Group #43252697 — subgroups

Subgroups — #43252697

Order 28544.

396 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 65 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	#28383436	C₂	1	2	yes	maximal
2	#379×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₂₂₃	—	2	2	no	—
4	C₂×C₂²⋊Dic₂₂₃	—	2	2	no	—
5	C₂×C₂²⋊Dic₂₂₃	C₄	1	4	yes	complemented
6	C₂⁵×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₂₂₃	—	2	2	no	—
8	C₂×C₂²⋊C₄×C₂₂₃	C₄	1	2	yes	complemented
9	C₂²×Dic₂₂₃	—	2	4	no	—
10	C₂²⋊Dic₂₂₃	—	4	4	no	—
11	C₂²⋊Dic₂₂₃	—	4	4	no	—
12	C₂²⋊Dic₂₂₃	—	4	4	no	—
13	C₂⁴×C₂₂₃	—	4	2	no	—
14	C₂⁴×C₂₂₃	—	4	4	no	—
15	C₂⁴×C₂₂₃	C₂×C₄	1	2	yes	—
16	C₂⁴×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂⁴×C₂₂₃	D₄	1	2	yes	—
18	C₂⁴×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂⁴×C₂₂₃	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₂₂₃	—	2	2	no	—
21	C₂²⋊C₄×C₂₂₃	—	4	4	no	—
22	C₂²⋊C₄×C₂₂₃	—	4	2	no	—
23	C₂×Dic₂₂₃	—	4	4	no	—
24	C₂×Dic₂₂₃	—	4	4	no	—
25	C₂×Dic₂₂₃	—	4	4	no	—
26	C₂³×C₂₂₃	—	2	2	no	—
27	C₂³×C₂₂₃	—	2	4	no	—
28	C₂³×C₂₂₃	—	2	4	no	—
29	C₂³×C₂₂₃	—	2	4	no	—
30	C₂³×C₂₂₃	—	2	4	no	—
31	C₂³×C₂₂₃	—	4	2	no	—
32	C₂³×C₂₂₃	—	4	2	no	—
33	C₂³×C₂₂₃	—	4	2	no	—
34	C₂³×C₂₂₃	—	4	2	no	—
35	C₂³×C₂₂₃	—	4	1	no	—
36	C₂³×C₂₂₃	—	4	2	no	—
37	C₂³×C₂₂₃	—	4	1	no	—
38	C₂³×C₂₂₃	—	4	16	no	—
39	C₂³×C₂₂₃	C₄²	1	1	yes	characteristic
40	C₂³×C₂₂₃	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂³×C₂₂₃	C₄⋊C₄	1	2	yes	—
43	C₂³×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₂₃	—	4	4	no	—
45	C₂×C₄×C₂₂₃	—	4	2	no	—
46	C₂²×C₂₂₃	—	2	2	no	—
47	C₂²×C₂₂₃	—	2	2	no	—
48	C₂²×C₂₂₃	—	2	2	no	—
49	C₂²×C₂₂₃	—	2	2	no	—
50	C₂²×C₂₂₃	—	2	1	no	—
51	C₂²×C₂₂₃	—	2	2	no	—
52	C₂²×C₂₂₃	—	2	2	no	—
53	C₂²×C₂₂₃	—	2	2	no	—
54	C₂²×C₂₂₃	—	2	2	no	—
55	C₂²×C₂₂₃	—	2	1	no	—
56	C₂²×C₂₂₃	—	4	4	no	—
57	C₂²×C₂₂₃	—	4	8	no	—
58	C₂²×C₂₂₃	—	4	8	no	—
59	C₂²×C₂₂₃	—	4	4	no	—
60	C₂²×C₂₂₃	—	4	4	no	—
61	C₂²×C₂₂₃	#95	1	1	yes	characteristic
62	C₂²×C₂₂₃	#99	1	4	yes	—
63	C₂²×C₂₂₃	#99	1	2	yes	—
