Subgroups — C₇×#12058 #13189274

Order 1792.

218 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2183	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#2169	C₂	1	1	yes	maximal, characteristic
3	C₇×C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic
4	C₇×#1370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₇×#1312	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₇×M₃₂	—	2	1	no	—
9	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic
10	C₄×C₇×SD₁₆	C₄	1	1	yes	characteristic
11	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₁₆	—	2	1	no	—
13	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₇×C₂²⋊C₁₆	—	2	1	no	—
16	C₇×#346	—	2	1	no	—
17	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
18	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
19	C₇×#334	C₄	1	1	yes	characteristic
20	C₇×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₇×D₄⋊C₈	C₄	1	1	yes	characteristic
22	C₇×C₈²	C₂²	1	1	yes	characteristic
23	#12058	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₇×SD₁₆	C₈	1	1	yes	characteristic
25	C₂²×C₇×C₈	—	2	1	no	—
26	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₇×D₄	C₂×C₄	1	1	yes	characteristic
28	C₇×M₃₂	—	4	2	no	—
29	C₂×C₇×C₁₆	—	2	1	no	—
30	C₂×C₇×C₁₆	—	2	1	no	—
31	C₂×C₇×C₁₆	—	2	1	no	—
32	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
33	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
34	C₇×#106	C₈	1	1	yes	characteristic
35	C₇×C₄⋊C₈	—	2	1	no	—
36	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₇×#103	C₈	1	1	yes	characteristic
39	C₇×#102	C₈	1	1	yes	characteristic
40	C₇×C₂²⋊C₈	—	2	1	no	—
41	C₄×C₇×C₈	—	2	1	no	—
42	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
45	#2183	C₂×C₇	1	1	yes	characteristic, complemented
46	#2169	C₂×C₇	1	1	yes	characteristic
47	C₈×SD₁₆	C₂×C₇	1	1	yes	characteristic
48	#1370	C₂×C₇	1	1	yes	characteristic, complemented
49	Q₈⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
50	D₄⋊C₁₆	C₂×C₇	1	1	yes	characteristic
51	#1312	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂×C₇×Q₈	C₂×C₈	1	1	yes	characteristic
53	C₂×C₇×D₄	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₇×SD₁₆	—	2	2	no	—
56	C₂×C₇×C₈	—	2	1	no	—
57	C₂×C₇×C₈	—	2	1	no	—
58	C₂×C₇×C₈	—	2	1	no	—
59	C₂×C₇×C₈	—	4	1	no	—
60	C₂×C₇×C₈	—	4	1	no	—
61	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂×C₈	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₇×C₈	#41	1	1	yes	characteristic
65	C₇×C₄⋊C₄	—	2	1	no	—
66	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
67	C₇×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
68	C₇×C₂²⋊C₄	—	2	1	no	—
69	C₄²×C₇	—	2	1	no	—
70	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₇×C₁₆	—	2	1	no	—
72	C₇×C₁₆	—	2	1	no	—
73	C₇×C₁₆	—	4	1	no	—
74	C₇×C₁₆	—	4	1	no	—
75	C₇×C₁₆	—	4	1	no	—
76	C₂×M₃₂	—	2	1	no	—
77	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic
78	C₄×SD₁₆	C₄×C₇	1	1	yes	characteristic
79	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
80	C₄⋊C₁₆	—	2	1	no	—
81	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
82	C₄⋊C₁₆	C₂²×C₇	1	1	yes	characteristic
83	C₂²⋊C₁₆	—	2	1	no	—
84	#346	—	2	1	no	—
85	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
86	C₄×C₁₆	C₂²×C₇	1	1	yes	characteristic
87	#334	C₄×C₇	1	1	yes	characteristic
88	Q₈⋊C₈	C₄×C₇	1	1	yes	characteristic
89	D₄⋊C₈	C₄×C₇	1	1	yes	characteristic
90	C₈²	C₂²×C₇	1	1	yes	characteristic
91	C₂³×C₇	—	2	1	no	—
92	C₇×Q₈	—	2	1	no	—
93	C₇×Q₈	—	2	1	no	—
94	C₇×D₄	—	2	1	no	—
95	C₇×D₄	—	2	1	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	4	1	no	—
100	C₂×C₄×C₇	—	4	1	no	—
101	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₇	#131	1	1	yes	characteristic
