Subgroups — C₇×#11935 #13189151

Order 1792.

302 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₇×#1363	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#125	—	2	1	no	—
6	C₂×C₇×M₃₂	—	2	1	no	—
7	C₇×#499	—	2	1	no	—
8	C₇×#499	—	2	1	no	—
9	C₇×#499	C₄	1	2	yes	—
10	C₇×#499	C₂²	1	2	yes	—
11	C₇×#431	—	2	1	no	—
12	C₂×C₇×#107	C₄	1	1	yes	characteristic
13	C₂×C₇×#107	C₄	1	1	yes	characteristic
14	C₂×C₇×#106	—	2	1	no	—
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×C₄⋊C₁₆	—	2	1	no	—
17	C₇×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₇×#335	—	2	1	no	—
19	C₇×#334	C₂²	1	2	yes	—
20	#11935	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
23	C₇×#125	—	2	2	no	—
24	C₇×#125	—	2	2	no	—
25	C₇×#125	—	4	1	no	—
26	C₇×#125	—	4	1	no	—
27	C₂×C₇×C₄⋊C₄	—	2	1	no	—
28	C₂×C₇×C₄⋊C₄	—	2	1	no	—
29	C₂×C₇×C₄⋊C₄	—	4	1	no	—
30	C₂×C₇×C₄⋊C₄	—	4	1	no	—
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₇×M₃₂	—	4	2	no	—
33	C₂×C₇×C₁₆	—	2	2	no	—
34	C₇×#107	—	2	1	no	—
35	C₇×#107	—	2	1	no	—
36	C₇×#107	C₂×C₄	1	2	yes	—
37	C₇×#107	C₂×C₄	1	2	yes	—
38	C₇×#106	—	2	2	no	—
39	C₇×#106	—	2	2	no	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₂²⋊C₈	—	4	1	no	—
42	C₄×C₇×C₈	—	4	1	no	—
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	D₄	1	1	yes	characteristic
45	C₄×C₇×C₈	D₄	1	1	yes	characteristic
46	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
47	C₂×#499	C₂×C₇	1	1	yes	characteristic
48	#2150	C₂×C₇	1	1	yes	characteristic
49	#1564	C₂×C₇	1	1	yes	characteristic
50	#1363	C₂×C₇	1	4	yes	complemented
51	C₂²×C₄×C₇	—	4	1	no	—
52	C₂²×C₄×C₇	—	4	1	no	—
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	4	2	no	—
58	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	SD₁₆	1	2	yes	—
61	C₂×C₇×C₈	Q₁₆	1	2	yes	—
62	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₇×C₄⋊C₄	—	2	2	no	—
65	C₇×C₄⋊C₄	—	2	2	no	—
66	C₇×C₄⋊C₄	—	4	1	no	—
67	C₇×C₄⋊C₄	—	4	1	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₇×C₄⋊C₄	—	4	2	no	—
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇	D₈	1	1	yes	characteristic
74	C₄²×C₇	SD₁₆	1	1	yes	characteristic
75	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
76	C₇×C₁₆	—	4	2	no	—
77	C₂×#125	—	2	1	no	—
78	C₂×M₃₂	—	2	1	no	—
79	#499	—	2	1	no	—
80	#499	—	2	1	no	—
81	#499	C₄×C₇	1	2	yes	—
82	#499	C₂²×C₇	1	2	yes	—
83	#431	—	2	1	no	—
84	C₂×#107	C₄×C₇	1	1	yes	characteristic
85	C₂×#107	C₄×C₇	1	1	yes	characteristic
86	C₂×#106	—	2	1	no	—
87	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	1	no	—
89	C₄⋊C₁₆	C₂²×C₇	1	2	yes	—
90	#335	—	2	1	no	—
91	#334	C₂²×C₇	1	2	yes	—
92	C₂³×C₇	#102	1	1	yes	characteristic
93	C₂×C₄×C₇	—	4	1	no	—
94	C₂×C₄×C₇	—	4	1	no	—
95	C₂×C₄×C₇	—	4	2	no	—
96	C₂×C₄×C₇	—	4	2	no	—
97	C₂×C₄×C₇	—	8	2	no	—
98	C₂×C₄×C₇	—	8	2	no	—
99	C₂×C₄×C₇	#101	1	2	yes	—
100	C₂×C₄×C₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₇	#103	1	2	yes	—
104	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
107	C₇×C₈	—	2	1	no	—
108	C₇×C₈	—	2	1	no	—
109	C₇×C₈	—	8	2	no	—
110	C₇×C₈	#103	1	2	yes	—
111	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
112	C₇×C₈	C₂×Q₁₆	1	1	yes	characteristic
113	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
