Group #15674178 — subgroups

Subgroups — #15674178

Order 2944.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13092459	C₂	1	2	yes	maximal
2	C₂×#10624999	C₂	1	1	yes	maximal, characteristic
3	#13092372	C₂	1	1	yes	maximal, characteristic
4	#13092185	C₂	1	2	yes	maximal
5	C₂×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	#10625097	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
10	C₂×#67267	C₂²	1	1	yes	characteristic
11	C₂₃×#108	C₄	1	4	yes	complemented
12	#10625000	—	2	2	no	—
13	#10624999	—	2	1	no	—
14	#10624999	—	2	1	no	—
15	#10624999	—	2	2	no	—
16	#10624997	—	2	1	no	—
17	C₈×Dic₂₃	—	2	1	no	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₃	—	2	2	no	—
20	M₁₆×C₂₃	—	2	2	no	—
21	M₁₆×C₂₃	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₂₃	Q₈	1	2	yes	—
27	C₂²⋊Dic₂₃	—	2	1	no	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄⋊Dic₂₃	—	2	2	no	—
30	C₄⋊Dic₂₃	D₄	1	2	yes	—
31	C₄×Dic₂₃	—	2	1	no	—
32	#67267	—	2	1	no	—
33	#67267	Q₈	1	2	yes	—
34	C₂×C₂₃⋊C₈	—	2	1	no	—
35	C₂³×C₂₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂×D₄	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×Dic₂₃	—	2	2	no	—
43	C₂×Dic₂₃	—	4	2	no	—
44	C₂×Dic₂₃	—	8	2	no	—
45	C₈×C₂₃	—	2	4	no	—
46	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
47	C₈×C₂₃	C₄⋊C₄	1	2	yes	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
56	C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₃	—	4	2	no	—
59	Dic₂₃	—	8	2	no	—
60	C₂×#108	D₂₃	1	1	yes	characteristic
61	#428	—	23	2	no	—
62	C₂×#106	—	23	1	no	—
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	C₂×D₂₃	1	2	yes	—
71	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
72	#117	—	23	1	no	—
73	C₂×C₄⋊C₄	—	23	2	no	—
74	#108	Dic₂₃	1	4	yes	complemented
75	#107	—	46	2	no	—
76	#106	—	46	2	no	—
77	#106	—	46	1	no	—
78	#106	—	46	1	no	—
79	#97	—	46	1	no	—
80	C₄×C₈	—	46	1	no	—
81	C₂₃	#1951	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	46	2	no	—
83	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	23	2	no	—
86	M₁₆	—	46	1	no	—
87	M₁₆	C₂×Dic₂₃	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	46	1	no	—
90	C₂×C₈	Dic₄₆	1	2	yes	—
91	C₂×C₈	D₉₂	1	1	yes	characteristic
92	C₂×C₈	D₉₂	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
94	C₄⋊C₄	—	23	2	no	—
95	C₄⋊C₄	—	46	2	no	—
96	C₄⋊C₄	—	46	2	no	—
97	C₂²⋊C₄	—	46	1	no	—
98	C₄²	—	46	1	no	—
99	C₂³	#67292	1	1	yes	characteristic
100	C₂×C₄	—	46	2	no	—
101	C₂×C₄	—	92	2	no	—
102	C₂×C₄	—	184	2	no	—
103	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
104	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
105	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
106	C₂×C₄	#67290	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₂₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	46	2	no	—
111	C₈	C₄⋊Dic₂₃	1	2	yes	—
112	C₈	C₄⋊Dic₂₃	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	#10625061	1	1	yes	characteristic
115	C₂²	#10625066	1	1	yes	characteristic
116	C₂²	Q₈×Dic₂₃	1	1	yes	characteristic
117	C₄	—	92	2	no	—
118	C₄	—	184	2	no	—
119	C₄	#10625059	1	1	yes	characteristic
120	C₄	#10625068	1	1	yes	characteristic
121	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
122	C₄	D₄×Dic₂₃	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#13092316	1	1	yes	characteristic
126	C₂	#13092681	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092459

C₂

yes

maximal

C₂×#10624999

C₂

yes

maximal, characteristic

#13092372

C₂

yes

maximal, characteristic

#13092185

C₂

yes

maximal

C₂×C₂₃×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂×#67267

C₂²

yes

characteristic

C₂₃×#108

C₄

yes

—

—

—

—

—

C₈×Dic₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

Q₈

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

—

C₄×Dic₂₃

—

#67267

—

#67267

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

#41

yes

characteristic

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

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₂₃⋊C₈

—

#1951

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×Q₈

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

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

—

Dic₂₃

—

C₂×#108

D₂₃

yes

—

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#389

yes

characteristic

C₂×C₂₃

#471

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₂₃

yes

—

—

—

—

—

C₄×C₈

—

C₂₃

#1951

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×Dic₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

D₉₂

yes

characteristic

C₂×C₈

D₉₂

yes

characteristic

C₂×C₈

C₂×Dic₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

#67292

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

184

—

103

C₂×C₄

C₂×Dic₄₆

yes

characteristic

104

C₂×C₄

C₂×D₉₂

yes

characteristic

105

C₂×C₄

D₄×D₂₃

yes

characteristic

106

C₂×C₄

#67290

yes

characteristic

107

C₂×C₄

Q₈×D₂₃

yes

characteristic

108

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

C₄⋊Dic₂₃

yes

—

112

C₈

C₄⋊Dic₂₃

yes

—

113

C₂²

—

114

C₂²

#10625061

yes

characteristic

115

C₂²

#10625066

yes

characteristic

116

C₂²

Q₈×Dic₂₃

yes

characteristic

117

C₄

—

118

C₄

—

184

—

119

C₄

#10625059

yes

characteristic

120

C₄

#10625068

yes

characteristic

121

C₄

C₂×C₄⋊Dic₂₃

yes

characteristic

122

C₄

D₄×Dic₂₃

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#13092316

yes

characteristic

126

C₂

#13092681

yes

—

	subgroup	G/H
center	C₂²	#10625066
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₂₃×#108	C₂
socle	C₂²×C₂₃	#121
prime radical	C₂³×C₂₃	#41
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×#108	D₂₃
hypocenter	C₂₃	#1951
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#10625066

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₂₃×#108

C₂

C₂²×C₂₃

C₂³×C₂₃

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
11. C₂₃×#108	C₄	semidirect
74. #108	Dic₂₃	semidirect
81. C₂₃	#1951	semidirect

H ≅ G/N

type

11. C₂₃×#108

C₄

semidirect

74. #108

Dic₂₃

semidirect

81. C₂₃

#1951

semidirect

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

subgroup	index
#13092459	2
C₂×#10624999	2
#13092372	2
#13092185	2
C₂×C₂₃×#108	2
#1951	23

subgroup

index

C₂×C₂₃×#108

Subgroups — #15674178

Conjugacy classes of subgroups

Complements

Maximal subgroups