Subgroups — C₇×#19655 #13196871

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3227	C₂	1	2	yes	maximal, complemented
2	C₇×#3206	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₇×#1714	C₂	1	4	yes	maximal, complemented
6	C₇×#1707	C₂	1	2	yes	maximal, complemented
7	C₇×#1706	C₂	1	2	yes	maximal, complemented
8	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₇×#135	—	2	2	no	—
10	C₇×#532	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₇×#500	—	2	2	no	—
13	C₇×#500	C₂²	1	2	yes	—
14	C₇×#500	C₂²	1	2	yes	—
15	C₇×#498	C₂²	1	4	yes	—
16	C₇×#495	—	2	2	no	—
17	C₇×#494	C₂²	1	1	yes	characteristic
18	C₇×#493	C₂²	1	1	yes	characteristic
19	C₇×#492	C₂²	1	2	yes	—
20	C₇×#492	C₂²	1	4	yes	—
21	C₇×#487	C₂²	1	4	yes	—
22	C₇×#486	C₂²	1	4	yes	—
23	C₇×#481	—	2	4	no	—
24	C₇×#476	—	2	4	no	—
25	C₇×#461	—	2	4	no	—
26	C₇×#448	—	2	4	no	—
27	C₇×#431	C₂²	1	1	yes	characteristic
28	C₇×#431	C₂²	1	1	yes	characteristic
29	C₂×C₇×#107	—	2	2	no	—
30	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
31	C₇×#322	C₂²	1	2	yes	—
32	C₇×C₈²	C₂²	1	2	yes	—
33	#19655	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₇×#41	—	2	2	no	—
35	C₂²×C₇×Q₈	—	2	2	no	—
36	C₇×#135	—	4	8	no	—
37	C₂×C₇×Q₁₆	—	2	2	no	—
38	C₂×C₇×SD₁₆	—	2	4	no	—
39	C₂×C₇×SD₁₆	—	2	8	no	—
40	C₂×C₇×D₈	—	2	2	no	—
41	C₂²×C₇×C₈	D₄	1	2	yes	—
42	C₇×C₄⋊Q₈	—	2	2	no	—
43	C₇×C₄⋊Q₈	—	2	2	no	—
44	C₇×C₄⋊Q₈	—	2	2	no	—
45	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₇×#127	C₂³	1	2	yes	—
48	C₇×#124	—	2	2	no	—
49	C₇×#121	—	2	2	no	—
50	C₇×#121	—	2	2	no	—
51	C₄×C₇×D₄	—	2	2	no	—
52	C₄×C₇×D₄	—	2	2	no	—
53	C₂×C₇×C₄⋊C₄	—	2	2	no	—
54	C₂×C₇×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
56	C₇×#107	—	2	4	no	—
57	C₇×#107	—	2	4	no	—
58	C₇×#106	—	2	8	no	—
59	C₇×C₄⋊C₈	D₄	1	2	yes	—
60	C₇×C₄⋊C₈	D₄	1	2	yes	—
61	C₇×#103	—	2	8	no	—
62	C₇×#102	—	2	8	no	—
63	C₇×C₂²⋊C₈	D₄	1	2	yes	—
64	C₇×C₂²⋊C₈	D₄	1	2	yes	—
65	C₄×C₇×C₈	D₄	1	2	yes	—
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	2	yes	—
68	C₄×C₇×C₈	C₂³	1	2	yes	—
69	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
70	#3227	C₂×C₇	1	2	yes	complemented
71	#3206	C₂×C₇	1	2	yes	complemented
72	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
73	#3147	C₂×C₇	1	1	yes	characteristic
74	#1714	C₂×C₇	1	4	yes	complemented
75	#1707	C₂×C₇	1	2	yes	complemented
76	#1706	C₂×C₇	1	2	yes	complemented
77	#1450	C₂×C₇	1	1	yes	characteristic, complemented
78	C₇×#41	—	4	8	no	—
79	C₂×C₇×Q₈	—	2	4	no	—
80	C₂×C₇×Q₈	—	2	2	no	—
81	C₂×C₇×Q₈	—	4	2	no	—
82	C₂×C₇×Q₈	—	8	4	no	—
83	C₂×C₇×D₄	—	2	2	no	—
84	C₂×C₇×D₄	—	2	2	no	—
85	C₂×C₇×D₄	—	2	2	no	—
86	C₂×C₇×D₄	—	4	2	no	—
87	C₂²×C₄×C₇	—	4	2	no	—
88	C₂²×C₄×C₇	—	4	2	no	—
89	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₇×Q₁₆	—	4	4	no	—
93	C₇×SD₁₆	—	4	8	no	—
94	C₇×SD₁₆	—	8	8	no	—
95	C₇×D₈	—	4	4	no	—
96	C₂×C₇×C₈	SD₁₆	1	8	yes	complemented
97	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
