Group #15682799 — subgroups

Subgroups — #15682799

Order 2944.

772 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 161 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	#13092981	C₂	1	3	yes	maximal, complemented
2	#13092604	C₂	1	8	yes	maximal, complemented
3	#13092391	C₂	1	3	yes	maximal, complemented
4	C₂₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×2₋¹⁺⁴	—	2	3	no	—
6	C₂₃×H(5,𝔽₂)	—	2	3	no	—
7	C₂₃×H(5,𝔽₂)	C₂²	1	4	yes	—
8	C₂×#41×C₂₃	—	2	6	no	—
9	C₂×#41×C₂₃	C₂²	1	3	yes	—
10	C₂²×D₉₂	C₂²	1	1	yes	characteristic, complemented
11	#10625118	—	2	3	no	—
12	#10625111	—	2	3	no	—
13	#10625108	C₂²	1	12	yes	complemented
14	#10625101	—	2	6	no	—
15	#10625097	C₂²	1	3	yes	complemented
16	#10625016	C₂²	1	12	yes	complemented
17	C₂³×D₂₃	C₂³	1	2	yes	—
18	#41×C₂₃	—	2	4	no	—
19	#41×C₂₃	—	2	12	no	—
20	#41×C₂₃	—	2	6	no	—
21	#41×C₂₃	—	2	12	no	—
22	#41×C₂₃	—	2	6	no	—
23	C₂×Q₈×C₂₃	—	2	6	no	—
24	C₂×Q₈×C₂₃	D₄	1	3	yes	complemented
25	C₂×D₄×C₂₃	—	2	6	no	—
26	C₂×D₄×C₂₃	—	2	12	no	—
27	C₂×D₄×C₂₃	D₄	1	3	yes	complemented
28	C₂×D₄×C₂₃	C₂³	1	6	yes	—
29	C₂²×C₄×C₂₃	—	2	4	no	—
30	C₂²×C₄×C₂₃	D₄	1	6	yes	complemented
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂×C₂²⋊D₂₃	—	2	12	no	—
33	C₂×D₉₂	—	2	3	no	—
34	C₂×D₉₂	—	2	3	no	—
35	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
36	(C₂×D₂₃)⋊C₄	—	2	12	no	—
37	C₄⋊Dic₂₃	—	2	3	no	—
38	C₄×Dic₂₃	—	2	3	no	—
39	C₂³×C₂₃	—	2	4	no	—
40	C₂³×C₂₃	C₂×D₄	1	6	yes	—
41	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
42	C₂²×D₂₃	—	2	6	no	—
43	C₂²×D₂₃	—	8	2	no	—
44	Q₈×C₂₃	—	2	4	no	—
45	Q₈×C₂₃	—	2	6	no	—
46	D₄×C₂₃	—	2	12	no	—
47	D₄×C₂₃	—	2	12	no	—
48	D₄×C₂₃	—	2	6	no	—
49	C₂×C₄×C₂₃	—	2	12	no	—
50	C₂×C₄×C₂₃	—	2	6	no	—
51	C₂×C₄×C₂₃	—	2	6	no	—
52	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
53	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
55	C₂²⋊D₂₃	—	8	12	no	—
56	C₂×Dic₂₃	—	2	6	no	—
57	D₉₂	—	4	1	no	—
58	D₉₂	—	4	3	no	—
59	#3027	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	6	no	—
61	C₂²×C₂₃	—	4	1	no	—
62	C₂²×C₂₃	—	4	6	no	—
63	C₂²×C₂₃	—	8	4	no	—
64	C₂²×C₂₃	C₂²×D₄	1	3	yes	—
65	C₂×D₂₃	—	4	12	no	—
66	C₂×D₂₃	—	4	2	no	—
67	C₄×C₂₃	—	2	6	no	—
