Group #15671685 — subgroups

Subgroups — #15671685

Order 2944.

208 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 101 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	#13092468	C₂	1	1	yes	maximal, characteristic
2	C₄×#67262	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092139	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092139	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092139	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67262	C₄	1	2	yes	—
9	C₂×C₈×D₂₃	C₄	1	2	yes	—
10	C₄²×D₂₃	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	(C₂×D₂₃)⋊C₈	C₄	1	2	yes	—
15	#10624997	C₄	1	2	yes	—
16	Dic₂₃⋊C₈	C₄	1	2	yes	—
17	C₈×Dic₂₃	C₄	1	2	yes	—
18	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
34	#67262	—	2	4	no	—
35	C₈×D₂₃	—	2	4	no	—
36	C₂²×D₂₃	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₂₃	C₄²	1	1	yes	characteristic
41	C₂×Dic₂₃	C₄²	1	1	yes	characteristic
42	C₂×Dic₂₃	C₄²	1	1	yes	characteristic
43	C₄×D₂₃	—	2	2	no	—
44	C₄×D₂₃	—	2	2	no	—
45	C₄×D₂₃	M₁₆	1	4	yes	—
46	C₈×C₂₃	—	2	2	no	—
47	C₈×C₂₃	—	2	2	no	—
48	C₈×C₂₃	—	2	2	no	—
49	C₂₃⋊C₈	—	2	2	no	—
50	C₂₃⋊C₈	—	2	2	no	—
51	C₂₃⋊C₈	M₁₆	1	4	yes	complemented
52	#1451	—	23	1	no	maximal, Sylow
53	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
54	C₂×D₂₃	—	2	1	no	—
55	C₂×D₂₃	—	2	1	no	—
56	C₂×D₂₃	#97	1	2	yes	—
57	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
58	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
59	C₄×C₂₃	#131	1	2	yes	—
60	Dic₂₃	—	2	1	no	—
61	Dic₂₃	—	2	1	no	—
62	Dic₂₃	#97	1	2	yes	—
63	#431	—	23	1	no	—
64	C₄×M₁₆	—	23	1	no	—
65	C₂×C₄×C₈	—	23	1	no	—
66	#322	—	23	1	no	—
67	#322	—	23	1	no	—
68	#322	—	23	1	no	—
69	#322	D₂₃	1	1	yes	characteristic, complemented
70	C₂×C₂₃	C₂×#97	1	1	yes	characteristic
71	C₂×C₂₃	C₄×M₁₆	1	1	yes	characteristic
72	C₂×C₂₃	#405	1	1	yes	characteristic
73	D₂₃	—	2	2	no	—
74	C₂×M₁₆	—	23	2	no	—
75	C₂²×C₈	—	23	2	no	—
76	C₂×C₄²	—	23	1	no	—
77	C₄⋊C₈	—	23	2	no	—
78	C₂²⋊C₈	—	23	2	no	—
79	#97	—	23	2	no	—
80	C₄×C₈	—	23	2	no	—
81	C₄×C₈	—	23	1	no	—
82	C₄×C₈	—	23	1	no	—
83	C₄×C₈	—	23	1	no	—
84	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
85	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
86	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
87	C₂₃	#1451	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₄	—	23	1	no	—
89	C₂²×C₄	—	23	1	no	—
90	C₂²×C₄	—	23	1	no	—
91	M₁₆	—	46	4	no	—
92	C₂×C₈	—	23	2	no	—
93	C₂×C₈	—	23	2	no	—
94	C₂×C₈	—	23	2	no	—
95	C₂×C₈	—	46	4	no	—
96	C₂×C₈	C₄×D₂₃	1	2	yes	—
97	C₂×C₈	C₄×D₂₃	1	2	yes	—
98	C₂×C₈	C₄×D₂₃	1	2	yes	—
99	C₄²	—	23	1	no	—
100	C₄²	—	23	1	no	—
101	C₄²	—	23	1	no	—
102	C₄²	C₂²×D₂₃	1	1	yes	characteristic
103	C₂³	—	23	1	no	—
104	C₂×C₄	—	23	1	no	—
105	C₂×C₄	—	23	4	no	—
106	C₂×C₄	—	23	1	no	—
107	C₂×C₄	—	23	1	no	—
108	C₂×C₄	—	46	2	no	—
109	C₂×C₄	—	46	2	no	—
110	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	23	4	no	—
117	C₈	—	46	2	no	—
118	C₈	—	46	2	no	—
119	C₂²	—	23	2	no	—
120	C₂²	—	46	1	no	—
121	C₂²	—	46	1	no	—
122	C₂²	C₄²×D₂₃	1	1	yes	characteristic
123	C₄	—	23	2	no	—
124	C₄	—	46	1	no	—
125	C₄	—	46	1	no	—
126	C₄	#10625072	1	2	yes	—
127	C₄	M₁₆×D₂₃	1	2	yes	—
128	C₄	M₁₆×D₂₃	1	2	yes	—
129	C₂	—	46	2	no	—
130	C₂	#97×D₂₃	1	1	yes	characteristic
131	C₂	#13092369	1	1	yes	characteristic
132	C₂	#13092371	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092468

