Subgroups — C₇×#13645 #13190861

Order 1792.

204 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₇×C₈×D₄	C₄	1	1	yes	characteristic
11	C₇×M₆₄	—	2	2	no	—
12	C₂×C₇×C₃₂	—	2	2	no	—
13	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₇×C₃₂	C₂²	1	1	yes	characteristic
17	C₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic
18	C₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
19	C₇×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
20	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
21	#13645	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×D₄	C₈	1	1	yes	characteristic
25	C₂×C₇×C₁₆	—	2	1	no	—
26	C₂×C₇×C₁₆	—	2	1	no	—
27	C₂×C₇×C₁₆	—	2	1	no	—
28	C₂×C₇×C₁₆	—	2	1	no	—
29	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
33	C₇×C₄⋊C₈	C₈	1	1	yes	characteristic
34	C₇×C₂²⋊C₈	C₈	1	1	yes	characteristic
35	C₇×C₂²⋊C₈	C₈	1	1	yes	characteristic
36	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
37	C₇×C₃₂	—	2	1	no	—
38	C₇×C₃₂	—	2	1	no	—
39	C₇×C₃₂	—	2	1	no	—
40	C₇×C₃₂	D₄	1	2	yes	complemented
41	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic, complemented
42	C₂²×C₃₂	C₂×C₇	1	1	yes	characteristic, complemented
43	D₄×C₁₆	C₂×C₇	1	1	yes	characteristic
44	C₄⋊C₃₂	C₂×C₇	1	1	yes	characteristic, complemented
45	C₂²⋊C₃₂	C₂×C₇	1	1	yes	characteristic, complemented
46	C₂²⋊C₃₂	C₂×C₇	1	1	yes	characteristic, complemented
47	#1397	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂×C₇×D₄	C₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₇×C₈	—	2	1	no	—
52	C₂×C₇×C₈	—	2	1	no	—
53	C₂×C₇×C₈	—	2	1	no	—
54	C₂×C₇×C₈	—	2	1	no	—
55	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	C₁₆	1	1	yes	characteristic
60	C₇×C₂²⋊C₄	C₁₆	1	1	yes	characteristic
61	C₇×C₂²⋊C₄	C₁₆	1	1	yes	characteristic
62	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
63	C₇×C₁₆	—	2	1	no	—
64	C₇×C₁₆	—	2	1	no	—
65	C₇×C₁₆	—	2	1	no	—
66	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
67	C₇×C₁₆	#41	1	1	yes	characteristic
68	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
69	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
70	C₈×D₄	C₄×C₇	1	1	yes	characteristic
71	M₆₄	—	2	2	no	—
72	C₂×C₃₂	—	2	2	no	—
73	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
74	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
75	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
76	C₂×C₃₂	C₂²×C₇	1	1	yes	characteristic
77	C₄⋊C₁₆	C₄×C₇	1	1	yes	characteristic
78	C₂²⋊C₁₆	C₄×C₇	1	1	yes	characteristic
79	C₂²⋊C₁₆	C₄×C₇	1	1	yes	characteristic
80	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
81	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
82	C₂³×C₇	C₂×C₁₆	1	1	yes	characteristic
83	C₇×D₄	—	2	2	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	1	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂×C₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
93	C₇×C₈	—	2	1	no	—
94	C₇×C₈	—	2	1	no	—
95	C₇×C₈	—	2	1	no	—
96	C₇×C₈	C₄×D₄	1	1	yes	characteristic
97	C₇×C₈	#131	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
99	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
100	C₄×D₄	C₇×C₈	1	1	yes	characteristic
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	1	no	—
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
106	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
