Group #34092409 — subgroups

Subgroups — #34092409

Order 19904.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#23755708	C₂	1	1	yes	maximal, characteristic, complemented
2	#23755694	C₂	1	4	yes	maximal, complemented
3	#23755682	C₂	1	2	yes	maximal, complemented
4	#23755679	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₂₄₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#23755667	C₂	1	2	yes	maximal, complemented
7	#23755666	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₃₁₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₆₂₂	C₂²	1	2	yes	—
10	C₂×D₁₂₄₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₃₁₁	C₂²	1	4	yes	—
12	C₂×C₄×D₃₁₁	C₂²	1	2	yes	—
13	C₄⋊Dic₃₁₁	C₂²	1	2	yes	—
14	C₄⋊Dic₃₁₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₁₁	C₂²	1	2	yes	—
16	C₂²⋊Dic₃₁₁	C₂²	1	4	yes	—
17	(C₂×D₃₁₁)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₃₁₁)⋊C₄	C₂²	1	2	yes	—
19	C₄×Dic₃₁₁	C₂²	1	4	yes	—
20	C₂×D₄×C₃₁₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₃₁₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₃₁₁	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₃₁₁	C₂²	1	2	yes	—
24	C₄²×C₃₁₁	C₂²	1	1	yes	characteristic
25	C₂²×D₃₁₁	C₂³	1	2	yes	—
26	Dic₆₂₂	—	4	2	no	—
27	D₁₂₄₄	—	2	2	no	—
28	C₂²⋊D₃₁₁	—	4	4	no	—
29	C₂×Dic₃₁₁	C₂³	1	2	yes	—
30	C₂×Dic₃₁₁	C₂³	1	4	yes	—
31	C₄×D₃₁₁	—	2	4	no	—
32	C₂³×C₃₁₁	C₂³	1	2	yes	—
33	D₄×C₃₁₁	—	2	2	no	—
34	C₂×C₄×C₃₁₁	—	2	4	no	—
35	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₃₁₁	C₂³	1	2	yes	—
37	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₃₁₁	C₂³	1	1	yes	characteristic
39	C₂²×C₃₁₁	—	2	2	no	—
40	C₂²×C₃₁₁	—	2	2	no	—
41	C₂²×C₃₁₁	—	2	2	no	—
42	C₂²×C₃₁₁	C₂⁴	1	1	yes	characteristic
43	C₂×D₃₁₁	—	2	2	no	—
44	C₂×D₃₁₁	—	2	2	no	—
45	C₂×D₃₁₁	—	2	2	no	—
46	C₄×C₃₁₁	—	2	2	no	—
47	C₄×C₃₁₁	—	2	1	no	—
48	C₄×C₃₁₁	#41	1	2	yes	—
49	C₄×C₃₁₁	#41	1	2	yes	—
50	Dic₃₁₁	—	2	2	no	—
51	Dic₃₁₁	—	2	4	no	—
52	C₂×C₃₁₁	—	4	2	no	—
53	C₂×C₃₁₁	C₂×#41	1	1	yes	characteristic
54	C₂×C₃₁₁	C₂×#41	1	1	yes	characteristic
55	C₂×C₃₁₁	H(5,𝔽₂)	1	1	yes	characteristic
56	D₃₁₁	—	4	2	no	—
57	C₃₁₁	#555	1	1	yes	characteristic, complemented, Sylow
58	#555	—	311	1	no	maximal, Sylow
59	C₄⋊Q₈	—	311	1	no	—
60	#124	—	311	4	no	—
61	#121	—	311	2	no	—
62	#121	—	311	2	no	—
63	C₄×D₄	—	311	1	no	—
64	C₄×D₄	D₃₁₁	1	1	yes	characteristic, complemented
65	#117	—	311	2	no	—
66	#117	—	311	2	no	—
67	C₂×Q₈	—	311	2	no	—
68	C₂×D₄	—	311	4	no	—
69	C₂×D₄	—	311	1	no	—
70	C₂×D₄	C₂×D₃₁₁	1	1	yes	characteristic
71	C₂²×C₄	—	311	2	no	—
72	C₂²×C₄	C₂×D₃₁₁	1	2	yes	—
73	C₄⋊C₄	—	311	1	no	—
74	C₄⋊C₄	—	311	2	no	—
75	C₄⋊C₄	—	311	2	no	—
76	C₄⋊C₄	C₂×D₃₁₁	1	1	yes	characteristic
77	C₂²⋊C₄	—	311	4	no	—
78	C₂²⋊C₄	—	311	4	no	—
79	C₂²⋊C₄	—	311	2	no	—
80	C₂²⋊C₄	C₂×D₃₁₁	1	2	yes	—
81	C₄²	—	311	4	no	—
82	C₄²	C₂×D₃₁₁	1	1	yes	characteristic
83	C₂³	—	311	2	no	—
84	C₂³	C₂²×D₃₁₁	1	2	yes	—
85	Q₈	—	1244	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	622	2	no	—
88	D₄	—	1244	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	311	2	no	—
91	C₂×C₄	—	311	4	no	—
92	C₂×C₄	—	622	4	no	—
93	C₂×C₄	C₂²×D₃₁₁	1	2	yes	—
94	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₃₁₁	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	622	2	no	—
101	C₂²	—	622	2	no	—
102	C₂²	—	622	2	no	—
103	C₂²	C₂³×D₃₁₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	622	2	no	—
107	C₄	—	622	4	no	—
108	C₄	#19097314	1	2	yes	—
109	C₄	#19097315	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	1244	2	no	—
112	C₂	C₂×#19097314	1	1	yes	characteristic
113	C₂	C₂×#19097315	1	1	yes	characteristic
114	C₂	#23755754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23755708

