Group #11729121 — subgroups

Subgroups — #11729121

Order 896.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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×#500	C₂	1	1	yes	maximal, characteristic
2	#89865	C₂	1	1	yes	maximal, characteristic
3	#89864	C₂	1	1	yes	maximal, characteristic
4	#89853	C₂	1	1	yes	maximal, characteristic
5	#89850	C₂	1	1	yes	maximal, characteristic
6	#89452	C₂	1	1	yes	maximal, characteristic
7	#89250	C₂	1	1	yes	maximal, characteristic
8	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×#1144	—	2	2	no	—
11	#6432	C₂²	1	1	yes	characteristic
12	#6432	C₂²	1	1	yes	characteristic
13	C₇×#107	—	2	2	no	—
14	C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	#6409	—	2	2	no	—
16	#6391	—	2	2	no	—
17	#6384	—	2	2	no	—
18	#6381	—	2	2	no	—
19	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	#1723	—	7	1	no	maximal, Sylow
22	C₂×C₇×Q₈	—	2	1	no	—
23	C₂×C₇×Q₈	—	2	1	no	—
24	C₂×Dic₁₄	—	2	1	no	—
25	C₂×Dic₁₄	—	2	1	no	—
26	C₂×C₇×C₈	D₄	1	2	yes	—
27	C₇×C₄⋊C₄	—	2	1	no	—
28	C₇×C₄⋊C₄	—	2	1	no	—
29	C₇×C₄⋊C₄	—	2	1	no	—
30	C₇×C₄⋊C₄	—	2	1	no	—
31	C₄²×C₇	C₂³	1	1	yes	characteristic
32	#1144	—	4	4	no	—
33	C₄⋊Dic₇	—	2	1	no	—
34	C₄⋊Dic₇	—	2	1	no	—
35	C₄⋊Dic₇	—	2	1	no	—
36	C₄⋊Dic₇	—	2	1	no	—
37	C₂×C₇⋊C₈	D₄	1	2	yes	—
38	C₂×C₇⋊C₈	D₄	1	2	yes	—
39	#500	—	7	1	no	—
40	#500	D₇	1	1	yes	characteristic
41	#498	—	7	1	no	—
42	#494	—	7	1	no	—
43	#487	—	7	1	no	—
44	#487	—	7	1	no	—
45	#322	—	7	1	no	—
46	C₇×Q₈	—	4	2	no	—
47	C₇×Q₈	—	4	2	no	—
48	C₂×C₄×C₇	—	4	1	no	—
49	C₂×C₄×C₇	—	4	1	no	—
50	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₇	—	4	1	no	—
54	C₂×Dic₇	—	4	1	no	—
55	Dic₁₄	—	4	2	no	—
56	Dic₁₄	—	4	2	no	—
57	C₇×C₈	Q₁₆	1	4	yes	complemented
58	C₇⋊C₈	—	2	2	no	—
59	C₇⋊C₈	—	2	2	no	—
60	C₂×Q₁₆	—	14	2	no	—
61	C₄⋊Q₈	—	7	1	no	—
62	C₄⋊Q₈	—	7	1	no	—
63	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
64	C₄⋊Q₈	C₂×D₇	1	1	yes	characteristic
65	#107	—	2	2	no	—
66	#107	—	14	2	no	—
67	#106	—	14	2	no	—
68	#103	—	14	2	no	—
69	#103	—	14	2	no	—
70	C₄×C₈	—	7	1	no	—
71	C₄×C₈	—	7	1	no	—
72	C₄×C₈	C₂×D₇	1	1	yes	characteristic
73	C₂²×C₇	#127	1	1	yes	characteristic
74	C₄×C₇	—	8	1	no	—
75	C₄×C₇	—	8	1	no	—
76	C₄×C₇	C₂×Q₁₆	1	2	yes	—
77	C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₄×C₇	#137	1	2	yes	—
79	Dic₇	—	8	1	no	—
80	Dic₇	—	8	1	no	—
81	C₂×Q₈	—	2	1	no	—
82	C₂×Q₈	—	2	1	no	—
83	C₂×Q₈	—	14	1	no	—
84	C₂×Q₈	—	14	1	no	—
85	Q₁₆	—	28	4	no	—
86	C₂×C₈	—	7	2	no	—
87	C₂×C₈	—	7	2	no	—
88	C₂×C₈	C₂²⋊D₇	1	2	yes	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	14	1	no	—
94	C₄⋊C₄	—	14	1	no	—
95	C₄⋊C₄	—	14	1	no	—
96	C₄⋊C₄	—	14	1	no	—
97	C₄²	C₂²×D₇	1	1	yes	characteristic
98	C₂×C₇	#494	1	1	yes	characteristic
99	C₂×C₇	#496	1	1	yes	characteristic
100	C₂×C₇	#497	1	1	yes	characteristic
101	Q₈	—	4	2	no	—
102	Q₈	—	4	2	no	—
103	Q₈	—	28	2	no	—
104	Q₈	—	28	2	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	28	1	no	—
108	C₂×C₄	—	28	1	no	—
109	C₂×C₄	D₄×D₇	1	1	yes	characteristic
110	C₂×C₄	D₄×D₇	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
112	C₈	—	14	2	no	—
113	C₈	—	14	2	no	—
114	C₈	#1144	1	4	yes	complemented
115	C₇	#1723	1	1	yes	characteristic, complemented, Sylow
116	C₂²	#6503	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	56	1	no	—
120	C₄	—	56	1	no	—
121	C₄	#6474	1	2	yes	—
122	C₄	#6481	1	2	yes	—
123	C₄	C₂×#1144	1	2	yes	—
124	C₂	#89864	1	1	yes	characteristic
125	C₂	#89926	1	1	yes	characteristic
126	C₂	#89962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#500

