Subgroups — C₇×#19712 #13196928

Order 1792.

704 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3228	C₂	1	2	yes	maximal, complemented
2	C₇×#3210	C₂	1	2	yes	maximal, complemented
3	C₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3169	C₂	1	1	yes	maximal, characteristic
5	C₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₇×#1722	C₂	1	2	yes	maximal, complemented
7	C₇×#1720	C₂	1	2	yes	maximal, complemented
8	C₇×#1717	C₂	1	2	yes	maximal, complemented
9	C₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#533	C₂²	1	2	yes	—
11	C₇×#532	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#500	C₂²	1	1	yes	characteristic
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#498	C₂²	1	2	yes	—
16	C₇×#494	C₂²	1	1	yes	characteristic
17	C₇×#494	C₂²	1	2	yes	—
18	C₇×#493	C₂²	1	1	yes	characteristic
19	C₇×#492	C₂²	1	2	yes	—
20	C₇×#487	C₂²	1	2	yes	—
21	C₇×#487	C₂²	1	2	yes	—
22	C₇×#486	C₂²	1	2	yes	—
23	C₇×#486	C₂²	1	2	yes	—
24	C₇×#486	C₂²	1	2	yes	—
25	C₇×#486	C₂²	1	2	yes	—
26	C₇×#482	—	2	4	no	—
27	C₇×#478	—	2	4	no	—
28	C₇×#467	—	2	2	no	—
29	C₇×#466	—	2	2	no	—
30	C₇×#464	—	2	4	no	—
31	C₇×#452	—	2	4	no	—
32	C₄×C₇×Q₁₆	—	2	2	no	—
33	C₄×C₇×D₈	—	2	2	no	—
34	C₇×#431	C₂²	1	1	yes	characteristic
35	C₇×#431	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
37	C₇×#322	C₂²	1	2	yes	—
38	C₇×#322	C₂²	1	2	yes	—
39	#19712	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇×#41	—	2	2	no	—
41	C₂×C₇×Q₁₆	—	2	2	no	—
42	C₂×C₇×Q₁₆	—	2	4	no	—
43	C₂×C₇×SD₁₆	—	2	4	no	—
44	C₂×C₇×D₈	—	2	2	no	—
45	C₂²×C₇×C₈	D₄	1	2	yes	—
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₇×#127	C₂³	1	2	yes	—
50	C₇×#125	—	2	2	no	—
51	C₇×#124	—	2	2	no	—
52	C₇×C₂²⋊Q₈	—	2	2	no	—
53	C₇×C₂²⋊Q₈	—	2	2	no	—
54	C₇×#121	—	2	2	no	—
55	C₇×#121	—	2	2	no	—
56	C₄×C₇×Q₈	—	2	2	no	—
57	C₄×C₇×Q₈	—	2	2	no	—
58	C₄×C₇×D₄	—	2	2	no	—
59	C₄×C₇×D₄	—	2	2	no	—
60	C₇×#117	—	2	2	no	—
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×#107	—	2	2	no	—
63	C₇×#107	—	2	2	no	—
64	C₇×#107	—	2	4	no	—
65	C₇×#106	—	2	4	no	—
66	C₇×C₄⋊C₈	D₄	1	2	yes	—
67	C₇×C₄⋊C₈	D₄	1	2	yes	—
68	C₇×#103	—	2	4	no	—
69	C₇×#103	—	2	4	no	—
70	C₇×#102	—	2	4	no	—
71	C₇×#102	—	2	4	no	—
72	C₇×#102	—	2	4	no	—
73	C₇×#102	—	2	4	no	—
74	C₇×C₂²⋊C₈	D₄	1	2	yes	—
75	C₇×C₂²⋊C₈	D₄	1	2	yes	—
76	C₄×C₇×C₈	D₄	1	2	yes	—
77	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
78	C₄×C₇×C₈	C₂³	1	2	yes	—
79	C₄×C₇×C₈	C₂³	1	2	yes	—
80	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
81	#3228	C₂×C₇	1	2	yes	complemented
82	#3210	C₂×C₇	1	2	yes	complemented
83	#3170	C₂×C₇	1	1	yes	characteristic, complemented
84	#3169	C₂×C₇	1	1	yes	characteristic
85	#1723	C₂×C₇	1	2	yes	complemented
86	#1722	C₂×C₇	1	2	yes	complemented
87	#1720	C₂×C₇	1	2	yes	complemented
88	#1717	C₂×C₇	1	2	yes	complemented
89	#1452	C₂×C₇	1	1	yes	characteristic, complemented
90	C₇×#41	—	8	4	no	—
91	C₂×C₇×Q₈	—	2	2	no	—
92	C₂×C₇×Q₈	—	2	2	no	—
93	C₂×C₇×Q₈	—	2	2	no	—
94	C₂×C₇×D₄	—	2	2	no	—
