Group #34092201 — subgroups

Subgroups — #34092201

Order 19904.

330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂²×C₄×D₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#23755689	C₂	1	2	yes	maximal, complemented
3	#23755684	C₂	1	2	yes	maximal, complemented
4	#23755679	C₂	1	2	yes	maximal, complemented
5	#23755676	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₃₁₁	C₂	1	4	yes	maximal, complemented
7	#23755667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₃₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₃₁₁	C₂²	1	4	yes	—
11	C₂²×Dic₃₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃₁₁	—	2	4	no	—
13	C₂×C₄×D₃₁₁	C₂²	1	2	yes	—
14	C₂×C₄×D₃₁₁	C₂²	1	2	yes	—
15	C₄⋊Dic₃₁₁	C₂²	1	2	yes	complemented
16	Dic₃₁₁⋊C₄	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₃₁₁	C₂²	1	2	yes	—
18	C₂²⋊Dic₃₁₁	C₂²	1	4	yes	complemented
19	(C₂×D₃₁₁)⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₃₁₁	C₂²	1	2	yes	complemented
21	#41×C₃₁₁	—	2	4	no	—
22	C₂×Q₈×C₃₁₁	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₄×C₃₁₁	C₂²	1	2	yes	—
24	C₂×D₄×C₃₁₁	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×C₃₁₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₃₁₁	C₂²	1	2	yes	complemented
27	C₂²×D₃₁₁	—	2	2	no	—
28	C₂²×D₃₁₁	—	2	4	no	—
29	C₂²×D₃₁₁	C₂³	1	2	yes	—
30	C₂²⋊D₃₁₁	—	2	8	no	—
31	C₂×Dic₃₁₁	—	2	2	no	—
32	C₂×Dic₃₁₁	C₂³	1	4	yes	—
33	C₂×Dic₃₁₁	C₂³	1	2	yes	—
34	C₄×D₃₁₁	—	2	4	no	—
35	C₄×D₃₁₁	—	2	4	no	—
36	C₂³×C₃₁₁	C₂³	1	2	yes	—
37	C₂³×C₃₁₁	C₂³	1	1	yes	characteristic
38	Q₈×C₃₁₁	—	2	2	no	—
39	D₄×C₃₁₁	—	2	4	no	—
40	D₄×C₃₁₁	—	2	2	no	—
41	C₂×C₄×C₃₁₁	—	2	4	no	—
42	C₂×C₄×C₃₁₁	D₄	1	4	yes	complemented
43	C₂×C₄×C₃₁₁	C₂³	1	2	yes	—
44	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
46	C₂²×C₃₁₁	—	2	2	no	—
47	C₂²×C₃₁₁	—	2	2	no	—
48	C₂²×C₃₁₁	—	2	4	no	—
49	C₂²×C₃₁₁	C₂×D₄	1	2	yes	—
50	C₂²×C₃₁₁	C₂⁴	1	1	yes	characteristic
51	C₂×D₃₁₁	—	2	8	no	—
52	C₂×D₃₁₁	—	2	2	no	—
53	C₂×D₃₁₁	—	2	2	no	—
54	C₂×D₃₁₁	#41	1	4	yes	complemented
55	C₄×C₃₁₁	—	2	2	no	—