C₂

yes

maximal, characteristic

C₄×#67262

C₂

yes

maximal, characteristic

C₄×C₈×D₂₃

C₂

yes

maximal, characteristic

#13092139

C₂

yes

maximal, characteristic, complemented

#13092139

C₂

yes

maximal, characteristic, complemented

#13092139

C₂

yes

maximal, characteristic, complemented

C₂₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×#67262

C₄

yes

—

C₂×C₈×D₂₃

C₄

yes

—

C₄²×D₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

(C₂×D₂₃)⋊C₈

C₄

yes

—

#10624997

C₄

yes

—

Dic₂₃⋊C₈

C₄

yes

—

C₈×Dic₂₃

C₄

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₄×Dic₂₃

C₂×C₄

yes

characteristic

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

#67262

—

C₈×D₂₃

—

C₂²×D₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₃

C₄²

yes

characteristic

C₂×Dic₂₃

C₄²

yes

characteristic

C₂×Dic₂₃

C₄²

yes

characteristic

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

M₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

M₁₆

yes

complemented

#1451

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄²

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

#97

yes

—

C₄×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

C₂×M₁₆

yes

—

C₄×C₂₃

#131

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

#97

yes

—

#431

—

C₄×M₁₆

—

C₂×C₄×C₈

—

—

—

—

D₂₃

yes

characteristic, complemented

C₂×C₂₃

C₂×#97

yes

characteristic

C₂×C₂₃

C₄×M₁₆

yes

characteristic

C₂×C₂₃

#405

yes

characteristic

D₂₃

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₄×C₈

C₂×D₂₃

yes

characteristic

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1451

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₂×C₈

C₄×D₂₃

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×D₂₃

yes

characteristic

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

111

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

112

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₄²×D₂₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#10625072

yes

—

127

C₄

M₁₆×D₂₃

yes

—

128

C₄

M₁₆×D₂₃

yes

—

129

C₂

—

130

C₂

#97×D₂₃

yes

characteristic

131

C₂

#13092369

yes

characteristic

132

C₂

#13092371

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂×C₄²
Fitting	C₂₃×#322	C₂
prime radical	C₂²×D₂₃	C₄²
cosocle	C₄²×C₂₃	C₂³
hypercenter	#322	D₂₃
hypocenter	C₂₃	#1451
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₂₃

commutator, socle

C₂²×C₂₃

C₂×C₄²

Fitting

C₂₃×#322

C₂

prime radical

C₂²×D₂₃

C₄²

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #13092139	C₂	semidirect
5. #13092139	C₂	semidirect
6. #13092139	C₂	semidirect
7. C₂₃×#322	C₂	semidirect
51. C₂₃⋊C₈	M₁₆	semidirect
69. #322	D₂₃	semidirect
87. C₂₃	#1451	semidirect

H ≅ G/N

type

4. #13092139

C₂

semidirect

5. #13092139

C₂

semidirect

6. #13092139

C₂

semidirect

7. C₂₃×#322

C₂

semidirect

51. C₂₃⋊C₈

M₁₆

semidirect

69. #322

D₂₃

semidirect

87. C₂₃

#1451

semidirect

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

subgroup	index
#13092468	2
C₄×#67262	2
C₄×C₈×D₂₃	2
#13092139	2
C₂₃×#322	2
#1451	23

subgroup

index

C₄×C₈×D₂₃

C₂₃×#322

Subgroups — #15671685

Conjugacy classes of subgroups

Complements

Maximal subgroups