Group #15673381 — subgroups

Subgroups — #15673381

Order 2944.

312 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
3	#13092172	C₂	1	2	yes	maximal, complemented
4	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#117	—	2	4	no	—
7	C₂×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₂×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
9	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
12	C₂²⋊(C₂₃⋊C₈)	—	2	4	no	—
13	C₄⋊(C₂₃⋊C₈)	—	2	4	no	—
14	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₃	—	2	4	no	—
24	C₂²⋊C₄×C₂₃	—	2	4	no	—
25	C₄²×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	D₄	1	4	yes	—
27	C₂×C₂₃⋊C₈	—	2	4	no	—
28	C₂×C₂₃⋊C₈	—	2	4	no	—
29	C₂×C₂₃⋊C₈	—	2	4	no	—
30	C₂³×C₂₃	—	2	1	no	—
31	C₂³×C₂₃	—	2	1	no	—
32	C₂³×C₂₃	—	2	4	no	—
33	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
34	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	1	no	—
36	C₂×C₄×C₂₃	—	2	1	no	—
37	C₂×C₄×C₂₃	—	2	4	no	—
38	C₂×C₄×C₂₃	—	2	4	no	—
39	C₂×C₄×C₂₃	—	2	4	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₂₃⋊C₈	—	4	4	no	—
49	#1798	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	—	2	4	no	—
52	C₂²×C₂₃	—	4	2	no	—
53	C₂²×C₂₃	—	4	2	no	—
54	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₃	#117	1	1	yes	characteristic
57	C₂²×C₂₃	#131	1	4	yes	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	—	2	4	no	—
60	C₄×C₂₃	—	4	2	no	—
61	C₄×C₂₃	#121	1	2	yes	—
62	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
63	C₂×#117	D₂₃	1	1	yes	characteristic
64	C₂×C₄⋊C₈	—	23	2	no	—
65	C₂×C₂²⋊C₈	—	23	2	no	—
66	#336	—	23	2	no	—
67	C₂×C₂₃	—	4	2	no	—
68	C₂×C₂₃	#380	1	1	yes	characteristic
69	C₂×C₂₃	#408	1	2	yes	—
70	C₂×C₂₃	#424	1	2	yes	—
71	C₂×C₂₃	#433	1	2	yes	—
72	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
73	C₂²×C₈	—	23	4	no	—
74	#117	—	2	4	no	—
75	C₂×C₄⋊C₄	Dic₂₃	1	2	yes	—
76	C₂×C₂²⋊C₄	Dic₂₃	1	2	yes	—
77	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
79	C₄⋊C₈	—	46	4	no	—
80	C₂²⋊C₈	—	46	4	no	—
81	C₂₃	#1798	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	Dic₄₆	1	2	yes	—
86	C₂²×C₄	D₉₂	1	2	yes	—
87	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
88	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
91	C₂×C₈	—	46	4	no	—
92	C₂×C₈	—	46	4	no	—
93	C₂×C₈	—	46	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²⋊D₂₃	1	4	yes	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	1	no	—
101	C₂³	C₄⋊Dic₂₃	1	2	yes	—
102	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
112	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
113	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
114	C₂×C₄	#67288	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
116	C₈	—	92	4	no	—
117	C₂²	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	2	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
122	C₂²	#10625097	1	1	yes	characteristic
123	C₂²	#10625119	1	4	yes	—
124	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
125	C₄	—	2	4	no	—
126	C₄	—	2	2	no	—
127	C₄	—	4	2	no	—
128	C₄	#10625095	1	2	yes	—
129	C₄	#10625101	1	2	yes	—
130	C₂	—	4	2	no	—
131	C₂	#13092277	1	1	yes	characteristic
132	C₂	#13092384	1	2	yes	—
133	C₂	#13092444	1	2	yes	—
134	C₂	#13092478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₂₃⋊C₈)

C₂

yes

maximal

#13092172

C₂

yes

maximal, complemented

C₂×C₂₃×#117

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂₃×#117

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

—

C₂×C₂²⋊C₄×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

—

C₂⁴×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂₃⋊C₈

—

#1798

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#121

yes

—

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₂×#117

D₂₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

#336

—

C₂×C₂₃

—

C₂×C₂₃

#380

yes

characteristic

C₂×C₂₃

#408

yes

—

C₂×C₂₃

#424

yes

—

C₂×C₂₃

#433

yes

—

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

Dic₂₃

yes

—

C₂×C₂²⋊C₄

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1798

yes

characteristic, complemented, Sylow

C₂⁴

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₄₆

yes

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₄⋊Dic₂₃

yes

—

102

C₂³

C₂²×Dic₂₃

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

C₄⋊Dic₂₃

yes

—

111

C₂×C₄

C₂²⋊Dic₂₃

yes

—

112

C₂×C₄

C₂×Dic₄₆

yes

characteristic

113

C₂×C₄

C₂×D₉₂

yes

characteristic

114

C₂×C₄

#67288

yes

—

115

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

122

C₂²

#10625097

yes

characteristic

123

C₂²

#10625119

yes

—

124

C₂²

C₂×C₂²⋊Dic₂₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#10625095

yes

—

129

C₄

#10625101

yes

—

130

C₂

—

131

C₂

#13092277

yes

characteristic

132

C₂

#13092384

yes

—

133

C₂

#13092444

yes

—

134

C₂

#13092478

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×#117	C₂
prime radical	C₂⁴×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#1798
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₂₃×#117

C₂

prime radical

C₂⁴×C₂₃

C₂×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
3. #13092172	C₂	semidirect
81. C₂₃	#1798	semidirect

H ≅ G/N

type

1. C₂×C₄⋊(C₂₃⋊C₈)

C₂

semidirect

3. #13092172

C₂

semidirect

81. C₂₃

#1798

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₂²⋊(C₂₃⋊C₈)	2
#13092172	2
C₂×C₂₃×#117	2
#1798	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×C₂²⋊(C₂₃⋊C₈)

#13092172

C₂×C₂₃×#117

#1798

Subgroups — #15673381

Conjugacy classes of subgroups

Complements

Maximal subgroups