Group #15674348 — subgroups

Subgroups — #15674348

Order 2944.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#10625040	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	#13092180	C₂	1	4	yes	maximal
4	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	—	2	1	no	—
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	—	2	1	no	—
9	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₃	—	2	1	no	—
12	#10625040	—	2	2	no	—
13	#10625040	C₄	1	4	yes	complemented
14	C₂₃×C₄⋊C₈	—	2	2	no	—
15	C₂₃⋊#95	—	2	1	no	—
16	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	—	2	2	no	—
20	C₂²×Dic₂₃	—	2	2	no	—
21	C₂×Dic₄₆	—	2	4	no	—
22	C₂×Dic₄₆	—	4	2	no	—
23	C₂×Dic₄₆	—	4	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	4	no	—
26	C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₄⋊Dic₂₃	—	2	4	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	—	2	2	no	—
31	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₂₃	Q₈	1	4	yes	complemented
33	C₄×Dic₂₃	—	4	2	no	—
34	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	2	no	—
36	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₃	D₈	1	2	yes	—
39	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
40	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	C₂×Dic₂₃	—	2	4	no	—
45	C₂×Dic₂₃	—	2	4	no	—
46	C₂×Dic₂₃	—	4	2	no	—
47	C₂×Dic₂₃	—	4	2	no	—
48	C₂×Dic₂₃	—	4	2	no	—
49	C₂×Dic₂₃	—	4	2	no	—
50	Dic₄₆	—	4	8	no	—
51	C₈×C₂₃	—	4	2	no	—
52	#1988	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
55	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
56	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₂₃	#137	1	2	yes	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	#102	1	4	yes	—
60	C₄×C₂₃	C₄×Q₈	1	2	yes	—
61	C₄×C₂₃	#124	1	1	yes	characteristic
62	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₂₃	—	4	4	no	—
64	Dic₂₃	—	4	4	no	—
65	C₂×C₄⋊Q₈	—	23	1	no	—
66	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
67	C₄×C₄⋊C₄	—	23	1	no	—
68	#340	—	23	4	no	—
69	C₂×C₂₃	#391	1	1	yes	characteristic
70	C₂×C₂₃	C₂×#102	1	1	yes	characteristic
71	C₂×C₂₃	#419	1	1	yes	characteristic
72	C₂×C₂₃	#474	1	2	yes	—
73	C₂×C₂₃	#475	1	2	yes	—
74	C₂²×Q₈	—	46	1	no	—
75	C₂²×C₈	C₂×D₂₃	1	2	yes	—
76	C₄⋊Q₈	—	23	4	no	—
77	C₄⋊Q₈	—	46	2	no	—
78	C₂×C₄⋊C₄	—	23	2	no	—
79	C₂×C₄⋊C₄	—	23	2	no	—
80	C₂×C₄⋊C₄	—	46	1	no	—
81	C₂×C₄²	—	46	1	no	—
82	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
83	C₄⋊C₈	—	2	2	no	—
84	#95	—	46	1	no	—
85	C₂₃	#1988	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	46	4	no	—
87	C₂×Q₈	—	92	2	no	—
88	C₂×Q₈	—	92	2	no	—
89	C₂²×C₄	—	46	2	no	—
90	C₂²×C₄	—	46	2	no	—
91	C₂²×C₄	D₉₂	1	1	yes	characteristic
92	C₂²×C₄	D₉₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₄⋊C₄	—	23	4	no	—
97	C₄⋊C₄	—	23	4	no	—
98	C₄⋊C₄	—	46	4	no	—
99	C₄⋊C₄	—	46	2	no	—
100	C₄⋊C₄	—	46	2	no	—
101	C₄²	—	92	2	no	—
102	C₄²	C₄×D₂₃	1	2	yes	—
103	C₄²	C₂²⋊D₂₃	1	2	yes	—
104	C₂³	C₂×D₉₂	1	1	yes	characteristic
105	Q₈	—	92	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	46	4	no	—
108	C₂×C₄	—	46	4	no	—
109	C₂×C₄	—	92	2	no	—
110	C₂×C₄	—	92	2	no	—
111	C₂×C₄	—	92	2	no	—
112	C₂×C₄	—	92	2	no	—
113	C₂×C₄	#67263	1	2	yes	—
114	C₂×C₄	D₁₈₄	1	2	yes	—
115	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
116	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
118	C₂×C₄	#67290	1	2	yes	—
119	C₂×C₄	Q₈×D₂₃	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₂²	#10625068	1	2	yes	—
123	C₂²	C₂×#67263	1	1	yes	characteristic
124	C₂²	C₂×D₁₈₄	1	1	yes	characteristic
125	C₂²	#10625080	1	2	yes	—
126	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	92	4	no	—
129	C₄	—	92	4	no	—
130	C₄	#10625003	1	4	yes	—
131	C₄	#10625058	1	2	yes	—
132	C₄	#10625107	1	1	yes	characteristic
133	C₄	#10625115	1	1	yes	characteristic
134	C₂	#13092322	1	1	yes	characteristic
135	C₂	C₂×#10625003	1	1	yes	characteristic
136	C₂	#13092427	1	1	yes	characteristic
137	C₂	#13092698	1	2	yes	—
138	C₂	#13092703	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₂₃

C₂

yes

maximal, characteristic

#13092180

C₂

yes

maximal

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×C₄×Dic₂₃

—

#10625040

—

#10625040

C₄

yes

complemented

C₂₃×C₄⋊C₈

—

C₂₃⋊#95

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

Q₈

yes

complemented

C₄×Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

D₈

yes

—

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

—

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₈×C₂₃

—

#1988

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂²×C₂₃

C₂×D₈

yes

characteristic

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

#137

yes

—

C₄×C₂₃

—

C₄×C₂₃

#102

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

#124

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#475

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#95

—

C₂₃

#1988

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₄×D₂₃

yes

—

103

C₄²

C₂²⋊D₂₃

yes

—

104

C₂³

C₂×D₉₂

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#67263

yes

—

114

C₂×C₄

D₁₈₄

yes

—

115

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

116

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

117

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

118

C₂×C₄

#67290

yes

—

119

C₂×C₄

Q₈×D₂₃

yes

—

120

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

121

C₈

—

122

C₂²

#10625068

yes

—

123

C₂²

C₂×#67263

yes

characteristic

124

C₂²

C₂×D₁₈₄

yes

characteristic

125

C₂²

#10625080

yes

—

126

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#10625003

yes

—

131

C₄

#10625058

yes

—

132

C₄

#10625107

yes

characteristic

133

C₄

#10625115

yes

characteristic

134

C₂

#13092322

yes

characteristic

135

C₂

C₂×#10625003

yes

characteristic

136

C₂

#13092427

yes

characteristic

137

C₂

#13092698

yes

—

138

C₂

#13092703

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#1988
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×C₂₃×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₃

hypocenter

C₂₃

#1988

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
13. #10625040	C₄	semidirect
32. C₄⋊Dic₂₃	Q₈	semidirect
85. C₂₃	#1988	semidirect

H ≅ G/N

type

13. #10625040

C₄

semidirect

32. C₄⋊Dic₂₃

Q₈

semidirect

85. C₂₃

#1988

semidirect

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

subgroup	index
C₂×#10625040	2
C₄×C₄⋊Dic₂₃	2
#13092180	2
C₂×C₂₃×C₄⋊C₈	2
#1988	23

subgroup

index

C₂×#10625040

C₄×C₄⋊Dic₂₃

#13092180

C₂×C₂₃×C₄⋊C₈

#1988

Subgroups — #15674348

Conjugacy classes of subgroups

Complements

Maximal subgroups