Group #15674175 — subgroups

Subgroups — #15674175

Order 2944.

196 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 89 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×#10625001	C₂	1	1	yes	maximal, characteristic
2	#13092457	C₂	1	2	yes	maximal
3	#13092372	C₂	1	1	yes	maximal, characteristic
4	#13092186	C₂	1	2	yes	maximal
5	C₂×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	#10625097	C₂²	1	1	yes	characteristic
9	C₂×#67267	C₂²	1	2	yes	—
10	C₂×#67267	C₂²	1	1	yes	characteristic
11	C₂₃×#106	—	2	1	no	—
12	C₂₃×#106	—	2	1	no	—
13	#10625001	C₄	1	4	yes	complemented
14	#10624997	—	2	1	no	—
15	C₈×Dic₂₃	—	2	1	no	—
16	#10624990	—	2	2	no	—
17	#10624989	—	2	2	no	—
18	C₂²×C₄×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₂₃	Q₈	1	2	yes	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂²⋊Dic₂₃	—	2	1	no	—
27	C₄⋊Dic₂₃	D₄	1	2	yes	—
28	C₄×Dic₂₃	—	2	1	no	—
29	#67267	—	2	2	no	—
30	#67267	—	2	1	no	—
31	#67267	C₂×C₄	1	4	yes	—
32	#67267	Q₈	1	2	yes	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	—	2	1	no	—
35	C₂³×C₂₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	—	8	2	no	—
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
44	C₂×Dic₂₃	—	2	2	no	—
45	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₂₃⋊C₈	—	2	4	no	—
48	C₂₃⋊C₈	—	2	2	no	—
49	#1951	—	23	1	no	maximal, Sylow
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	C₄×Q₈	1	1	yes	characteristic
52	C₂²×C₂₃	#121	1	1	yes	characteristic
53	C₂²×C₂₃	#125	1	1	yes	characteristic
54	C₄×C₂₃	—	8	2	no	—
55	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
57	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
58	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₂₃	—	4	2	no	—
60	C₂×#108	—	23	1	no	—
61	#428	—	23	2	no	—
62	C₂×#106	D₂₃	1	1	yes	characteristic
63	#405	—	23	1	no	—
64	#343	—	23	2	no	—
65	C₂×C₂₃	—	2	1	no	—
66	C₂×C₂₃	—	2	1	no	—
67	C₂×C₂₃	#389	1	1	yes	characteristic
68	C₂×C₂₃	#471	1	2	yes	—
69	C₂×M₁₆	—	23	1	no	—
70	C₂×M₁₆	—	23	2	no	—
71	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
72	#117	—	23	1	no	—
73	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
74	#108	—	23	4	no	—
75	#107	—	46	2	no	—
76	#106	—	2	1	no	—
77	#106	—	2	1	no	—
78	#106	—	46	2	no	—
79	#97	—	46	1	no	—
80	C₄×C₈	—	46	1	no	—
81	C₂₃	#1951	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
84	M₁₆	—	23	4	no	—
85	M₁₆	—	23	2	no	—
86	M₁₆	—	46	2	no	—
87	M₁₆	—	46	1	no	—
88	C₂×C₈	—	46	2	no	—
89	C₂×C₈	—	46	1	no	—
90	C₂×C₈	Dic₄₆	1	2	yes	—
91	C₂×C₈	C₄×D₂₃	1	2	yes	—
92	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	23	2	no	—
97	C₂²⋊C₄	—	46	1	no	—
98	C₄²	—	46	1	no	—
99	C₂³	#67290	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	46	2	no	—
103	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
105	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
107	C₂×C₄	#67292	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
109	C₈	—	46	4	no	—
110	C₈	—	46	2	no	—
111	C₈	Dic₂₃⋊C₄	1	2	yes	—
112	C₈	Dic₂₃⋊C₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	#10625058	1	1	yes	characteristic
115	C₂²	#10625061	1	1	yes	characteristic
116	C₂²	#10625109	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	92	2	no	—
119	C₄	#10625059	1	1	yes	characteristic
120	C₄	#10625064	1	1	yes	characteristic
121	C₄	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
122	C₄	#10625117	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#13092313	1	1	yes	characteristic
126	C₂	#13092677	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625001

C₂

yes

maximal, characteristic

#13092457

C₂

yes

maximal

#13092372

C₂

yes

maximal, characteristic

#13092186

C₂

yes

maximal

C₂×C₂₃×#106

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

#10625097

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂×#67267

C₂²

yes

characteristic

C₂₃×#106

—

C₂₃×#106

—

#10625001

C₄

yes

complemented

#10624997

—

C₈×Dic₂₃

—

#10624990

—

#10624989

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

Q₈

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

#67267

—

#67267

—

#67267

C₂×C₄

yes

—

#67267

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

#41

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1951

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×#108

—

#428

—

C₂×#106

D₂₃

yes

characteristic

#405

—

#343

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₂₃

#1951

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

#67290

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×Dic₄₆

yes

characteristic

104

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

105

C₂×C₄

D₄×D₂₃

yes

characteristic

106

C₂×C₄

Q₈×D₂₃

yes

characteristic

107

C₂×C₄

#67292

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

Dic₂₃⋊C₄

yes

—

112

C₈

Dic₂₃⋊C₄

yes

—

113

C₂²

—

114

C₂²

#10625058

yes

characteristic

115

C₂²

#10625061

yes

characteristic

116

C₂²

#10625109

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#10625059

yes

characteristic

120

C₄

#10625064

yes

characteristic

121

C₄

C₂×Dic₂₃⋊C₄

yes

characteristic

122

C₄

#10625117

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#13092313

yes

characteristic

126

C₂

#13092677

yes

—

	subgroup	G/H
center	C₂²	#10625109
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#106	C₂
socle	C₂²×C₂₃	#121
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#106	D₂₃
hypocenter	C₂₃	#1951
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#10625109

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#106

C₂

C₂²×C₂₃

C₂³×C₂₃

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
13. #10625001	C₄	semidirect
81. C₂₃	#1951	semidirect

H ≅ G/N

type

13. #10625001

C₄

semidirect

81. C₂₃

#1951

semidirect

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

subgroup	index
C₂×#10625001	2
#13092457	2
#13092372	2
#13092186	2
C₂×C₂₃×#106	2
#1951	23

subgroup

index

C₂×C₂₃×#106

Subgroups — #15674175

Conjugacy classes of subgroups

Complements

Maximal subgroups