Group #44450500 — subgroups

Subgroups — #44450500

Order 29824.

216 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 97 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	#28416459	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#14272338	C₂	1	1	yes	maximal, characteristic
3	#28416184	C₂	1	2	yes	maximal, complemented
4	#28416181	C₂	1	2	yes	maximal, complemented
5	#533×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#16836578	—	2	1	no	—
7	C₂²×#14272338	—	2	1	no	—
8	C₄⋊#14272338	—	2	1	no	—
9	C₄⋊#14272338	—	2	1	no	—
10	C₄⋊#14272338	C₂²	1	2	yes	—
11	C₄⋊#14272338	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₂₃₃	C₂²	1	1	yes	characteristic
14	#125×C₂₃₃	—	2	1	no	—
15	C₂²⋊Q₈×C₂₃₃	—	2	1	no	—
16	C₂²⋊Q₈×C₂₃₃	C₄	1	2	yes	—
17	C₄×Q₈×C₂₃₃	—	2	1	no	—
18	C₄×Q₈×C₂₃₃	C₄	1	2	yes	—
19	#117×C₂₃₃	—	2	1	no	—
20	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
21	#16836578	—	4	2	no	—
22	C₂×#14272338	—	2	2	no	—
23	C₂×#14272338	—	2	2	no	—
24	C₂×#14272338	—	4	2	no	—
25	C₂×Q₈×C₂₃₃	—	2	1	no	—
26	C₂×Q₈×C₂₃₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₂₃₃	—	2	2	no	—
31	C₄⋊C₄×C₂₃₃	—	2	2	no	—
32	C₄⋊C₄×C₂₃₃	—	2	1	no	—
33	C₄⋊C₄×C₂₃₃	—	2	1	no	—
34	C₄⋊C₄×C₂₃₃	—	2	1	no	—
35	C₄⋊C₄×C₂₃₃	C₂×C₄	1	2	yes	—
36	C₂²⋊C₄×C₂₃₃	—	2	2	no	—
37	C₄²×C₂₃₃	—	2	2	no	—
38	C₄²×C₂₃₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
40	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
41	C₄²×C₂₃₃	C₂³	1	1	yes	characteristic
42	#14272338	—	4	2	no	—
43	#14272338	—	4	2	no	—
44	C₂³×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
45	Q₈×C₂₃₃	—	4	2	no	—
46	Q₈×C₂₃₃	—	4	1	no	—
47	Q₈×C₂₃₃	—	4	1	no	—
48	C₂×C₄×C₂₃₃	—	2	1	no	—
49	C₂×C₄×C₂₃₃	—	2	1	no	—
50	C₂×C₄×C₂₃₃	—	2	2	no	—
51	C₂×C₄×C₂₃₃	—	2	2	no	—
52	C₂×C₄×C₂₃₃	—	4	1	no	—
53	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₃₃	SD₁₆	1	2	yes	—
57	C₂×C₄×C₂₃₃	Q₁₆	1	2	yes	—
58	C₂×C₄×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₂₃₃	—	2	1	no	—
62	C₂²×C₂₃₃	—	2	1	no	—
63	C₂²×C₂₃₃	#103	1	2	yes	—
64	C₂²×C₂₃₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₃₃	—	2	1	no	—
66	C₄×C₂₃₃	—	2	1	no	—
67	C₄×C₂₃₃	—	2	1	no	—
68	C₄×C₂₃₃	—	4	2	no	—
69	C₄×C₂₃₃	—	4	2	no	—
70	C₄×C₂₃₃	#103	1	2	yes	—
71	C₄×C₂₃₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₂₃₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₂₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₂₃₃	#137	1	1	yes	characteristic
