Group #44450494 — subgroups

Subgroups — #44450494

Order 29824.

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

normal subgroups only

This group has 240 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	C₂×#21418535	C₂	1	1	yes	maximal, characteristic
2	#28416459	C₂	1	1	yes	maximal, characteristic
3	#28416182	C₂	1	4	yes	maximal, complemented
4	#431×C₂₃₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆₆	—	2	1	no	—
6	C₂×#16836578	—	2	1	no	—
7	#21418535	—	2	1	no	—
8	#21418535	—	2	1	no	—
9	#21418535	C₄	1	2	yes	—
10	#21418535	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₂₃₃	—	2	1	no	—
12	C₂×C₄⋊Dic₂₃₃	C₄	1	2	yes	—
13	C₄⋊#14272338	—	2	1	no	—
14	C₄⋊#14272338	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₃₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₂₃₃	—	2	1	no	—
18	C₄×C₈×C₂₃₃	—	2	1	no	—
19	C₂×Dic₄₆₆	—	2	2	no	—
20	C₂×Dic₄₆₆	—	2	2	no	—
21	C₂×Dic₄₆₆	—	4	4	no	—
22	C₂²×Dic₂₃₃	—	2	2	no	—
23	#16836578	—	4	2	no	—
24	C₂×#14272338	—	2	2	no	—
25	C₄⋊Dic₂₃₃	—	2	2	no	—
26	C₄⋊Dic₂₃₃	—	2	2	no	—
27	C₄⋊Dic₂₃₃	—	2	2	no	—
28	C₄⋊Dic₂₃₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃₃	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃₃	—	2	2	no	—
33	C₄²×C₂₃₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
35	C₄²×C₂₃₃	D₄	1	1	yes	characteristic
36	C₄²×C₂₃₃	C₂³	1	1	yes	characteristic
37	Dic₄₆₆	—	4	4	no	—
38	Dic₄₆₆	—	4	4	no	—
39	C₂×Dic₂₃₃	—	2	4	no	—
40	C₂×Dic₂₃₃	—	4	4	no	—
41	#14272338	—	4	2	no	—
42	C₂³×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃₃	—	2	2	no	—
44	C₂×C₄×C₂₃₃	—	2	2	no	—
45	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₃₃	SD₁₆	1	2	yes	—
49	C₂×C₄×C₂₃₃	Q₁₆	1	2	yes	—
50	C₂×C₄×C₂₃₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃₃	C₂×D₄	1	1	yes	characteristic
53	C₈×C₂₃₃	—	4	2	no	—
54	C₂²×C₂₃₃	—	2	1	no	—
55	C₂²×C₂₃₃	—	2	1	no	—
56	C₂²×C₂₃₃	#103	1	2	yes	—
57	C₂²×C₂₃₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₃₃	—	2	1	no	—
59	C₄×C₂₃₃	—	2	1	no	—
60	C₄×C₂₃₃	#101	1	2	yes	—
61	C₄×C₂₃₃	#103	1	2	yes	—
62	C₄×C₂₃₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₂₃₃	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₂₃₃	AGL(1,ℤ/8)	1	2	yes	—
65	Dic₂₃₃	—	4	4	no	—
66	C₂×C₂₃₃	—	2	2	no	—
67	C₂×C₂₃₃	C₂×#101	1	1	yes	characteristic
68	C₂×C₂₃₃	C₂×#103	1	1	yes	characteristic
69	C₂×C₂₃₃	#418	1	1	yes	characteristic
70	C₂₃₃	#1549	1	1	yes	characteristic, complemented, Sylow
71	#1549	—	233	1	no	maximal, Sylow
72	C₂×C₄⋊Q₈	—	233	1	no	—
73	#431	D₂₃₃	1	1	yes	characteristic
74	#423	—	233	1	no	—
75	#332	—	233	4	no	—
76	C₂²×Q₈	—	466	1	no	—
77	C₂×M₁₆	—	466	1	no	—
78	C₄⋊Q₈	—	466	1	no	—
79	C₄⋊Q₈	—	466	1	no	—
80	C₄⋊Q₈	—	233	2	no	—
81	C₄⋊Q₈	—	233	2	no	—
82	C₂×C₄⋊C₄	—	466	1	no	—
83	C₂×C₄⋊C₄	—	233	2	no	—
84	C₂×C₄²	C₂×D₂₃₃	1	1	yes	characteristic
85	C₄⋊C₈	—	466	1	no	—
86	C₄⋊C₈	—	233	2	no	—
87	C₄⋊C₈	C₂×D₂₃₃	1	2	yes	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₂×Q₈	—	932	4	no	—
91	C₂×Q₈	—	466	2	no	—
92	C₂×Q₈	—	466	2	no	—
93	C₂²×C₄	—	466	2	no	—
94	C₂²×C₄	C₄×D₂₃₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
96	C₂²×C₄	D₉₃₂	1	1	yes	characteristic
97	M₁₆	—	932	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	466	2	no	—
100	C₄⋊C₄	—	466	2	no	—
101	C₄⋊C₄	—	466	2	no	—
102	C₄⋊C₄	—	466	2	no	—
103	C₄⋊C₄	—	233	4	no	—
104	C₄²	C₄×D₂₃₃	1	1	yes	characteristic
105	C₄²	C₂²⋊D₂₃₃	1	1	yes	characteristic
106	C₄²	D₉₃₂	1	1	yes	characteristic
107	C₄²	C₂²×D₂₃₃	1	1	yes	characteristic
108	C₂³	#16836571	1	1	yes	characteristic
109	Q₈	—	932	4	no	—
110	Q₈	—	932	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	932	4	no	—
114	C₂×C₄	—	466	4	no	—
115	C₂×C₄	#16836571	1	1	yes	characteristic
116	C₂×C₄	#16836571	1	1	yes	characteristic
117	C₂×C₄	#16836571	1	1	yes	characteristic
118	C₂×C₄	#16836583	1	2	yes	—
119	C₂×C₄	Dic₉₃₂	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₂₃₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
122	C₂×C₄	C₂×D₉₃₂	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	932	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#21418460	1	2	yes	—
128	C₂²	C₂×#16836571	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	932	4	no	—
132	C₄	#21418453	1	2	yes	—
133	C₄	#21418460	1	2	yes	—
134	C₄	C₂×#16836583	1	1	yes	characteristic
135	C₄	C₂×Dic₉₃₂	1	1	yes	characteristic
136	C₄	#21418560	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#21418453	1	1	yes	characteristic
139	C₂	C₂×#21418460	1	1	yes	characteristic
140	C₂	#28416439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21418535

