Group #15674220 — subgroups

Subgroups — #15674220

Order 2944.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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×#10625045	C₂	1	1	yes	maximal, characteristic
2	C₂×#10624991	C₂	1	4	yes	maximal, complemented
3	C₂×#10624985	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₉₂	—	2	2	no	—
6	C₂²×D₉₂	C₂²	1	2	yes	—
7	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
10	#10625045	—	2	2	no	—
11	#10625045	C₄	1	4	yes	complemented
12	C₂₃×#95	—	2	2	no	—
13	#10624991	—	2	8	no	—
14	#10624985	—	2	2	no	—
15	C₂³×D₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×D₉₂	—	2	4	no	—
20	C₂×D₉₂	—	2	4	no	—
21	C₂×D₉₂	—	2	8	no	—
22	C₂×D₉₂	—	2	2	no	—
23	C₂×D₉₂	—	4	8	no	—
24	C₂×D₉₂	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₄²×C₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₃	D₄	1	2	yes	—
28	C₂×C₂₃⋊C₈	—	2	2	no	—
29	C₂×C₂₃⋊C₈	—	2	2	no	—
30	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
31	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₂₃	—	2	4	no	—
33	C₂²×D₂₃	—	4	2	no	—
34	C₂²×D₂₃	—	4	2	no	—
35	C₂²×D₂₃	—	4	8	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	4	4	no	—
39	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	#41	1	2	yes	—
44	D₉₂	—	2	8	no	—
45	D₉₂	—	4	16	no	—
46	D₉₂	—	4	4	no	—
47	C₂₃⋊C₈	—	2	4	no	—
48	#1961	—	23	1	no	maximal, Sylow
49	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₃	#124	1	1	yes	characteristic
51	C₂²×C₂₃	#127	1	1	yes	characteristic
52	C₂²×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
53	C₂×D₂₃	—	4	4	no	—
54	C₂×D₂₃	—	4	8	no	—
55	C₂×D₂₃	—	8	4	no	—
56	C₂×D₂₃	—	8	4	no	—
57	C₄×C₂₃	—	2	4	no	—
58	C₄×C₂₃	—	8	2	no	—
59	C₄×C₂₃	C₄×D₄	1	2	yes	—
60	C₄×C₂₃	#121	1	2	yes	—
61	C₂×#127	—	23	1	no	—
62	C₂×#102	—	23	4	no	—
63	C₂×#97	—	23	1	no	—
64	#382	D₂₃	1	1	yes	characteristic, complemented
65	C₂×C₂₃	#390	1	1	yes	characteristic
66	C₂×C₂₃	#418	1	2	yes	—
67	C₂×C₂₃	#490	1	2	yes	—
68	C₂×C₂₃	#496	1	2	yes	—
69	D₂₃	—	8	4	no	—
70	C₂²×D₄	—	23	2	no	—
71	C₂²×D₄	—	46	2	no	—
72	C₂²×C₈	—	23	2	no	—
73	#127	—	23	4	no	—
74	#127	—	46	2	no	—
75	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
76	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
77	#102	—	46	8	no	—
78	#97	—	46	2	no	—
79	#95	—	2	2	no	—
80	C₂₃	#1961	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	46	2	no	—
82	C₂×D₄	—	23	4	no	—
83	C₂×D₄	—	46	8	no	—
84	C₂×D₄	—	46	2	no	—
85	C₂×D₄	—	46	4	no	—
86	C₂×D₄	—	46	4	no	—
87	C₂×D₄	—	92	8	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
91	C₂×C₈	—	23	4	no	—
92	C₂×C₈	—	46	2	no	—
93	C₂×C₈	—	46	2	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄²	C₄×D₂₃	1	2	yes	—
96	C₄²	D₉₂	1	2	yes	—
97	C₂³	—	46	4	no	—
98	C₂³	—	92	2	no	—
99	C₂³	—	92	2	no	—
100	C₂³	—	92	8	no	—
101	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
102	D₄	—	46	8	no	—
103	D₄	—	92	16	no	—
104	D₄	—	92	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	(C₂×D₂₃)⋊C₄	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
110	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
111	C₂×C₄	D₄×D₂₃	1	2	yes	—
112	C₂×C₄	#67292	1	2	yes	—
113	C₈	—	46	4	no	—
114	C₂²	—	92	8	no	—
115	C₂²	—	92	4	no	—
116	C₂²	—	184	4	no	—
117	C₂²	—	184	4	no	—
118	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
119	C₂²	#10625111	1	1	yes	characteristic
120	C₂²	#10625118	1	1	yes	characteristic
121	C₂²	#10625120	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	8	2	no	—
124	C₄	#10625064	1	2	yes	—
125	C₄	#10625066	1	2	yes	—
126	C₂	—	184	4	no	—
127	C₂	#13092318	1	1	yes	characteristic
128	C₂	#13092421	1	2	yes	—
129	C₂	#13092783	1	2	yes	—
130	C₂	#13092807	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625045

