Group #15673685 — subgroups

Subgroups — #15673685

Order 2944.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13092444	C₂	1	2	yes	maximal, complemented
3	#13092384	C₂	1	1	yes	maximal, characteristic
4	#13092187	C₂	1	2	yes	maximal, complemented
5	C₂₃×#410	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#117	C₂²	1	2	yes	—
8	#10625119	—	2	4	no	—
9	C₂×#67267	—	2	2	no	—
10	C₂×#67267	C₂²	1	1	yes	characteristic
11	C₂×#67267	C₂²	1	2	yes	—
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂₃×#99	C₄	1	4	yes	—
14	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
15	C₄⋊(C₂₃⋊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	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₄²×C₂₃	—	2	2	no	—
26	#67267	—	2	4	no	—
27	#67267	—	2	2	no	—
28	#67267	—	2	2	no	—
29	#67267	—	2	2	no	—
30	C₂×C₂₃⋊C₈	—	2	4	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	C₂×C₂₃⋊C₈	—	2	1	no	—
33	C₂×C₂₃⋊C₈	—	2	1	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	—	2	4	no	—
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	—	2	4	no	—
49	C₂₃⋊C₈	—	4	1	no	—
50	C₂₃⋊C₈	—	4	1	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	—	4	4	no	—
59	C₄×C₂₃	#120	1	1	yes	characteristic
60	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₃	#123	1	1	yes	characteristic
62	#568	—	23	1	no	—
63	#424	—	23	2	no	—
64	#410	D₂₃	1	1	yes	characteristic
65	#408	—	23	1	no	—
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	—	23	1	no	—
75	C₂×M₁₆	—	46	2	no	—
76	C₂²×C₈	—	23	1	no	—
77	#117	C₂×D₂₃	1	2	yes	—
78	C₄⋊C₈	—	46	4	no	—
79	#99	Dic₂₃	1	4	yes	—
80	C₂²⋊C₈	—	46	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	C₂×Dic₂₃	1	2	yes	—
87	C₂²×C₄	D₉₂	1	2	yes	—
88	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
89	M₁₆	—	46	4	no	—
90	M₁₆	—	46	2	no	—
91	M₁₆	—	46	2	no	—
92	M₁₆	—	46	2	no	—
93	C₂×C₈	—	23	4	no	—
94	C₂×C₈	—	46	4	no	—
95	C₂×C₈	—	46	1	no	—
96	C₂×C₈	—	46	1	no	—
97	C₂×C₈	—	46	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₂³	C₄⋊Dic₂₃	1	2	yes	—
102	C₂³	C₂²×Dic₂₃	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	—	2	4	no	—
109	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
110	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
111	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
112	C₂×C₄	D₄×D₂₃	1	2	yes	—
113	C₂×C₄	#67290	1	2	yes	—
114	C₈	—	46	4	no	—
115	C₈	—	92	1	no	—
116	C₈	—	92	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
121	C₂²	D₄×Dic₂₃	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	#10625049	1	2	yes	—
125	C₄	#10625053	1	1	yes	characteristic
126	C₄	#10625057	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#13092292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093184

C₂

yes

maximal, characteristic

#13092444

C₂

yes

maximal, complemented

#13092384

C₂

yes

maximal, characteristic

#13092187

C₂

yes

maximal, complemented

C₂₃×#410

C₂

yes

maximal, characteristic

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625119

—

C₂×#67267

—

C₂×#67267

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂₃×#99

C₄

yes

—

C₂²⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊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

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

—

—

—

—

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

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

—

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

—

C₂₃⋊C₈

—

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

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

yes

—

—

D₂₃

yes

characteristic

#408

—

#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

—

C₂²×C₈

—

#117

C₂×D₂₃

yes

—

C₄⋊C₈

—

#99

Dic₂₃

yes

—

C₂²⋊C₈

—

C₂₃

#1868

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

Dic₄₆

yes

characteristic

C₂×D₄

Dic₄₆

yes

characteristic

C₂×D₄

C₂×Dic₂₃

yes

—

C₂²×C₄

D₉₂

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

C₂×Dic₂₃

yes

—

100

C₄²

—

101

C₂³

C₄⋊Dic₂₃

yes

—

102

C₂³

C₂²×Dic₂₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊Dic₂₃

yes

—

110

C₂×C₄

C₂×Dic₄₆

yes

characteristic

111

C₂×C₄

C₂×D₉₂

yes

characteristic

112

C₂×C₄

D₄×D₂₃

yes

—

113

C₂×C₄

#67290

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

121

C₂²

D₄×Dic₂₃

yes

—

122

C₄

—

123

C₄

—

124

C₄

#10625049

yes

—

125

C₄

#10625053

yes

characteristic

126

C₄

#10625057

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#13092292

yes

characteristic

	subgroup	G/H
center	C₄	#10625053
commutator	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#410	C₂
socle	C₂×C₂₃	#385
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#410	D₂₃
hypocenter	C₂₃	#1868
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #410

subgroup

G/H

center

C₄

#10625053

commutator

C₂³×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#410

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 ◁ #410

N	H ≅ G/N	type
2. #13092444	C₂	semidirect
4. #13092187	C₂	semidirect
81. C₂₃	#1868	semidirect

H ≅ G/N

type

2. #13092444

C₂

semidirect

4. #13092187

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
#13092444	2
#13092384	2
#13092187	2
C₂₃×#410	2
#1868	23

subgroup

index

C₂₃×#410

Subgroups — #15673685

Conjugacy classes of subgroups

Complements

Maximal subgroups