Group #27213858 — subgroups

Subgroups — #27213858

Order 13504.

236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20269975	C₂	1	1	yes	maximal, characteristic, complemented
2	#20269966	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269963	C₂	1	2	yes	maximal, complemented
4	#20269950	C₂	1	2	yes	maximal, complemented
5	#20269949	C₂	1	2	yes	maximal, complemented
6	(C₂²⋊D₂₁₁)⋊C₄	C₂	1	2	yes	maximal, complemented
7	#20269934	C₂	1	2	yes	maximal, complemented
8	#20269933	C₂	1	2	yes	maximal, complemented
9	#124×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₂₂	C₂²	1	1	yes	characteristic
11	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₂₁₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₂₁₁	C₂²	1	2	yes	—
14	C₂²×Dic₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄×D₂₁₁	C₂²	1	2	yes	—
16	Dic₂₁₁⋊C₄	C₂²	1	2	yes	—
17	Dic₂₁₁⋊C₄	C₂²	1	2	yes	—
18	Dic₂₁₁⋊C₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₂₁₁	C₂²	1	2	yes	—
20	(C₂×D₂₁₁)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₂₁₁)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₂₁₁)⋊C₄	C₂²	1	2	yes	—
23	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
24	C₄×Dic₂₁₁	C₂²	1	2	yes	—
25	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
30	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
31	C₂²×D₂₁₁	C₂³	1	2	yes	—
32	Dic₄₂₂	—	4	1	no	—
33	D₈₄₄	—	4	1	no	—
34	C₂²⋊D₂₁₁	—	2	4	no	—
35	C₂²⋊D₂₁₁	—	4	2	no	—
36	C₂×Dic₂₁₁	—	2	4	no	—
37	C₂×Dic₂₁₁	C₂³	1	2	yes	—
38	C₂×Dic₂₁₁	C₂³	1	2	yes	—
39	C₂×Dic₂₁₁	C₂³	1	2	yes	—
40	C₄×D₂₁₁	—	2	4	no	—
41	C₂³×C₂₁₁	C₂³	1	2	yes	—
42	Q₈×C₂₁₁	—	4	1	no	—
43	D₄×C₂₁₁	—	4	1	no	—
44	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
48	C₂²×C₂₁₁	—	2	2	no	—
49	C₂²×C₂₁₁	—	2	2	no	—
50	C₂²×C₂₁₁	—	2	2	no	—
51	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₂₁₁	—	2	2	no	—
53	C₂×D₂₁₁	—	2	2	no	—
54	C₂×D₂₁₁	—	2	2	no	—
55	C₄×C₂₁₁	—	2	2	no	—
56	C₄×C₂₁₁	—	2	1	no	—
57	C₄×C₂₁₁	—	2	1	no	—
58	C₄×C₂₁₁	—	2	1	no	—
59	Dic₂₁₁	—	2	2	no	—
60	Dic₂₁₁	—	2	2	no	—
61	Dic₂₁₁	#41	1	4	yes	—
62	C₂×C₂₁₁	—	4	2	no	—
63	C₂×C₂₁₁	C₂×#41	1	2	yes	—
64	C₂×C₂₁₁	H(5,𝔽₂)	1	1	yes	characteristic
65	D₂₁₁	—	4	2	no	—
66	C₂₁₁	#555	1	1	yes	characteristic, complemented, Sylow
67	#555	—	211	1	no	maximal, Sylow
68	C₄⋊Q₈	—	211	1	no	—
69	#124	—	211	1	no	—
70	#124	—	211	2	no	—
71	#124	D₂₁₁	1	1	yes	characteristic, complemented
72	#121	—	211	2	no	—
73	#121	—	211	2	no	—
74	C₄×D₄	—	211	2	no	—
75	#117	—	211	2	no	—
76	#117	—	211	2	no	—
77	C₂×Q₈	—	211	1	no	—
78	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
79	C₂×D₄	—	211	2	no	—
80	C₂×D₄	—	211	2	no	—
81	C₂×D₄	—	211	1	no	—
82	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
83	C₂²×C₄	—	211	2	no	—
84	C₂²×C₄	—	211	2	no	—
85	C₄⋊C₄	—	211	2	no	—
86	C₄⋊C₄	—	211	2	no	—
87	C₄⋊C₄	—	211	2	no	—
88	C₂²⋊C₄	—	211	2	no	—
89	C₂²⋊C₄	—	211	2	no	—
90	C₂²⋊C₄	—	211	2	no	—
91	C₂²⋊C₄	—	211	2	no	—
92	C₂²⋊C₄	C₂×D₂₁₁	1	2	yes	—
93	C₂²⋊C₄	C₂×D₂₁₁	1	2	yes	—
94	C₄²	—	211	1	no	—
95	C₄²	—	211	2	no	—
96	C₄²	—	211	1	no	—
97	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
98	C₂³	—	211	2	no	—
99	C₂³	C₂²×D₂₁₁	1	2	yes	—
100	Q₈	—	4	1	no	—
101	Q₈	—	844	1	no	—
102	D₄	—	4	1	no	—
103	D₄	—	844	1	no	—
104	D₄	—	844	2	no	—
105	D₄	—	422	4	no	—
106	C₂×C₄	—	422	4	no	—
107	C₂×C₄	—	422	4	no	—
108	C₂×C₄	—	211	2	no	—
109	C₂×C₄	—	211	2	no	—
110	C₂×C₄	—	211	2	no	—
111	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
112	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	422	2	no	—
119	C₂²	—	422	2	no	—
120	C₂²	—	422	2	no	—
121	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	422	2	no	—
127	C₄	—	422	2	no	—
128	C₄	—	211	4	no	—
129	C₂	—	4	2	no	—
130	C₂	—	844	2	no	—
131	C₂	#41×D₂₁₁	1	2	yes	—
132	C₂	#20270022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20269975