56	C₄×C₃₁₁	C₂×D₄	1	2	yes	—
57	C₄×C₃₁₁	C₂×D₄	1	2	yes	—
58	Dic₃₁₁	—	2	2	no	—
59	Dic₃₁₁	—	2	4	no	—
60	C₂×C₃₁₁	—	2	2	no	—
61	C₂×C₃₁₁	—	4	2	no	—
62	C₂×C₃₁₁	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₃₁₁	C₂×#41	1	2	yes	—
64	D₃₁₁	—	2	4	no	—
65	C₃₁₁	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	311	1	no	maximal, Sylow
67	C₂×#41	D₃₁₁	1	1	yes	characteristic, complemented
68	C₂³×C₄	—	311	1	no	—
69	#123	—	311	2	no	—
70	C₂²⋊Q₈	—	311	2	no	—
71	#121	—	311	2	no	—
72	#120	—	311	2	no	—
73	C₄×D₄	—	311	4	no	—
74	#117	—	311	1	no	—
75	C₂⁴	—	311	1	no	—
76	#41	—	2	4	no	—
77	C₂×Q₈	C₂×D₃₁₁	1	1	yes	characteristic, complemented
78	C₂×D₄	—	311	4	no	—
79	C₂×D₄	C₂×D₃₁₁	1	2	yes	—
80	C₂×D₄	C₂×D₃₁₁	1	1	yes	characteristic, complemented
81	C₂²×C₄	—	311	1	no	—
82	C₂²×C₄	—	311	2	no	—
83	C₂²×C₄	—	311	2	no	—
84	C₂²×C₄	—	622	4	no	—
85	C₂²×C₄	C₂×D₃₁₁	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₃₁₁	1	2	yes	complemented
87	C₄⋊C₄	—	311	2	no	—
88	C₄⋊C₄	—	311	4	no	—
89	C₂²⋊C₄	—	311	2	no	—
90	C₂²⋊C₄	—	311	4	no	—
91	C₂²⋊C₄	—	311	4	no	—
92	C₄²	—	311	2	no	—
93	C₂³	—	311	2	no	—
94	C₂³	—	622	4	no	—
95	C₂³	—	622	2	no	—
96	C₂³	C₂²×D₃₁₁	1	2	yes	—
97	C₂³	C₂²×D₃₁₁	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	4	no	—
100	D₄	—	2	2	no	—
101	D₄	—	622	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	311	4	no	—
104	C₂×C₄	—	311	2	no	—
105	C₂×C₄	—	622	2	no	—
106	C₂×C₄	—	622	4	no	—
107	C₂×C₄	—	622	4	no	—
108	C₂×C₄	C₂²⋊D₃₁₁	1	4	yes	complemented
109	C₂×C₄	C₂²×D₃₁₁	1	2	yes	—
110	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	311	4	no	—
116	C₂²	—	622	8	no	—
117	C₂²	—	622	2	no	—
118	C₂²	—	622	2	no	—
119	C₂²	C₂×C₂²⋊D₃₁₁	1	2	yes	—
120	C₂²	C₂³×D₃₁₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	622	2	no	—
123	C₄	—	622	4	no	—
124	C₄	C₂×C₂²⋊D₃₁₁	1	2	yes	—
125	C₄	C₂×C₂²⋊D₃₁₁	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	4	2	no	—
128	C₂	—	622	4	no	—
129	C₂	C₂²×C₂²⋊D₃₁₁	1	1	yes	characteristic
130	C₂	#41×D₃₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₃₁₁

