Group #13092986 — subgroups

Subgroups — #13092986

Order 1472.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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×#67288	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625066	C₂	1	4	yes	maximal, complemented
5	#10625053	C₂	1	4	yes	maximal, complemented
6	#10625046	C₂	1	2	yes	maximal, complemented
7	#10625045	C₂	1	2	yes	maximal, complemented
8	C₂³×D₂₃	C₂²	1	2	yes	—
9	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
11	#67288	—	2	4	no	—
12	C₂×D₉₂	—	2	4	no	—
13	C₂×D₉₂	C₂²	1	2	yes	—
14	C₂×D₉₂	C₂²	1	2	yes	—
15	C₂×D₉₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×D₉₂	C₂²	1	8	yes	complemented
17	C₂×C₄×D₂₃	C₂²	1	2	yes	complemented
18	C₂×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₄²×C₂₃	C₂²	1	2	yes	complemented
22	(C₂×D₂₃)⋊C₄	C₂²	1	8	yes	complemented
23	C₂³×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×D₂₃	—	2	4	no	—
25	C₂²×D₂₃	—	4	4	no	—
26	C₂²×D₂₃	C₂³	1	4	yes	—
27	C₂²×D₂₃	C₂³	1	2	yes	—
28	C₂×C₄×C₂₃	D₄	1	4	yes	complemented
29	C₂×C₄×C₂₃	C₂³	1	4	yes	—
30	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²⋊D₂₃	—	2	4	no	—
33	C₂×Dic₂₃	C₂³	1	2	yes	—
34	D₉₂	—	2	4	no	—
35	D₉₂	—	2	4	no	—
36	D₉₂	—	2	2	no	—
37	D₉₂	—	4	8	no	—
38	C₄×D₂₃	—	2	4	no	—
39	Dic₄₆	—	2	2	no	—
40	C₂²×C₂₃	—	2	2	no	—
41	C₂²×C₂₃	C₂×D₄	1	2	yes	—
42	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
43	C₂×D₂₃	—	2	2	no	—
44	C₂×D₂₃	—	2	4	no	—
45	C₂×D₂₃	—	2	4	no	—
46	C₂×D₂₃	—	2	8	no	—
47	C₂×D₂₃	—	4	8	no	—
48	C₄×C₂₃	—	2	4	no	—
49	C₄×C₂₃	C₂×D₄	1	2	yes	—
50	C₄×C₂₃	C₂×D₄	1	2	yes	—
51	Dic₂₃	—	2	2	no	—
52	#535	—	23	1	no	maximal, Sylow
53	C₂×C₂₃	—	2	2	no	—
54	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
56	D₂₃	—	4	2	no	—
57	D₂₃	—	4	4	no	—
58	C₂×#41	—	23	1	no	—
59	C₂²×D₄	—	23	1	no	—
60	#127	—	23	2	no	—
61	#124	—	23	2	no	—
62	#121	—	23	4	no	—
63	#120	—	23	4	no	—
64	#117	D₂₃	1	1	yes	characteristic, complemented
65	C₂₃	#535	1	1	yes	characteristic, complemented, Sylow
66	C₂⁴	—	23	2	no	—
67	#41	—	46	4	no	—
68	C₂×Q₈	—	23	1	no	—
69	C₂×D₄	—	23	2	no	—
70	C₂×D₄	—	23	2	no	—
71	C₂×D₄	—	23	2	no	—
72	C₂×D₄	—	23	1	no	—
73	C₂×D₄	—	23	8	no	—
74	C₂×D₄	—	46	4	no	—
75	C₂²×C₄	—	23	2	no	—
76	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
78	C₂²⋊C₄	—	23	8	no	—
79	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
80	C₄²	C₂×D₂₃	1	2	yes	complemented
81	C₂³	—	23	4	no	—
82	C₂³	—	23	2	no	—
83	C₂³	—	46	4	no	—
84	C₂³	—	92	4	no	—
85	C₂³	C₂²×D₂₃	1	1	yes	characteristic
86	Q₈	—	46	2	no	—
87	D₄	—	46	4	no	—
88	D₄	—	46	4	no	—
89	D₄	—	46	4	no	—
90	D₄	—	46	2	no	—
91	D₄	—	92	8	no	—
92	C₂×C₄	—	23	2	no	—
93	C₂×C₄	—	46	4	no	—
94	C₂×C₄	D₉₂	1	4	yes	complemented
95	C₂×C₄	C₂²×D₂₃	1	4	yes	—
96	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	46	2	no	—
100	C₂²	—	46	4	no	—
101	C₂²	—	46	4	no	—
102	C₂²	—	46	8	no	—
103	C₂²	—	92	8	no	—
104	C₂²	C₂×D₉₂	1	2	yes	—
105	C₂²	C₂³×D₂₃	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	46	2	no	—
108	C₄	C₂×D₉₂	1	2	yes	—
109	C₄	C₂×D₉₂	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	92	4	no	—
112	C₂	—	92	2	no	—
113	C₂	C₂²×D₉₂	1	1	yes	characteristic
114	C₂	#10625160	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67288

