Group #13186008 — subgroups

Subgroups — C₇×#8792 #13186008

Order 1792.

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1837	C₂	1	1	yes	maximal, characteristic
2	C₇×#1835	C₂	1	1	yes	maximal, characteristic
3	C₇×#1450	C₂	1	1	yes	maximal, characteristic
4	C₇×#1286	C₂	1	4	yes	maximal
5	C₇×#431	C₂²	1	2	yes	—
6	C₇×#423	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×#382	C₂²	1	2	yes	—
9	C₇×#341	—	2	2	no	—
10	C₇×#340	—	2	2	no	—
11	C₇×#322	C₄	1	2	yes	complemented
12	C₇×C₈²	C₄	1	2	yes	complemented
13	#8792	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂×C₇×M₁₆	—	2	2	no	—
15	C₂²×C₇×C₈	D₄	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	—	2	2	no	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	C₇×C₄⋊C₈	—	2	2	no	—
19	C₇×C₄⋊C₈	—	2	4	no	—
20	C₇×C₂²⋊C₈	—	2	2	no	—
21	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	Q₈	1	2	yes	—
25	C₇×#95	—	2	4	no	—
26	#1837	C₂×C₇	1	1	yes	characteristic
27	#1835	C₂×C₇	1	1	yes	characteristic
28	#1450	C₂×C₇	1	1	yes	characteristic
29	#1286	C₂×C₇	1	4	yes	—
30	C₂²×C₄×C₇	—	4	2	no	—
31	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₇	#41	1	2	yes	—
33	C₇×M₁₆	—	4	4	no	—
34	C₂×C₇×C₈	—	2	4	no	—
35	C₂×C₇×C₈	—	2	4	no	—
36	C₂×C₇×C₈	—	4	2	no	—
37	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
38	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
39	C₇×C₄⋊C₄	—	4	4	no	—
40	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₇	#41	1	2	yes	—
43	#431	C₂²×C₇	1	2	yes	—
44	#423	C₂²×C₇	1	2	yes	—
45	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
46	#382	C₂²×C₇	1	2	yes	—
47	#341	—	2	2	no	—
48	#340	—	2	2	no	—
49	#322	C₄×C₇	1	2	yes	complemented
50	C₈²	C₄×C₇	1	2	yes	complemented
51	C₂³×C₇	#124	1	1	yes	characteristic
52	C₂×C₄×C₇	—	4	2	no	—
53	C₂×C₄×C₇	—	8	4	no	—
54	C₂×C₄×C₇	—	8	2	no	—
55	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇	#117	1	2	yes	—
57	C₂×C₄×C₇	#124	1	1	yes	characteristic
58	C₂×C₄×C₇	#125	1	2	yes	—
59	C₂×C₄×C₇	#135	1	4	yes	—
60	C₇×C₈	—	2	2	no	—
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	—	4	4	no	—
63	C₇×C₈	—	8	2	no	—
64	C₂×M₁₆	—	2	2	no	—
65	C₂²×C₈	C₇×D₄	1	2	yes	—
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	4	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
72	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
73	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
74	C₄×C₈	C₇×Q₈	1	2	yes	—
75	#95	—	2	4	no	—
76	C₂²×C₇	—	2	2	no	—
77	C₂²×C₇	#382	1	1	yes	characteristic
78	C₂²×C₇	#488	1	2	yes	—
79	C₄×C₇	—	2	2	no	—
80	C₄×C₇	—	2	2	no	—
81	C₄×C₇	—	16	2	no	—
82	C₄×C₇	#427	1	2	yes	—
83	C₄×C₇	#488	1	2	yes	—
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₇×#41	1	2	yes	—
87	M₁₆	—	4	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
92	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄	—	4	4	no	—
94	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
95	C₄²	C₂×C₇×Q₈	1	1	yes	characteristic
96	C₄²	C₇×#41	1	2	yes	—
97	C₂×C₇	—	2	2	no	—
98	C₂×C₇	#1833	1	2	yes	—
99	C₂×C₇	#1835	1	1	yes	characteristic
100	C₂³	C₇×#124	1	1	yes	characteristic
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	4	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄	C₇×#117	1	2	yes	—
106	C₂×C₄	C₇×#124	1	1	yes	characteristic
107	C₂×C₄	C₇×#125	1	2	yes	—
108	C₂×C₄	C₇×#135	1	4	yes	—
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	4	4	no	—
112	C₈	—	8	2	no	—
113	C₇	#8792	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	C₇×#382	1	1	yes	characteristic
116	C₂²	C₇×#488	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	16	2	no	—
120	C₄	C₇×#427	1	2	yes	—
121	C₄	C₇×#488	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	C₇×#1833	1	2	yes	—
124	C₂	C₇×#1835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1837