C₂

yes

maximal, characteristic

C₂×#10624991

C₂

yes

maximal, complemented

C₂×#10624985

C₂

yes

maximal, characteristic, complemented

C₂₃×#382

C₂

yes

maximal, characteristic, complemented

C₂²×D₉₂

—

C₂²×D₉₂

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10625045

—

#10625045

C₄

yes

complemented

C₂₃×#95

—

#10624991

—

#10624985

—

C₂³×D₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂×D₉₂

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

complemented

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

D₉₂

—

D₉₂

—

D₉₂

—

C₂₃⋊C₈

—

#1961

—

maximal, Sylow

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

#121

yes

—

C₂×#127

—

C₂×#102

—

C₂×#97

—

#382

D₂₃

yes

characteristic, complemented

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#418

yes

—

C₂×C₂₃

#490

yes

—

C₂×C₂₃

#496

yes

—

D₂₃

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

—

—

—

C₂₃

#1961

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

C₄×D₂₃

yes

—

C₄²

D₉₂

yes

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

110

C₂×C₄

C₂×D₉₂

yes

characteristic

111

C₂×C₄

D₄×D₂₃

yes

—

112

C₂×C₄

#67292

yes

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

184

—

117

C₂²

—

184

—

118

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

119

C₂²

#10625111

yes

characteristic

120

C₂²

#10625118

yes

characteristic

121

C₂²

#10625120

yes

—

122

C₄

—

123

C₄

—

124

C₄

#10625064

yes

—

125

C₄

#10625066

yes

—

126

C₂

—

184

—

127

C₂

#13092318

yes

characteristic

128

C₂

#13092421

yes

—

129

C₂

#13092783

yes

—

130

C₂

#13092807

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#382	C₂
socle	C₂³×C₂₃	C₂×D₄
prime radical	C₂×#10625045	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#382	D₂₃
hypocenter	C₂₃	#1961
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#382

C₂

socle

C₂³×C₂₃

C₂×D₄

prime radical

C₂×#10625045

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
2. C₂×#10624991	C₂	semidirect
3. C₂×#10624985	C₂	semidirect
4. C₂₃×#382	C₂	semidirect
11. #10625045	C₄	semidirect
30. C₂×C₂₃⋊C₈	D₄	semidirect
64. #382	D₂₃	semidirect
80. C₂₃	#1961	semidirect

H ≅ G/N

type

2. C₂×#10624991

C₂

semidirect

3. C₂×#10624985

C₂

semidirect

4. C₂₃×#382

C₂

semidirect

11. #10625045

C₄

semidirect

30. C₂×C₂₃⋊C₈

D₄

semidirect

64. #382

D₂₃

semidirect

80. C₂₃

#1961

semidirect

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

subgroup	index
C₂×#10625045	2
C₂×#10624991	2
C₂×#10624985	2
C₂₃×#382	2
#1961	23

subgroup

index

C₂₃×#382

Subgroups — #15674220

Conjugacy classes of subgroups

Complements

Maximal subgroups