Subgroups — C₇×#19731 #13196947

Order 1792.

730 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3230	C₂	1	2	yes	maximal, complemented
2	C₇×#3228	C₂	1	2	yes	maximal
3	C₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3152	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₇×#1722	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1719	C₂	1	2	yes	maximal, complemented
8	C₇×#1716	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1708	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#137	—	2	2	no	—
13	C₇×#533	C₂²	1	2	yes	—
14	C₇×#532	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₇×#500	C₂²	1	2	yes	—
17	C₇×#500	C₂²	1	2	yes	—
18	C₇×#500	C₂²	1	2	yes	—
19	C₇×#498	C₂²	1	1	yes	characteristic
20	C₇×#498	C₂²	1	2	yes	—
21	C₇×#497	—	2	2	no	—
22	C₇×#494	C₂²	1	1	yes	characteristic
23	C₇×#494	C₂²	1	1	yes	characteristic
24	C₇×#494	C₂²	1	1	yes	characteristic
25	C₇×#492	C₂²	1	1	yes	characteristic
26	C₇×#492	C₂²	1	1	yes	characteristic
27	C₇×#487	C₂²	1	2	yes	—
28	C₇×#487	C₂²	1	2	yes	—
29	C₇×#487	C₂²	1	2	yes	—
30	C₇×#486	C₂²	1	2	yes	—
31	C₇×#486	C₂²	1	2	yes	—
32	C₇×#484	—	2	4	no	—
33	C₇×#482	—	2	4	no	—
34	C₇×#478	—	2	4	no	—
35	C₇×#477	—	2	4	no	—
36	C₇×#470	—	2	2	no	—
37	C₇×#441	—	2	2	no	—
38	C₇×#431	C₂²	1	2	yes	—
39	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
40	C₇×#322	C₂²	1	1	yes	characteristic
41	C₇×#322	C₂²	1	2	yes	—
42	C₇×C₈²	C₂²	1	1	yes	characteristic
43	#19731	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	—	2	1	no	—
45	C₂²×C₇×Q₈	—	2	1	no	—
46	C₇×#137	—	4	8	no	—
47	C₂×C₇×Q₁₆	—	2	2	no	—
48	C₂×C₇×Q₁₆	—	2	2	no	—
49	C₂×C₇×Q₁₆	—	2	2	no	—
50	C₂×C₇×SD₁₆	—	2	2	no	—
51	C₂×C₇×SD₁₆	—	2	2	no	—
52	C₂×C₇×M₁₆	D₄	1	2	yes	—
53	C₇×C₄⋊Q₈	—	2	2	no	—
54	C₇×C₄⋊Q₈	—	2	1	no	—
55	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
58	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₇×#127	C₂³	1	1	yes	characteristic
61	C₇×#125	—	2	2	no	—
62	C₇×#124	—	2	1	no	—
63	C₇×C₂²⋊Q₈	—	2	2	no	—
64	C₇×C₂²⋊Q₈	—	2	2	no	—
65	C₇×#121	—	2	2	no	—
66	C₄×C₇×Q₈	—	2	2	no	—
67	C₄×C₇×Q₈	—	2	2	no	—
68	C₄×C₇×D₄	—	2	2	no	—
69	C₇×#117	—	2	2	no	—
70	C₂×C₇×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
72	C₇×#107	—	2	4	no	—
73	C₇×#107	—	2	4	no	—
74	C₇×#107	—	2	4	no	—
75	C₇×#106	—	2	2	no	—
76	C₇×#106	—	2	4	no	—
77	C₇×C₄⋊C₈	D₄	1	4	yes	—
78	C₇×#103	—	2	4	no	—
79	C₇×#103	—	2	4	no	—
80	C₇×#103	—	2	4	no	—
81	C₇×#102	—	2	4	no	—
82	C₇×#102	—	2	4	no	—
83	C₇×C₂²⋊C₈	D₄	1	4	yes	—
84	C₇×#97	D₄	1	2	yes	—
85	C₄×C₇×C₈	C₂³	1	2	yes	—
86	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
87	C₄×C₇×C₈	C₂³	1	2	yes	—
88	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
89	#3230	C₂×C₇	1	2	yes	complemented
90	#3228	C₂×C₇	1	2	yes	—
91	#3185	C₂×C₇	1	1	yes	characteristic, complemented
92	#3152	C₂×C₇	1	1	yes	characteristic
93	#1723	C₂×C₇	1	2	yes	complemented
94	#1722	C₂×C₇	1	1	yes	characteristic, complemented
95	#1719	C₂×C₇	1	2	yes	complemented
96	#1716	C₂×C₇	1	1	yes	characteristic, complemented
97	#1713	C₂×C₇	1	1	yes	characteristic, complemented
98	#1708	C₂×C₇	1	1	yes	characteristic, complemented
99	#1453	C₂×C₇	1	1	yes	characteristic, complemented
100	C₇×#41	—	4	4	no	—
101	C₂×C₇×Q₈	—	2	2	no	—
102	C₂×C₇×Q₈	—	2	2	no	—
103	C₂×C₇×Q₈	—	2	1	no	—
104	C₂×C₇×Q₈	—	2	1	no	—
