Group #15683906 — subgroups

Subgroups — #15683906

Order 2944.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#13092972	C₂	1	2	yes	maximal
2	#13092820	C₂	1	2	yes	maximal, complemented
3	#13092817	C₂	1	4	yes	maximal, complemented
4	#13092815	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092452	C₂	1	4	yes	maximal
6	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	#10625117	—	2	2	no	—
10	Q₈×Dic₂₃	—	2	2	no	—
11	C₂×C₈×D₂₃	C₂²	1	2	yes	—
12	#10625068	—	2	2	no	—
13	#10625063	C₂²	1	4	yes	—
14	#10625061	C₂²	1	4	yes	—
15	#10625058	—	2	2	no	—
16	C₄²×D₂₃	C₂²	1	1	yes	characteristic
17	#10625040	C₂²	1	2	yes	—
18	C₂₃×#106	C₂²	1	4	yes	—
19	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	#10624999	C₂²	1	4	yes	—
21	C₈×Dic₂₃	C₂²	1	2	yes	—
22	#10624989	C₂²	1	8	yes	—
23	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₂₃	—	2	2	no	—
25	C₂×C₄×D₂₃	D₄	1	2	yes	—
26	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₄₆	—	2	2	no	—
28	C₂×C₈×C₂₃	C₂³	1	2	yes	—
29	C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
31	C₄²×C₂₃	C₂³	1	1	yes	characteristic
32	(C₂×D₂₃)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₂₃	—	2	2	no	—
34	C₄⋊Dic₂₃	—	2	4	no	—
35	C₄⋊Dic₂₃	C₂³	1	4	yes	—
36	Dic₂₃⋊C₄	—	2	4	no	—
37	C₄×Dic₂₃	—	2	4	no	—
38	C₄×Dic₂₃	D₄	1	2	yes	—
39	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
41	C₈×D₂₃	Q₈	1	8	yes	complemented
42	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₂₃	—	4	4	no	—
44	C₂×C₄×C₂₃	—	2	4	no	—
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₂₃	—	2	4	no	—
48	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
49	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₂₃	—	2	4	no	—
51	C₄×D₂₃	C₂×Q₈	1	4	yes	—
52	Dic₄₆	—	4	4	no	—
53	C₈×C₂₃	C₂×Q₈	1	4	yes	—
54	C₂₃⋊C₈	C₂×Q₈	1	4	yes	—
55	#3160	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₂₃	—	2	2	no	—
58	C₂×D₂₃	C₄⋊Q₈	1	2	yes	—
59	C₄×C₂₃	—	4	4	no	—
60	C₄×C₂₃	#135	1	4	yes	—
61	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
62	Dic₂₃	—	2	2	no	—
63	Dic₂₃	—	4	4	no	—
64	Dic₂₃	C₄⋊Q₈	1	2	yes	—
65	#533	—	23	2	no	—
66	#500	—	23	2	no	—
67	#499	—	23	4	no	—
68	#498	—	23	1	no	—
69	#498	D₂₃	1	1	yes	characteristic, complemented
70	#427	—	23	4	no	—
71	C₂×C₄×C₈	—	23	1	no	—
72	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₂₃	C₂×#135	1	2	yes	—
74	D₂₃	—	2	2	no	—
75	C₂²×C₈	—	23	2	no	—
76	C₄⋊Q₈	—	23	2	no	—
77	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
78	#125	—	23	4	no	—
79	C₂²⋊Q₈	—	46	2	no	—
80	C₂²⋊Q₈	—	46	2	no	—
81	C₄×Q₈	—	46	2	no	—
82	C₄×Q₈	—	46	2	no	—
83	#117	—	23	4	no	—
84	C₂×C₄²	—	23	1	no	—
85	#107	—	23	8	no	—
86	#106	—	23	4	no	—
87	#106	C₂×D₂₃	1	4	yes	—
88	C₄×C₈	—	23	2	no	—
89	C₄×C₈	—	23	1	no	—
90	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
91	C₂₃	#3160	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	46	2	no	—
94	C₂²×C₄	—	23	1	no	—
95	C₂²×C₄	—	23	2	no	—
96	C₂×C₈	—	23	2	no	—
97	C₂×C₈	—	23	8	no	—
98	C₂×C₈	C₂²×D₂₃	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	23	4	no	—
101	C₄⋊C₄	—	46	4	no	—
102	C₄⋊C₄	—	46	4	no	—
103	C₄⋊C₄	—	46	2	no	—
104	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
105	C₂²⋊C₄	—	46	4	no	—
106	C₄²	—	23	2	no	—
107	C₄²	—	23	1	no	—
108	C₄²	—	46	4	no	—
109	C₄²	C₂²×D₂₃	1	1	yes	characteristic
110	C₂³	—	23	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	92	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	23	1	no	—
115	C₂×C₄	—	23	4	no	—
116	C₂×C₄	—	23	2	no	—
117	C₂×C₄	—	46	4	no	—
118	C₂×C₄	—	46	4	no	—
119	C₂×C₄	D₄×D₂₃	1	2	yes	—
120	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
121	C₈	—	23	4	no	—
122	C₈	Q₈×D₂₃	1	4	yes	—
123	C₂²	—	23	2	no	—
124	C₂²	—	46	2	no	—
125	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	23	2	no	—
128	C₄	—	46	2	no	—
129	C₄	—	92	4	no	—
130	C₄	#10625087	1	4	yes	—
131	C₄	C₂×Q₈×D₂₃	1	2	yes	—
132	C₂	—	46	2	no	—
133	C₂	C₄⋊Q₈×D₂₃	1	1	yes	characteristic
134	C₂	C₂×#10625087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092972