C₂

yes

maximal, characteristic, complemented

#20269966

C₂

yes

maximal, characteristic, complemented

#20269963

C₂

yes

maximal, complemented

#20269950

C₂

yes

maximal, complemented

#20269949

C₂

yes

maximal, complemented

(C₂²⋊D₂₁₁)⋊C₄

C₂

yes

maximal, complemented

#20269934

C₂

yes

maximal, complemented

#20269933

C₂

yes

maximal, complemented

#124×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₂₂

C₂²

yes

characteristic

C₂×D₈₄₄

C₂²

yes

characteristic

C₂×C₂²⋊D₂₁₁

C₂²

yes

—

C₂×C₂²⋊D₂₁₁

C₂²

yes

—

C₂²×Dic₂₁₁

C₂²

yes

—

C₂×C₄×D₂₁₁

C₂²

yes

—

Dic₂₁₁⋊C₄

C₂²

yes

—

Dic₂₁₁⋊C₄

C₂²

yes

—

Dic₂₁₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₁₁

C₂²

yes

—

(C₂×D₂₁₁)⋊C₄

C₂²

yes

—

(C₂×D₂₁₁)⋊C₄

C₂²

yes

—

(C₂×D₂₁₁)⋊C₄

C₂²

yes

—

C₄×Dic₂₁₁

C₂²

yes

characteristic

C₄×Dic₂₁₁

C₂²

yes

—

C₄×Dic₂₁₁

C₂²

yes

characteristic

C₂×Q₈×C₂₁₁

C₂²

yes

characteristic

C₂×D₄×C₂₁₁

C₂²

yes

characteristic

C₂²⋊C₄×C₂₁₁

C₂²

yes

—

C₂²⋊C₄×C₂₁₁

C₂²

yes

—

C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×D₂₁₁

C₂³

yes

—

Dic₄₂₂

—

D₈₄₄

—

C₂²⋊D₂₁₁

—

C₂²⋊D₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

C₂³

yes

—

C₂×Dic₂₁₁

C₂³

yes

—

C₂×Dic₂₁₁

C₂³

yes

—

C₄×D₂₁₁

—

C₂³×C₂₁₁

C₂³

yes

—

Q₈×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂³

yes

—

C₂×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₄×C₂₁₁

C₂³

yes

characteristic

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂⁴

yes

characteristic

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

—

Dic₂₁₁

#41

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₂×#41

yes

—

C₂×C₂₁₁

H(5,𝔽₂)

