Group #11742126 — subgroups

Subgroups — C₇×#2817 #11742126

Order 896.

444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 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	C₂×C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₇×#400	C₂	1	4	yes	maximal, complemented
3	C₇×#397	C₂	1	4	yes	maximal
4	C₇×#395	C₂	1	4	yes	maximal, complemented
5	C₇×#381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	2	yes	—
7	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×C₂²⋊Q₈	—	2	8	no	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
14	C₇×#95	C₂²	1	4	yes	—
15	C₇×#95	C₂²	1	8	yes	—
16	C₇×#95	C₂²	1	4	yes	—
17	#2817	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂⁴×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₇×Q₈	—	2	4	no	—
20	C₂×C₇×Q₈	—	4	4	no	—
21	C₂²×C₄×C₇	—	4	2	no	—
22	C₂²×C₄×C₇	D₄	1	4	yes	—
23	C₂²×C₄×C₇	C₂³	1	4	yes	—
24	C₂²×C₄×C₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₇	C₂³	1	2	yes	—
27	C₇×C₄⋊C₄	—	2	4	no	—
28	C₇×C₄⋊C₄	—	2	8	no	—
29	C₇×C₄⋊C₄	—	4	4	no	—
30	C₇×C₂²⋊C₄	—	2	8	no	—
31	C₇×C₂²⋊C₄	—	4	2	no	—
32	C₂×C₂²⋊Q₈	C₂×C₇	1	2	yes	—
33	#400	C₂×C₇	1	4	yes	complemented
34	#397	C₂×C₇	1	4	yes	—
35	#395	C₂×C₇	1	4	yes	complemented
36	#381	C₂×C₇	1	1	yes	characteristic
37	C₂³×C₇	—	2	2	no	—
38	C₂³×C₇	—	2	4	no	—
39	C₂³×C₇	C₂×D₄	1	2	yes	—
40	C₂³×C₇	C₂⁴	1	1	yes	characteristic
41	C₇×Q₈	—	4	8	no	—
42	C₂×C₄×C₇	—	2	4	no	—
43	C₂×C₄×C₇	—	2	4	no	—
44	C₂×C₄×C₇	—	2	4	no	—
45	C₂×C₄×C₇	—	2	4	no	—
46	C₂×C₄×C₇	—	2	8	no	—
47	C₂×C₄×C₇	—	2	8	no	—
48	C₂×C₄×C₇	—	2	8	no	—
49	C₂×C₄×C₇	—	4	4	no	—
50	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
51	C₂²×Q₈	C₂²×C₇	1	2	yes	—
52	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
53	C₂²⋊Q₈	—	2	8	no	—
54	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
55	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
56	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
57	C₂×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
58	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
59	#95	C₂²×C₇	1	4	yes	—
60	#95	C₂²×C₇	1	8	yes	—
61	#95	C₂²×C₇	1	4	yes	—
62	C₂²×C₇	—	2	2	no	—
63	C₂²×C₇	—	2	4	no	—
64	C₂²×C₇	—	4	4	no	—
65	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
67	C₂²×C₇	2₋¹⁺⁴	1	4	yes	—
68	C₄×C₇	—	2	4	no	—
69	C₄×C₇	—	4	4	no	—
70	C₄×C₇	—	4	4	no	—
71	C₄×C₇	—	4	4	no	—
72	C₂⁴	C₂³×C₇	1	1	yes	characteristic
73	C₂×Q₈	—	2	4	no	—
74	C₂×Q₈	—	4	4	no	—
75	C₂²×C₄	—	4	2	no	—
76	C₂²×C₄	C₇×D₄	1	4	yes	—
77	C₂²×C₄	C₂³×C₇	1	4	yes	—
78	C₂²×C₄	C₂³×C₇	1	4	yes	—
79	C₂²×C₄	C₂³×C₇	1	4	yes	—
80	C₂²×C₄	C₂³×C₇	1	2	yes	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	8	no	—
83	C₄⋊C₄	—	4	4	no	—
84	C₂²⋊C₄	—	2	8	no	—
85	C₂²⋊C₄	—	4	2	no	—
86	C₂×C₇	—	4	2	no	—
87	C₂×C₇	#534	1	1	yes	characteristic
88	C₂×C₇	#536	1	2	yes	—
89	C₂×C₇	#563	1	4	yes	—
90	C₂³	—	2	2	no	—
91	C₂³	—	2	4	no	—
92	C₂³	C₂×C₇×D₄	1	2	yes	—
93	C₂³	C₂⁴×C₇	1	1	yes	characteristic
94	Q₈	—	4	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
104	C₇	#2817	1	1	yes	characteristic, direct product factor, Sylow
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	4	4	no	—
108	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
109	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
110	C₂²	C₇×2₋¹⁺⁴	1	4	yes	—
111	C₄	—	2	4	no	—
112	C₄	—	4	4	no	—
113	C₄	—	4	4	no	—
114	C₄	—	4	4	no	—
115	C₂	—	4	2	no	—
116	C₂	C₇×#534	1	1	yes	characteristic
117	C₂	C₇×#536	1	2	yes	—
118	C₂	C₇×#563	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×C₂²⋊Q₈