64	C₄×C₂₂₃	—	8	2	no	—
65	Dic₂₂₃	—	8	4	no	—
66	C₂×C₂₂₃	—	2	4	no	—
67	C₂×C₂₂₃	—	2	2	no	—
68	C₂×C₂₂₃	—	4	1	no	—
69	C₂×C₂₂₃	—	4	2	no	—
70	C₂×C₂₂₃	—	4	1	no	—
71	C₂×C₂₂₃	#342	1	2	yes	—
72	C₂×C₂₂₃	#342	1	1	yes	characteristic
73	C₂₂₃	#1304	1	1	yes	characteristic, complemented, Sylow
74	#1304	—	223	1	no	maximal, Sylow
75	#379	—	223	2	no	—
76	#379	D₂₂₃	1	1	yes	characteristic
77	C₂⁵	C₂×D₂₂₃	1	1	yes	characteristic
78	C₂×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₂²⋊C₄	—	446	2	no	—
80	C₂×C₂²⋊C₄	—	446	2	no	—
81	C₂×C₂²⋊C₄	—	223	4	no	—
82	C₂×C₂²⋊C₄	Dic₂₂₃	1	2	yes	complemented
83	C₂⁴	—	4	4	no	—
84	C₂⁴	—	4	2	no	—
85	C₂⁴	C₄×D₂₂₃	1	2	yes	—
86	C₂⁴	C₂×Dic₂₂₃	1	1	yes	characteristic
87	C₂⁴	C₂²⋊D₂₂₃	1	2	yes	—
88	C₂⁴	D₈₉₂	1	1	yes	characteristic
89	C₂⁴	Dic₄₄₆	1	1	yes	characteristic
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	446	4	no	—
92	C₂²⋊C₄	—	4	4	no	—
93	C₂²⋊C₄	—	4	2	no	—
94	C₂²⋊C₄	—	892	4	no	—
95	C₂²⋊C₄	—	892	4	no	—
96	C₂²⋊C₄	—	892	4	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	4	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	4	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	16	no	—
110	C₂³	C₄×Dic₂₂₃	1	1	yes	characteristic
111	C₂³	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
112	C₂³	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
113	C₂³	Dic₂₂₃⋊C₄	1	2	yes	—
114	C₂³	C₄⋊Dic₂₂₃	1	1	yes	characteristic
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	892	4	no	—
118	C₂×C₄	—	892	4	no	—
119	C₂×C₄	—	892	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	4	8	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	4	no	—
133	C₂²	—	4	8	no	—
134	C₂²	—	4	4	no	—
135	C₂²	C₂₂₃⋊#95	1	1	yes	characteristic
136	C₂²	#20297218	1	4	yes	—
137	C₂²	#20297219	1	2	yes	—
138	C₄	—	8	2	no	—
139	C₄	—	1784	4	no	—
140	C₂	—	2	2	no	—
141	C₂	—	2	4	no	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	—	4	1	no	—
145	C₂	#28383346	1	2	yes	—
146	C₂	#28383347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28383436

