	1	1	yes	characteristic
196	#117	—	2	1	no	—
197	#117	C₂×C₄×C₇	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
200	M₃₂	—	2	2	no	—
201	M₃₂	—	2	2	no	—
202	C₂×C₁₆	—	2	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	#97	—	2	1	no	—
207	#97	—	2	1	no	—
208	#97	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	C₄×C₈	—	2	1	no	—
212	C₄×C₈	C₇×D₄	1	1	yes	characteristic
213	C₄×C₈	C₇×D₄	1	1	yes	characteristic
214	C₄×C₈	C₇×D₄	1	1	yes	characteristic
215	C₄×C₈	C₇×D₄	1	1	yes	characteristic
216	C₂²×C₇	—	2	1	no	—
217	C₂²×C₇	—	2	1	no	—
218	C₂²×C₇	—	2	2	no	—
219	C₂²×C₇	—	4	1	no	—
220	C₂²×C₇	—	4	1	no	—
221	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
222	C₂²×C₇	#423	1	1	yes	characteristic
223	C₂²×C₇	#432	1	1	yes	characteristic
224	C₄×C₇	—	2	2	no	—
225	C₄×C₇	—	2	1	no	—
226	C₄×C₇	—	2	1	no	—
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	4	1	no	—
229	C₄×C₇	—	4	1	no	—
230	C₄×C₇	#350	1	2	yes	complemented
231	C₄×C₇	#380	1	1	yes	characteristic
232	C₄×C₇	#407	1	1	yes	characteristic
233	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
234	C₄×C₇	#431	1	1	yes	characteristic
235	#41	—	2	1	no	—
236	#41	—	2	1	no	—
237	#41	C₇×C₄⋊C₄	1	2	yes	—
238	#41	C₇×M₁₆	1	2	yes	—
239	C₂×Q₈	C₂×C₇×C₈	1	1	yes	characteristic
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×C₇×C₈	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
246	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
248	M₁₆	—	2	4	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	C₇×C₄⋊C₄	1	2	yes	—
254	M₁₆	C₇×M₁₆	1	2	yes	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
265	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
268	C₂×C₈	C₇×#41	1	1	yes	characteristic
269	C₂×C₈	C₇×#41	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
273	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
281	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
282	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
283	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
284	C₁₆	—	4	2	no	—
285	C₁₆	—	4	2	no	—
286	C₂×C₇	—	2	1	no	—
287	C₂×C₇	—	2	1	no	—
288	C₂×C₇	—	4	2	no	—
289	C₂×C₇	#1797	1	1	yes	characteristic
290	C₂×C₇	C₂×#350	1	1	yes	characteristic
291	C₂×C₇	#2117	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
294	Q₈	—	2	1	no	—
295	Q₈	C₇×C₄⋊C₈	1	2	yes	—
296	D₄	—	2	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	2	1	no	—
299	D₄	C₇×C₄⋊C₈	1	2	yes	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
310	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
311	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
312	C₂×C₄	C₇×C₂²⋊C₈	1	1	yes	characteristic
313	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
315	C₂×C₄	C₇×#117	1	1	yes	characteristic
316	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
318	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	1	no	—
323	C₈	C₇×#121	1	1	yes	characteristic
324	C₈	C₇×#121	1	1	yes	characteristic
325	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
326	C₈	C₇×C₂²⋊Q₈	1	1	yes	characteristic
327	C₇	#12787	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
334	C₂²	C₇×#423	1	1	yes	characteristic
335	C₂²	C₇×#432	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	C₇×#350	1	2	yes	complemented
343	C₄	C₇×#380	1	1	yes	characteristic
344	C₄	C₇×#407	1	1	yes	characteristic
345	C₄	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
346	C₄	C₇×#431	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	2	no	—
350	C₂	C₇×#1797	1	1	yes	characteristic
351	C₂	C₂×C₇×#350	1	1	yes	characteristic
352	C₂	C₇×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂×C₇×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2150	C₂	semidirect
3. C₂×C₇×C₄⋊C₁₆	C₂	semidirect
5. C₇×#1379	C₂	semidirect
24. #12787	C₇	direct (N × H)
67. #2150	C₂×C₇	semidirect
68. C₂×C₄⋊C₁₆	C₂×C₇	semidirect
70. #1379	C₂×C₇	semidirect
230. C₄×C₇	#350	semidirect
327. C₇	#12787	direct (N × H)
342. C₄	C₇×#350	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×#131	2
C₇×#2150	2
C₂×C₇×C₄⋊C₁₆	2
C₂×C₇×#350	2
C₇×#1379	2
#12787	7

Subgroups — C7×#12787 #13190003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12787 #13190003