yes

characteristic

D₂₁₁

—

C₂₁₁

#555

yes

characteristic, complemented, Sylow

#555

—

211

maximal, Sylow

C₄⋊Q₈

—

211

—

#124

—

211

—

#124

—

211

—

#124

D₂₁₁

yes

characteristic, complemented

#121

—

211

—

#121

—

211

—

C₄×D₄

—

211

—

#117

—

211

—

#117

—

211

—

C₂×Q₈

—

211

—

C₂×Q₈

C₂×D₂₁₁

yes

characteristic

C₂×D₄

—

211

—

C₂×D₄

—

211

—

C₂×D₄

—

211

—

C₂×D₄

C₂×D₂₁₁

yes

characteristic

C₂²×C₄

—

211

—

C₂²×C₄

—

211

—

C₄⋊C₄

—

211

—

C₄⋊C₄

—

211

—

C₄⋊C₄

—

211

—

C₂²⋊C₄

—

211

—

C₂²⋊C₄

—

211

—

C₂²⋊C₄

—

211

—

C₂²⋊C₄

—

211

—

C₂²⋊C₄

C₂×D₂₁₁

yes

—

C₂²⋊C₄

C₂×D₂₁₁

yes

—

C₄²

—

211

—

C₄²

—

211

—

C₄²

—

211

—

C₄²

C₂×D₂₁₁

yes

characteristic

C₂³

—

211

—

C₂³

C₂²×D₂₁₁

yes

—

100

Q₈

—

101

Q₈

—

844

—

102

D₄

—

103

D₄

—

844

—

104

D₄

—

844

—

105

D₄

—

422

—

106

C₂×C₄

—

422

—

107

C₂×C₄

—

422

—

108

C₂×C₄

—

211

—

109

C₂×C₄

—

211

—

110

C₂×C₄

—

211

—

111

C₂×C₄

C₂²×D₂₁₁

yes

—

112

C₂×C₄

C₂²×D₂₁₁

yes

characteristic

113

C₂×C₄

C₂²×D₂₁₁

yes

characteristic

114

C₂×C₄

C₂²×D₂₁₁

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

422

—

119

C₂²

—

422

—

120

C₂²

—

422

—

121

C₂²

C₂³×D₂₁₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

422

—

127

C₄

—

422

—

128

C₄

—

211

—

129

C₂

—

130

C₂

—

844

—

131

C₂

#41×D₂₁₁

yes

—

132

C₂

#20270022

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	#124×C₂₁₁	C₂
hypercenter	#124	D₂₁₁
hypocenter	C₂₁₁	#555
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₁₁

commutator, socle, cosocle

C₂²×C₂₁₁

C₂⁴

Fitting

#124×C₂₁₁

C₂

D₂₁₁

C₂₁₁

G ▷ C₂²×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #20269975	C₂	semidirect
2. #20269966	C₂	semidirect
3. #20269963	C₂	semidirect
4. #20269950	C₂	semidirect
5. #20269949	C₂	semidirect
6. (C₂²⋊D₂₁₁)⋊C₄	C₂	semidirect
7. #20269934	C₂	semidirect
8. #20269933	C₂	semidirect
9. #124×C₂₁₁	C₂	semidirect
66. C₂₁₁	#555	semidirect
71. #124	D₂₁₁	semidirect

H ≅ G/N

type

1. #20269975

C₂

semidirect

2. #20269966

C₂

semidirect

3. #20269963

C₂

semidirect

4. #20269950

C₂

semidirect

5. #20269949

C₂

semidirect

6. (C₂²⋊D₂₁₁)⋊C₄

C₂

semidirect

7. #20269934

C₂

semidirect

8. #20269933

C₂

semidirect

9. #124×C₂₁₁

C₂

semidirect

66. C₂₁₁

#555

semidirect

71. #124

D₂₁₁

semidirect

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

subgroup	index
#20269975	2
#20269966	2
#20269963	2
#20269950	2
#20269949	2
(C₂²⋊D₂₁₁)⋊C₄	2
#20269934	2
#20269933	2
#124×C₂₁₁	2
#555	211

subgroup

index

(C₂²⋊D₂₁₁)⋊C₄

#124×C₂₁₁

211

Subgroups — #27213858

Conjugacy classes of subgroups

Complements

Maximal subgroups