Subgroups — C₇×#19661 #13196877

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3205	C₂	1	4	yes	maximal, complemented
2	C₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1721	C₂	1	4	yes	maximal, complemented
5	C₇×#1712	C₂	1	2	yes	maximal, complemented
6	C₇×#1709	C₂	1	2	yes	maximal, complemented
7	C₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×#135	—	2	2	no	—
9	C₂²×C₇×D₈	—	2	2	no	—
10	C₇×#532	C₂²	1	2	yes	—
11	C₂×C₇×#127	C₂²	1	2	yes	—
12	C₇×#495	—	2	2	no	—
13	C₇×#494	C₂²	1	1	yes	characteristic, complemented
14	C₇×#493	—	2	2	no	—
15	C₇×#493	C₂²	1	2	yes	—
16	C₇×#493	C₂²	1	2	yes	—
17	C₇×#493	C₂²	1	1	yes	characteristic, complemented
18	C₇×#493	C₂²	1	8	yes	complemented
19	C₇×#492	C₂²	1	2	yes	complemented
20	C₇×#486	C₂²	1	8	yes	complemented
21	C₇×#459	—	2	8	no	—
22	C₇×#447	—	2	8	no	—
23	C₇×#431	C₂²	1	2	yes	—
24	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
25	C₇×#322	C₂²	1	2	yes	complemented
26	C₇×C₈²	C₂²	1	2	yes	complemented
27	#19661	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₇×#41	—	2	2	no	—
29	C₂²×C₇×D₄	—	2	2	no	—
30	C₂²×C₇×D₄	—	2	4	no	—
31	C₇×#135	—	4	8	no	—
32	C₂×C₇×Q₁₆	—	2	2	no	—
33	C₂×C₇×SD₁₆	—	2	4	no	—
34	C₂×C₇×D₈	—	2	4	no	—
35	C₂×C₇×D₈	—	2	4	no	—
36	C₂×C₇×D₈	—	2	2	no	—
37	C₂×C₇×D₈	—	2	16	no	—
38	C₂×C₇×D₈	—	4	8	no	—
39	C₂²×C₇×C₈	D₄	1	2	yes	—
40	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₇×#127	—	2	2	no	—
42	C₇×#127	—	2	4	no	—
43	C₇×#127	C₂³	1	4	yes	—
44	C₇×#127	C₂³	1	2	yes	—
45	C₇×#124	—	2	2	no	—
46	C₇×#121	—	2	4	no	—
47	C₄×C₇×D₄	—	2	4	no	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×C₄⋊C₈	D₄	1	4	yes	—
50	C₇×#102	—	2	16	no	—
51	C₇×C₂²⋊C₈	D₄	1	4	yes	—
52	C₄×C₇×C₈	D₄	1	2	yes	—
53	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂³	1	4	yes	—
55	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
56	#3205	C₂×C₇	1	4	yes	complemented
57	#3147	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂×#493	C₂×C₇	1	1	yes	characteristic, complemented
59	#1721	C₂×C₇	1	4	yes	complemented
60	#1712	C₂×C₇	1	2	yes	complemented
61	#1709	C₂×C₇	1	2	yes	complemented
62	#1450	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂⁴×C₇	—	4	2	no	—
64	C₇×#41	—	4	8	no	—
65	C₂×C₇×Q₈	—	2	2	no	—
66	C₂×C₇×D₄	—	2	8	no	—
67	C₂×C₇×D₄	—	2	4	no	—
68	C₂×C₇×D₄	—	2	2	no	—
69	C₂×C₇×D₄	—	2	4	no	—
70	C₂×C₇×D₄	—	4	2	no	—
71	C₂×C₇×D₄	—	4	4	no	—
72	C₂×C₇×D₄	—	4	2	no	—
73	C₂×C₇×D₄	—	4	8	no	—
74	C₂×C₇×D₄	—	8	8	no	—
75	C₂²×C₄×C₇	—	4	2	no	—
76	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
78	C₇×Q₁₆	—	4	4	no	—
79	C₇×SD₁₆	—	4	8	no	—
80	C₇×D₈	—	4	8	no	—
81	C₇×D₈	—	4	8	no	—
82	C₇×D₈	—	4	4	no	—
83	C₇×D₈	—	8	16	no	—
84	C₂×C₇×C₈	D₈	1	8	yes	complemented
85	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
86	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
88	C₇×C₄⋊C₄	—	2	4	no	—