107	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
108	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
109	C₄⋊C₈	C₇×C₈	1	1	yes	characteristic
110	C₂²⋊C₈	C₇×C₈	1	1	yes	characteristic
111	C₂²⋊C₈	C₇×C₈	1	1	yes	characteristic
112	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
113	C₃₂	—	2	1	no	—
114	C₃₂	—	2	1	no	—
115	C₃₂	—	2	1	no	—
116	C₃₂	C₇×D₄	1	2	yes	complemented
117	C₂²×C₇	—	2	1	no	—
118	C₂²×C₇	—	2	1	no	—
119	C₂²×C₇	—	2	1	no	—
120	C₂²×C₇	—	2	1	no	—
121	C₂²×C₇	—	2	1	no	—
122	C₂²×C₇	M₆₄	1	2	yes	complemented
123	C₂²×C₇	C₂²×C₁₆	1	1	yes	characteristic
124	C₄×C₇	—	2	1	no	—
125	C₄×C₇	—	2	1	no	—
126	C₄×C₇	—	2	1	no	—
127	C₄×C₇	—	2	1	no	—
128	C₄×C₇	C₈×D₄	1	1	yes	characteristic
129	C₄×C₇	#504	1	1	yes	characteristic
130	C₂×D₄	C₇×C₁₆	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₇×C₈	1	1	yes	characteristic
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
138	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
139	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
141	C₄⋊C₄	C₇×C₁₆	1	1	yes	characteristic
142	C₂²⋊C₄	C₇×C₁₆	1	1	yes	characteristic
143	C₂²⋊C₄	C₇×C₁₆	1	1	yes	characteristic
144	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
145	C₁₆	—	2	1	no	—
146	C₁₆	—	2	1	no	—
147	C₁₆	—	2	1	no	—
148	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
149	C₁₆	C₇×#41	1	1	yes	characteristic
150	C₂×C₇	—	2	2	no	—
151	C₂×C₇	—	4	1	no	—
152	C₂×C₇	D₄×C₁₆	1	1	yes	characteristic
153	C₂×C₇	C₂×M₆₄	1	1	yes	characteristic
154	C₂×C₇	#2258	1	1	yes	characteristic
155	C₂³	C₂×C₇×C₁₆	1	1	yes	characteristic
156	C₂³	C₂×C₇×C₁₆	1	1	yes	characteristic
157	D₄	—	2	2	no	—
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
163	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
164	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
165	C₂×C₄	C₂×C₇×C₁₆	1	1	yes	characteristic
166	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	C₄×C₇×D₄	1	1	yes	characteristic
171	C₈	C₇×#131	1	1	yes	characteristic
172	C₇	#13645	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	C₇×M₆₄	1	2	yes	complemented
179	C₂²	C₂²×C₇×C₁₆	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₇×C₈×D₄	1	1	yes	characteristic
185	C₄	C₇×#504	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	1	no	—
188	C₂	C₇×D₄×C₁₆	1	1	yes	characteristic
189	C₂	C₂×C₇×M₆₄	1	1	yes	characteristic
190	C₂	C₇×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₁₆	C₂³
commutator	C₂²	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂²×C₇	C₂²×C₁₆
prime radical	C₂×C₇×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×M₆₄	C₂	semidirect
2. C₂²×C₇×C₃₂	C₂	semidirect
4. C₇×C₄⋊C₃₂	C₂	semidirect
5. C₇×C₂²⋊C₃₂	C₂	semidirect
6. C₇×C₂²⋊C₃₂	C₂	semidirect
7. C₇×#1397	C₂	semidirect
21. #13645	C₇	direct (N × H)
40. C₇×C₃₂	D₄	semidirect
41. C₂×M₆₄	C₂×C₇	semidirect
42. C₂²×C₃₂	C₂×C₇	semidirect
44. C₄⋊C₃₂	C₂×C₇	semidirect
45. C₂²⋊C₃₂	C₂×C₇	semidirect
46. C₂²⋊C₃₂	C₂×C₇	semidirect
47. #1397	C₂×C₇	semidirect
116. C₃₂	C₇×D₄	semidirect
122. C₂²×C₇	M₆₄	semidirect
172. C₇	#13645	direct (N × H)
178. C₂²	C₇×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×M₆₄	2
C₂²×C₇×C₃₂	2
C₇×D₄×C₁₆	2
C₇×C₄⋊C₃₂	2
C₇×C₂²⋊C₃₂	2
C₇×#1397	2
#13645	7

Subgroups — C7×#13645 #13190861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13645 #13190861