C₂

yes

maximal, characteristic

#28416459

C₂

yes

maximal, characteristic

#28416182

C₂

yes

maximal, complemented

#431×C₂₃₃

C₂

yes

maximal, characteristic

C₂²×Dic₄₆₆

—

—

—

—

C₄

yes

—

#21418535

C₂²

yes

—

C₂×C₄⋊Dic₂₃₃

—

C₂×C₄⋊Dic₂₃₃

C₄

yes

—

C₄⋊#14272338

—

C₄⋊#14272338

C₂²

yes

—

C₂×C₄²×C₂₃₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₃

C₂²

yes

—

C₂²⋊C₈×C₂₃₃

—

C₄×C₈×C₂₃₃

—

C₂×Dic₄₆₆

—

C₂×Dic₄₆₆

—

C₂×Dic₄₆₆

—

C₂²×Dic₂₃₃

—

#16836578

—

C₂×#14272338

—

C₄⋊Dic₂₃₃

—

C₄⋊Dic₂₃₃

—

C₄⋊Dic₂₃₃

—

C₄⋊Dic₂₃₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

C₂²×C₄×C₂₃₃

D₄

yes

characteristic

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

#101

yes

—

C₄×C₂₃₃

#103

yes

—

C₄×C₂₃₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃₃

C₂×Q₁₆

yes

characteristic

C₄×C₂₃₃

AGL(1,ℤ/8)