89	C₇×C₂²⋊C₄	—	4	4	no	—
90	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₇	C₂×D₄	1	2	yes	—
92	C₄²×C₇	C₂⁴	1	1	yes	characteristic
93	C₂×#135	—	2	2	no	—
94	C₂²×D₈	—	2	2	no	—
95	#532	C₂²×C₇	1	2	yes	—
96	C₂×#127	C₂²×C₇	1	2	yes	—
97	#495	—	2	2	no	—
98	#494	C₂²×C₇	1	1	yes	characteristic, complemented
99	#493	—	2	2	no	—
100	#493	C₂²×C₇	1	2	yes	—
101	#493	C₂²×C₇	1	2	yes	—
102	#493	C₂²×C₇	1	1	yes	characteristic, complemented
103	#493	C₂²×C₇	1	8	yes	complemented
104	#492	C₂²×C₇	1	2	yes	complemented
105	#486	C₂²×C₇	1	8	yes	complemented
106	#459	—	2	8	no	—
107	#447	—	2	8	no	—
108	#431	C₂²×C₇	1	2	yes	—
109	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
110	#322	C₂²×C₇	1	2	yes	complemented
111	C₈²	C₂²×C₇	1	2	yes	complemented
112	C₂³×C₇	—	4	2	no	—
113	C₂³×C₇	—	4	4	no	—
114	C₂³×C₇	—	8	8	no	—
115	C₂³×C₇	—	8	4	no	—
116	C₂³×C₇	#127	1	1	yes	characteristic
117	C₇×Q₈	—	4	4	no	—
118	C₇×D₄	—	4	8	no	—
119	C₇×D₄	—	4	4	no	—
120	C₇×D₄	—	4	8	no	—
121	C₇×D₄	—	4	16	no	—
122	C₇×D₄	—	8	8	no	—
123	C₇×D₄	—	8	4	no	—
124	C₇×D₄	—	8	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	4	2	no	—
127	C₂×C₄×C₇	—	8	4	no	—
128	C₂×C₄×C₇	#127	1	2	yes	—
129	C₂×C₄×C₇	#127	1	1	yes	characteristic
130	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
131	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
133	C₇×C₈	—	2	8	no	—
134	C₇×C₈	C₂×D₈	1	4	yes	—
135	C₇×C₈	C₂×D₈	1	4	yes	—
136	C₂×#41	—	2	2	no	—
137	C₂²×D₄	—	2	2	no	—
138	C₂²×D₄	—	2	4	no	—
139	#135	—	4	8	no	—
140	C₂×Q₁₆	—	2	2	no	—
141	C₂×SD₁₆	—	2	4	no	—
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	—	2	4	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	2	16	no	—
146	C₂×D₈	—	4	8	no	—
147	C₂²×C₈	C₇×D₄	1	2	yes	—
148	C₄⋊Q₈	C₂³×C₇	1	2	yes	—
149	#127	—	2	4	no	—
150	#127	—	2	2	no	—
151	#127	C₂³×C₇	1	4	yes	—
152	#127	C₂³×C₇	1	2	yes	—
153	#124	—	2	2	no	—
154	#121	—	2	4	no	—
155	C₄×D₄	—	2	4	no	—
156	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
157	C₄⋊C₈	C₇×D₄	1	4	yes	—
158	#102	—	2	16	no	—
159	C₂²⋊C₈	C₇×D₄	1	4	yes	—
160	C₄×C₈	C₇×D₄	1	2	yes	—
161	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₇	1	4	yes	—
163	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
164	C₂²×C₇	—	2	2	no	—
165	C₂²×C₇	—	8	2	no	—
166	C₂²×C₇	—	8	4	no	—
167	C₂²×C₇	—	8	8	no	—
168	C₂²×C₇	—	8	4	no	—
169	C₂²×C₇	—	16	8	no	—
170	C₂²×C₇	#493	1	2	yes	—
171	C₂²×C₇	C₂×#127	1	1	yes	characteristic
172	C₄×C₇	—	2	2	no	—
173	C₄×C₇	—	8	2	no	—
174	C₄×C₇	#493	1	2	yes	—
175	C₄×C₇	C₂²×D₈	1	2	yes	—
176	C₄×C₇	#576	1	4	yes	—
177	C₂⁴	—	4	2	no	—
178	#41	—	4	8	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×D₄	—	2	8	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	8	no	—
188	C₂×D₄	—	8	8	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
191	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
192	Q₁₆	—	4	4	no	—
