Group #44450266 — subgroups

Subgroups — #44450266

Order 29824.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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	#28416992	C₂	1	1	yes	maximal, characteristic
2	#28416459	C₂	1	1	yes	maximal, characteristic
3	#28416167	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#16836578	—	2	1	no	—
6	#21418535	C₄	1	1	yes	characteristic
7	#21418535	C₄	1	1	yes	characteristic
8	#21418527	—	2	1	no	—
9	#21418514	—	2	1	no	—
10	#21418514	C₄	1	2	yes	—
11	C₄×Dic₄₆₆	—	2	1	no	—
12	C₄×Dic₄₆₆	C₂²	1	2	yes	—
13	#21418494	—	2	1	no	—
14	C₄⋊#14272338	—	2	1	no	—
15	C₄⋊#14272338	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₃₃	—	2	1	no	—
17	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₂₃₃	—	2	1	no	—
19	C₄×C₈×C₂₃₃	C₂²	1	2	yes	—
20	C₂×Dic₄₆₆	—	2	1	no	—
21	C₂×Dic₄₆₆	C₂×C₄	1	2	yes	—
22	#16836578	—	4	2	no	—
23	C₂×#14272338	—	2	2	no	—
24	C₄⋊Dic₂₃₃	—	2	1	no	—
25	C₄⋊Dic₂₃₃	—	2	1	no	—
26	C₄⋊Dic₂₃₃	—	2	1	no	—
27	C₄⋊Dic₂₃₃	C₂×C₄	1	2	yes	—
28	#16836573	—	2	2	no	—
29	#16836573	—	2	2	no	—
30	C₂²⋊Dic₂₃₃	—	2	2	no	—
31	C₄×Dic₂₃₃	—	2	2	no	—
32	C₂²×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃₃	—	2	4	no	—
36	C₂×C₈×C₂₃₃	—	2	2	no	—
37	C₄²×C₂₃₃	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
39	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
40	C₄²×C₂₃₃	C₂³	1	1	yes	characteristic
41	Dic₄₆₆	—	2	4	no	—
42	Dic₄₆₆	—	4	2	no	—
43	C₂×Dic₂₃₃	—	2	2	no	—
44	C₂×Dic₂₃₃	—	2	2	no	—
45	#14272338	—	4	2	no	—
46	C₂³×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃₃	—	2	2	no	—
48	C₂×C₄×C₂₃₃	—	2	2	no	—
49	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃₃	SD₁₆	1	2	yes	—
53	C₂×C₄×C₂₃₃	Q₁₆	1	2	yes	—
54	C₂×C₄×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
57	C₈×C₂₃₃	—	2	4	no	—
58	C₂²×C₂₃₃	—	2	1	no	—
59	C₂²×C₂₃₃	—	2	1	no	—
60	C₂²×C₂₃₃	#103	1	2	yes	—
61	C₂²×C₂₃₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₂₃₃	—	2	1	no	—
63	C₄×C₂₃₃	—	2	1	no	—
64	C₄×C₂₃₃	#103	1	2	yes	—
65	C₄×C₂₃₃	#104	1	2	yes	—
66	C₄×C₂₃₃	#131	1	2	yes	—
67	C₄×C₂₃₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₂₃₃	C₂×Q₁₆	1	1	yes	characteristic
69	Dic₂₃₃	—	4	2	no	—
70	Dic₂₃₃	—	4	2	no	—
71	C₂×C₂₃₃	—	2	2	no	—
72	C₂×C₂₃₃	#408	1	1	yes	characteristic
73	C₂×C₂₃₃	C₂×#103	1	1	yes	characteristic
