Group #13093166 — subgroups

Subgroups — #13093166

Order 1472.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625118	C₂	1	1	yes	maximal, characteristic
3	#10625117	C₂	1	2	yes	maximal, complemented
4	#10625115	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625069	C₂	1	2	yes	maximal, complemented
6	#10625067	C₂	1	2	yes	maximal, complemented
7	#10625065	C₂	1	2	yes	maximal
8	#10625060	C₂	1	2	yes	maximal, complemented
9	#10625047	C₂	1	2	yes	maximal, complemented
10	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×D₉₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₂₃	C₂²	1	2	yes	—
14	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
19	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
23	C₄⋊Dic₂₃	C₂²	1	2	yes	—
24	Dic₂₃⋊C₄	C₂²	1	2	yes	—
25	Dic₂₃⋊C₄	C₂²	1	2	yes	—
26	Dic₂₃⋊C₄	C₂²	1	2	yes	—
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂²×D₂₃	C₂³	1	2	yes	—
32	Q₈×C₂₃	—	4	1	no	—
33	Q₈×C₂₃	—	4	1	no	—
34	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×Dic₂₃	C₂³	1	2	yes	—
40	C₂×Dic₂₃	C₂³	1	2	yes	—
41	C₂×Dic₂₃	C₂³	1	2	yes	—
42	D₉₂	—	4	1	no	—
43	C₄×D₂₃	—	4	2	no	—
44	Dic₄₆	—	4	1	no	—
45	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
46	C₂×D₂₃	—	2	2	no	—
47	C₂×D₂₃	—	2	4	no	—
48	C₄×C₂₃	—	2	1	no	—
49	C₄×C₂₃	—	2	2	no	—
50	C₄×C₂₃	—	2	1	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	—	2	2	no	—
53	Dic₂₃	—	2	2	no	—
54	Dic₂₃	—	2	2	no	—
55	Dic₂₃	—	2	2	no	—
56	#563	—	23	1	no	maximal, Sylow
57	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
58	C₂×C₂₃	2₋¹⁺⁴	1	2	yes	—
59	D₂₃	—	4	2	no	—
60	C₄⋊Q₈	—	23	1	no	—
61	C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
62	#126	—	23	2	no	—
63	#126	—	23	2	no	—
64	#125	—	23	2	no	—
65	#124	—	23	1	no	—
66	#123	—	23	2	no	—
67	C₂²⋊Q₈	—	23	2	no	—
68	C₂²⋊Q₈	—	23	2	no	—
69	C₂₃	#563	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₈	—	23	1	no	—
71	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
73	C₂×D₄	—	23	1	no	—
74	C₂²×C₄	—	23	2	no	—
75	C₄⋊C₄	—	23	2	no	—
76	C₄⋊C₄	—	23	2	no	—
77	C₄⋊C₄	—	23	2	no	—
78	C₄⋊C₄	—	23	2	no	—
79	C₄⋊C₄	—	23	2	no	—
80	C₄⋊C₄	—	23	2	no	—
81	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
82	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
83	C₂²⋊C₄	—	23	2	no	—
84	C₂²⋊C₄	—	23	2	no	—
85	C₂²⋊C₄	—	23	2	no	—
86	C₂²⋊C₄	—	23	2	no	—
87	C₂²⋊C₄	—	23	2	no	—
88	C₄²	—	23	1	no	—
89	C₄²	—	23	1	no	—
90	C₄²	C₂×D₂₃	1	1	yes	characteristic
91	C₂³	—	23	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	92	1	no	—
95	D₄	—	92	1	no	—
96	C₂×C₄	—	23	2	no	—
97	C₂×C₄	—	23	2	no	—
98	C₂×C₄	—	23	2	no	—
99	C₂×C₄	—	92	2	no	—
100	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₂₃	1	2	yes	—
103	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₂₃	1	2	yes	—
105	C₂²	—	46	2	no	—
106	C₂²	—	46	4	no	—
107	C₂²	C₂³×D₂₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	46	2	no	—
114	C₄	—	46	2	no	—
115	C₄	—	46	2	no	—
116	C₂	—	92	2	no	—
117	C₂	#10625155	1	1	yes	characteristic
118	C₂	#10625158	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#10625118

C₂

yes

maximal, characteristic

#10625117

C₂

yes

maximal, complemented

#10625115

C₂

yes

maximal, characteristic, complemented

#10625069

C₂

yes

maximal, complemented

#10625067

C₂

yes

maximal, complemented

#10625065

C₂

yes

maximal

#10625060

C₂

yes

maximal, complemented

#10625047

C₂

yes

maximal, complemented

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×D₂₃

C₂³

yes

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

C₂×C₂₃

yes

C₂×C₂₃

2₋¹⁺⁴

yes

—

D₂₃

—

C₄⋊Q₈

—

C₄⋊Q₈

D₂₃

yes

characteristic, complemented

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂₃

#563

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

—

C₄⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₂₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²×D₂₃

yes

characteristic

101

C₂×C₄

C₂²×D₂₃

yes

characteristic

102

C₂×C₄

C₂²×D₂₃

yes

—

103

C₂×C₄

C₂²×D₂₃

yes

characteristic

104

C₂×C₄

C₂²×D₂₃

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂³×D₂₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₂

—

117

C₂

#10625155

yes

characteristic

118

C₂

#10625158

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₄⋊Q₈	C₂
prime radical	C₂×D₉₂	C₂²
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#563
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×C₄⋊Q₈

C₂

prime radical

C₂×D₉₂

C₂²

hypercenter

C₄⋊Q₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂₃×C₄⋊Q₈	C₂	semidirect
3. #10625117	C₂	semidirect
4. #10625115	C₂	semidirect
5. #10625069	C₂	semidirect
6. #10625067	C₂	semidirect
8. #10625060	C₂	semidirect
9. #10625047	C₂	semidirect
61. C₄⋊Q₈	D₂₃	semidirect
69. C₂₃	#563	semidirect

H ≅ G/N

type

1. C₂₃×C₄⋊Q₈

C₂

semidirect

3. #10625117

C₂

semidirect

4. #10625115

C₂

semidirect

5. #10625069

C₂

semidirect

6. #10625067

C₂

semidirect

8. #10625060

C₂

semidirect

9. #10625047

C₂

semidirect

61. C₄⋊Q₈

D₂₃

semidirect

69. C₂₃

#563

semidirect

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

subgroup	index
C₂₃×C₄⋊Q₈	2
#10625118	2
#10625117	2
#10625115	2
#10625069	2
#10625067	2
#10625065	2
#10625060	2
#10625047	2
#563	23

subgroup

index

C₂₃×C₄⋊Q₈

Subgroups — #13093166

Conjugacy classes of subgroups

Complements

Maximal subgroups