Subgroups — C₇×#19660 #13196876

Order 1792.

914 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3226	C₂	1	2	yes	maximal, complemented
2	C₇×#3205	C₂	1	2	yes	maximal, complemented
3	C₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1722	C₂	1	2	yes	maximal, complemented
6	C₇×#1721	C₂	1	2	yes	maximal, complemented
7	C₇×#1706	C₂	1	4	yes	maximal, complemented
8	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#135	—	2	2	no	—
10	C₂²×C₇×SD₁₆	—	2	2	no	—
11	C₇×#532	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₇×#127	C₂²	1	1	yes	characteristic
14	C₇×#500	C₂²	1	4	yes	complemented
15	C₇×#495	—	2	2	no	—
16	C₇×#494	C₂²	1	1	yes	characteristic, complemented
17	C₇×#493	C₂²	1	4	yes	—
18	C₇×#493	C₂²	1	1	yes	characteristic, complemented
19	C₇×#492	—	2	2	no	—
20	C₇×#492	C₂²	1	2	yes	—
21	C₇×#492	C₂²	1	2	yes	—
22	C₇×#492	C₂²	1	2	yes	complemented
23	C₇×#486	C₂²	1	4	yes	—
24	C₇×#486	C₂²	1	4	yes	complemented
25	C₇×#480	—	2	4	no	—
26	C₇×#474	—	2	4	no	—
27	C₇×#459	—	2	4	no	—
28	C₇×#447	—	2	4	no	—
29	C₇×#431	C₂²	1	2	yes	—
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	2	yes	complemented
32	C₇×C₈²	C₂²	1	2	yes	complemented
33	#19660	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	2	no	—
35	C₂²×C₇×Q₈	—	2	1	no	—
36	C₂²×C₇×D₄	—	2	1	no	—
37	C₂²×C₇×D₄	—	2	2	no	—
38	C₇×#135	—	4	8	no	—
39	C₂×C₇×Q₁₆	—	2	2	no	—
40	C₂×C₇×SD₁₆	—	2	4	no	—
41	C₂×C₇×SD₁₆	—	2	4	no	—
42	C₂×C₇×SD₁₆	—	2	4	no	—
43	C₂×C₇×SD₁₆	—	4	8	no	—
44	C₂×C₇×D₈	—	2	2	no	—
45	C₂×C₇×D₈	—	2	8	no	—
46	C₂²×C₇×C₈	D₄	1	2	yes	—
47	C₇×C₄⋊Q₈	—	2	2	no	—
48	C₇×C₄⋊Q₈	—	2	1	no	—
49	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₇×#127	—	2	1	no	—
52	C₇×#127	—	2	2	no	—
53	C₇×#127	C₂³	1	2	yes	—
54	C₇×#127	C₂³	1	2	yes	—
55	C₇×#124	—	2	2	no	—
56	C₇×#121	—	2	2	no	—
57	C₇×#121	—	2	2	no	—
58	C₄×C₇×D₄	—	2	2	no	—
59	C₄×C₇×D₄	—	2	2	no	—
60	C₂×C₇×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₇×#107	—	2	8	no	—
63	C₇×C₄⋊C₈	D₄	1	4	yes	—
64	C₇×#102	—	2	8	no	—
65	C₇×#102	—	2	8	no	—
66	C₇×C₂²⋊C₈	D₄	1	4	yes	—
67	C₄×C₇×C₈	D₄	1	2	yes	—
68	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₇×C₈	C₂³	1	4	yes	—
70	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
71	#3226	C₂×C₇	1	2	yes	complemented
72	#3205	C₂×C₇	1	2	yes	complemented
73	#3147	C₂×C₇	1	1	yes	characteristic, complemented
74	C₂×#492	C₂×C₇	1	1	yes	characteristic, complemented
75	#1722	C₂×C₇	1	2	yes	complemented
76	#1721	C₂×C₇	1	2	yes	complemented
77	#1706	C₂×C₇	1	4	yes	complemented
78	#1450	C₂×C₇	1	1	yes	characteristic, complemented
79	C₂⁴×C₇	—	4	1	no	—
80	C₇×#41	—	4	8	no	—
81	C₂×C₇×Q₈	—	2	2	no	—
82	C₂×C₇×Q₈	—	2	2	no	—
83	C₂×C₇×Q₈	—	4	1	no	—
84	C₂×C₇×Q₈	—	4	4	no	—
85	C₂×C₇×D₄	—	2	2	no	—
86	C₂×C₇×D₄	—	2	4	no	—
87	C₂×C₇×D₄	—	2	2	no	—
88	C₂×C₇×D₄	—	2	2	no	—
89	C₂×C₇×D₄	—	2	2	no	—
90	C₂×C₇×D₄	—	4	2	no	—
91	C₂×C₇×D₄	—	4	2	no	—
92	C₂×C₇×D₄	—	4	1	no	—
93	C₂×C₇×D₄	—	4	4	no	—
94	C₂×C₇×D₄	—	8	4	no	—
95	C₂²×C₄×C₇	—	4	1	no	—
96	C₂²×C₄×C₇	—	4	2	no	—
97	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
99	C₇×Q₁₆	—	4	4	no	—
100	C₇×SD₁₆	—	4	8	no	—
101	C₇×SD₁₆	—	4	8	no	—
102	C₇×SD₁₆	—	4	8	no	—
103	C₇×D₈	—	4	4	no	—
104	C₇×D₈	—	8	8	no	—
105	C₂×C₇×C₈	D₈	1	8	yes	complemented