74	C₂×C₂₃₃	C₂×#104	1	1	yes	characteristic
75	C₂₃₃	#1493	1	1	yes	characteristic, complemented, Sylow
76	#1493	—	233	1	no	maximal, Sylow
77	#533	—	233	1	no	—
78	#423	—	233	1	no	—
79	C₂×C₄×C₈	D₂₃₃	1	1	yes	characteristic
80	Q₈⋊C₈	—	233	4	no	—
81	C₂×M₁₆	—	466	1	no	—
82	C₂²×C₈	—	2	1	no	—
83	C₄⋊Q₈	—	233	1	no	—
84	C₄⋊Q₈	—	233	1	no	—
85	#125	—	466	1	no	—
86	C₂²⋊Q₈	—	466	1	no	—
87	C₂²⋊Q₈	—	233	2	no	—
88	C₄×Q₈	—	466	1	no	—
89	C₄×Q₈	—	233	2	no	—
90	#117	—	466	1	no	—
91	C₂×C₄²	C₂×D₂₃₃	1	1	yes	characteristic
92	C₄⋊C₈	—	466	1	no	—
93	C₄⋊C₈	—	233	2	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	C₂×D₂₃₃	1	2	yes	—
96	C₂×Q₈	—	466	1	no	—
97	C₂×Q₈	—	233	2	no	—
98	C₂²×C₄	C₄×D₂₃₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
100	C₂²×C₄	D₉₃₂	1	1	yes	characteristic
101	M₁₆	—	932	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	466	2	no	—
105	C₄⋊C₄	—	466	1	no	—
106	C₄⋊C₄	—	466	1	no	—
107	C₄⋊C₄	—	466	1	no	—
108	C₄⋊C₄	—	466	2	no	—
109	C₄⋊C₄	—	466	2	no	—
110	C₄⋊C₄	—	233	2	no	—
111	C₂²⋊C₄	—	466	2	no	—
112	C₄²	—	466	2	no	—
113	C₄²	C₄×D₂₃₃	1	1	yes	characteristic
114	C₄²	C₂²⋊D₂₃₃	1	1	yes	characteristic
115	C₄²	D₉₃₂	1	1	yes	characteristic
116	C₄²	C₂²×D₂₃₃	1	1	yes	characteristic
117	C₂³	#16836571	1	1	yes	characteristic
118	Q₈	—	932	2	no	—
119	Q₈	—	466	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	466	2	no	—
123	C₂×C₄	—	466	2	no	—
124	C₂×C₄	#16836571	1	1	yes	characteristic
125	C₂×C₄	#16836571	1	1	yes	characteristic
126	C₂×C₄	#16836571	1	1	yes	characteristic
127	C₂×C₄	#16836583	1	2	yes	—
128	C₂×C₄	Dic₉₃₂	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₂₃₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
131	C₂×C₄	C₂×D₉₃₂	1	1	yes	characteristic
132	C₈	—	2	4	no	—
133	C₈	—	932	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#21418460	1	2	yes	—
137	C₂²	C₂×#16836571	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	932	2	no	—
141	C₄	—	932	2	no	—
142	C₄	#21418460	1	2	yes	—
143	C₄	#21418463	1	2	yes	—
144	C₄	#21418541	1	2	yes	—
145	C₄	C₂×#16836583	1	1	yes	characteristic
146	C₄	C₂×Dic₉₃₂	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#28416401	1	1	yes	characteristic
149	C₂	C₂×#21418460	1	1	yes	characteristic
150	C₂	C₂×#21418463	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28416992