114	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
115	#125	—	2	2	no	—
116	#125	—	2	2	no	—
117	#125	—	4	1	no	—
118	#125	—	4	1	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	2	1	no	—
121	C₂×C₄⋊C₄	—	4	1	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
124	M₃₂	—	4	2	no	—
125	C₂×C₁₆	—	2	2	no	—
126	#107	—	2	1	no	—
127	#107	—	2	1	no	—
128	#107	C₂×C₄×C₇	1	2	yes	—
129	#107	C₂×C₄×C₇	1	2	yes	—
130	#106	—	2	2	no	—
131	#106	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
136	C₄×C₈	C₇×D₄	1	1	yes	characteristic
137	C₄×C₈	C₇×D₄	1	1	yes	characteristic
138	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
139	C₂²×C₇	—	2	1	no	—
140	C₂²×C₇	—	2	1	no	—
141	C₂²×C₇	#332	1	2	yes	—
142	C₂²×C₇	C₂×#102	1	1	yes	characteristic
143	C₄×C₇	—	2	1	no	—
144	C₄×C₇	—	2	1	no	—
145	C₄×C₇	—	2	1	no	—
146	C₄×C₇	—	2	1	no	—
147	C₄×C₇	—	8	2	no	—
148	C₄×C₇	—	8	2	no	—
149	C₄×C₇	#332	1	2	yes	—
150	C₄×C₇	C₂×#101	1	1	yes	characteristic
151	C₄×C₇	C₂×#103	1	1	yes	characteristic
152	C₂²×C₄	—	4	1	no	—
153	C₂²×C₄	—	4	1	no	—
154	C₂²×C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
156	C₂²×C₄	C₇×SD₁₆	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₈	C₇×SD₁₆	1	2	yes	—
162	C₂×C₈	C₇×Q₁₆	1	2	yes	—
163	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
164	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₄⋊C₄	—	4	1	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄²	C₇×C₂²⋊C₄	1	1	yes	characteristic
174	C₄²	C₇×D₈	1	1	yes	characteristic
175	C₄²	C₇×SD₁₆	1	1	yes	characteristic
176	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
177	C₁₆	—	4	2	no	—
178	C₂×C₇	—	2	2	no	—
179	C₂×C₇	C₂×#332	1	1	yes	characteristic
180	C₂×C₇	#2143	1	1	yes	characteristic
181	C₂×C₇	#2145	1	1	yes	characteristic
182	C₂³	C₇×#102	1	1	yes	characteristic
183	C₂×C₄	—	4	1	no	—
184	C₂×C₄	—	4	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	C₇×#101	1	2	yes	—
190	C₂×C₄	C₇×#102	1	1	yes	characteristic
191	C₂×C₄	C₇×#102	1	1	yes	characteristic
192	C₂×C₄	C₇×#102	1	1	yes	characteristic
193	C₂×C₄	C₇×#103	1	2	yes	—
194	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
197	C₈	—	2	1	no	—
198	C₈	—	2	1	no	—
199	C₈	—	8	2	no	—
200	C₈	C₇×#103	1	2	yes	—
201	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
202	C₈	C₂×C₇×Q₁₆	1	1	yes	characteristic
203	C₇	#11935	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	C₇×#332	1	2	yes	—
207	C₂²	C₂×C₇×#102	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	2	no	—
213	C₄	—	8	2	no	—
214	C₄	C₇×#332	1	2	yes	—
215	C₄	C₂×C₇×#101	1	1	yes	characteristic
216	C₄	C₂×C₇×#103	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	C₂×C₇×#332	1	1	yes	characteristic
219	C₂	C₇×#2143	1	1	yes	characteristic
220	C₂	C₇×#2145	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₂³×C₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1363	C₂	semidirect
20. #11935	C₇	direct (N × H)
50. #1363	C₂×C₇	semidirect
203. C₇	#11935	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×#499	2
C₇×#2150	2
C₇×#1564	2
C₇×#1363	2
#11935	7

Subgroups — C7×#11935 #13189151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#11935 #13189151