Group #13076974 — subgroups

Subgroups — #13076974

Order 1408.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₁₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#67000	C₂	1	4	yes	maximal, complemented
3	C₂×#66927	C₂	1	1	yes	maximal, characteristic
4	#10622601	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₁	C₂²	1	2	yes	—
7	C₂²×D₄₄	C₂²	1	2	yes	—
8	C₂²×D₄₄	C₂²	1	4	yes	—
9	C₁₁×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₁₁×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₁	C₂²	1	1	yes	characteristic
12	#67000	—	2	16	no	—
13	C₂×(C₂×D₁₁)⋊C₄	C₂²	1	16	yes	—
14	C₂×C₄×Dic₁₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₁	C₂²	1	2	yes	—
16	#66927	—	2	4	no	—
17	C₂³×D₁₁	C₂³	1	4	yes	—
18	C₂×Q₈×C₁₁	—	2	8	no	—
19	C₂×Q₈×C₁₁	—	2	4	no	—
20	C₂²×C₄×C₁₁	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₁	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₁	C₂³	1	2	yes	—
23	C₂²×Dic₁₁	C₂³	1	4	yes	—
24	C₂×D₄₄	—	2	8	no	—
25	C₂×D₄₄	—	2	4	no	—
26	C₂×D₄₄	—	2	8	no	—
27	C₂×D₄₄	—	2	16	no	—
28	C₁₁×C₄⋊C₄	—	2	8	no	—
29	C₄²×C₁₁	D₄	1	4	yes	—
30	(C₂×D₁₁)⋊C₄	—	2	32	no	—
31	C₄×Dic₁₁	—	2	8	no	—
32	C₄×Dic₁₁	D₄	1	8	yes	complemented
33	#2858	—	11	1	no	maximal, Sylow
34	C₂³×C₁₁	C₂⁴	1	1	yes	characteristic
35	C₂²×D₁₁	—	2	8	no	—
36	C₂²×D₁₁	—	2	4	no	—
37	C₂²×D₁₁	—	2	16	no	—
38	Q₈×C₁₁	—	4	8	no	—
39	C₂×C₄×C₁₁	—	2	8	no	—
40	C₂×C₄×C₁₁	—	2	4	no	—
41	C₂×C₄×C₁₁	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₁	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₁	#41	1	8	yes	—
45	C₂×Dic₁₁	—	2	8	no	—
46	C₂×Dic₁₁	C₂×D₄	1	8	yes	—
47	D₄₄	—	4	16	no	—
48	D₄₄	—	4	8	no	—
49	C₂×C₄⋊Q₈	D₁₁	1	1	yes	characteristic, complemented
50	C₂×#127	—	11	1	no	—
51	C₂×#124	—	11	4	no	—
52	#390	—	11	8	no	—
53	C₄³	—	11	1	no	—
54	C₂²×C₁₁	C₂²×D₄	1	2	yes	—
55	C₂²×C₁₁	C₂²×D₄	1	1	yes	characteristic
56	C₂²×C₁₁	C₂×#41	1	4	yes	—
57	C₂×D₁₁	—	4	8	no	—
58	C₂×D₁₁	—	4	4	no	—
59	C₂×D₁₁	—	4	16	no	—
60	C₄×C₁₁	—	4	4	no	—
61	C₄×C₁₁	#124	1	8	yes	complemented
62	C₄×C₁₁	#127	1	4	yes	—
63	Dic₁₁	—	2	8	no	—
64	C₂²×Q₈	C₂×D₁₁	1	2	yes	—
65	C₂²×D₄	—	11	2	no	—
66	C₂²×D₄	—	11	4	no	—
67	C₄⋊Q₈	—	2	4	no	—
68	#127	—	22	4	no	—
69	#124	—	22	16	no	—
70	C₂×C₄⋊C₄	C₂×D₁₁	1	4	yes	—
71	C₂×C₂²⋊C₄	—	11	16	no	—
72	C₂×C₄²	—	11	2	no	—
73	C₂×C₄²	—	11	4	no	—
74	C₂×C₄²	C₂×D₁₁	1	1	yes	characteristic
75	C₂×C₁₁	C₂×#124	1	2	yes	—
76	C₂×C₁₁	C₂×#127	1	1	yes	characteristic
77	C₂×C₁₁	#532	1	4	yes	—
78	D₁₁	—	8	4	no	—
79	C₂⁴	—	11	4	no	—
80	C₂×Q₈	—	2	8	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×D₄	—	22	8	no	—
83	C₂×D₄	—	22	16	no	—
84	C₂×D₄	—	22	4	no	—
85	C₂×D₄	—	22	8	no	—
86	C₂²×C₄	—	11	4	no	—
87	C₂²×C₄	C₂²×D₁₁	1	4	yes	—
88	C₂²×C₄	C₂²×D₁₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₁	1	2	yes	—
90	C₄⋊C₄	—	2	8	no	—
91	C₂²⋊C₄	—	22	32	no	—
92	C₄²	—	11	8	no	—
93	C₄²	—	22	8	no	—
94	C₄²	C₂²⋊D₁₁	1	4	yes	—
95	C₁₁	#2858	1	1	yes	characteristic, complemented, Sylow
96	C₂³	—	22	8	no	—
97	C₂³	—	22	16	no	—
98	C₂³	—	22	4	no	—
99	C₂³	C₂³×D₁₁	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	44	8	no	—
102	D₄	—	44	16	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	11	8	no	—
106	C₂×C₄	—	22	8	no	—
107	C₂×C₄	D₄×D₁₁	1	4	yes	—
108	C₂×C₄	#4343	1	8	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₁	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₁	1	4	yes	—
111	C₂²	—	44	16	no	—
112	C₂²	—	44	4	no	—
113	C₂²	—	44	8	no	—
114	C₂²	C₂×D₄×D₁₁	1	2	yes	—
115	C₂²	C₂×#4343	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	22	8	no	—
119	C₄	#66993	1	4	yes	—
120	C₄	#67000	1	8	yes	complemented
121	C₂	—	88	4	no	—
122	C₂	#10623256	1	4	yes	—
123	C₂	C₂×#66993	1	1	yes	characteristic
124	C₂	C₂×#67000	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₁×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×#67000