C₂

yes

maximal, characteristic

#28416459

C₂

yes

maximal, characteristic

#28416167

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₂₃₃

C₂

yes

maximal, characteristic

C₂×#16836578

—

#21418535

C₄

yes

characteristic

#21418535

C₄

yes

—

—

C₄

yes

—

C₄×Dic₄₆₆

—

C₄×Dic₄₆₆

C₂²

yes

—

#21418494

—

C₄⋊#14272338

—

C₄⋊#14272338

C₂²

yes

—

C₂²×C₈×C₂₃₃

—

C₂×C₄²×C₂₃₃

C₂²

yes

characteristic

C₄×C₈×C₂₃₃

—

C₄×C₈×C₂₃₃

C₂²

yes

—

C₂×Dic₄₆₆

—

C₂×Dic₄₆₆

C₂×C₄

yes

—

#16836578

—

C₂×#14272338

—

C₄⋊Dic₂₃₃

—

C₄⋊Dic₂₃₃

—

C₄⋊Dic₂₃₃

—

C₄⋊Dic₂₃₃

C₂×C₄

yes

—

#16836573

—

#16836573

—

C₂²⋊Dic₂₃₃

—

C₄×Dic₂₃₃

—

C₂²×C₄×C₂₃₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

C₂×C₈×C₂₃₃

—

C₂×C₈×C₂₃₃

—

C₄²×C₂₃₃

C₂×C₄

yes

characteristic

C₄²×C₂₃₃

D₄

yes

characteristic

C₄²×C₂₃₃

D₄

yes

characteristic

C₄²×C₂₃₃

C₂³

yes

characteristic

Dic₄₆₆

—

Dic₄₆₆

—

C₂×Dic₂₃₃

—

C₂×Dic₂₃₃

—

#14272338

—

C₂³×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₃

SD₁₆

yes

—

C₂×C₄×C₂₃₃

Q₁₆

yes

—

C₂×C₄×C₂₃₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₃

C₂×D₄

yes

characteristic

C₈×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

#103

yes

—

C₂²×C₂₃₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

#103

yes

—

C₄×C₂₃₃

#104

yes

—

C₄×C₂₃₃

#131

yes

—

C₄×C₂₃₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃₃

C₂×Q₁₆

yes

characteristic

Dic₂₃₃

—

Dic₂₃₃

—

C₂×C₂₃₃

—

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂₃₃

yes

characteristic, complemented, Sylow

#1493

—

233

maximal, Sylow

#533

—

233

—

#423

—

233

—

C₂×C₄×C₈

D₂₃₃

yes

characteristic

Q₈⋊C₈

—

233

—

C₂×M₁₆

—

466

—

C₂²×C₈

—

C₄⋊Q₈

—

233

—

C₄⋊Q₈

—

233

—

#125

—

466

—

C₂²⋊Q₈

—

466

—

C₂²⋊Q₈

—

233

—

C₄×Q₈

—

466

—

C₄×Q₈

—

233

—

#117

—

466

—

C₂×C₄²

C₂×D₂₃₃

yes

characteristic

C₄⋊C₈

—

466

—

C₄⋊C₈

—

233

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃₃

yes

—

C₂×Q₈

—

466

—

C₂×Q₈

—

233

—

C₂²×C₄

C₄×D₂₃₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃₃

yes

characteristic

100

C₂²×C₄

D₉₃₂

yes

characteristic

101

M₁₆

—

932

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

466

—

105

C₄⋊C₄

—

466

—

106

C₄⋊C₄

—

466

—

107

C₄⋊C₄

—

466

—

108

C₄⋊C₄

—

466

—

109

C₄⋊C₄

—

466

—

110

C₄⋊C₄

—

233

—

111

C₂²⋊C₄

—

466

—

112

C₄²

—

466

—

113

C₄²

C₄×D₂₃₃

yes

characteristic

114

C₄²

C₂²⋊D₂₃₃

yes

characteristic

115

C₄²

D₉₃₂

yes

characteristic

116

C₄²

C₂²×D₂₃₃

yes

characteristic

117

C₂³

#16836571

yes

characteristic

118

Q₈

—

932

—

119

Q₈

—

466

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

466

—

123

C₂×C₄

—

466

—

124

C₂×C₄

#16836571

yes

characteristic

125

C₂×C₄

#16836571

yes

characteristic

126

C₂×C₄

#16836571

yes

characteristic

127

C₂×C₄

#16836583

yes

—

128

C₂×C₄

Dic₉₃₂

yes

—

129

C₂×C₄

C₂×C₄×D₂₃₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₂₃₃

yes

characteristic

131

C₂×C₄

C₂×D₉₃₂

yes

characteristic

132

C₈

—

133

C₈

—

932

—

134

C₂²

—

135

C₂²

—

136

C₂²

#21418460

yes

—

137

C₂²

C₂×#16836571

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

932

—

141

C₄

—

932

—

142

C₄

#21418460

yes

—

143

C₄

#21418463

yes

—

144

C₄

#21418541

yes

—

145

C₄

C₂×#16836583

yes

characteristic

146

C₄

C₂×Dic₉₃₂

yes

characteristic

147

C₂

—

148

C₂

#28416401

yes

characteristic

149

C₂

C₂×#21418460

yes

characteristic

150

C₂

C₂×#21418463

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₉₃₂
commutator	C₂×C₄×C₂₃₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃₃
Fitting	C₂×C₄×C₈×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃₃	C₂²⋊C₄
cosocle	C₄²×C₂₃₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃₃
hypocenter	C₂₃₃	#1493
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₉₃₂

commutator

C₂×C₄×C₂₃₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃₃

Fitting

C₂×C₄×C₈×C₂₃₃

C₂

socle

C₂²×C₂₃₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃₃

C₂²⋊C₄

cosocle

C₄²×C₂₃₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₃₃

hypocenter

C₂₃₃

#1493

LCS

G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #28416167	C₂	semidirect
75. C₂₃₃	#1493	semidirect

H ≅ G/N

type

3. #28416167

C₂

semidirect

75. C₂₃₃

#1493

semidirect

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

subgroup	index
#28416992	2
#28416459	2
#28416167	2
C₂×C₄×C₈×C₂₃₃	2
#1493	233

subgroup

index

C₂×C₄×C₈×C₂₃₃

233

Subgroups — #44450266

Conjugacy classes of subgroups

Complements

Maximal subgroups