C₂

yes

maximal, characteristic

C₇×#1835

C₂

yes

maximal, characteristic

C₇×#1450

C₂

yes

maximal, characteristic

C₇×#1286

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₇×#423

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#382

C₂²

yes

—

C₇×#341

—

C₇×#340

—

C₇×#322

C₄

yes

complemented

C₇×C₈²

C₄

yes

complemented

#8792

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×M₁₆

—

C₂²×C₇×C₈

D₄

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

Q₈

yes

—

C₇×#95

—

#1837

C₂×C₇

yes

characteristic

#1835

C₂×C₇

yes

characteristic

#1450

C₂×C₇

yes

characteristic

#1286

C₂×C₇

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

—

#431

C₂²×C₇

yes

—

#423

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

#382

C₂²×C₇

yes

—

#341

—

#340

—

#322

C₄×C₇

yes

complemented

C₈²

C₄×C₇

yes

complemented

C₂³×C₇

#124

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#125

yes

—

C₂×C₄×C₇

#135

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₂×M₁₆

—

C₂²×C₈

C₇×D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×C₄×C₇

yes

—

C₄×C₈

C₂×C₄×C₇

yes

characteristic

C₄×C₈

C₂×C₄×C₇

yes

characteristic

C₄×C₈

C₇×Q₈

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₂²×C₇

#488

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#427

yes

—

C₄×C₇

#488

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₇×D₄

yes

characteristic

C₂²×C₄

C₇×#41

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×C₄⋊C₄

yes

—

C₂×C₈

C₇×C₄⋊C₄

yes

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₇

yes

characteristic

C₄²

C₂×C₇×Q₈

yes

characteristic

C₄²

C₇×#41

yes

—

C₂×C₇

—

C₂×C₇

#1833

yes

—

C₂×C₇

#1835

yes

characteristic

100

C₂³

C₇×#124

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₇×C₄⋊C₄

yes

characteristic

105

C₂×C₄

C₇×#117

yes

—

106

C₂×C₄

C₇×#124

yes

characteristic

107

C₂×C₄

C₇×#125

yes

—

108

C₂×C₄

C₇×#135

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₇

#8792

yes

characteristic, direct product factor, Sylow

114

C₂²

—

115

C₂²

C₇×#382

yes

characteristic

116

C₂²

C₇×#488

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₇×#427

yes

—

121

C₄

C₇×#488

yes

—

122

C₂

—

123

C₂

C₇×#1833

yes

—

124

C₂

C₇×#1835

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#382

commutator

C₄²

C₂²×C₄×C₇

Frattini, cosocle

C₂×C₄²

C₂³×C₇

C₂³×C₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
11. C₇×#322	C₄	semidirect
12. C₇×C₈²	C₄	semidirect
13. #8792	C₇	direct (N × H)
49. #322	C₄×C₇	semidirect
50. C₈²	C₄×C₇	semidirect
113. C₇	#8792	direct (N × H)

H ≅ G/N

type

11. C₇×#322

C₄

semidirect

12. C₇×C₈²

C₄

semidirect

13. #8792

C₇

direct (N × H)

49. #322

C₄×C₇

semidirect

50. C₈²

C₄×C₇

semidirect

113. C₇

#8792

direct (N × H)

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

subgroup	index
C₇×#1837	2
C₇×#1835	2
C₇×#1450	2
C₇×#1286	2
#8792	7

subgroup

index

Subgroups — C₇×#8792 #13186008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#8792 #13186008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8792 #13186008