98	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
99	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
100	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
101	C₇×C₄⋊C₄	—	2	4	no	—
102	C₇×C₄⋊C₄	—	2	2	no	—
103	C₇×C₄⋊C₄	—	2	2	no	—
104	C₇×C₄⋊C₄	—	2	4	no	—
105	C₇×C₄⋊C₄	—	4	2	no	—
106	C₇×C₄⋊C₄	—	4	2	no	—
107	C₇×C₂²⋊C₄	—	4	2	no	—
108	C₇×C₂²⋊C₄	—	4	2	no	—
109	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇	C₂⁴	1	1	yes	characteristic
113	C₂×#135	—	2	2	no	—
114	#532	C₂²×C₇	1	2	yes	—
115	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
116	#500	—	2	2	no	—
117	#500	C₂²×C₇	1	2	yes	—
118	#500	C₂²×C₇	1	2	yes	—
119	#498	C₂²×C₇	1	4	yes	—
120	#495	—	2	2	no	—
121	#494	C₂²×C₇	1	1	yes	characteristic
122	#493	C₂²×C₇	1	1	yes	characteristic
123	#492	C₂²×C₇	1	2	yes	—
124	#492	C₂²×C₇	1	4	yes	—
125	#487	C₂²×C₇	1	4	yes	—
126	#486	C₂²×C₇	1	4	yes	—
127	#481	—	2	4	no	—
128	#476	—	2	4	no	—
129	#461	—	2	4	no	—
130	#448	—	2	4	no	—
131	#431	C₂²×C₇	1	1	yes	characteristic
132	#431	C₂²×C₇	1	1	yes	characteristic
133	C₂×#107	—	2	2	no	—
134	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
135	#322	C₂²×C₇	1	2	yes	—
136	C₈²	C₂²×C₇	1	2	yes	—
137	C₂³×C₇	—	4	2	no	—
138	C₂³×C₇	#127	1	1	yes	characteristic
139	C₇×Q₈	—	4	4	no	—
140	C₇×Q₈	—	4	8	no	—
141	C₇×Q₈	—	8	4	no	—
142	C₇×D₄	—	4	4	no	—
143	C₇×D₄	—	4	4	no	—
144	C₇×D₄	—	4	4	no	—
145	C₇×D₄	—	8	4	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	2	2	no	—
148	C₂×C₄×C₇	—	4	2	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	—	8	4	no	—
151	C₂×C₄×C₇	—	8	4	no	—
152	C₂×C₄×C₇	#127	1	1	yes	characteristic
153	C₂×C₄×C₇	#127	1	1	yes	characteristic
154	C₂×C₄×C₇	#127	1	1	yes	characteristic
155	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
156	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
159	C₇×C₈	—	2	4	no	—
160	C₇×C₈	—	2	4	no	—
161	C₇×C₈	C₂×SD₁₆	1	4	yes	—
162	C₇×C₈	C₂×SD₁₆	1	4	yes	—
163	C₂×#41	—	2	2	no	—
164	C₂²×Q₈	—	2	2	no	—
165	#135	—	4	8	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×SD₁₆	—	2	4	no	—
168	C₂×SD₁₆	—	2	8	no	—
169	C₂×D₈	—	2	2	no	—
170	C₂²×C₈	C₇×D₄	1	2	yes	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
175	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
176	#127	C₂³×C₇	1	2	yes	—
177	#124	—	2	2	no	—
178	#121	—	2	2	no	—
179	#121	—	2	2	no	—
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
185	#107	—	2	4	no	—
186	#107	—	2	4	no	—
187	#106	—	2	8	no	—
188	C₄⋊C₈	C₇×D₄	1	2	yes	—
189	C₄⋊C₈	C₇×D₄	1	2	yes	—
190	#103	—	2	8	no	—
191	#102	—	2	8	no	—
192	C₂²⋊C₈	C₇×D₄	1	2	yes	—
193	C₂²⋊C₈	C₇×D₄	1	2	yes	—
194	C₄×C₈	C₇×D₄	1	2	yes	—
195	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₇	1	2	yes	—
197	C₄×C₈	C₂³×C₇	1	2	yes	—
198	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
199	C₂²×C₇	—	2	1	no	—
200	C₂²×C₇	—	2	1	no	—
201	C₂²×C₇	—	8	2	no	—
202	C₂²×C₇	—	8	2	no	—