75	C₂×C₂₃₃	—	2	2	no	—
76	C₂×C₂₃₃	#413	1	1	yes	characteristic
77	C₂×C₂₃₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₂₃₃	#417	1	1	yes	characteristic
79	C₂₃₃	#1550	1	1	yes	characteristic, complemented, Sylow
80	#1550	—	233	1	no	maximal, Sylow
81	#533	D₂₃₃	1	1	yes	characteristic
82	#423	—	233	1	no	—
83	C₂×C₄⋊C₈	—	233	1	no	—
84	#333	—	233	2	no	—
85	#332	—	233	2	no	—
86	C₂×M₁₆	—	466	1	no	—
87	C₂²×C₈	—	466	1	no	—
88	C₄⋊Q₈	C₂×D₂₃₃	1	1	yes	characteristic
89	C₄⋊Q₈	C₂×D₂₃₃	1	1	yes	characteristic
90	#125	—	2	1	no	—
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	Dic₂₃₃	1	2	yes	—
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	Dic₂₃₃	1	2	yes	—
95	#117	—	2	1	no	—
96	C₂×C₄²	C₂×D₂₃₃	1	1	yes	characteristic
97	C₄⋊C₈	—	466	1	no	—
98	C₄⋊C₈	—	466	1	no	—
99	C₄⋊C₈	—	233	2	no	—
100	C₄⋊C₈	—	233	2	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	C₂×Dic₂₃₃	1	2	yes	—
103	C₂²×C₄	C₂×Dic₂₃₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
106	M₁₆	—	932	2	no	—
107	C₂×C₈	—	932	2	no	—
108	C₂×C₈	—	466	2	no	—
109	C₂×C₈	—	466	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	C₂×Dic₂₃₃	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	C₂×Dic₂₃₃	1	1	yes	characteristic
119	C₄²	C₂²⋊D₂₃₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₂₃₃	1	1	yes	characteristic
121	C₄²	C₂²×D₂₃₃	1	1	yes	characteristic
122	C₂³	C₂²⋊Dic₂₃₃	1	1	yes	characteristic
123	Q₈	—	4	2	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	C₂²⋊Dic₂₃₃	1	1	yes	characteristic
132	C₂×C₄	C₂²⋊Dic₂₃₃	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₂₃₃	1	1	yes	characteristic
134	C₂×C₄	#16836581	1	2	yes	—
135	C₂×C₄	#16836584	1	2	yes	—
136	C₂×C₄	C₂²×Dic₂₃₃	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
139	C₈	—	932	2	no	—
140	C₈	—	932	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#21418459	1	2	yes	—
144	C₂²	C₂×C₂²⋊Dic₂₃₃	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	4	2	no	—
150	C₄	#21418459	1	2	yes	—
151	C₄	C₂×#16836581	1	1	yes	characteristic
152	C₄	C₂×#16836584	1	1	yes	characteristic
153	C₄	#21418560	1	1	yes	characteristic
154	C₄	#21418565	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#28416419	1	1	yes	characteristic
157	C₂	C₂×#21418459	1	1	yes	characteristic
158	C₂	#28416436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#28416459

