Group #15673249 — subgroups

Subgroups — #15673249

Order 2944.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×Dic₂₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×#10624997	C₂	1	2	yes	maximal
3	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092172	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₂₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
13	#10624997	—	2	4	no	—
14	Dic₂₃⋊C₈	C₄	1	8	yes	complemented
15	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
19	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₂×C₈×C₂₃	—	2	2	no	—
22	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
23	C₄²×C₂₃	—	2	2	no	—
24	C₄×Dic₂₃	—	2	4	no	—
25	C₄×Dic₂₃	—	2	4	no	—
26	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
27	C₄×Dic₂₃	D₄	1	2	yes	—
28	C₄×Dic₂₃	Q₈	1	2	yes	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	2	no	—
31	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
32	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	—	2	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
36	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₂₃	#41	1	2	yes	—
40	C₂×Dic₂₃	—	2	2	no	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	—	2	2	no	—
43	C₂×Dic₂₃	C₄²	1	2	yes	—
44	C₂×Dic₂₃	C₄²	1	2	yes	—
45	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₂₃	M₁₆	1	4	yes	—
47	C₈×C₂₃	—	2	4	no	—
48	C₂₃⋊C₈	—	2	4	no	—
49	#1767	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
51	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₃	#117	1	1	yes	characteristic
53	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
54	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
55	C₄×C₂₃	—	2	4	no	—
56	C₄×C₂₃	C₄×D₄	1	2	yes	—
57	C₄×C₂₃	C₄×Q₈	1	2	yes	—
58	Dic₂₃	—	2	2	no	—
59	Dic₂₃	—	2	4	no	—
60	Dic₂₃	#97	1	4	yes	complemented
61	C₂×C₄⋊C₈	—	23	2	no	—
62	C₂×#97	—	23	2	no	—
63	C₄³	—	23	1	no	—
64	#336	—	23	1	no	—
65	#336	D₂₃	1	1	yes	characteristic
66	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₂₃	C₂×#97	1	1	yes	characteristic
68	C₂×C₂₃	C₄×M₁₆	1	1	yes	characteristic
69	C₂×C₂₃	#423	1	2	yes	—
70	C₂×C₂₃	#432	1	2	yes	—
71	C₂²×C₈	—	23	2	no	—
72	C₂²×C₈	C₂×D₂₃	1	2	yes	—
73	C₂×C₄²	—	23	1	no	—
74	C₂×C₄²	—	23	1	no	—
75	C₂×C₄²	—	23	2	no	—
76	C₂×C₄²	—	23	2	no	—
77	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
78	C₄⋊C₈	—	23	8	no	—
79	#97	—	46	4	no	—
80	C₂₃	#1767	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₄	—	23	1	no	—
82	C₂²×C₄	—	23	1	no	—
83	C₂²×C₄	—	23	2	no	—
84	C₂²×C₄	C₄×D₂₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	23	4	no	—
89	C₂×C₈	—	46	2	no	—
90	C₂×C₈	—	46	2	no	—
91	C₂×C₈	C₄×D₂₃	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	23	4	no	—
94	C₄²	—	23	2	no	—
95	C₄²	—	23	2	no	—
96	C₄²	—	46	4	no	—
97	C₄²	—	46	4	no	—
98	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	23	2	no	—
102	C₂×C₄	—	23	4	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	23	4	no	—
105	C₂×C₄	—	46	2	no	—
106	C₂×C₄	—	46	2	no	—
107	C₂×C₄	—	46	2	no	—
108	C₂×C₄	#67262	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
110	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
111	C₂×C₄	#67290	1	2	yes	—
112	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
113	C₈	—	2	4	no	—
114	C₈	—	46	4	no	—
115	C₂²	C₄²×D₂₃	1	1	yes	characteristic
116	C₂²	#10625048	1	1	yes	characteristic
117	C₂²	C₄⋊C₄×D₂₃	1	1	yes	characteristic
118	C₂²	C₂×#67262	1	2	yes	—
119	C₂²	M₁₆×D₂₃	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	23	4	no	—
122	C₄	—	46	2	no	—
123	C₄	—	46	4	no	—
124	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
125	C₄	#10625058	1	2	yes	—
126	C₂	#13092268	1	1	yes	characteristic
127	C₂	#97×D₂₃	1	1	yes	characteristic
128	C₂	C₄×#67262	1	1	yes	characteristic
129	C₂	#13092439	1	2	yes	—
130	C₂	#13092471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₂₃⋊C₈

C₂

yes

maximal

C₂×#10624997

C₂

yes

maximal

C₄²×Dic₂₃

C₂

yes

maximal, characteristic

#13092172

C₂

yes

maximal, characteristic

C₂₃×#336

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₄

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10624997

—

Dic₂₃⋊C₈

C₄

yes

complemented

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂²×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

C₂×C₄

yes

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄²

yes

—

C₂×Dic₂₃

C₄²

yes

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₂×Dic₂₃

M₁₆

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1767

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#117

yes

characteristic

C₂²×C₂₃

C₂×M₁₆

yes

—

C₂²×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

C₄×Q₈

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

#97

yes

complemented

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

—

#336

—

#336

D₂₃

yes

characteristic

C₂×C₂₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₃

C₂×#97

yes

characteristic

C₂×C₂₃

C₄×M₁₆

yes

characteristic

C₂×C₂₃

#423

yes

—

C₂×C₂₃

#432

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#97

—

C₂₃

#1767

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#67262

yes

—

109

C₂×C₄

C₂×C₄×D₂₃

yes

—

110

C₂×C₄

D₄×D₂₃

yes

characteristic

111

C₂×C₄

#67290

yes

—

112

C₂×C₄

Q₈×D₂₃

yes

characteristic

113

C₈

—

114

C₈

—

115

C₂²

C₄²×D₂₃

yes

characteristic

116

C₂²

#10625048

yes

characteristic

117

C₂²

C₄⋊C₄×D₂₃

yes

characteristic

118

C₂²

C₂×#67262

yes

—

119

C₂²

M₁₆×D₂₃

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

(C₂²⋊D₂₃)⋊C₄

yes

—

125

C₄

#10625058

yes

—

126

C₂

#13092268

yes

characteristic

127

C₂

#97×D₂₃

yes

characteristic

128

C₂

C₄×#67262

yes

characteristic

129

C₂

#13092439

yes

—

130

C₂

#13092471

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂₃×#336	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#336	D₂₃
hypocenter	C₂₃	#1767
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator

C₂²×C₂₃

C₂×C₄²

Fitting

C₂₃×#336

C₂

socle, prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
14. Dic₂₃⋊C₈	C₄	semidirect
60. Dic₂₃	#97	semidirect
80. C₂₃	#1767	semidirect

H ≅ G/N

type

14. Dic₂₃⋊C₈

C₄

semidirect

60. Dic₂₃

#97

semidirect

80. C₂₃

#1767

semidirect

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

subgroup	index
C₂×Dic₂₃⋊C₈	2
C₂×#10624997	2
C₄²×Dic₂₃	2
#13092172	2
C₂₃×#336	2
#1767	23

subgroup

index

C₂×Dic₂₃⋊C₈

C₂×#10624997

C₄²×Dic₂₃

#13092172

C₂₃×#336

#1767

Subgroups — #15673249

Conjugacy classes of subgroups

Complements

Maximal subgroups