C₂

yes

maximal, characteristic, complemented

#23755694

C₂

yes

maximal, complemented

#23755682

C₂

yes

maximal, complemented

#23755679

C₂

yes

maximal, complemented

C₄×D₁₂₄₄

C₂

yes

maximal, characteristic, complemented

#23755667

C₂

yes

maximal, complemented

#23755666

C₂

yes

maximal, complemented

C₄×D₄×C₃₁₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₆₂₂

C₂²

yes

—

C₂×D₁₂₄₄

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁₁

C₂²

yes

—

C₂×C₄×D₃₁₁

C₂²

yes

—

C₄⋊Dic₃₁₁

C₂²

yes

—

C₄⋊Dic₃₁₁

C₂²

yes

characteristic

C₄⋊Dic₃₁₁

C₂²

yes

—

C₂²⋊Dic₃₁₁

C₂²

yes

—

(C₂×D₃₁₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁₁)⋊C₄

C₂²

yes

—

C₄×Dic₃₁₁

C₂²

yes

—

C₂×D₄×C₃₁₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁₁

C₂²

yes

—

C₄⋊C₄×C₃₁₁

C₂²

yes

characteristic

C₂²⋊C₄×C₃₁₁

C₂²

yes

—

C₄²×C₃₁₁

C₂²

yes

characteristic

C₂²×D₃₁₁

C₂³

yes

—

Dic₆₂₂

—

D₁₂₄₄

—

C₂²⋊D₃₁₁

—

C₂×Dic₃₁₁

C₂³

yes

—

C₂×Dic₃₁₁

C₂³

yes

—

C₄×D₃₁₁

—

C₂³×C₃₁₁

C₂³

yes

—

D₄×C₃₁₁

—

C₂×C₄×C₃₁₁

—

C₂×C₄×C₃₁₁

C₂³

yes

characteristic

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

C₂⁴

yes

characteristic

C₂×D₃₁₁

—

C₂×D₃₁₁

—

C₂×D₃₁₁

—

C₄×C₃₁₁

—

C₄×C₃₁₁

—

C₄×C₃₁₁

#41

yes

—

C₄×C₃₁₁

#41

yes

—

Dic₃₁₁

—

Dic₃₁₁

—

C₂×C₃₁₁

—

C₂×C₃₁₁

C₂×#41

yes

characteristic

C₂×C₃₁₁

C₂×#41

yes

characteristic

C₂×C₃₁₁

H(5,𝔽₂)