104	C₇×C₈	—	2	1	no	—
105	C₇×C₈	—	2	1	no	—
106	C₇×C₈	—	4	1	no	—
107	C₇×C₈	—	4	1	no	—
108	C₇×C₈	—	4	1	no	—
109	C₇×C₈	M₃₂	1	2	yes	complemented
110	C₇×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₇×C₈	#137	1	1	yes	characteristic
112	C₂×SD₁₆	C₇×C₈	1	1	yes	characteristic
113	C₂²×C₈	—	2	1	no	—
114	C₄×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
115	C₄×D₄	C₂×C₄×C₇	1	1	yes	characteristic
116	M₃₂	—	4	2	no	—
117	C₂×C₁₆	—	2	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
121	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
122	#106	C₇×C₈	1	1	yes	characteristic
123	C₄⋊C₈	—	2	1	no	—
124	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
125	C₄⋊C₈	C₂×C₄×C₇	1	1	yes	characteristic
126	#103	C₇×C₈	1	1	yes	characteristic
127	#102	C₇×C₈	1	1	yes	characteristic
128	C₂²⋊C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
131	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
132	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
133	C₂²×C₇	—	4	1	no	—
134	C₂²×C₇	—	4	1	no	—
135	C₂²×C₇	—	4	1	no	—
136	C₂²×C₇	C₈×D₄	1	1	yes	characteristic
137	C₄×C₇	—	2	1	no	—
138	C₄×C₇	—	4	1	no	—
139	C₄×C₇	—	4	1	no	—
140	C₄×C₇	—	4	1	no	—
141	C₄×C₇	#440	1	1	yes	characteristic
142	C₄×C₇	#444	1	1	yes	characteristic
143	C₄×C₇	C₂×M₃₂	1	1	yes	characteristic
144	C₄×C₇	#504	1	1	yes	characteristic
145	C₂×Q₈	C₂×C₇×C₈	1	1	yes	characteristic
146	C₂×D₄	C₂×C₇×C₈	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	SD₁₆	—	2	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
155	C₂×C₈	C₂×C₇×C₈	1	1	yes	characteristic
156	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
157	C₂×C₈	C₇×#41	1	1	yes	characteristic
158	C₄⋊C₄	—	2	1	no	—
159	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
160	C₄⋊C₄	C₂×C₇×C₈	1	1	yes	characteristic
161	C₂²⋊C₄	—	2	1	no	—
162	C₄²	—	2	1	no	—
163	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	—	4	1	no	—
167	C₁₆	—	4	1	no	—
168	C₁₆	—	4	1	no	—
169	C₂×C₇	—	8	1	no	—
170	C₂×C₇	SD₁₆⋊C₈	1	1	yes	characteristic
171	C₂×C₇	#2169	1	1	yes	characteristic
172	C₂×C₇	#2171	1	1	yes	characteristic
173	C₂³	—	2	1	no	—
174	Q₈	—	2	1	no	—
175	Q₈	—	2	1	no	—
176	D₄	—	2	1	no	—
177	D₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
185	C₂×C₄	C₇×#131	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	4	1	no	—
189	C₈	—	4	1	no	—
190	C₈	—	4	1	no	—
191	C₈	C₇×M₃₂	1	2	yes	complemented
192	C₈	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
193	C₈	C₇×#137	1	1	yes	characteristic
194	C₇	#12058	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₇×C₈×D₄	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	4	1	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	1	no	—
203	C₄	C₇×#440	1	1	yes	characteristic
204	C₄	C₇×#444	1	1	yes	characteristic
205	C₄	C₂×C₇×M₃₂	1	1	yes	characteristic
206	C₄	C₇×#504	1	1	yes	characteristic
207	C₂	—	8	1	no	—
208	C₂	C₇×SD₁₆⋊C₈	1	1	yes	characteristic
209	C₂	C₇×#2169	1	1	yes	characteristic
210	C₂	C₇×#2171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₇
socle	C₂²×C₇	C₈×D₄
prime radical	C₂×C₇×D₄	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#2183	C₂	semidirect
4. C₇×#1370	C₂	semidirect
5. C₇×Q₈⋊C₁₆	C₂	semidirect
7. C₇×#1312	C₂	semidirect
23. #12058	C₇	direct (N × H)
45. #2183	C₂×C₇	semidirect
48. #1370	C₂×C₇	semidirect
49. Q₈⋊C₁₆	C₂×C₇	semidirect
51. #1312	C₂×C₇	semidirect
109. C₇×C₈	M₃₂	semidirect
191. C₈	C₇×M₃₂	semidirect
194. C₇	#12058	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×#2183	2
C₇×#2169	2
C₇×C₈×SD₁₆	2
C₇×#1370	2
C₇×Q₈⋊C₁₆	2
C₇×D₄⋊C₁₆	2
C₇×#1312	2
#12058	7

Subgroups — C7×#12058 #13189274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12058 #13189274