95	C₂×C₇×D₄	—	2	2	no	—
96	C₂×C₇×D₄	—	2	2	no	—
97	C₂×C₇×D₄	—	4	2	no	—
98	C₂²×C₄×C₇	—	4	2	no	—
99	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
102	C₇×Q₁₆	—	4	4	no	—
103	C₇×Q₁₆	—	8	4	no	—
104	C₇×SD₁₆	—	8	4	no	—
105	C₇×D₈	—	4	4	no	—
106	C₂×C₇×C₈	—	4	2	no	—
107	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
110	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	—	2	2	no	—
113	C₇×C₄⋊C₄	—	2	2	no	—
114	C₇×C₄⋊C₄	—	2	2	no	—
115	C₇×C₄⋊C₄	—	2	2	no	—
116	C₇×C₄⋊C₄	—	2	2	no	—
117	C₇×C₄⋊C₄	—	4	2	no	—
118	C₇×C₄⋊C₄	—	4	2	no	—
119	C₇×C₂²⋊C₄	—	4	2	no	—
120	C₇×C₂²⋊C₄	—	4	2	no	—
121	C₇×C₂²⋊C₄	—	4	2	no	—
122	C₄²×C₇	—	4	2	no	—
123	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₇	C₂⁴	1	1	yes	characteristic
127	#533	C₂²×C₇	1	2	yes	—
128	#532	C₂²×C₇	1	2	yes	—
129	#500	C₂²×C₇	1	1	yes	characteristic
130	#500	C₂²×C₇	1	1	yes	characteristic
131	#500	C₂²×C₇	1	2	yes	—
132	#498	C₂²×C₇	1	2	yes	—
133	#494	C₂²×C₇	1	1	yes	characteristic
134	#494	C₂²×C₇	1	2	yes	—
135	#493	C₂²×C₇	1	1	yes	characteristic
136	#492	C₂²×C₇	1	2	yes	—
137	#487	C₂²×C₇	1	2	yes	—
138	#487	C₂²×C₇	1	2	yes	—
139	#486	C₂²×C₇	1	2	yes	—
140	#486	C₂²×C₇	1	2	yes	—
141	#486	C₂²×C₇	1	2	yes	—
142	#486	C₂²×C₇	1	2	yes	—
143	#482	—	2	4	no	—
144	#478	—	2	4	no	—
145	#467	—	2	2	no	—
146	#466	—	2	2	no	—
147	#464	—	2	4	no	—
148	#452	—	2	4	no	—
149	C₄×Q₁₆	—	2	2	no	—
150	C₄×D₈	—	2	2	no	—
151	#431	C₂²×C₇	1	1	yes	characteristic
152	#431	C₂²×C₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
154	#322	C₂²×C₇	1	2	yes	—
155	#322	C₂²×C₇	1	2	yes	—
156	C₂³×C₇	—	4	2	no	—
157	C₂³×C₇	#127	1	1	yes	characteristic
158	C₇×Q₈	—	4	4	no	—
159	C₇×Q₈	—	4	4	no	—
160	C₇×Q₈	—	4	4	no	—
161	C₇×D₄	—	4	4	no	—
162	C₇×D₄	—	4	4	no	—
163	C₇×D₄	—	4	4	no	—
164	C₇×D₄	—	8	4	no	—
165	C₂×C₄×C₇	—	2	2	no	—
166	C₂×C₄×C₇	—	2	2	no	—
167	C₂×C₄×C₇	—	2	2	no	—
168	C₂×C₄×C₇	—	4	2	no	—
169	C₂×C₄×C₇	—	4	2	no	—
170	C₂×C₄×C₇	—	4	2	no	—
171	C₂×C₄×C₇	—	8	4	no	—
172	C₂×C₄×C₇	#127	1	1	yes	characteristic
173	C₂×C₄×C₇	#127	1	1	yes	characteristic
174	C₂×C₄×C₇	#127	1	1	yes	characteristic
175	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
178	C₇×C₈	—	2	2	no	—
179	C₇×C₈	—	2	2	no	—
180	C₇×C₈	—	2	4	no	—
181	C₇×C₈	—	2	4	no	—
182	C₂×#41	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	4	no	—
185	C₂×SD₁₆	—	2	4	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂²×C₈	C₇×D₄	1	2	yes	—
188	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
189	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
190	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
191	#127	C₂³×C₇	1	2	yes	—
192	#125	—	2	2	no	—
193	#124	—	2	2	no	—
194	C₂²⋊Q₈	—	2	2	no	—
195	C₂²⋊Q₈	—	2	2	no	—
196	#121	—	2	2	no	—
197	#121	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	#117	—	2	2	no	—
203	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
204	#107	—	2	2	no	—
205	#107	—	2	2	no	—
206	#107	—	2	4	no	—
207	#106	—	2	4	no	—
208	C₄⋊C₈	C₇×D₄	1	2	yes	—
209	C₄⋊C₈	C₇×D₄	1	2	yes	—
