Group #13093088 — subgroups

Subgroups — #13093088

Order 1472.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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	2	yes	maximal, complemented
2	C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625111	C₂	1	2	yes	maximal, complemented
4	#10625101	C₂	1	2	yes	maximal, complemented
5	#10625064	C₂	1	2	yes	maximal, complemented
6	#10625055	C₂	1	4	yes	maximal, complemented
7	#10625045	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
13	#67288	—	2	8	no	—
14	C₂×D₉₂	C₂²	1	2	yes	—
15	C₂×D₉₂	C₂²	1	2	yes	—
16	C₂×D₉₂	C₂²	1	2	yes	—
17	C₂×C₄×D₂₃	C₂²	1	4	yes	—
18	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₄²×C₂₃	C₂²	1	1	yes	characteristic
22	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
23	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	Dic₂₃⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₂₃	C₂²	1	2	yes	—
26	C₂³×C₂₃	C₂³	1	2	yes	—
27	C₂²×D₂₃	C₂³	1	4	yes	—
28	D₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	—	2	4	no	—
30	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²⋊D₂₃	—	2	8	no	—
35	C₂²⋊D₂₃	—	4	4	no	—
36	C₂×Dic₂₃	C₂³	1	2	yes	—
37	C₂×Dic₂₃	C₂³	1	2	yes	—
38	D₉₂	—	2	4	no	—
39	D₉₂	—	2	4	no	—
40	D₉₂	—	4	2	no	—
41	C₄×D₂₃	—	2	8	no	—
42	Dic₄₆	D₄	1	4	yes	complemented
43	C₂²×C₂₃	—	2	2	no	—
44	C₂²×C₂₃	—	2	2	no	—
45	C₂²×C₂₃	—	2	2	no	—
46	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×D₂₃	—	2	4	no	—
48	C₂×D₂₃	—	2	4	no	—
49	C₂×D₂₃	—	2	4	no	—
50	C₄×C₂₃	—	2	2	no	—
51	C₄×C₂₃	—	2	1	no	—
52	C₄×C₂₃	C₂×D₄	1	2	yes	—
53	C₄×C₂₃	#41	1	2	yes	complemented
54	Dic₂₃	—	2	2	no	—
55	Dic₂₃	C₂×D₄	1	4	yes	—
56	#550	—	23	1	no	maximal, Sylow
57	C₂×C₂₃	—	4	2	no	—
58	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
59	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
60	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
61	D₂₃	—	4	4	no	—
62	C₂×#41	—	23	2	no	—
63	#127	—	23	2	no	—
64	#127	—	23	1	no	—
65	#121	—	23	2	no	—
66	#121	—	23	4	no	—
67	C₄×Q₈	—	23	1	no	—
68	C₄×D₄	—	23	2	no	—
69	C₄×D₄	D₂₃	1	1	yes	characteristic, complemented
70	C₂₃	#550	1	1	yes	characteristic, complemented, Sylow
71	#41	—	46	8	no	—
72	C₂×Q₈	—	23	1	no	—
73	C₂×D₄	—	23	4	no	—
74	C₂×D₄	—	23	4	no	—
75	C₂×D₄	—	23	2	no	—
76	C₂×D₄	—	23	2	no	—
77	C₂×D₄	—	23	2	no	—
78	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
79	C₂²×C₄	—	23	4	no	—
80	C₂²×C₄	C₂×D₂₃	1	2	yes	—
81	C₄⋊C₄	—	23	2	no	—
82	C₄⋊C₄	—	23	1	no	—
83	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
84	C₂²⋊C₄	—	23	4	no	—
85	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
86	C₄²	—	23	2	no	—
87	C₄²	C₂×D₂₃	1	1	yes	characteristic
88	C₂³	—	23	4	no	—
89	C₂³	C₂²×D₂₃	1	2	yes	—
90	Q₈	—	23	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	46	8	no	—
93	D₄	—	46	4	no	—
94	D₄	—	46	4	no	—
95	D₄	—	92	2	no	—
96	D₄	—	92	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	23	2	no	—
99	C₂×C₄	—	23	2	no	—
100	C₂×C₄	—	46	8	no	—
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	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	46	4	no	—
109	C₂²	—	46	4	no	—
110	C₂²	—	46	4	no	—
111	C₂²	C₂³×D₂₃	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	23	4	no	—
115	C₄	—	46	2	no	—
116	C₄	#67288	1	2	yes	complemented
117	C₄	D₄×D₂₃	1	2	yes	—
118	C₂	—	4	2	no	—
119	C₂	—	92	4	no	—
120	C₂	C₂×#67288	1	1	yes	characteristic
121	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic
122	C₂	#10625160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67288

