Group #15674228 — subgroups

Subgroups — #15674228

Order 2944.

320 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 89 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	#13092967	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625007	C₂	1	4	yes	maximal, complemented
3	#13092263	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#67288	C₂²	1	2	yes	—
6	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	#10625101	—	2	2	no	—
9	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
10	#10625046	C₄	1	2	yes	complemented
11	#10625045	C₄	1	1	yes	characteristic, complemented
12	C₄×D₉₂	—	2	2	no	—
13	#10625040	C₄	1	1	yes	characteristic, complemented
14	C₂₃×C₄⋊C₈	—	2	2	no	—
15	C₂₃⋊#95	—	2	2	no	—
16	#10625007	—	2	8	no	—
17	C₂²×C₄×C₂₃	D₄	1	2	yes	—
18	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₃	—	2	2	no	—
20	C₂²×Dic₂₃	—	2	2	no	—
21	#67288	—	2	4	no	—
22	#67288	—	2	4	no	—
23	C₂×D₉₂	—	2	2	no	—
24	C₂×D₉₂	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₂₃	—	2	2	no	—
26	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
27	M₁₆×C₂₃	—	2	4	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₃	D₄	1	2	yes	—
32	(C₂×D₂₃)⋊C₄	—	2	4	no	—
33	C₄⋊Dic₂₃	—	2	2	no	—
34	C₄×Dic₂₃	—	2	2	no	—
35	C₄×Dic₂₃	D₄	1	4	yes	complemented
36	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×D₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	#41	1	2	yes	—
45	C₂²⋊D₂₃	—	4	4	no	—
46	C₂×Dic₂₃	—	2	2	no	—
47	C₂×Dic₂₃	—	2	4	no	—
48	C₂×Dic₂₃	—	4	4	no	—
49	D₉₂	—	2	4	no	—
50	D₉₂	—	4	4	no	—
51	C₄×D₂₃	—	4	2	no	—
52	C₄×D₂₃	—	4	2	no	—
53	Dic₄₆	—	2	4	no	—
54	C₈×C₂₃	—	4	2	no	—
55	#1963	—	23	1	no	maximal, Sylow
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₂₃	#121	1	2	yes	—
59	C₂×D₂₃	—	4	2	no	—
60	C₂×D₂₃	—	4	4	no	—
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	C₄×D₄	1	2	yes	—
64	C₄×C₂₃	#124	1	1	yes	characteristic
65	C₄×C₂₃	#127	1	1	yes	characteristic
66	Dic₂₃	—	4	4	no	—
67	Dic₂₃	—	4	2	no	—
68	#532	—	23	1	no	—
69	#423	D₂₃	1	1	yes	characteristic, complemented
70	C₂×#104	—	23	4	no	—
71	#376	—	23	1	no	—
72	C₂×C₂₃	—	2	2	no	—
73	C₂×C₂₃	#390	1	1	yes	characteristic
74	C₂×C₂₃	#421	1	2	yes	—
75	D₂₃	—	8	2	no	—
76	C₂×#41	—	23	2	no	—
77	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
78	C₄⋊Q₈	—	23	1	no	—
79	#127	—	23	1	no	—
80	#124	—	23	2	no	—
81	#121	—	46	2	no	—
82	C₄×D₄	—	46	2	no	—
83	C₂×C₄²	—	23	2	no	—
84	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	#104	—	46	8	no	—
87	#95	—	46	2	no	—
88	C₂₃	#1963	1	1	yes	characteristic, complemented, Sylow
89	#41	—	46	4	no	—
90	#41	—	46	4	no	—
91	C₂×Q₈	—	23	2	no	—
92	C₂×D₄	—	23	2	no	—
93	C₂×D₄	—	46	2	no	—
94	C₂×D₄	—	46	2	no	—
95	C₂²×C₄	—	46	2	no	—
96	C₂²×C₄	—	46	2	no	—
97	C₂²×C₄	D₉₂	1	2	yes	—
98	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	46	2	no	—
102	C₂²⋊C₄	—	46	4	no	—
103	C₄²	—	23	4	no	—
104	C₄²	—	46	2	no	—
105	C₄²	C₄×D₂₃	1	1	yes	characteristic
106	C₄²	C₄×D₂₃	1	1	yes	characteristic
107	C₄²	C₂²⋊D₂₃	1	2	yes	—
108	C₂³	—	46	2	no	—
109	C₂³	C₂×D₉₂	1	1	yes	characteristic
110	Q₈	—	46	4	no	—
111	D₄	—	46	4	no	—
112	D₄	—	92	4	no	—
113	D₄	—	92	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	46	4	no	—
116	C₂×C₄	—	46	2	no	—
117	C₂×C₄	—	92	4	no	—
118	C₂×C₄	—	92	2	no	—
119	C₂×C₄	—	92	2	no	—
120	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
121	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
122	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
123	C₂×C₄	D₄×D₂₃	1	2	yes	—
124	C₂×C₄	#67292	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	92	4	no	—
129	C₂²	—	92	2	no	—
130	C₂²	#10625066	1	2	yes	—
131	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	92	4	no	—
135	C₄	—	92	2	no	—
136	C₄	#10625064	1	2	yes	—
137	C₄	#10625111	1	1	yes	characteristic
138	C₄	#10625118	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	184	2	no	—
141	C₂	#13092318	1	1	yes	characteristic
142	C₂	#13092433	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092967