203	C₂²×C₇	—	8	2	no	—
204	C₂²×C₇	#492	1	2	yes	—
205	C₂²×C₇	C₂×#127	1	1	yes	characteristic
206	C₄×C₇	—	2	1	no	—
207	C₄×C₇	—	2	1	no	—
208	C₄×C₇	—	8	2	no	—
209	C₄×C₇	—	8	4	no	—
210	C₄×C₇	#492	1	2	yes	—
211	C₄×C₇	C₂²×SD₁₆	1	2	yes	—
212	C₄×C₇	#576	1	2	yes	—
213	C₄×C₇	#578	1	2	yes	—
214	#41	—	4	8	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	8	4	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
228	Q₁₆	—	4	4	no	—
229	SD₁₆	—	4	8	no	—
230	SD₁₆	—	8	8	no	—
231	D₈	—	4	4	no	—
232	C₂×C₈	C₇×SD₁₆	1	8	yes	complemented
233	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
234	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
235	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
236	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
246	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
247	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
248	C₄²	C₂⁴×C₇	1	1	yes	characteristic
249	C₂×C₇	—	2	2	no	—
250	C₂×C₇	—	16	2	no	—
251	C₂×C₇	C₂×#492	1	1	yes	characteristic
252	C₂×C₇	#3155	1	1	yes	characteristic
253	C₂×C₇	#3156	1	1	yes	characteristic
254	C₂³	—	4	2	no	—
255	C₂³	C₇×#127	1	1	yes	characteristic
256	Q₈	—	4	4	no	—
257	Q₈	—	4	8	no	—
258	Q₈	—	8	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	8	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₇×#127	1	1	yes	characteristic
270	C₂×C₄	C₇×#127	1	1	yes	characteristic
271	C₂×C₄	C₇×#127	1	1	yes	characteristic
272	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
273	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₈	C₂×C₇×SD₁₆	1	4	yes	—
279	C₈	C₂×C₇×SD₁₆	1	4	yes	—
280	C₇	#19655	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	C₇×#492	1	2	yes	—
287	C₂²	C₂×C₇×#127	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	4	no	—
291	C₄	—	8	2	no	—
292	C₄	C₇×#492	1	2	yes	—
293	C₄	C₂²×C₇×SD₁₆	1	2	yes	—
294	C₄	C₇×#576	1	2	yes	—
295	C₄	C₇×#578	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×C₇×#492	1	1	yes	characteristic
299	C₂	C₇×#3155	1	1	yes	characteristic
300	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×#3147	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×#3227	C₂	semidirect
2. C₇×#3206	C₂	semidirect
3. C₂×C₇×#500	C₂	semidirect
5. C₇×#1714	C₂	semidirect
6. C₇×#1707	C₂	semidirect
7. C₇×#1706	C₂	semidirect
8. C₇×#1450	C₂	semidirect
33. #19655	C₇	direct (N × H)
70. #3227	C₂×C₇	semidirect
71. #3206	C₂×C₇	semidirect
72. C₂×#500	C₂×C₇	semidirect
74. #1714	C₂×C₇	semidirect
75. #1707	C₂×C₇	semidirect
76. #1706	C₂×C₇	semidirect
77. #1450	C₂×C₇	semidirect
96. C₂×C₇×C₈	SD₁₆	semidirect
232. C₂×C₈	C₇×SD₁₆	semidirect
280. C₇	#19655	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×#3227	2
C₇×#3206	2
C₂×C₇×#500	2
C₇×#3147	2
C₇×#1714	2
C₇×#1707	2
C₇×#1706	2
C₇×#1450	2
#19655	7

Subgroups — C7×#19655 #13196871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19655 #13196871