106	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
107	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
109	C₇×C₄⋊C₄	—	2	2	no	—
110	C₇×C₄⋊C₄	—	2	4	no	—
111	C₇×C₄⋊C₄	—	2	2	no	—
112	C₇×C₄⋊C₄	—	4	2	no	—
113	C₇×C₂²⋊C₄	—	4	2	no	—
114	C₇×C₂²⋊C₄	—	4	2	no	—
115	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇	C₂×D₄	1	2	yes	—
117	C₄²×C₇	C₂⁴	1	1	yes	characteristic
118	C₂×#135	—	2	2	no	—
119	C₂²×SD₁₆	—	2	2	no	—
120	#532	C₂²×C₇	1	2	yes	—
121	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
122	C₂×#127	C₂²×C₇	1	1	yes	characteristic
123	#500	C₂²×C₇	1	4	yes	complemented
124	#495	—	2	2	no	—
125	#494	C₂²×C₇	1	1	yes	characteristic, complemented
126	#493	C₂²×C₇	1	4	yes	—
127	#493	C₂²×C₇	1	1	yes	characteristic, complemented
128	#492	—	2	2	no	—
129	#492	C₂²×C₇	1	2	yes	—
130	#492	C₂²×C₇	1	2	yes	—
131	#492	C₂²×C₇	1	2	yes	complemented
132	#486	C₂²×C₇	1	4	yes	—
133	#486	C₂²×C₇	1	4	yes	complemented
134	#480	—	2	4	no	—
135	#474	—	2	4	no	—
136	#459	—	2	4	no	—
137	#447	—	2	4	no	—
138	#431	C₂²×C₇	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
140	#322	C₂²×C₇	1	2	yes	complemented
141	C₈²	C₂²×C₇	1	2	yes	complemented
142	C₂³×C₇	—	4	2	no	—
143	C₂³×C₇	—	4	2	no	—
144	C₂³×C₇	—	8	4	no	—
145	C₂³×C₇	—	8	2	no	—
146	C₂³×C₇	#127	1	1	yes	characteristic
147	C₇×Q₈	—	4	4	no	—
148	C₇×Q₈	—	4	4	no	—
149	C₇×Q₈	—	8	2	no	—
150	C₇×D₄	—	4	4	no	—
151	C₇×D₄	—	4	4	no	—
152	C₇×D₄	—	4	4	no	—
153	C₇×D₄	—	4	8	no	—
154	C₇×D₄	—	4	4	no	—
155	C₇×D₄	—	8	4	no	—
156	C₇×D₄	—	8	4	no	—
157	C₇×D₄	—	8	2	no	—
158	C₂×C₄×C₇	—	2	4	no	—
159	C₂×C₄×C₇	—	4	2	no	—
160	C₂×C₄×C₇	—	4	2	no	—
161	C₂×C₄×C₇	—	8	2	no	—
162	C₂×C₄×C₇	—	8	4	no	—
163	C₂×C₄×C₇	#127	1	2	yes	—
164	C₂×C₄×C₇	#127	1	1	yes	characteristic
165	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
166	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
168	C₇×C₈	—	2	8	no	—
169	C₇×C₈	C₂×D₈	1	4	yes	—
170	C₇×C₈	C₂×D₈	1	4	yes	—
171	C₂×#41	—	2	2	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×D₄	—	2	1	no	—
174	C₂²×D₄	—	2	2	no	—
175	#135	—	4	8	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	2	4	no	—
178	C₂×SD₁₆	—	2	4	no	—
179	C₂×SD₁₆	—	2	4	no	—
180	C₂×SD₁₆	—	4	8	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	—	2	8	no	—
183	C₂²×C₈	C₇×D₄	1	2	yes	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	1	no	—
186	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
187	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
188	#127	—	2	2	no	—
189	#127	—	2	1	no	—
190	#127	C₂³×C₇	1	2	yes	—
191	#127	C₂³×C₇	1	2	yes	—
192	#124	—	2	2	no	—
193	#121	—	2	2	no	—
194	#121	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	C₄×D₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
199	#107	—	2	8	no	—
200	C₄⋊C₈	C₇×D₄	1	4	yes	—
201	#102	—	2	8	no	—
202	#102	—	2	8	no	—
203	C₂²⋊C₈	C₇×D₄	1	4	yes	—
204	C₄×C₈	C₇×D₄	1	2	yes	—
205	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₇	1	4	yes	—
207	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
208	C₂²×C₇	—	2	2	no	—
209	C₂²×C₇	—	8	2	no	—
210	C₂²×C₇	—	8	2	no	—
211	C₂²×C₇	—	8	4	no	—
212	C₂²×C₇	—	8	2	no	—
213	C₂²×C₇	—	8	2	no	—
214	C₂²×C₇	—	16	4	no	—
215	C₂²×C₇	#493	1	2	yes	—
216	C₂²×C₇	C₂×#127	1	1	yes	characteristic