C₂

yes

maximal, characteristic, complemented

#23755689

C₂

yes

maximal, complemented

#23755684

C₂

yes

maximal, complemented

#23755679

C₂

yes

maximal, complemented

#23755676

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₁₁

C₂

yes

maximal, complemented

#23755667

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₃₁₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₃₁₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁₁

C₂²

yes

—

C₂²×Dic₃₁₁

C₂²

yes

characteristic

C₂×C₄×D₃₁₁

—

C₂×C₄×D₃₁₁

C₂²

yes

—

C₂×C₄×D₃₁₁

C₂²

yes

—

C₄⋊Dic₃₁₁

C₂²

yes

complemented

Dic₃₁₁⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₃₁₁

C₂²

yes

—

C₂²⋊Dic₃₁₁

C₂²

yes

complemented

(C₂×D₃₁₁)⋊C₄

C₂²

yes

—

C₄×Dic₃₁₁

C₂²

yes

complemented

#41×C₃₁₁

—

C₂×Q₈×C₃₁₁

C₂²

yes

characteristic, complemented

C₂×D₄×C₃₁₁

C₂²

yes

—

C₂×D₄×C₃₁₁

C₂²

yes

characteristic, complemented

C₂²×C₄×C₃₁₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁₁

C₂²

yes

complemented

C₂²×D₃₁₁

—

C₂²×D₃₁₁

—

C₂²×D₃₁₁

C₂³

yes

—

C₂²⋊D₃₁₁

—

C₂×Dic₃₁₁

—

C₂×Dic₃₁₁

C₂³

yes

—

C₂×Dic₃₁₁

C₂³

yes

—

C₄×D₃₁₁

—

C₄×D₃₁₁

—

C₂³×C₃₁₁

C₂³

yes

—

C₂³×C₃₁₁

C₂³

yes

characteristic

Q₈×C₃₁₁

—

D₄×C₃₁₁

—

D₄×C₃₁₁

—

C₂×C₄×C₃₁₁

—

C₂×C₄×C₃₁₁

D₄

yes

complemented

C₂×C₄×C₃₁₁

C₂³

yes

—

C₂×C₄×C₃₁₁

C₂³

yes

characteristic

C₂×C₄×C₃₁₁

C₂³

yes

characteristic

C₂²×C₃₁₁

—

C₂²×C₃₁₁

—

C₂²×C₃₁₁

—

C₂²×C₃₁₁

C₂×D₄

yes

—

C₂²×C₃₁₁

C₂⁴

yes

characteristic

C₂×D₃₁₁

—

C₂×D₃₁₁

—

C₂×D₃₁₁

—

C₂×D₃₁₁

#41

yes

complemented

C₄×C₃₁₁

—

C₄×C₃₁₁

C₂×D₄

yes

—

C₄×C₃₁₁

C₂×D₄

yes

—

Dic₃₁₁

—

Dic₃₁₁

—

C₂×C₃₁₁

—

C₂×C₃₁₁

—

C₂×C₃₁₁

C₂²×D₄

yes

characteristic

C₂×C₃₁₁

C₂×#41

yes

—

D₃₁₁

—

C₃₁₁

#525

yes

characteristic, complemented, Sylow

#525

—

311

maximal, Sylow

C₂×#41

D₃₁₁

yes

characteristic, complemented

C₂³×C₄

—

311

—

#123

—

311

—

C₂²⋊Q₈

—

311

—

#121

—

311

—

#120

—

311

—

C₄×D₄

—

311

—

#117

—

311

—

C₂⁴

—

311

—

#41

—

C₂×Q₈

C₂×D₃₁₁

yes

characteristic, complemented

C₂×D₄

—

311

—

C₂×D₄

C₂×D₃₁₁

yes

—

C₂×D₄

C₂×D₃₁₁

yes

characteristic, complemented

C₂²×C₄

—

311

—

C₂²×C₄

—

311

—

C₂²×C₄

—

311

—

C₂²×C₄

—

622

—

C₂²×C₄

C₂×D₃₁₁

yes

characteristic

C₂²×C₄

C₂×D₃₁₁

yes

complemented

C₄⋊C₄

—

311

—

C₄⋊C₄

—

311

—

C₂²⋊C₄

—

311

—

C₂²⋊C₄

—

311

—

C₂²⋊C₄

—

311

—

C₄²

—

311

—

C₂³

—

311

—

C₂³

—

622

—

C₂³

—

622

—

C₂³

C₂²×D₃₁₁

yes

—

C₂³

C₂²×D₃₁₁

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

D₄

—

622

—

102

C₂×C₄

—

103

C₂×C₄

—

311

—

104

C₂×C₄

—

311

—

105

C₂×C₄

—

622

—

106

C₂×C₄

—

622

—

107

C₂×C₄

—

622

—

108

C₂×C₄

C₂²⋊D₃₁₁

yes

complemented

109

C₂×C₄

C₂²×D₃₁₁

yes

—

110

C₂×C₄

C₂²×D₃₁₁

yes

characteristic

111

C₂×C₄

C₂²×D₃₁₁

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

311

—

116

C₂²

—

622

—

117

C₂²

—

622

—

118

C₂²

—

622

—

119

C₂²

C₂×C₂²⋊D₃₁₁

yes

—

120

C₂²

C₂³×D₃₁₁

yes

characteristic

121

C₄

—

122

C₄

—

622

—

123

C₄

—

622

—

124

C₄

C₂×C₂²⋊D₃₁₁

yes

—

125

C₄

C₂×C₂²⋊D₃₁₁

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

622

—

129

C₂

C₂²×C₂²⋊D₃₁₁

yes

characteristic

130

C₂

#41×D₃₁₁

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₁
commutator, socle, cosocle	C₂²×C₃₁₁	C₂⁴
Frattini	C₂²	C₂³×D₃₁₁
Fitting	C₂×#41×C₃₁₁	C₂
hypercenter	C₂×#41	D₃₁₁
hypocenter	C₃₁₁	#525
LCS	G ▷ C₂²×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂×C₄