yes

characteristic

D₃₁₁

—

C₃₁₁

#555

yes

characteristic, complemented, Sylow

#555

—

311

maximal, Sylow

C₄⋊Q₈

—

311

—

#124

—

311

—

#121

—

311

—

#121

—

311

—

C₄×D₄

—

311

—

C₄×D₄

D₃₁₁

yes

characteristic, complemented

#117

—

311

—

#117

—

311

—

C₂×Q₈

—

311

—

C₂×D₄

—

311

—

C₂×D₄

—

311

—

C₂×D₄

C₂×D₃₁₁

yes

characteristic

C₂²×C₄

—

311

—

C₂²×C₄

C₂×D₃₁₁

yes

—

C₄⋊C₄

—

311

—

C₄⋊C₄

—

311

—

C₄⋊C₄

—

311

—

C₄⋊C₄

C₂×D₃₁₁

yes

characteristic

C₂²⋊C₄

—

311

—

C₂²⋊C₄

—

311

—

C₂²⋊C₄

—

311

—

C₂²⋊C₄

C₂×D₃₁₁

yes

—

C₄²

—

311

—

C₄²

C₂×D₃₁₁

yes

characteristic

C₂³

—

311

—

C₂³

C₂²×D₃₁₁

yes

—

Q₈

—

1244

—

D₄

—

D₄

—

622

—

D₄

—

1244

—

C₂×C₄

—

C₂×C₄

—

311

—

C₂×C₄

—

311

—

C₂×C₄

—

622

—

C₂×C₄

C₂²×D₃₁₁

yes

—

C₂×C₄

C₂²×D₃₁₁

yes

characteristic

C₂×C₄

C₂²×D₃₁₁

yes

characteristic

C₂×C₄

C₂²×D₃₁₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

622

—

101

C₂²

—

622

—

102

C₂²

—

622

—

103

C₂²

C₂³×D₃₁₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

622

—

107

C₄

—

622

—

108

C₄

#19097314

yes

—

109

C₄

#19097315

yes

—

110

C₂

—

111

C₂

—

1244

—

112

C₂

C₂×#19097314

yes

characteristic

113

C₂

C₂×#19097315

yes

characteristic

114

C₂

#23755754

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₁
commutator, socle, cosocle	C₂²×C₃₁₁	C₂⁴
Fitting	C₄×D₄×C₃₁₁	C₂
hypercenter	C₄×D₄	D₃₁₁
hypocenter	C₃₁₁	#555
LCS	G ▷ C₂²×C₃₁₁ ▷ C₃₁₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁₁

commutator, socle, cosocle

C₂²×C₃₁₁

C₂⁴

Fitting

C₄×D₄×C₃₁₁

C₂

hypercenter

C₄×D₄

D₃₁₁

C₃₁₁

G ▷ C₂²×C₃₁₁ ▷ C₃₁₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #23755708	C₂	semidirect
2. #23755694	C₂	semidirect
3. #23755682	C₂	semidirect
4. #23755679	C₂	semidirect
5. C₄×D₁₂₄₄	C₂	semidirect
6. #23755667	C₂	semidirect
7. #23755666	C₂	semidirect
8. C₄×D₄×C₃₁₁	C₂	semidirect
57. C₃₁₁	#555	semidirect
64. C₄×D₄	D₃₁₁	semidirect

H ≅ G/N

type

1. #23755708

C₂

semidirect

2. #23755694

C₂

semidirect

3. #23755682

C₂

semidirect

4. #23755679

C₂

semidirect

5. C₄×D₁₂₄₄

C₂

semidirect

6. #23755667

C₂

semidirect

7. #23755666

C₂

semidirect

8. C₄×D₄×C₃₁₁

C₂

semidirect

57. C₃₁₁

#555

semidirect

64. C₄×D₄

D₃₁₁

semidirect

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

subgroup	index
#23755708	2
#23755694	2
#23755682	2
#23755679	2
C₄×D₁₂₄₄	2
#23755667	2
#23755666	2
C₄×D₄×C₃₁₁	2
#555	311

subgroup

index

C₄×D₁₂₄₄

C₄×D₄×C₃₁₁

311

Subgroups — #34092409

Conjugacy classes of subgroups

Complements

Maximal subgroups