217	C₄×C₇	—	2	2	no	—
218	C₄×C₇	—	8	2	no	—
219	C₄×C₇	—	8	2	no	—
220	C₄×C₇	#493	1	2	yes	—
221	C₄×C₇	C₂²×D₈	1	2	yes	—
222	C₄×C₇	#577	1	4	yes	—
223	C₂⁴	—	4	1	no	—
224	#41	—	4	8	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	8	4	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
242	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
243	Q₁₆	—	4	4	no	—
244	SD₁₆	—	4	8	no	—
245	SD₁₆	—	4	8	no	—
246	SD₁₆	—	4	8	no	—
247	D₈	—	4	4	no	—
248	D₈	—	8	8	no	—
249	C₂×C₈	C₇×D₈	1	8	yes	complemented
250	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
251	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
252	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
260	C₄²	C₂×C₇×D₄	1	2	yes	—
261	C₄²	C₂⁴×C₇	1	1	yes	characteristic
262	C₂×C₇	—	2	2	no	—
263	C₂×C₇	—	16	2	no	—
264	C₂×C₇	—	16	2	no	—
265	C₂×C₇	C₂×#493	1	1	yes	characteristic
266	C₂×C₇	#3154	1	2	yes	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	8	4	no	—
270	C₂³	—	8	2	no	—
271	C₂³	C₇×#127	1	1	yes	characteristic
272	Q₈	—	4	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	8	2	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	8	no	—
279	D₄	—	4	4	no	—
280	D₄	—	8	4	no	—
281	D₄	—	8	4	no	—
282	D₄	—	8	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₇×#127	1	2	yes	—
289	C₂×C₄	C₇×#127	1	1	yes	characteristic
290	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
291	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
293	C₈	—	2	8	no	—
294	C₈	C₂×C₇×D₈	1	4	yes	—
295	C₈	C₂×C₇×D₈	1	4	yes	—
296	C₇	#19660	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	16	4	no	—
304	C₂²	C₇×#493	1	2	yes	—
305	C₂²	C₂×C₇×#127	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	C₇×#493	1	2	yes	—
310	C₄	C₂²×C₇×D₈	1	2	yes	—
311	C₄	C₇×#577	1	4	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	16	2	no	—
314	C₂	—	16	2	no	—
315	C₂	C₂×C₇×#493	1	1	yes	characteristic
316	C₂	C₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
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×#3226	C₂	semidirect
2. C₇×#3205	C₂	semidirect
3. C₇×#3147	C₂	semidirect
4. C₂×C₇×#492	C₂	semidirect
5. C₇×#1722	C₂	semidirect
6. C₇×#1721	C₂	semidirect
7. C₇×#1706	C₂	semidirect
8. C₇×#1450	C₂	semidirect
14. C₇×#500	C₂²	semidirect
16. C₇×#494	C₂²	semidirect
18. C₇×#493	C₂²	semidirect
22. C₇×#492	C₂²	semidirect
24. C₇×#486	C₂²	semidirect
31. C₇×#322	C₂²	semidirect
32. C₇×C₈²	C₂²	semidirect
33. #19660	C₇	direct (N × H)
71. #3226	C₂×C₇	semidirect
72. #3205	C₂×C₇	semidirect
73. #3147	C₂×C₇	semidirect
74. C₂×#492	C₂×C₇	semidirect
75. #1722	C₂×C₇	semidirect
76. #1721	C₂×C₇	semidirect
77. #1706	C₂×C₇	semidirect
78. #1450	C₂×C₇	semidirect
105. C₂×C₇×C₈	D₈	semidirect
123. #500	C₂²×C₇	semidirect
125. #494	C₂²×C₇	semidirect
127. #493	C₂²×C₇	semidirect
131. #492	C₂²×C₇	semidirect
133. #486	C₂²×C₇	semidirect
140. #322	C₂²×C₇	semidirect
141. C₈²	C₂²×C₇	semidirect
249. C₂×C₈	C₇×D₈	semidirect
296. C₇	#19660	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×#3226	2
C₇×#3205	2
C₇×#3147	2
C₂×C₇×#492	2
C₇×#1722	2
C₇×#1721	2
C₇×#1706	2
C₇×#1450	2
#19660	7

Subgroups — C7×#19660 #13196876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19660 #13196876