68	C₄×C₂₃	#120	1	1	yes	characteristic
69	C₄×C₂₃	#120	1	3	yes	—
70	Dic₂₃	—	4	6	no	—
71	#585	D₂₃	1	1	yes	characteristic, complemented
72	#535	—	23	3	no	—
73	B(4,𝔽₂)	—	23	8	no	—
74	#410	—	23	3	no	—
75	C₂×C₂₃	—	2	3	no	—
76	C₂×C₂₃	—	4	6	no	—
77	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
78	D₂₃	—	8	2	no	—
79	2₋¹⁺⁴	—	2	3	no	—
80	H(5,𝔽₂)	—	2	3	no	—
81	H(5,𝔽₂)	C₂×D₂₃	1	4	yes	—
82	C₂×#41	—	2	6	no	—
83	C₂×#41	C₂×D₂₃	1	3	yes	—
84	C₂²×D₄	—	23	1	no	—
85	#127	—	46	3	no	—
86	#124	—	46	3	no	—
87	#121	—	46	6	no	—
88	#120	—	23	12	no	—
89	#117	—	23	3	no	—
90	#99	—	23	12	no	—
91	C₂₃	#3027	1	1	yes	characteristic, complemented, Sylow
92	C₂⁴	—	23	2	no	—
93	#41	—	2	4	no	—
94	#41	—	2	12	no	—
95	#41	—	2	6	no	—
96	#41	—	2	12	no	—
97	#41	—	2	6	no	—
98	C₂×Q₈	—	2	6	no	—
99	C₂×Q₈	C₂²⋊D₂₃	1	3	yes	complemented
100	C₂×D₄	—	2	6	no	—
101	C₂×D₄	—	2	12	no	—
102	C₂×D₄	—	46	12	no	—
103	C₂×D₄	—	46	3	no	—
104	C₂×D₄	—	46	3	no	—
105	C₂×D₄	C₂²⋊D₂₃	1	3	yes	complemented
106	C₂×D₄	C₂²×D₂₃	1	6	yes	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	C₂²⋊D₂₃	1	6	yes	complemented
109	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
110	C₄⋊C₄	—	46	3	no	—
111	C₂²⋊C₄	—	23	6	no	—
112	C₂²⋊C₄	—	46	12	no	—
113	C₄²	—	46	3	no	—
114	C₂³	—	2	4	no	—
115	C₂³	—	46	6	no	—
116	C₂³	—	184	2	no	—
117	C₂³	C₂×C₂²⋊D₂₃	1	6	yes	—
118	C₂³	C₂³×D₂₃	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	6	no	—
121	D₄	—	2	12	no	—
122	D₄	—	2	12	no	—
123	D₄	—	2	6	no	—
124	D₄	—	92	1	no	—
125	D₄	—	92	3	no	—
126	D₄	—	184	12	no	—
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	46	6	no	—
131	C₂×C₄	C₂×C₂²⋊D₂₃	1	6	yes	—
132	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
133	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
134	C₂²	—	2	6	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	6	no	—
137	C₂²	—	8	4	no	—
138	C₂²	—	92	12	no	—
139	C₂²	—	92	2	no	—
140	C₂²	C₂²×C₂²⋊D₂₃	1	3	yes	—
141	C₄	—	2	6	no	—
142	C₄	—	92	6	no	—
143	C₄	#10625124	1	1	yes	characteristic
144	C₄	#10625124	1	3	yes	—
145	C₂	—	2	3	no	—
146	C₂	—	4	6	no	—
147	C₂	—	184	2	no	—
148	C₂	C₂×#10625124	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092981