105	C₂×C₇×Q₈	—	2	1	no	—
106	C₂×C₇×Q₈	—	4	4	no	—
107	C₂×C₇×Q₈	—	4	1	no	—
108	C₂×C₇×D₄	—	2	2	no	—
109	C₂×C₇×D₄	—	2	1	no	—
110	C₂×C₇×D₄	—	4	1	no	—
111	C₂²×C₄×C₇	—	4	1	no	—
112	C₂²×C₄×C₇	—	4	1	no	—
113	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
115	C₇×Q₁₆	—	4	4	no	—
116	C₇×Q₁₆	—	4	4	no	—
117	C₇×Q₁₆	—	4	4	no	—
118	C₇×SD₁₆	—	4	4	no	—
119	C₇×SD₁₆	—	4	4	no	—
120	C₇×M₁₆	—	2	4	no	—
121	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
122	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
123	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
124	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
125	C₇×C₄⋊C₄	—	2	2	no	—
126	C₇×C₄⋊C₄	—	2	2	no	—
127	C₇×C₄⋊C₄	—	2	2	no	—
128	C₇×C₄⋊C₄	—	2	2	no	—
129	C₇×C₄⋊C₄	—	2	2	no	—
130	C₇×C₄⋊C₄	—	2	2	no	—
131	C₇×C₄⋊C₄	—	2	2	no	—
132	C₇×C₄⋊C₄	—	4	2	no	—
133	C₇×C₄⋊C₄	—	4	2	no	—
134	C₇×C₄⋊C₄	—	4	2	no	—
135	C₇×C₂²⋊C₄	—	4	2	no	—
136	C₇×C₂²⋊C₄	—	4	2	no	—
137	C₄²×C₇	—	4	2	no	—
138	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₇	C₂×D₄	1	2	yes	—
140	C₄²×C₇	C₂⁴	1	1	yes	characteristic
141	C₂×#137	—	2	2	no	—
142	#533	C₂²×C₇	1	2	yes	—
143	#532	C₂²×C₇	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
145	#500	C₂²×C₇	1	2	yes	—
146	#500	C₂²×C₇	1	2	yes	—
147	#500	C₂²×C₇	1	2	yes	—
148	#498	C₂²×C₇	1	1	yes	characteristic
149	#498	C₂²×C₇	1	2	yes	—
150	#497	—	2	2	no	—
151	#494	C₂²×C₇	1	1	yes	characteristic
152	#494	C₂²×C₇	1	1	yes	characteristic
153	#494	C₂²×C₇	1	1	yes	characteristic
154	#492	C₂²×C₇	1	1	yes	characteristic
155	#492	C₂²×C₇	1	1	yes	characteristic
156	#487	C₂²×C₇	1	2	yes	—
157	#487	C₂²×C₇	1	2	yes	—
158	#487	C₂²×C₇	1	2	yes	—
159	#486	C₂²×C₇	1	2	yes	—
160	#486	C₂²×C₇	1	2	yes	—
161	#484	—	2	4	no	—
162	#482	—	2	4	no	—
163	#478	—	2	4	no	—
164	#477	—	2	4	no	—
165	#470	—	2	2	no	—
166	#441	—	2	2	no	—
167	#431	C₂²×C₇	1	2	yes	—
168	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
169	#322	C₂²×C₇	1	1	yes	characteristic
170	#322	C₂²×C₇	1	2	yes	—
171	C₈²	C₂²×C₇	1	1	yes	characteristic
172	C₂³×C₇	—	4	1	no	—
173	C₂³×C₇	#127	1	1	yes	characteristic
174	C₇×Q₈	—	4	2	no	—
175	C₇×Q₈	—	4	2	no	—
176	C₇×Q₈	—	4	2	no	—
177	C₇×Q₈	—	4	4	no	—
178	C₇×Q₈	—	4	4	no	—
179	C₇×Q₈	—	8	2	no	—
180	C₇×D₄	—	4	2	no	—
181	C₇×D₄	—	4	4	no	—
182	C₇×D₄	—	8	2	no	—
183	C₂×C₄×C₇	—	2	2	no	—
184	C₂×C₄×C₇	—	2	4	no	—
185	C₂×C₄×C₇	—	4	2	no	—
186	C₂×C₄×C₇	—	4	2	no	—
187	C₂×C₄×C₇	—	4	1	no	—
188	C₂×C₄×C₇	—	4	1	no	—
189	C₂×C₄×C₇	—	4	1	no	—
190	C₂×C₄×C₇	—	8	2	no	—
191	C₂×C₄×C₇	—	8	2	no	—
192	C₂×C₄×C₇	#127	1	2	yes	—
193	C₂×C₄×C₇	#127	1	1	yes	characteristic
194	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
196	C₇×C₈	—	2	2	no	—
197	C₇×C₈	—	2	4	no	—
198	C₇×C₈	—	2	4	no	—
199	C₇×C₈	AGL(1,ℤ/8)	1	4	yes	—
200	C₂×#41	—	2	1	no	—
201	C₂²×Q₈	—	2	1	no	—
202	#137	—	4	8	no	—
203	C₂×Q₁₆	—	2	2	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	2	2	no	—
208	C₂×M₁₆	C₇×D₄	1	2	yes	—
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
212	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
213	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
214	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
216	#127	C₂³×C₇	1	1	yes	characteristic
217	#125	—	2	2	no	—
218	#124	—	2	1	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	#117	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