C₂

yes

maximal, characteristic

#89865

C₂

yes

maximal, characteristic

#89864

C₂

yes

maximal, characteristic

#89853

C₂

yes

maximal, characteristic

#89850

C₂

yes

maximal, characteristic

#89452

C₂

yes

maximal, characteristic

#89250

C₂

yes

maximal, characteristic

C₇×C₄⋊Q₈

C₂²

yes

characteristic

C₇×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#1144

—

#6432

C₂²

yes

characteristic

#6432

C₂²

yes

characteristic

C₇×#107

—

C₄×C₇×C₈

C₂²

yes

—

—

—

—

C₄×C₇⋊C₈

C₂²

yes

characteristic

C₄×C₇⋊C₈

C₂²

yes

characteristic

#1723

—

maximal, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂³

yes

characteristic

#1144

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

D₄

yes

—

C₂×C₇⋊C₈

D₄

yes

—

#500

—

#500

D₇

yes

—

—

—

—

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

Q₁₆

yes

complemented

C₇⋊C₈

—

C₇⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇

yes

characteristic

C₄⋊Q₈

C₂×D₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇

yes

characteristic

C₂²×C₇

#127

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂×Q₁₆

yes

—

C₄×C₇

AGL(1,ℤ/8)

yes

—

C₄×C₇

#137

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₇

yes

C₂×C₇

yes

C₂×C₇

yes

100

C₂×C₇

#497

yes

characteristic

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

D₄×D₇

yes

characteristic

110

C₂×C₄

D₄×D₇

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

#1144

yes

complemented

115

C₇

#1723

yes

characteristic, complemented, Sylow

116

C₂²

#6503

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#6474

yes

—

122

C₄

#6481

yes

—

123

C₄

C₂×#1144

yes

—

124

C₂

#89864

yes

characteristic

125

C₂

#89926

yes

characteristic

126

C₂

#89962

yes

characteristic

	subgroup	G/H
center	C₂²	#6503
commutator, cosocle	C₄²×C₇	C₂³
Frattini	C₄²	C₂²×D₇
Fitting	C₇×#500	C₂
socle, prime radical	C₂²×C₇	#127
hypercenter	#500	D₇
hypocenter	C₇	#1723
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#6503

commutator, cosocle

C₄²×C₇

C₂³

Frattini

C₄²

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
57. C₇×C₈	Q₁₆	semidirect
114. C₈	#1144	semidirect
115. C₇	#1723	semidirect

H ≅ G/N

type

57. C₇×C₈

Q₁₆

semidirect

114. C₈

#1144

semidirect

115. C₇

#1723

semidirect

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

subgroup	index
C₇×#500	2
#89865	2
#89864	2
#89853	2
#89850	2
#89452	2
#89250	2
#1723	7

subgroup

index

Subgroups — #11729121

Conjugacy classes of subgroups

Complements

Maximal subgroups