Group #15673682 — subgroups

Subgroups — #15673682

Order 2944.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	#13093184	C₂	1	1	yes	maximal, characteristic
2	#13092443	C₂	1	2	yes	maximal, complemented
3	#13092391	C₂	1	1	yes	maximal, characteristic
4	#13092189	C₂	1	2	yes	maximal, complemented
5	C₂₃×#408	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	#10625119	—	2	4	no	—
10	#10625097	C₂²	1	2	yes	—
11	C₂×#67267	—	2	2	no	—
12	C₂×#67267	C₂²	1	2	yes	—
13	C₂₃×C₂²⋊C₈	—	2	2	no	—
14	#10625016	C₄	1	4	yes	—
15	Dic₂₃⋊C₈	—	2	4	no	—
16	#41×C₂₃	—	2	2	no	—
17	#41×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₂₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	2	no	—
24	C₂×C₈×C₂₃	—	2	2	no	—
25	C₂×C₈×C₂₃	—	2	1	no	—
26	C₂×C₈×C₂₃	—	2	1	no	—
27	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₂₃	—	2	2	no	—
29	C₄×Dic₂₃	—	2	2	no	—
30	#67267	—	2	4	no	—
31	#67267	—	2	2	no	—
32	#67267	—	2	2	no	—
33	C₂×C₂₃⋊C₈	—	2	4	no	—
34	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
35	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
36	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₂₃	—	2	2	no	—
38	D₄×C₂₃	—	2	4	no	—
39	D₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×Dic₂₃	—	2	4	no	—
48	C₈×C₂₃	—	4	1	no	—
49	C₈×C₂₃	—	4	1	no	—
50	C₂₃⋊C₈	—	2	4	no	—
51	#1868	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	—	4	1	no	—
54	C₂²×C₂₃	—	4	2	no	—
55	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₃	C₄×D₄	1	2	yes	—
57	C₄×C₂₃	—	2	2	no	—
58	C₄×C₂₃	#120	1	1	yes	characteristic
59	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₂₃	#123	1	1	yes	characteristic
61	Dic₂₃	—	4	4	no	—
62	#568	—	23	1	no	—
63	#424	—	23	2	no	—
64	#410	—	23	1	no	—
65	#408	D₂₃	1	1	yes	characteristic
66	#344	—	23	2	no	—
67	C₂×C₂₃	—	2	2	no	—
68	C₂×C₂₃	—	2	1	no	—
69	C₂×C₂₃	—	4	2	no	—
70	C₂×C₂₃	#385	1	1	yes	characteristic
71	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
72	#131	—	46	4	no	—
73	C₂×M₁₆	—	23	2	no	—
74	C₂×M₁₆	—	46	2	no	—
75	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
77	#117	—	23	2	no	—
78	C₄⋊C₈	—	46	4	no	—
79	#99	—	23	4	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂₃	#1868	1	1	yes	characteristic, complemented, Sylow
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₄₆	1	1	yes	characteristic
85	C₂×D₄	Dic₄₆	1	1	yes	characteristic
86	C₂×D₄	C₄×D₂₃	1	2	yes	—
87	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
88	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
89	M₁₆	—	2	2	no	—
90	M₁₆	—	46	4	no	—
91	M₁₆	—	46	2	no	—
92	M₁₆	—	46	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	23	4	no	—
97	C₂×C₈	—	46	4	no	—
98	C₄⋊C₄	—	46	2	no	—
99	C₂²⋊C₄	—	23	4	no	—
100	C₄²	—	46	2	no	—
101	C₂³	Dic₂₃⋊C₄	1	2	yes	—
102	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	46	4	no	—
109	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₃	1	2	yes	—
112	C₂×C₄	#67290	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	46	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
121	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	92	4	no	—
124	C₄	#10625049	1	2	yes	—
125	C₄	#10625106	1	1	yes	characteristic
126	C₄	#10625108	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#13092289	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093184

C₂

yes

maximal, characteristic

#13092443

C₂

yes

maximal, complemented

#13092391

C₂

yes

maximal, characteristic

#13092189

C₂

yes

maximal, complemented

C₂₃×#408

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625119

—

#10625097

C₂²

yes

—

C₂×#67267

—

C₂×#67267

C₂²

yes

—

C₂₃×C₂²⋊C₈

—

#10625016

C₄

yes

—

Dic₂₃⋊C₈

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂²⋊Dic₂₃

C₂×C₄

yes

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

#67267

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂×Dic₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1868

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#123

yes

characteristic

Dic₂₃

—

—

—

—

D₂₃

yes

characteristic

#344

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#385

yes

characteristic

C₂×#41

C₂×D₂₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

#117

—

C₄⋊C₈

—

#99

—

C₂²⋊C₈

—

C₂₃

#1868

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

Dic₄₆

yes

characteristic

C₂×D₄

Dic₄₆

yes

characteristic

C₂×D₄

C₄×D₂₃

yes

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₂³

Dic₂₃⋊C₄

yes

—

102

C₂³

C₂×C₄×D₂₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₂₃⋊C₄

yes

—

110

C₂×C₄

C₂×Dic₄₆

yes

characteristic

111

C₂×C₄

D₄×D₂₃

yes

—

112

C₂×C₄

#67290

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

(C₂²⋊D₂₃)⋊C₄

yes

—

121

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#10625049

yes

—

125

C₄

#10625106

yes

characteristic

126

C₄

#10625108

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#13092289

yes

characteristic

	subgroup	G/H
center	C₄	#10625108
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#408	C₂
socle	C₂×C₂₃	#385
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#408	D₂₃
hypocenter	C₂₃	#1868
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#10625108

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#408

C₂

socle

C₂×C₂₃

#385

prime radical

C₂×#41×C₂₃

C₂²

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #13092443	C₂	semidirect
4. #13092189	C₂	semidirect
81. C₂₃	#1868	semidirect

H ≅ G/N

type

2. #13092443

C₂

semidirect

4. #13092189

C₂

semidirect

81. C₂₃

#1868

semidirect

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

subgroup	index
#13093184	2
#13092443	2
#13092391	2
#13092189	2
C₂₃×#408	2
#1868	23

subgroup

index

C₂₃×#408

Subgroups — #15673682

Conjugacy classes of subgroups

Complements

Maximal subgroups