193	SD₁₆	—	4	8	no	—
194	D₈	—	4	8	no	—
195	D₈	—	4	8	no	—
196	D₈	—	4	4	no	—
197	D₈	—	8	16	no	—
198	C₂×C₈	C₇×D₈	1	8	yes	complemented
199	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
200	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
201	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₂²⋊C₄	—	4	4	no	—
204	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
205	C₄²	C₂×C₇×D₄	1	2	yes	—
206	C₄²	C₂⁴×C₇	1	1	yes	characteristic
207	C₂×C₇	—	2	2	no	—
208	C₂×C₇	—	16	2	no	—
209	C₂×C₇	—	16	4	no	—
210	C₂×C₇	C₂×#493	1	1	yes	characteristic
211	C₂×C₇	#3155	1	2	yes	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	8	8	no	—
215	C₂³	—	8	4	no	—
216	C₂³	C₇×#127	1	1	yes	characteristic
217	Q₈	—	4	4	no	—
218	D₄	—	4	8	no	—
219	D₄	—	4	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	16	no	—
222	D₄	—	8	8	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	4	no	—
228	C₂×C₄	C₇×#127	1	2	yes	—
229	C₂×C₄	C₇×#127	1	1	yes	characteristic
230	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
231	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
233	C₈	—	2	8	no	—
234	C₈	C₂×C₇×D₈	1	4	yes	—
235	C₈	C₂×C₇×D₈	1	4	yes	—
236	C₇	#19661	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²	—	2	2	no	—
238	C₂²	—	8	8	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	16	8	no	—
243	C₂²	C₇×#493	1	2	yes	—
244	C₂²	C₂×C₇×#127	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	2	no	—
247	C₄	C₇×#493	1	2	yes	—
248	C₄	C₂²×C₇×D₈	1	2	yes	—
249	C₄	C₇×#576	1	4	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	16	4	no	—
252	C₂	—	16	2	no	—
253	C₂	C₂×C₇×#493	1	1	yes	characteristic
254	C₂	C₇×#3155	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×#3205	C₂	semidirect
2. C₇×#3147	C₂	semidirect
3. C₂×C₇×#493	C₂	semidirect
4. C₇×#1721	C₂	semidirect
5. C₇×#1712	C₂	semidirect
6. C₇×#1709	C₂	semidirect
7. C₇×#1450	C₂	semidirect
13. C₇×#494	C₂²	semidirect
17. C₇×#493	C₂²	semidirect
18. C₇×#493	C₂²	semidirect
19. C₇×#492	C₂²	semidirect
20. C₇×#486	C₂²	semidirect
25. C₇×#322	C₂²	semidirect
26. C₇×C₈²	C₂²	semidirect
27. #19661	C₇	direct (N × H)
56. #3205	C₂×C₇	semidirect
57. #3147	C₂×C₇	semidirect
58. C₂×#493	C₂×C₇	semidirect
59. #1721	C₂×C₇	semidirect
60. #1712	C₂×C₇	semidirect
61. #1709	C₂×C₇	semidirect
62. #1450	C₂×C₇	semidirect
84. C₂×C₇×C₈	D₈	semidirect
98. #494	C₂²×C₇	semidirect
102. #493	C₂²×C₇	semidirect
103. #493	C₂²×C₇	semidirect
104. #492	C₂²×C₇	semidirect
105. #486	C₂²×C₇	semidirect
110. #322	C₂²×C₇	semidirect
111. C₈²	C₂²×C₇	semidirect
198. C₂×C₈	C₇×D₈	semidirect
236. C₇	#19661	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×#3205	2
C₇×#3147	2
C₂×C₇×#493	2
C₇×#1721	2
C₇×#1712	2
C₇×#1709	2
C₇×#1450	2
#19661	7

Subgroups — C7×#19661 #13196877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19661 #13196877