C₂

yes

maximal

C₇×#400

C₂

yes

maximal, complemented

C₇×#397

C₂

yes

maximal

C₇×#395

C₂

yes

maximal, complemented

C₇×#381

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₂³×C₄×C₇

C₂²

yes

characteristic

C₇×C₂²⋊Q₈

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₂²⋊C₄

C₂²

yes

—

C₂×C₇×C₂²⋊C₄

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇×#95

C₂²

yes

—

C₇×#95

C₂²

yes

—

#2817

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₇

C₂³

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₇×C₂²⋊C₄

—

C₂×C₂²⋊Q₈

C₂×C₇

yes

—

#400

C₂×C₇

yes

complemented

#397

C₂×C₇

yes

—

#395

C₂×C₇

yes

complemented

#381

C₂×C₇

yes

characteristic

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂×D₄

yes

—

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂²×Q₈

C₂²×C₇

yes

—

C₂³×C₄

C₂²×C₇

yes

characteristic

C₂²⋊Q₈

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₇

yes

—

#95

C₂²×C₇

yes

—

#95

C₂²×C₇

yes

—

#95

C₂²×C₇

yes

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₂⁴

C₂³×C₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₇×D₄

yes

—

C₂²×C₄

C₂³×C₇

yes

—

C₂²×C₄

C₂³×C₇

yes

—

C₂²×C₄

C₂³×C₇

yes

—

C₂²×C₄

C₂³×C₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂×C₇

—

C₂×C₇

#534

yes

characteristic

C₂×C₇

#536

yes

—

C₂×C₇

#563

yes

—

C₂³

—

C₂³

—

C₂³

C₂×C₇×D₄

yes

—

C₂³

C₂⁴×C₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₇×D₄

yes

—

104

C₇

#2817

yes

characteristic, direct product factor, Sylow

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²×C₇×D₄

yes

characteristic

109

C₂²

C₇×H(5,𝔽₂)

yes

—

110

C₂²

C₇×2₋¹⁺⁴

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₂

—

116

C₂

C₇×#534

yes

characteristic

117

C₂

C₇×#536

yes

—

118

C₂

C₇×#563

yes

—

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂⁴×C₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₇

prime radical

C₂⁴×C₇

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₇ ◁ G

N	H ≅ G/N	type
2. C₇×#400	C₂	semidirect
4. C₇×#395	C₂	semidirect
17. #2817	C₇	direct (N × H)
33. #400	C₂×C₇	semidirect
35. #395	C₂×C₇	semidirect
104. C₇	#2817	direct (N × H)

H ≅ G/N

type

2. C₇×#400

C₂

semidirect

4. C₇×#395

C₂

semidirect

17. #2817

C₇

direct (N × H)

33. #400

C₂×C₇

semidirect

35. #395

C₂×C₇

semidirect

104. C₇

#2817

direct (N × H)

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

subgroup	index
C₂×C₇×C₂²⋊Q₈	2
C₇×#400	2
C₇×#397	2
C₇×#395	2
C₇×#381	2
#2817	7

subgroup

index

C₂×C₇×C₂²⋊Q₈

Subgroups — C₇×#2817 #11742126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#2817 #11742126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2817 #11742126