C₂

yes

maximal, characteristic

C₂×#10625007

C₂

yes

maximal, complemented

#13092263

C₂

yes

maximal, characteristic, complemented

C₂₃×#423

C₂

yes

maximal, characteristic, complemented

C₂×#67288

C₂²

yes

—

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625101

—

C₂×C₄×Dic₂₃

C₂²

yes

—

#10625046

C₄

yes

complemented

#10625045

C₄

yes

characteristic, complemented

C₄×D₉₂

—

#10625040

C₄

yes

characteristic, complemented

C₂₃×C₄⋊C₈

—

C₂₃⋊#95

—

#10625007

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₂²⋊D₂₃

—

C₂²×Dic₂₃

—

#67288

—

#67288

—

C₂×D₉₂

—

C₂×D₉₂

C₂×C₄

yes

—

C₂×C₄×D₂₃

—

C₂×Dic₄₆

C₂×C₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

D₄

yes

—

(C₂×D₂₃)⋊C₄

—

C₄⋊Dic₂₃

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

complemented

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂²×D₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₈×C₂₃

—

#1963

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#121

yes

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

#532

—

#423

D₂₃

yes

characteristic, complemented

C₂×#104

—

#376

—

C₂×C₂₃

—

C₂×C₂₃

#390

yes

characteristic

C₂×C₂₃

#421

yes

—

D₂₃

—

C₂×#41

—

C₂×M₁₆

C₂×D₂₃

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

#104

—

#95

—

C₂₃

#1963

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₄×D₂₃

yes

characteristic

106

C₄²

C₄×D₂₃

yes

characteristic

107

C₄²

C₂²⋊D₂₃

yes

—

108

C₂³

—

109

C₂³

C₂×D₉₂

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

121

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

122

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

123

C₂×C₄

D₄×D₂₃

yes

—

124

C₂×C₄

#67292

yes

—

125

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

#10625066

yes

—

131

C₂²

C₂×(C₂×D₂₃)⋊C₄

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#10625064

yes

—

137

C₄

#10625111

yes

characteristic

138

C₄

#10625118

yes

characteristic

139

C₂

—

140

C₂

—

184

—

141

C₂

#13092318

yes

characteristic

142

C₂

#13092433

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#423	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical	#13092967	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#423	D₂₃
hypocenter	C₂₃	#1963
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#423

C₂

socle

C₂²×C₂₃

C₂×C₂²⋊C₄

prime radical

#13092967

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#10625007	C₂	semidirect
3. #13092263	C₂	semidirect
4. C₂₃×#423	C₂	semidirect
10. #10625046	C₄	semidirect
11. #10625045	C₄	semidirect
13. #10625040	C₄	semidirect
35. C₄×Dic₂₃	D₄	semidirect
69. #423	D₂₃	semidirect
88. C₂₃	#1963	semidirect

H ≅ G/N

type

2. C₂×#10625007

C₂

semidirect

3. #13092263

C₂

semidirect

4. C₂₃×#423

C₂

semidirect

10. #10625046

C₄

semidirect

11. #10625045

C₄

semidirect

13. #10625040

C₄

semidirect

35. C₄×Dic₂₃

D₄

semidirect

69. #423

D₂₃

semidirect

88. C₂₃

#1963

semidirect

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

subgroup	index
#13092967	2
C₂×#10625007	2
#13092263	2
C₂₃×#423	2
#1963	23

subgroup

index

C₂₃×#423

Subgroups — #15674228

Conjugacy classes of subgroups

Complements

Maximal subgroups