C₂

yes

maximal, characteristic

C₂×C₄⋊#14272338

C₂

yes

maximal, characteristic

#28416184

C₂

yes

maximal, complemented

#28416181

C₂

yes

maximal, complemented

#533×C₂₃₃

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#14272338

C₂²

yes

—

C₄⋊Q₈×C₂₃₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₃

C₂²

yes

characteristic

#125×C₂₃₃

—

C₂²⋊Q₈×C₂₃₃

—

C₂²⋊Q₈×C₂₃₃

C₄

yes

—

C₄×Q₈×C₂₃₃

—

C₄×Q₈×C₂₃₃

C₄

yes

—

#117×C₂₃₃

—

C₂×C₄²×C₂₃₃

C₂²

yes

—

—

—

—

C₂×Q₈×C₂₃₃

—

C₂×Q₈×C₂₃₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

—

C₄⋊C₄×C₂₃₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃₃

—

C₄²×C₂₃₃

—

C₄²×C₂₃₃

C₂×C₄

yes

characteristic

C₄²×C₂₃₃

D₄

yes

characteristic

C₄²×C₂₃₃

D₄

yes

characteristic

C₄²×C₂₃₃

C₂³

yes

characteristic

#14272338

—

#14272338

—

C₂³×C₂₃₃

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃₃

—

Q₈×C₂₃₃

—

Q₈×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

—

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

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

#103

yes

—

C₂²×C₂₃₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

#103

yes

—

C₄×C₂₃₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃₃

C₂×Q₁₆

yes

C₄×C₂₃₃

yes

C₄×C₂₃₃

yes

C₂×C₂₃₃

—

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂₃₃

yes

characteristic, complemented, Sylow

#1550

—

233

maximal, Sylow

#533

D₂₃₃

yes

characteristic

#423

—

233

—

C₂×C₄⋊C₈

—

233

—

#333

—

233

—

#332

—

233

—

C₂×M₁₆

—

466

—

C₂²×C₈

—

466

—

C₄⋊Q₈

C₂×D₂₃₃

yes

characteristic

C₄⋊Q₈

C₂×D₂₃₃

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₂₃₃

yes

—

C₄×Q₈

—

C₄×Q₈

Dic₂₃₃

yes

—

#117

—

C₂×C₄²

C₂×D₂₃₃

yes

characteristic

C₄⋊C₈

—

466

—

C₄⋊C₈

—

466

—

C₄⋊C₈

—

233

—

100

C₄⋊C₈

—

233

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₂₃₃

yes

—

103

C₂²×C₄

C₂×Dic₂₃₃

yes

characteristic

104

C₂²×C₄

C₂²⋊D₂₃₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₂₃₃

yes

characteristic

106

M₁₆

—

932

—

107

C₂×C₈

—

932

—

108

C₂×C₈

—

466

—

109

C₂×C₈

—

466

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×Dic₂₃₃

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂×Dic₂₃₃

yes

characteristic

119

C₄²

C₂²⋊D₂₃₃

yes

characteristic

120

C₄²

C₂²⋊D₂₃₃

yes

characteristic

121

C₄²

C₂²×D₂₃₃

yes

characteristic

122

C₂³

C₂²⋊Dic₂₃₃

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊Dic₂₃₃

yes

characteristic

132

C₂×C₄

C₂²⋊Dic₂₃₃

yes

characteristic

133

C₂×C₄

C₂²⋊Dic₂₃₃

yes

characteristic

134

C₂×C₄

#16836581

yes

—

135

C₂×C₄

#16836584

yes

—

136

C₂×C₄

C₂²×Dic₂₃₃

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₂₃₃

yes

characteristic

138

C₂×C₄

C₂×C₂²⋊D₂₃₃

yes

characteristic

139

C₈

—

932

—

140

C₈

—

932

—

141

C₂²

—

142

C₂²

—

143

C₂²

#21418459

yes

—

144

C₂²

C₂×C₂²⋊Dic₂₃₃

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#21418459

yes

—

151

C₄

C₂×#16836581

yes

characteristic

152

C₄

C₂×#16836584

yes

characteristic

153

C₄

#21418560

yes

characteristic

154

C₄

#21418565

yes

characteristic

155

C₂

—

156

C₂

#28416419

yes

characteristic

157

C₂

C₂×#21418459

yes

characteristic

158

C₂

#28416436

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₃₃
commutator	C₂×C₄×C₂₃₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃₃
Fitting	#533×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃₃	C₂²⋊C₄
cosocle	C₄²×C₂₃₃	C₂³
hypercenter	#533	D₂₃₃
hypocenter	C₂₃₃	#1550
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₃₃

commutator

C₂×C₄×C₂₃₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃₃

Fitting

#533×C₂₃₃

C₂

socle

C₂²×C₂₃₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃₃

C₂²⋊C₄

cosocle

C₄²×C₂₃₃

C₂³

D₂₃₃

C₂₃₃

G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #28416184	C₂	semidirect
4. #28416181	C₂	semidirect
79. C₂₃₃	#1550	semidirect

H ≅ G/N

type

3. #28416184

C₂

semidirect

4. #28416181

C₂

semidirect

79. C₂₃₃

#1550

semidirect

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

subgroup	index
#28416459	2
C₂×C₄⋊#14272338	2
#28416184	2
#28416181	2
#533×C₂₃₃	2
#1550	233

subgroup

index

#533×C₂₃₃

233

Subgroups — #44450500

Conjugacy classes of subgroups

Complements

Maximal subgroups