228	#107	—	2	4	no	—
229	#107	—	2	4	no	—
230	#107	—	2	4	no	—
231	#106	—	2	2	no	—
232	#106	—	2	4	no	—
233	C₄⋊C₈	C₇×D₄	1	4	yes	—
234	#103	—	2	4	no	—
235	#103	—	2	4	no	—
236	#103	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	C₂²⋊C₈	C₇×D₄	1	4	yes	—
240	#97	C₇×D₄	1	2	yes	—
241	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₇	1	2	yes	—
243	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₇	1	2	yes	—
245	C₂²×C₇	—	2	1	no	—
246	C₂²×C₇	—	2	2	no	—
247	C₂²×C₇	—	8	1	no	—
248	C₂²×C₇	—	8	2	no	—
249	C₂²×C₇	C₂×#127	1	1	yes	characteristic
250	C₄×C₇	—	2	1	no	—
251	C₄×C₇	—	2	2	no	—
252	C₄×C₇	—	8	1	no	—
253	C₄×C₇	—	8	1	no	—
254	C₄×C₇	—	8	1	no	—
255	C₄×C₇	—	8	2	no	—
256	C₄×C₇	—	8	2	no	—
257	C₄×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
258	C₄×C₇	#578	1	4	yes	—
259	#41	—	4	4	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	4	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
273	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
274	Q₁₆	—	4	4	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	4	4	no	—
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	4	no	—
279	M₁₆	—	2	4	no	—
280	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
281	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
282	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
283	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
298	C₄²	C₂×C₇×D₄	1	2	yes	—
299	C₄²	C₂⁴×C₇	1	1	yes	characteristic
300	C₂×C₇	—	4	1	no	—
301	C₂×C₇	—	16	1	no	—
302	C₂×C₇	#3151	1	1	yes	characteristic
303	C₂×C₇	#3156	1	2	yes	—
304	C₂³	—	4	1	no	—
305	C₂³	C₇×#127	1	1	yes	characteristic
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	4	no	—
314	D₄	—	8	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	C₇×#127	1	2	yes	—
325	C₂×C₄	C₇×#127	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
327	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
328	C₈	—	2	2	no	—
329	C₈	—	2	4	no	—
330	C₈	—	2	4	no	—
331	C₈	C₇×AGL(1,ℤ/8)	1	4	yes	—
332	C₇	#19731	1	1	yes	characteristic, direct product factor, Sylow
333	C₂²	—	2	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	C₂×C₇×#127	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	2	no	—
345	C₄	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
346	C₄	C₇×#578	1	4	yes	—
347	C₂	—	4	1	no	—
348	C₂	—	16	1	no	—
349	C₂	C₇×#3151	1	1	yes	characteristic
350	C₂	C₇×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3230	C₂	semidirect
3. C₇×#3185	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1722	C₂	semidirect
7. C₇×#1719	C₂	semidirect
8. C₇×#1716	C₂	semidirect
9. C₇×#1713	C₂	semidirect
10. C₇×#1708	C₂	semidirect
11. C₇×#1453	C₂	semidirect
43. #19731	C₇	direct (N × H)
89. #3230	C₂×C₇	semidirect
91. #3185	C₂×C₇	semidirect
93. #1723	C₂×C₇	semidirect
94. #1722	C₂×C₇	semidirect
95. #1719	C₂×C₇	semidirect
96. #1716	C₂×C₇	semidirect
97. #1713	C₂×C₇	semidirect
98. #1708	C₂×C₇	semidirect
99. #1453	C₂×C₇	semidirect
332. C₇	#19731	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×#3230	2
C₇×#3228	2
C₇×#3185	2
C₇×#3152	2
C₇×#1723	2
C₇×#1722	2
C₇×#1719	2
C₇×#1716	2
C₇×#1713	2
C₇×#1708	2
C₇×#1453	2
#19731	7

Subgroups — C7×#19731 #13196947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19731 #13196947