C₂

yes

maximal, characteristic, complemented

C₂²×D₉₂

C₂

yes

maximal, characteristic, complemented

C₂₃×#117

C₂

yes

maximal, characteristic, complemented

#10625066

C₂

yes

maximal, complemented

#10625053

C₂

yes

maximal, complemented

#10625046

C₂

yes

maximal, complemented

#10625045

C₂

yes

maximal, complemented

C₂³×D₂₃

C₂²

yes

—

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

#67288

—

C₂×D₉₂

—

C₂×D₉₂

C₂²

yes

—

C₂×D₉₂

C₂²

yes

—

C₂×D₉₂

C₂²

yes

characteristic, complemented

C₂×D₉₂

C₂²

yes

complemented

C₂×C₄×D₂₃

C₂²

yes

complemented

C₂×Dic₄₆

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₂₃

C₂²

yes

complemented

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

complemented

(C₂×D₂₃)⋊C₄

C₂²

yes

complemented

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

—

C₂×C₄×C₂₃

D₄

yes

complemented

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

D₉₂

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₄

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₄

yes

—

C₄×C₂₃

C₂×D₄

yes

—

Dic₂₃

—

#535

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

—

D₂₃

—

D₂₃

—

C₂×#41

—

C₂²×D₄

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂₃

#535

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₄²

C₂×D₂₃

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₂₃

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₉₂

yes

complemented

C₂×C₄

C₂²×D₂₃

yes

—

C₂×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₄

C₂²×D₂₃

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×D₉₂

yes

—

105

C₂²

C₂³×D₂₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×D₉₂

yes

—

109

C₄

C₂×D₉₂

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

113

C₂

C₂²×D₉₂

yes

characteristic

114

C₂

#10625160

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#117	C₂
hypercenter	#117	D₂₃
hypocenter	C₂₃	#535
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#117

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#67288	C₂	semidirect
2. C₂²×D₉₂	C₂	semidirect
3. C₂₃×#117	C₂	semidirect
4. #10625066	C₂	semidirect
5. #10625053	C₂	semidirect
6. #10625046	C₂	semidirect
7. #10625045	C₂	semidirect
15. C₂×D₉₂	C₂²	semidirect
16. C₂×D₉₂	C₂²	semidirect
17. C₂×C₄×D₂₃	C₂²	semidirect
18. C₂×Dic₄₆	C₂²	semidirect
19. C₄⋊C₄×C₂₃	C₂²	semidirect
21. C₄²×C₂₃	C₂²	semidirect
22. (C₂×D₂₃)⋊C₄	C₂²	semidirect
28. C₂×C₄×C₂₃	D₄	semidirect
64. #117	D₂₃	semidirect
65. C₂₃	#535	semidirect
77. C₄⋊C₄	C₂×D₂₃	semidirect
80. C₄²	C₂×D₂₃	semidirect
94. C₂×C₄	D₉₂	semidirect

H ≅ G/N

type

1. C₂×#67288

C₂

semidirect

2. C₂²×D₉₂

C₂

semidirect

3. C₂₃×#117

C₂

semidirect

4. #10625066

C₂

semidirect

5. #10625053

C₂

semidirect

6. #10625046

C₂

semidirect

7. #10625045

C₂

semidirect

15. C₂×D₉₂

C₂²

semidirect

16. C₂×D₉₂

C₂²

semidirect

17. C₂×C₄×D₂₃

C₂²

semidirect

18. C₂×Dic₄₆

C₂²

semidirect

19. C₄⋊C₄×C₂₃

C₂²

semidirect

21. C₄²×C₂₃

C₂²

semidirect

22. (C₂×D₂₃)⋊C₄

C₂²

semidirect

28. C₂×C₄×C₂₃

D₄

semidirect

64. #117

D₂₃

semidirect

65. C₂₃

#535

semidirect

77. C₄⋊C₄

C₂×D₂₃

semidirect

80. C₄²

C₂×D₂₃

semidirect

94. C₂×C₄

D₉₂

semidirect

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

subgroup	index
C₂×#67288	2
C₂²×D₉₂	2
C₂₃×#117	2
#10625066	2
#10625053	2
#10625046	2
#10625045	2
#535	23

subgroup

index

C₂×#67288

C₂²×D₉₂

C₂₃×#117

Subgroups — #13092986

Conjugacy classes of subgroups

Complements

Maximal subgroups