Group #15683928 — subgroups

Subgroups — #15683928

Order 2944.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#13093045	C₂	1	2	yes	maximal
2	#13092826	C₂	1	4	yes	maximal, complemented
3	#13092819	C₂	1	2	yes	maximal, complemented
4	#13092815	C₂	1	2	yes	maximal, complemented
5	#13092459	C₂	1	2	yes	maximal
6	#13092455	C₂	1	1	yes	maximal, characteristic
7	C₂×#10624999	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#117	C₂²	1	1	yes	characteristic
12	#10625097	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	#10625061	—	2	2	no	—
15	#10625059	—	2	2	no	—
16	#10625049	—	2	4	no	—
17	#10625041	C₂²	1	4	yes	—
18	#10625040	C₂²	1	4	yes	—
19	C₂₃×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	#10625000	C₂²	1	2	yes	—
22	#10625000	C₂²	1	4	yes	—
23	#10624999	C₂²	1	2	yes	—
24	#10624999	C₂²	1	2	yes	—
25	#10624999	C₂²	1	1	yes	characteristic
26	#10624999	C₂²	1	1	yes	characteristic
27	#10624999	C₂²	1	4	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₃	—	2	2	no	—
30	C₂×Dic₄₆	—	2	4	no	—
31	M₁₆×C₂₃	Q₈	1	4	yes	complemented
32	C₂×C₈×C₂₃	Q₈	1	4	yes	complemented
33	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₂₃	C₂³	1	2	yes	—
36	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
37	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
38	C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂²⋊Dic₂₃	—	2	2	no	—
40	C₄⋊Dic₂₃	—	2	4	no	—
41	C₄⋊Dic₂₃	—	2	2	no	—
42	C₄⋊Dic₂₃	C₂³	1	2	yes	—
43	C₄⋊Dic₂₃	C₂³	1	4	yes	—
44	C₄⋊Dic₂₃	C₂³	1	2	yes	—
45	Dic₂₃⋊C₄	—	2	4	no	—
46	Dic₂₃⋊C₄	—	2	4	no	—
47	C₄×Dic₂₃	—	2	2	no	—
48	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₂₃	—	2	2	no	—
55	C₂×Dic₂₃	—	2	2	no	—
56	C₂×Dic₂₃	—	2	4	no	—
57	C₂×Dic₂₃	—	8	2	no	—
58	Dic₄₆	—	8	4	no	—
59	C₈×C₂₃	C₂×Q₈	1	2	yes	—
60	C₈×C₂₃	C₂×Q₈	1	2	yes	—
61	C₈×C₂₃	C₂×Q₈	1	4	yes	—
62	#3163	—	23	1	no	maximal, Sylow
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
65	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₂₃	—	2	4	no	—
67	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
68	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
70	Dic₂₃	—	4	2	no	—
71	Dic₂₃	—	4	4	no	—
72	Dic₂₃	—	4	2	no	—
73	#544	—	23	2	no	—
74	#501	—	23	4	no	—
75	#499	—	23	2	no	—
76	#498	—	23	2	no	—
77	#428	—	23	2	no	—
78	#427	—	23	1	no	—
79	C₂×#106	—	23	1	no	—
80	#405	D₂₃	1	1	yes	characteristic
81	C₂×C₂₃	—	2	2	no	—
82	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₂₃	#577	1	2	yes	—
84	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
86	C₄⋊Q₈	—	23	4	no	—
87	C₄⋊Q₈	—	46	2	no	—
88	#125	—	23	4	no	—
89	#125	—	46	2	no	—
90	C₂²⋊Q₈	—	46	4	no	—
91	#117	—	23	2	no	—
92	#117	C₂×D₂₃	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	23	2	no	—
94	#107	—	23	2	no	—
95	#107	—	23	4	no	—
96	#106	—	23	2	no	—
97	#106	—	23	2	no	—
98	#106	—	23	1	no	—
99	#106	—	23	1	no	—
100	#106	—	23	4	no	—
101	#97	C₂×D₂₃	1	2	yes	—
102	C₄×C₈	C₂×D₂₃	1	2	yes	—
103	C₂₃	#3163	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	46	4	no	—
105	C₂²×C₄	—	46	2	no	—
106	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
107	M₁₆	Dic₄₆	1	4	yes	complemented
108	C₂×C₈	Dic₄₆	1	4	yes	complemented
109	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₂₃	1	2	yes	—
111	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
112	C₄⋊C₄	—	23	4	no	—
113	C₄⋊C₄	—	23	2	no	—
114	C₄⋊C₄	—	23	2	no	—
115	C₄⋊C₄	—	46	2	no	—
116	C₄⋊C₄	—	46	4	no	—
117	C₄⋊C₄	—	46	4	no	—
118	C₄⋊C₄	—	46	4	no	—
119	C₄⋊C₄	D₉₂	1	2	yes	—
120	C₂²⋊C₄	—	46	2	no	—
121	C₂²⋊C₄	D₉₂	1	2	yes	—
122	C₄²	—	46	2	no	—
123	C₄²	C₂²×D₂₃	1	2	yes	—
124	C₂³	C₂×D₉₂	1	1	yes	characteristic
125	Q₈	—	184	4	no	—
126	C₂×C₄	—	46	2	no	—
127	C₂×C₄	—	46	2	no	—
128	C₂×C₄	—	46	4	no	—
129	C₂×C₄	—	184	2	no	—
130	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
131	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
132	C₂×C₄	C₂×D₉₂	1	4	yes	—
133	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
135	C₈	C₂×Dic₄₆	1	2	yes	—
136	C₈	C₂×Dic₄₆	1	2	yes	—
137	C₈	C₂×Dic₄₆	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#10625040	1	2	yes	—
140	C₂²	C₂²×D₉₂	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	92	4	no	—
143	C₄	—	92	2	no	—
144	C₄	—	92	2	no	—
145	C₄	#10625040	1	2	yes	—
146	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
147	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#10625040	1	1	yes	characteristic
150	C₂	#13093229	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13093045

