Group #15674125 — subgroups

Subgroups — #15674125

Order 2944.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#10624999	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624999	C₂	1	1	yes	maximal, characteristic
3	#13092315	C₂	1	2	yes	maximal
4	#13092180	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂×Dic₂₃⋊C₄	—	2	2	no	—
12	C₄×C₈×C₂₃	—	2	2	no	—
13	C₂₃⋊#95	—	2	2	no	—
14	#10624999	—	2	2	no	—
15	#10624999	C₄	1	4	yes	complemented
16	C₂²×C₄×C₂₃	D₄	1	2	yes	—
17	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₃	—	2	2	no	—
19	C₂²×Dic₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	—	2	1	no	—
21	C₂×C₈×C₂₃	—	2	1	no	—
22	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₂₃	D₄	1	2	yes	—
24	C₂×C₈×C₂₃	Q₈	1	2	yes	—
25	C₂×C₈×C₂₃	Q₈	1	2	yes	—
26	C₄²×C₂₃	—	2	1	no	—
27	C₄²×C₂₃	—	2	1	no	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
32	Dic₂₃⋊C₄	—	4	4	no	—
33	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	SD₁₆	1	4	yes	—
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
43	C₂×Dic₂₃	—	2	4	no	—
44	C₂×Dic₂₃	—	4	2	no	—
45	C₂×Dic₂₃	—	4	2	no	—
46	C₂×Dic₂₃	—	4	2	no	—
47	C₂×Dic₂₃	—	4	2	no	—
48	C₂×Dic₂₃	—	4	2	no	—
49	C₈×C₂₃	—	2	2	no	—
50	C₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
51	#1941	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₃	#121	1	1	yes	characteristic
54	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₂₃	C₂×SD₁₆	1	2	yes	—
56	C₂²×C₂₃	#135	1	2	yes	—
57	C₄×C₂₃	—	2	4	no	—
58	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₂₃	#125	1	1	yes	characteristic
62	Dic₂₃	—	4	4	no	—
63	Dic₂₃	—	8	2	no	—
64	C₂×#106	—	23	1	no	—
65	C₂×#106	—	23	1	no	—
66	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
67	#389	—	23	2	no	—
68	#340	—	23	2	no	—
69	C₂×C₂₃	#389	1	1	yes	characteristic
70	C₂×C₂₃	C₄×SD₁₆	1	2	yes	—
71	C₂×C₂₃	#465	1	2	yes	—
72	C₂×C₂₃	#498	1	1	yes	characteristic
73	C₂×C₂₃	#499	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	23	2	no	—
77	C₂×C₄⋊C₄	—	23	2	no	—
78	C₂×C₄⋊C₄	—	46	2	no	—
79	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
80	#106	—	23	4	no	—
81	#106	—	46	2	no	—
82	C₄×C₈	—	2	2	no	—
83	#95	—	46	2	no	—
84	C₂₃	#1941	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	—	46	2	no	—
86	C₂²×C₄	—	46	2	no	—
87	C₂²×C₄	D₉₂	1	2	yes	—
88	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	Dic₄₆	1	2	yes	—
92	C₂×C₈	Dic₄₆	1	2	yes	—
93	C₂×C₈	C₄×D₂₃	1	2	yes	—
94	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
95	C₄⋊C₄	—	23	4	no	—
96	C₄⋊C₄	—	46	2	no	—
97	C₄⋊C₄	—	46	2	no	—
98	C₄⋊C₄	—	46	2	no	—
99	C₄⋊C₄	—	92	4	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₂³	C₂×D₉₂	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	46	4	no	—
106	C₂×C₄	—	92	2	no	—
107	C₂×C₄	—	92	2	no	—
108	C₂×C₄	—	92	2	no	—
109	C₂×C₄	—	92	2	no	—
110	C₂×C₄	—	92	2	no	—
111	C₂×C₄	#67263	1	4	yes	—
112	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
115	C₂×C₄	#67288	1	1	yes	characteristic
116	C₂×C₄	#67288	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	Dic₂₃⋊C₄	1	4	yes	complemented
120	C₂²	#10625040	1	1	yes	characteristic
121	C₂²	C₄×D₉₂	1	1	yes	characteristic
122	C₂²	C₂×#67263	1	2	yes	—
123	C₂²	#10625075	1	2	yes	—
124	C₂²	#10625101	1	1	yes	characteristic
125	C₄	—	2	4	no	—
126	C₄	—	92	4	no	—
127	C₄	—	184	2	no	—
128	C₄	C₄×Dic₄₆	1	1	yes	characteristic
129	C₄	#10625041	1	1	yes	characteristic
130	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
131	C₄	#10625095	1	1	yes	characteristic
132	C₂	#13092315	1	1	yes	characteristic
133	C₂	C₄×#67263	1	2	yes	—
134	C₂	#13092646	1	2	yes	—
135	C₂	#13092815	1	1	yes	characteristic
136	C₂	#13092819	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624999

C₂

yes

maximal, characteristic

C₂×#10624999

C₂

yes

maximal, characteristic

#13092315

C₂

yes

maximal

#13092180

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×Dic₂₃⋊C₄

—

C₄×C₈×C₂₃

—

C₂₃⋊#95

—

#10624999

—

#10624999

C₄

yes

complemented

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

C₂×C₄×C₂₃

yes

C₂×C₄×C₂₃

yes

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

complemented

#1941

—

maximal, Sylow

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₂²×C₂₃

#121

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

—

C₂²×C₂₃

#135

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×Q₈

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₃

#125

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

D₂₃

yes

characteristic

#389

—

#340

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

C₄×SD₁₆

yes

—

C₂×C₂₃

#465

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#106

—

#106

—

C₄×C₈

—

#95

—

C₂₃

#1941

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₂³

C₂×D₉₂

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#67263

yes

—

112

C₂×C₄

C₂×Dic₄₆

yes

characteristic

113

C₂×C₄

C₂×Dic₄₆

yes

characteristic

114

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

115

C₂×C₄

#67288

yes

characteristic

116

C₂×C₄

#67288

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

118

C₈

—

119

C₈

Dic₂₃⋊C₄

yes

complemented

120

C₂²

#10625040

yes

characteristic

121

C₂²

C₄×D₉₂

yes

characteristic

122

C₂²

C₂×#67263

yes

—

123

C₂²

#10625075

yes

—

124

C₂²

#10625101

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

184

—

128

C₄

C₄×Dic₄₆

yes

characteristic

129

C₄

#10625041

yes

characteristic

130

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

131

C₄

#10625095

yes

characteristic

132

C₂

#13092315

yes

characteristic

133

C₂

C₄×#67263

yes

—

134

C₂

#13092646

yes

—

135

C₂

#13092815

yes

characteristic

136

C₂

#13092819

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1941
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

#1941

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
15. #10624999	C₄	semidirect
50. C₈×C₂₃	C₄⋊C₄	semidirect
84. C₂₃	#1941	semidirect
119. C₈	Dic₂₃⋊C₄	semidirect

H ≅ G/N

type

15. #10624999

C₄

semidirect

50. C₈×C₂₃

C₄⋊C₄

semidirect

84. C₂₃

#1941

semidirect

119. C₈

Dic₂₃⋊C₄

semidirect

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

subgroup	index
C₂×#10624999	2
#13092315	2
#13092180	2
C₂×C₄×C₈×C₂₃	2
#1941	23

subgroup

index

C₂×C₄×C₈×C₂₃

Subgroups — #15674125

Conjugacy classes of subgroups

Complements

Maximal subgroups