C₂

yes

maximal

#379×C₂₂₃

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₂₂₃

—

C₂×C₂²⋊Dic₂₂₃

—

C₂×C₂²⋊Dic₂₂₃

C₄

yes

complemented

C₂⁵×C₂₂₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₂₃

—

C₂×C₂²⋊C₄×C₂₂₃

C₄

yes

complemented

C₂²×Dic₂₂₃

—

C₂²⋊Dic₂₂₃

—

C₂²⋊Dic₂₂₃

—

C₂²⋊Dic₂₂₃

—

C₂⁴×C₂₂₃

—

C₂⁴×C₂₂₃

—

C₂⁴×C₂₂₃

C₂×C₄

yes

—

C₂⁴×C₂₂₃

C₂×C₄

yes

characteristic

C₂⁴×C₂₂₃

D₄

yes

—

C₂⁴×C₂₂₃

D₄

yes

characteristic

C₂⁴×C₂₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₂×Dic₂₂₃

—

C₂×Dic₂₂₃

—

C₂×Dic₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

C₄²

yes

characteristic

C₂³×C₂₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₂₃

C₂²⋊C₄

yes

characteristic

C₂³×C₂₂₃

C₄⋊C₄

yes

—

C₂³×C₂₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

#95

yes

characteristic

C₂²×C₂₂₃

#99

yes

—

C₂²×C₂₂₃

#99

yes

—

C₄×C₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

#342

yes

—

C₂×C₂₂₃

#342

yes

characteristic

C₂₂₃

#1304

yes

characteristic, complemented, Sylow

#1304

—

223

maximal, Sylow

#379

—

223

—

#379

D₂₂₃

yes

characteristic

C₂⁵

C₂×D₂₂₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

446

—

C₂×C₂²⋊C₄

—

446

—

C₂×C₂²⋊C₄

—

223

—

C₂×C₂²⋊C₄

Dic₂₂₃

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

C₄×D₂₂₃

yes

—

C₂⁴

C₂×Dic₂₂₃

yes

characteristic

C₂⁴

C₂²⋊D₂₂₃

yes

—

C₂⁴

D₈₉₂

yes

characteristic

C₂⁴

Dic₄₄₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

446

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

892

—

C₂²⋊C₄

—

892

—

C₂²⋊C₄

—

892

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₄×Dic₂₂₃

yes

characteristic

111

C₂³

(C₂×D₂₂₃)⋊C₄

yes

—

112

C₂³

C₂²⋊Dic₂₂₃

yes

characteristic

113

C₂³

Dic₂₂₃⋊C₄

yes

—

114

C₂³

C₄⋊Dic₂₂₃

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

892

—

118

C₂×C₄

—

892

—

119

C₂×C₄

—

892

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂₂₃⋊#95

yes

characteristic

136

C₂²

#20297218

yes

—

137

C₂²

#20297219

yes

—

138

C₄

—

139

C₄

—

1784

—

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

#28383346

yes

—

146

C₂

#28383347

yes

characteristic

	subgroup	G/H
center	C₂²	C₂₂₃⋊#95
commutator	C₂³×C₂₂₃	C₄²
Frattini	C₂⁵	C₂×D₂₂₃
Fitting	#379×C₂₂₃	C₂
socle	C₂²×C₂₂₃	#95
prime radical, cosocle	C₂⁵×C₂₂₃	C₂²
hypercenter	#379	D₂₂₃
hypocenter	C₂₂₃	#1304
LCS	G ▷ C₂³×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #379

subgroup

G/H

center

C₂²

C₂₂₃⋊#95

commutator

C₂³×C₂₂₃

C₄²

Frattini

C₂⁵

C₂×D₂₂₃

Fitting

#379×C₂₂₃

C₂

socle

C₂²×C₂₂₃

#95

prime radical, cosocle

C₂⁵×C₂₂₃

C₂²

D₂₂₃

C₂₂₃

G ▷ C₂³×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃

UCS

C₁ ◁ C₂² ◁ C₂⁵ ◁ #379

N	H ≅ G/N	type
5. C₂×C₂²⋊Dic₂₂₃	C₄	semidirect
8. C₂×C₂²⋊C₄×C₂₂₃	C₄	semidirect
73. C₂₂₃	#1304	semidirect
82. C₂×C₂²⋊C₄	Dic₂₂₃	semidirect

H ≅ G/N

type

5. C₂×C₂²⋊Dic₂₂₃

C₄

semidirect

8. C₂×C₂²⋊C₄×C₂₂₃

C₄

semidirect

73. C₂₂₃

#1304

semidirect

82. C₂×C₂²⋊C₄

Dic₂₂₃

semidirect

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

subgroup	index
#28383436	2
#379×C₂₂₃	2
#1304	223

subgroup

index

#28383436

#379×C₂₂₃

#1304

223

Subgroups — #43252697

Conjugacy classes of subgroups

Complements

Maximal subgroups