C₂

yes

maximal, complemented

C₂×#66927

C₂

yes

maximal, characteristic

#10622601

C₂

yes

maximal, complemented

C₄²×Dic₁₁

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₁

C₂²

yes

—

C₂²×D₄₄

C₂²

yes

—

C₂²×D₄₄

C₂²

yes

—

C₁₁×C₄⋊Q₈

—

C₂×C₁₁×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₁

C₂²

yes

characteristic

#67000

—

C₂×(C₂×D₁₁)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

C₂×C₄×Dic₁₁

C₂²

yes

—

#66927

—

C₂³×D₁₁

C₂³

yes

—

C₂×Q₈×C₁₁

—

C₂×Q₈×C₁₁

—

C₂²×C₄×C₁₁

C₂³

yes

—

C₂²×C₄×C₁₁

C₂³

yes

characteristic

C₂²×C₄×C₁₁

C₂³

yes

—

C₂²×Dic₁₁

C₂³

yes

—

C₂×D₄₄

—

C₂×D₄₄

—

C₂×D₄₄

—

C₂×D₄₄

—

C₁₁×C₄⋊C₄

—

C₄²×C₁₁

D₄

yes

—

(C₂×D₁₁)⋊C₄

—

C₄×Dic₁₁

—

C₄×Dic₁₁

D₄

yes

complemented

#2858

—

maximal, Sylow

C₂³×C₁₁

C₂⁴

yes

characteristic

C₂²×D₁₁

—

C₂²×D₁₁

—

C₂²×D₁₁

—

Q₈×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

C₂×D₄

yes

—

C₂×C₄×C₁₁

#41

yes

—

C₂×Dic₁₁

—

C₂×Dic₁₁

C₂×D₄

yes

—

D₄₄

—

D₄₄

—

C₂×C₄⋊Q₈

D₁₁

yes

characteristic, complemented

C₂×#127

—

C₂×#124

—

#390

—

C₄³

—

C₂²×C₁₁

C₂²×D₄

yes

—

C₂²×C₁₁

C₂²×D₄

yes

characteristic

C₂²×C₁₁

C₂×#41

yes

—

C₂×D₁₁

—

C₂×D₁₁

—

C₂×D₁₁

—

C₄×C₁₁

—

C₄×C₁₁

#124

yes

complemented

C₄×C₁₁

#127

yes

—

Dic₁₁

—

C₂²×Q₈

C₂×D₁₁

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

#127

—

#124

—

C₂×C₄⋊C₄

C₂×D₁₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₁

yes

characteristic

C₂×C₁₁

C₂×#124

yes

—

C₂×C₁₁

C₂×#127

yes

characteristic

C₂×C₁₁

#532

yes

—

D₁₁

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₁

yes

—

C₂²×C₄

C₂²×D₁₁

yes

characteristic

C₂²×C₄

C₂²×D₁₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₁

yes

—

C₁₁

#2858

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₁

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₁₁

yes

—

108

C₂×C₄

#4343

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₁

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₁₁

yes

—

115

C₂²

C₂×#4343

yes

—

116

C₂²

C₂²×C₂²⋊D₁₁

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#66993

yes

—

120

C₄

#67000

yes

complemented

121

C₂

—

122

C₂

#10623256

yes

—

123

C₂

C₂×#66993

yes

characteristic

124

C₂

C₂×#67000

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₁
commutator, socle, cosocle	C₂³×C₁₁	C₂⁴
Fitting	C₂×C₁₁×C₄⋊Q₈	C₂
prime radical	C₂×#66927	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₁
hypocenter	C₁₁	#2858
LCS	G ▷ C₂³×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₁

commutator, socle, cosocle

C₂³×C₁₁

C₂⁴

Fitting

C₂×C₁₁×C₄⋊Q₈

C₂

prime radical

C₂×#66927

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₁

C₁₁

G ▷ C₂³×C₁₁ ▷ C₁₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×C₁₁×C₄⋊Q₈	C₂	semidirect
2. C₂×#67000	C₂	semidirect
4. #10622601	C₂	semidirect
5. C₄²×Dic₁₁	C₂	semidirect
32. C₄×Dic₁₁	D₄	semidirect
49. C₂×C₄⋊Q₈	D₁₁	semidirect
61. C₄×C₁₁	#124	semidirect
95. C₁₁	#2858	semidirect
120. C₄	#67000	semidirect

H ≅ G/N

type

1. C₂×C₁₁×C₄⋊Q₈

C₂

semidirect

2. C₂×#67000

C₂

semidirect

4. #10622601

C₂

semidirect

5. C₄²×Dic₁₁

C₂

semidirect

32. C₄×Dic₁₁

D₄

semidirect

49. C₂×C₄⋊Q₈

D₁₁

semidirect

61. C₄×C₁₁

#124

semidirect

95. C₁₁

#2858

semidirect

120. C₄

#67000

semidirect

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

subgroup	index
C₂×C₁₁×C₄⋊Q₈	2
C₂×#67000	2
C₂×#66927	2
#10622601	2
C₄²×Dic₁₁	2
#2858	11

subgroup

index

C₂×C₁₁×C₄⋊Q₈

C₄²×Dic₁₁

Subgroups — #13076974

Conjugacy classes of subgroups

Complements

Maximal subgroups