C₂²×D₃₁₁

commutator, socle, cosocle

C₂²×C₃₁₁

C₂⁴

Frattini

C₂²

C₂³×D₃₁₁

Fitting

C₂×#41×C₃₁₁

C₂

hypercenter

C₂×#41

D₃₁₁

C₃₁₁

G ▷ C₂²×C₃₁₁ ▷ C₃₁₁

C₁ ◁ C₂×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×C₄×D₃₁₁	C₂	semidirect
2. #23755689	C₂	semidirect
3. #23755684	C₂	semidirect
4. #23755679	C₂	semidirect
5. #23755676	C₂	semidirect
6. C₄×C₂²⋊D₃₁₁	C₂	semidirect
7. #23755667	C₂	semidirect
8. C₂×#41×C₃₁₁	C₂	semidirect
15. C₄⋊Dic₃₁₁	C₂²	semidirect
16. Dic₃₁₁⋊C₄	C₂²	semidirect
18. C₂²⋊Dic₃₁₁	C₂²	semidirect
20. C₄×Dic₃₁₁	C₂²	semidirect
22. C₂×Q₈×C₃₁₁	C₂²	semidirect
24. C₂×D₄×C₃₁₁	C₂²	semidirect
26. C₂²×C₄×C₃₁₁	C₂²	semidirect
42. C₂×C₄×C₃₁₁	D₄	semidirect
54. C₂×D₃₁₁	#41	semidirect
65. C₃₁₁	#525	semidirect
67. C₂×#41	D₃₁₁	semidirect
77. C₂×Q₈	C₂×D₃₁₁	semidirect
80. C₂×D₄	C₂×D₃₁₁	semidirect
86. C₂²×C₄	C₂×D₃₁₁	semidirect
108. C₂×C₄	C₂²⋊D₃₁₁	semidirect

H ≅ G/N

type

1. C₂²×C₄×D₃₁₁

C₂

semidirect

2. #23755689

C₂

semidirect

3. #23755684

C₂

semidirect

4. #23755679

C₂

semidirect

5. #23755676

C₂

semidirect

6. C₄×C₂²⋊D₃₁₁

C₂

semidirect

7. #23755667

C₂

semidirect

8. C₂×#41×C₃₁₁

C₂

semidirect

15. C₄⋊Dic₃₁₁

C₂²

semidirect

16. Dic₃₁₁⋊C₄

C₂²

semidirect

18. C₂²⋊Dic₃₁₁

C₂²

semidirect

20. C₄×Dic₃₁₁

C₂²

semidirect

22. C₂×Q₈×C₃₁₁

C₂²

semidirect

24. C₂×D₄×C₃₁₁

C₂²

semidirect

26. C₂²×C₄×C₃₁₁

C₂²

semidirect

42. C₂×C₄×C₃₁₁

D₄

semidirect

54. C₂×D₃₁₁

#41

semidirect

65. C₃₁₁

#525

semidirect

67. C₂×#41

D₃₁₁

semidirect

77. C₂×Q₈

C₂×D₃₁₁

semidirect

80. C₂×D₄

C₂×D₃₁₁

semidirect

86. C₂²×C₄

C₂×D₃₁₁

semidirect

108. C₂×C₄

C₂²⋊D₃₁₁

semidirect

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

subgroup	index
C₂²×C₄×D₃₁₁	2
#23755689	2
#23755684	2
#23755679	2
#23755676	2
C₄×C₂²⋊D₃₁₁	2
#23755667	2
C₂×#41×C₃₁₁	2
#525	311

subgroup

index

C₂²×C₄×D₃₁₁

C₄×C₂²⋊D₃₁₁

C₂×#41×C₃₁₁

311

Subgroups — #34092201

Conjugacy classes of subgroups

Complements

Maximal subgroups