yes

—

Dic₂₃₃

—

C₂×C₂₃₃

—

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂×C₂₃₃

yes

C₂₃₃

yes

characteristic, complemented, Sylow

#1549

—

233

maximal, Sylow

C₂×C₄⋊Q₈

—

233

—

#431

D₂₃₃

yes

characteristic

#423

—

233

—

#332

—

233

—

C₂²×Q₈

—

466

—

C₂×M₁₆

—

466

—

C₄⋊Q₈

—

466

—

C₄⋊Q₈

—

466

—

C₄⋊Q₈

—

233

—

C₄⋊Q₈

—

233

—

C₂×C₄⋊C₄

—

466

—

C₂×C₄⋊C₄

—

233

—

C₂×C₄²

C₂×D₂₃₃

yes

characteristic

C₄⋊C₈

—

466

—

C₄⋊C₈

—

233

—

C₄⋊C₈

C₂×D₂₃₃

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

932

—

C₂×Q₈

—

466

—

C₂×Q₈

—

466

—

C₂²×C₄

—

466

—

C₂²×C₄

C₄×D₂₃₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃₃

yes

characteristic

C₂²×C₄

D₉₃₂

yes

characteristic

M₁₆

—

932

—

C₂×C₈

—

C₂×C₈

—

466

—

100

C₄⋊C₄

—

466

—

101

C₄⋊C₄

—

466

—

102

C₄⋊C₄

—

466

—

103

C₄⋊C₄

—

233

—

104

C₄²

C₄×D₂₃₃

yes

characteristic

105

C₄²

C₂²⋊D₂₃₃

yes

characteristic

106

C₄²

D₉₃₂

yes

characteristic

107

C₄²

C₂²×D₂₃₃

yes

characteristic

108

C₂³

#16836571

yes

characteristic

109

Q₈

—

932

—

110

Q₈

—

932

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

932

—

114

C₂×C₄

—

466

—

115

C₂×C₄

#16836571

yes

characteristic

116

C₂×C₄

#16836571

yes

characteristic

117

C₂×C₄

#16836571

yes

characteristic

118

C₂×C₄

#16836583

yes

—

119

C₂×C₄

Dic₉₃₂

yes

—

120

C₂×C₄

C₂×C₄×D₂₃₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₂₃₃

yes

characteristic

122

C₂×C₄

C₂×D₉₃₂

yes

characteristic

123

C₈

—

124

C₈

—

932

—

125

C₂²

—

126

C₂²

—

127

C₂²

#21418460

yes

—

128

C₂²

C₂×#16836571

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

932

—

132

C₄

#21418453

yes

—

133

C₄

#21418460

yes

—

134

C₄

C₂×#16836583

yes

characteristic

135

C₄

C₂×Dic₉₃₂

yes

characteristic

136

C₄

#21418560

yes

—

137

C₂

—

138

C₂

C₂×#21418453

yes

characteristic

139

C₂

C₂×#21418460

yes

characteristic

140

C₂

#28416439

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#16836571
commutator	C₂×C₄×C₂₃₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃₃
Fitting	#431×C₂₃₃	C₂
socle	C₂²×C₂₃₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃₃	C₂²⋊C₄
cosocle	C₄²×C₂₃₃	C₂³
hypercenter	#431	D₂₃₃
hypocenter	C₂₃₃	#1549
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#16836571

commutator

C₂×C₄×C₂₃₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃₃

Fitting

#431×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 ◁ #431

N	H ≅ G/N	type
3. #28416182	C₂	semidirect
70. C₂₃₃	#1549	semidirect

H ≅ G/N

type

3. #28416182

C₂

semidirect

70. C₂₃₃

#1549

semidirect

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

subgroup	index
C₂×#21418535	2
#28416459	2
#28416182	2
#431×C₂₃₃	2
#1549	233

subgroup

index

#431×C₂₃₃

233

Subgroups — #44450494

Conjugacy classes of subgroups

Complements

Maximal subgroups