C₂

yes

maximal

#13092826

C₂

yes

maximal, complemented

#13092819

C₂

yes

maximal, complemented

#13092815

C₂

yes

maximal, complemented

#13092459

C₂

yes

maximal

#13092455

C₂

yes

maximal, characteristic

C₂×#10624999

C₂

yes

maximal, characteristic

C₂₃×#405

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

characteristic

#10625097

C₂²

yes

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

—

—

—

C₂²

yes

—

#10625040

C₂²

yes

—

C₂₃×#97

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

#10625000

C₂²

yes

—

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

#10624999

C₂²

yes

—

#10624999

C₂²

yes

characteristic

#10624999

C₂²

yes

characteristic

#10624999

C₂²

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

M₁₆×C₂₃

Q₈

yes

complemented

C₂×C₈×C₂₃

Q₈

yes

complemented

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

—

C₄⋊C₄×C₂₃

D₄

yes

—

C₂²⋊C₄×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

C₄⋊Dic₂₃

C₂³

yes

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

—

C₈×C₂₃

C₂×Q₈

yes

—

C₈×C₂₃

C₂×Q₈

yes

—

#3163

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

—

—

—

—

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

#577

yes

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

—

—

—

—

—

—

100

#106

—

101

#97

C₂×D₂₃

yes

—

102

C₄×C₈

C₂×D₂₃

yes

—

103

C₂₃

#3163

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₂₃

yes

characteristic

107

M₁₆

Dic₄₆

yes

complemented

108

C₂×C₈

Dic₄₆

yes

complemented

109

C₂×C₈

C₂²×D₂₃

yes

characteristic

110

C₂×C₈

C₂²×D₂₃

yes

—

111

C₂×C₈

C₂²×D₂₃

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₉₂

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

D₉₂

yes

—

122

C₄²

—

123

C₄²

C₂²×D₂₃

yes

—

124

C₂³

C₂×D₉₂

yes

characteristic

125

Q₈

—

184

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

184

—

130

C₂×C₄

C₂×Dic₄₆

yes

—

131

C₂×C₄

C₂×Dic₄₆

yes

—

132

C₂×C₄

C₂×D₉₂

yes

—

133

C₂×C₄

C₂×D₉₂

yes

characteristic

134

C₂×C₄

C₂³×D₂₃

yes

characteristic

135

C₈

C₂×Dic₄₆

yes

—

136

C₈

C₂×Dic₄₆

yes

—

137

C₈

C₂×Dic₄₆

yes

—

138

C₂²

—

139

C₂²

#10625040

yes

—

140

C₂²

C₂²×D₉₂

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#10625040

yes

—

146

C₄

C₂²×Dic₄₆

yes

characteristic

147

C₄

C₂²×Dic₄₆

yes

characteristic

148

C₂

—

149

C₂

C₂×#10625040

yes

characteristic

150

C₂

#13093229

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#405	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#405	D₂₃
hypocenter	C₂₃	#3163
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₉₂

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#405

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #13092826	C₂	semidirect
3. #13092819	C₂	semidirect
4. #13092815	C₂	semidirect
31. M₁₆×C₂₃	Q₈	semidirect
32. C₂×C₈×C₂₃	Q₈	semidirect
103. C₂₃	#3163	semidirect
107. M₁₆	Dic₄₆	semidirect
108. C₂×C₈	Dic₄₆	semidirect

H ≅ G/N

type

2. #13092826

C₂

semidirect

3. #13092819

C₂

semidirect

4. #13092815

C₂

semidirect

31. M₁₆×C₂₃

Q₈

semidirect

32. C₂×C₈×C₂₃

Q₈

semidirect

103. C₂₃

#3163

semidirect

107. M₁₆

Dic₄₆

semidirect

108. C₂×C₈

Dic₄₆

semidirect

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

subgroup	index
#13093045	2
#13092826	2
#13092819	2
#13092815	2
#13092459	2
#13092455	2
C₂×#10624999	2
C₂₃×#405	2
#3163	23

subgroup

index

C₂₃×#405

Subgroups — #15683928

Conjugacy classes of subgroups

Complements

Maximal subgroups