210	#103	—	2	4	no	—
211	#103	—	2	4	no	—
212	#102	—	2	4	no	—
213	#102	—	2	4	no	—
214	#102	—	2	4	no	—
215	#102	—	2	4	no	—
216	C₂²⋊C₈	C₇×D₄	1	2	yes	—
217	C₂²⋊C₈	C₇×D₄	1	2	yes	—
218	C₄×C₈	C₇×D₄	1	2	yes	—
219	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₇	1	2	yes	—
222	C₄×C₈	C₂³×C₇	1	2	yes	—
223	C₂²×C₇	—	2	1	no	—
224	C₂²×C₇	—	2	1	no	—
225	C₂²×C₇	—	2	1	no	—
226	C₂²×C₇	—	8	2	no	—
227	C₂²×C₇	—	8	2	no	—
228	C₂²×C₇	—	8	2	no	—
229	C₂²×C₇	C₂×#127	1	1	yes	characteristic
230	C₄×C₇	—	2	1	no	—
231	C₄×C₇	—	2	1	no	—
232	C₄×C₇	—	2	1	no	—
233	C₄×C₇	—	8	2	no	—
234	C₄×C₇	—	8	2	no	—
235	C₄×C₇	—	8	2	no	—
236	C₄×C₇	#575	1	2	yes	—
237	C₄×C₇	#576	1	2	yes	—
238	C₄×C₇	#578	1	2	yes	—
239	#41	—	8	4	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
251	Q₁₆	—	4	4	no	—
252	Q₁₆	—	8	4	no	—
253	SD₁₆	—	8	4	no	—
254	D₈	—	4	4	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
257	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
258	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
259	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
273	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
274	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
275	C₄²	C₂⁴×C₇	1	1	yes	characteristic
276	C₂×C₇	—	4	1	no	—
277	C₂×C₇	—	16	2	no	—
278	C₂×C₇	#3150	1	1	yes	characteristic
279	C₂×C₇	#3155	1	1	yes	characteristic
280	C₂×C₇	#3156	1	1	yes	characteristic
281	C₂³	—	4	2	no	—
282	C₂³	C₇×#127	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	4	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	8	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	C₇×#127	1	1	yes	characteristic
298	C₂×C₄	C₇×#127	1	1	yes	characteristic
299	C₂×C₄	C₇×#127	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	2	4	no	—
306	C₈	—	2	4	no	—
307	C₇	#19712	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×C₇×#127	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₇×#575	1	2	yes	—
322	C₄	C₇×#576	1	2	yes	—
323	C₄	C₇×#578	1	2	yes	—
324	C₂	—	4	1	no	—
325	C₂	—	16	2	no	—
326	C₂	C₇×#3150	1	1	yes	characteristic
327	C₂	C₇×#3155	1	1	yes	characteristic
328	C₂	C₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3169	C₂
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×#3228	C₂	semidirect
2. C₇×#3210	C₂	semidirect
3. C₇×#3170	C₂	semidirect
5. C₇×#1723	C₂	semidirect
6. C₇×#1722	C₂	semidirect
7. C₇×#1720	C₂	semidirect
8. C₇×#1717	C₂	semidirect
9. C₇×#1452	C₂	semidirect
39. #19712	C₇	direct (N × H)
81. #3228	C₂×C₇	semidirect
82. #3210	C₂×C₇	semidirect
83. #3170	C₂×C₇	semidirect
85. #1723	C₂×C₇	semidirect
86. #1722	C₂×C₇	semidirect
87. #1720	C₂×C₇	semidirect
88. #1717	C₂×C₇	semidirect
89. #1452	C₂×C₇	semidirect
307. C₇	#19712	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×#3228	2
C₇×#3210	2
C₇×#3170	2
C₇×#3169	2
C₇×#1723	2
C₇×#1722	2
C₇×#1720	2
C₇×#1717	2
C₇×#1452	2
#19712	7

Subgroups — C7×#19712 #13196928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19712 #13196928