C₂

yes

maximal, complemented

#13092604

C₂

yes

maximal, complemented

#13092391

C₂

yes

maximal, complemented

C₂₃×#585

C₂

yes

maximal, characteristic, complemented

C₂₃×2₋¹⁺⁴

—

C₂₃×H(5,𝔽₂)

—

C₂₃×H(5,𝔽₂)

C₂²

yes

—

C₂×#41×C₂₃

—

C₂×#41×C₂₃

C₂²

yes

—

C₂²×D₉₂

C₂²

yes

characteristic, complemented

#10625118

—

#10625111

—

#10625108

C₂²

yes

complemented

#10625101

—

#10625097

C₂²

yes

complemented

#10625016

C₂²

yes

complemented

C₂³×D₂₃

C₂³

yes

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

complemented

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

complemented

C₂×D₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

complemented

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

—

C₂×D₉₂

—

C₂×D₉₂

—

C₂²⋊Dic₂₃

C₂³

yes

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

#3027

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×D₄

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

#120

yes

—

Dic₂₃

—

#585

D₂₃

yes

characteristic, complemented

#535

—

B(4,𝔽₂)

—

#410

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#120

yes

characteristic

D₂₃

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₂₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₂₃

yes

—

C₂²×D₄

—

—

—

—

—

—

—

C₂₃

#3027

yes

characteristic, complemented, Sylow

C₂⁴

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

complemented

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂²⋊D₂₃

yes

complemented

106

C₂×D₄

C₂²×D₂₃

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊D₂₃

yes

complemented

109

C₂²×C₄

C₂²×D₂₃

yes

characteristic

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

184

—

117

C₂³

C₂×C₂²⋊D₂₃

yes

—

118

C₂³

C₂³×D₂₃

yes

characteristic

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

184

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

132

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

133

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×C₂²⋊D₂₃

yes

—

141

C₄

—

142

C₄

—

143

C₄

#10625124

yes

characteristic

144

C₄

#10625124

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

184

—

148

C₂

C₂×#10625124

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#10625124
commutator, cosocle	C₂³×C₂₃	C₂⁴
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂₃×#585	C₂
socle	C₂×C₂₃	C₂×#120
hypercenter	#585	D₂₃
hypocenter	C₂₃	#3027
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#10625124

commutator, cosocle

C₂³×C₂₃

C₂⁴

Frattini

C₂³

C₂³×D₂₃

Fitting

C₂₃×#585

C₂

C₂×C₂₃

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #13092981	C₂	semidirect
2. #13092604	C₂	semidirect
3. #13092391	C₂	semidirect
4. C₂₃×#585	C₂	semidirect
10. C₂²×D₉₂	C₂²	semidirect
13. #10625108	C₂²	semidirect
15. #10625097	C₂²	semidirect
16. #10625016	C₂²	semidirect
24. C₂×Q₈×C₂₃	D₄	semidirect
27. C₂×D₄×C₂₃	D₄	semidirect
30. C₂²×C₄×C₂₃	D₄	semidirect
71. #585	D₂₃	semidirect
91. C₂₃	#3027	semidirect
99. C₂×Q₈	C₂²⋊D₂₃	semidirect
105. C₂×D₄	C₂²⋊D₂₃	semidirect
108. C₂²×C₄	C₂²⋊D₂₃	semidirect

H ≅ G/N

type

1. #13092981

C₂

semidirect

2. #13092604

C₂

semidirect

3. #13092391

C₂

semidirect

4. C₂₃×#585

C₂

semidirect

10. C₂²×D₉₂

C₂²

semidirect

13. #10625108

C₂²

semidirect

15. #10625097

C₂²

semidirect

16. #10625016

C₂²

semidirect

24. C₂×Q₈×C₂₃

D₄

semidirect

27. C₂×D₄×C₂₃

D₄

semidirect

30. C₂²×C₄×C₂₃

D₄

semidirect

71. #585

D₂₃

semidirect

91. C₂₃

#3027

semidirect

99. C₂×Q₈

C₂²⋊D₂₃

semidirect

105. C₂×D₄

C₂²⋊D₂₃

semidirect

108. C₂²×C₄

C₂²⋊D₂₃

semidirect

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

subgroup	index
#13092981	2
#13092604	2
#13092391	2
C₂₃×#585	2
#3027	23

subgroup

index

C₂₃×#585

Subgroups — #15682799

Conjugacy classes of subgroups

Complements

Maximal subgroups