C₂

yes

maximal

#13092820

C₂

yes

maximal, complemented

#13092817

C₂

yes

maximal, complemented

#13092815

C₂

yes

maximal, characteristic, complemented

#13092452

C₂

yes

maximal

C₄×C₈×D₂₃

C₂

yes

maximal, characteristic

C₂₃×#498

C₂

yes

maximal, characteristic, complemented

C₂₃×C₄⋊Q₈

C₂²

yes

—

#10625117

—

Q₈×Dic₂₃

—

C₂×C₈×D₂₃

C₂²

yes

—

#10625068

—

#10625063

C₂²

yes

—

#10625061

C₂²

yes

—

#10625058

—

C₄²×D₂₃

C₂²

yes

characteristic

#10625040

C₂²

yes

—

C₂₃×#106

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624999

C₂²

yes

—

C₈×Dic₂₃

C₂²

yes

—

#10624989

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×C₄×D₂₃

D₄

yes

—

C₂×C₄×D₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

C₂³

yes

characteristic

C₂×C₂₃⋊C₈

C₂³

yes

—

C₈×D₂₃

Q₈

yes

complemented

C₂²×D₂₃

C₂×D₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

C₂×D₄

yes

characteristic

C₄×D₂₃

—

C₄×D₂₃

C₂×Q₈

yes

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

#135

yes

—

C₄×C₂₃

C₂²×Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

C₄⋊Q₈

yes

—

—

—

—

—

D₂₃

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

C₂×#135

yes

—

D₂₃

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₂₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂²×D₂₃

yes

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂²×D₂₃

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×D₂₃

yes

—

120

C₂×C₄

C₂³×D₂₃

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₂₃

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×D₄×D₂₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#10625087

yes

—

131

C₄

C₂×Q₈×D₂₃

yes

—

132

C₂

—

133

C₂

C₄⋊Q₈×D₂₃

yes

characteristic

134

C₂

C₂×#10625087

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#498	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂²×D₂₃	C₂×D₄
hypercenter	#498	D₂₃
hypocenter	C₂₃	#3160
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#498

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂²×D₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #13092820	C₂	semidirect
3. #13092817	C₂	semidirect
4. #13092815	C₂	semidirect
7. C₂₃×#498	C₂	semidirect
41. C₈×D₂₃	Q₈	semidirect
69. #498	D₂₃	semidirect
91. C₂₃	#3160	semidirect

H ≅ G/N

type

2. #13092820

C₂

semidirect

3. #13092817

C₂

semidirect

4. #13092815

C₂

semidirect

7. C₂₃×#498

C₂

semidirect

41. C₈×D₂₃

Q₈

semidirect

69. #498

D₂₃

semidirect

91. C₂₃

#3160

semidirect

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

subgroup	index
#13092972	2
#13092820	2
#13092817	2
#13092815	2
#13092452	2
C₄×C₈×D₂₃	2
C₂₃×#498	2
#3160	23

subgroup

index

C₄×C₈×D₂₃

C₂₃×#498

#3160

Subgroups — #15683906

Conjugacy classes of subgroups

Complements

Maximal subgroups