C₂

yes

maximal, complemented

C₄×D₄×C₂₃

C₂

yes

maximal, characteristic, complemented

#10625111

C₂

yes

maximal, complemented

#10625101

C₂

yes

maximal, complemented

#10625064

C₂

yes

maximal, complemented

#10625055

C₂

yes

maximal, complemented

#10625045

C₂

yes

maximal, characteristic, complemented

C₄×Dic₄₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

#67288

—

C₂×D₉₂

C₂²

yes

—

C₂×D₉₂

C₂²

yes

—

C₂×D₉₂

C₂²

yes

—

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

D₄

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₄

yes

—

C₄×C₂₃

#41

yes

complemented

Dic₂₃

—

Dic₂₃

C₂×D₄

yes

—

#550

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

D₂₃

—

C₂×#41

—

—

—

—

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₂₃

yes

characteristic, complemented

C₂₃

#550

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₄²

—

C₄²

C₂×D₂₃

yes

characteristic

C₂³

—

C₂³

C₂²×D₂₃

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

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

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₂₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#67288

yes

complemented

117

C₄

D₄×D₂₃

yes

—

118

C₂

—

119

C₂

—

120

C₂

C₂×#67288

yes

characteristic

121

C₂

C₂×D₄×D₂₃

yes

characteristic

122

C₂

#10625160

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₄×D₄×C₂₃	C₂
hypercenter	C₄×D₄	D₂₃
hypocenter	C₂₃	#550
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₄×D₄×C₂₃

C₂

hypercenter

C₄×D₄

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#67288	C₂	semidirect
2. C₄×D₄×C₂₃	C₂	semidirect
3. #10625111	C₂	semidirect
4. #10625101	C₂	semidirect
5. #10625064	C₂	semidirect
6. #10625055	C₂	semidirect
7. #10625045	C₂	semidirect
8. C₄×Dic₄₆	C₂	semidirect
42. Dic₄₆	D₄	semidirect
53. C₄×C₂₃	#41	semidirect
69. C₄×D₄	D₂₃	semidirect
70. C₂₃	#550	semidirect
116. C₄	#67288	semidirect

H ≅ G/N

type

1. C₂×#67288

C₂

semidirect

2. C₄×D₄×C₂₃

C₂

semidirect

3. #10625111

C₂

semidirect

4. #10625101

C₂

semidirect

5. #10625064

C₂

semidirect

6. #10625055

C₂

semidirect

7. #10625045

C₂

semidirect

8. C₄×Dic₄₆

C₂

semidirect

42. Dic₄₆

D₄

semidirect

53. C₄×C₂₃

#41

semidirect

69. C₄×D₄

D₂₃

semidirect

70. C₂₃

#550

semidirect

116. C₄

#67288

semidirect

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

subgroup	index
C₂×#67288	2
C₄×D₄×C₂₃	2
#10625111	2
#10625101	2
#10625064	2
#10625055	2
#10625045	2
C₄×Dic₄₆	2
#550	23

subgroup

index

C₄×D₄×C₂₃

C₄×Dic₄₆

Subgroups — #13093088

Conjugacy classes of subgroups

Complements

Maximal subgroups