Group #33922 — subgroups

Subgroups — #33922

Order 256.

244 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 99 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	#3339	C₂	1	2	yes	maximal, complemented
2	#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	#2234	C₂	1	2	yes	maximal, complemented
4	#2232	C₂	1	2	yes	maximal, complemented
5	#2228	C₂	1	2	yes	maximal, complemented
6	#2218	C₂	1	2	yes	maximal, complemented
7	C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#2169	C₂	1	1	yes	maximal, characteristic, complemented
9	#2159	C₂	1	2	yes	maximal, complemented
10	#553	—	2	2	no	—
11	C₂×D₁₆	C₂²	1	1	yes	characteristic
12	C₂×M₃₂	C₂²	1	2	yes	—
13	#499	C₂²	1	2	yes	—
14	#482	—	2	2	no	—
15	#481	—	2	2	no	—
16	#478	—	2	2	no	—
17	#466	C₂²	1	2	yes	—
18	#466	C₂²	1	2	yes	—
19	C₄×D₈	C₂²	1	2	yes	—
20	C₈×D₄	C₂²	1	1	yes	characteristic
21	#427	C₂²	1	2	yes	—
22	C₂×#107	C₂²	1	2	yes	—
23	#367	C₂²	1	2	yes	—
24	#367	C₂²	1	2	yes	—
25	#366	C₂²	1	1	yes	characteristic
26	#366	C₂²	1	2	yes	—
27	#366	C₂²	1	1	yes	characteristic
28	#366	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	#357	C₂²	1	2	yes	—
31	#357	C₂²	1	2	yes	—
32	#357	C₂²	1	2	yes	—
33	C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×D₈	C₂³	1	2	yes	—
36	C₂²×C₈	C₂³	1	2	yes	—
37	#126	—	4	2	no	—
38	#125	—	2	2	no	—
39	#123	—	4	2	no	—
40	#121	—	2	2	no	—
41	#121	—	2	2	no	—
42	#121	—	4	2	no	—
43	C₄×D₄	—	2	2	no	—
44	C₄×D₄	—	4	2	no	—
45	C₄×D₄	D₄	1	1	yes	characteristic
46	#117	—	2	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	D₁₆	—	2	2	no	—
49	M₃₂	—	2	4	no	—
50	C₂×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₆	C₂³	1	2	yes	—
52	C₂×C₁₆	C₂³	1	1	yes	characteristic
53	#107	—	2	2	no	—
54	#107	C₂³	1	2	yes	—
55	#107	C₂³	1	2	yes	—
56	#107	C₂³	1	2	yes	—
57	#106	—	2	2	no	—
58	#106	—	2	2	no	—
59	C₄⋊C₈	D₄	1	1	yes	characteristic
60	#102	—	2	2	no	—
61	#102	—	2	2	no	—
62	#102	—	2	2	no	—
63	C₂²⋊C₈	D₄	1	2	yes	—
64	C₄×C₈	C₂³	1	1	yes	characteristic
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	4	2	no	—
67	C₂×D₄	—	4	2	no	—
68	C₂×D₄	D₈	1	1	yes	characteristic
69	C₂²×C₄	—	4	2	no	—
70	C₂²×C₄	—	4	2	no	—
71	C₂²×C₄	C₂×D₄	1	2	yes	—
72	D₈	—	2	2	no	—
73	D₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₈	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	4	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	D₈	1	1	yes	characteristic
86	C₂²⋊C₄	—	4	2	no	—
87	C₂²⋊C₄	—	4	2	no	—
88	C₂²⋊C₄	—	4	2	no	—
89	C₂²⋊C₄	—	4	2	no	—
90	C₂²⋊C₄	D₈	1	2	yes	—
91	C₄²	—	4	2	no	—
92	C₄²	C₂×D₄	1	1	yes	characteristic
93	C₁₆	—	2	2	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	#41	1	2	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₂×D₈	1	2	yes	—
98	D₄	—	4	1	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	4	no	—
102	D₄	—	16	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	16	2	no	—
109	C₂×C₄	—	16	2	no	—
110	C₂×C₄	C₂×D₈	1	2	yes	—
111	C₂×C₄	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	2	1	no	—
116	C₈	C₂×#41	1	1	yes	characteristic
117	C₈	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	—	8	2	no	—
124	C₂²	C₂²×D₈	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	8	2	no	—
129	C₄	—	8	2	no	—
130	C₄	—	8	2	no	—
131	C₄	#547	1	1	yes	characteristic
132	C₄	#578	1	1	yes	characteristic
133	C₂	—	4	2	no	—
134	C₂	—	16	2	no	—
135	C₂	#3334	1	1	yes	characteristic
136	C₂	#3414	1	1	yes	characteristic
137	C₂	#3415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3339

C₂

yes

maximal, complemented

#2253

C₂

yes

maximal, characteristic, complemented

#2234

C₂

yes

maximal, complemented

#2232

C₂

yes

maximal, complemented

#2228

C₂

yes

maximal, complemented

#2218

C₂

yes

maximal, complemented

C₄×D₁₆

C₂

yes

maximal, characteristic, complemented

#2169

C₂

yes

maximal, characteristic, complemented

#2159

C₂

yes

maximal, complemented

#553

—

C₂×D₁₆

C₂²

yes

characteristic

C₂×M₃₂

C₂²

yes

—

#499

C₂²

yes

—

—

—

—

C₂²

yes

—

#466

C₂²

yes

—

C₄×D₈

C₂²

yes

—

C₈×D₄

C₂²

yes

characteristic

#427

C₂²

yes

—

C₂×#107

C₂²

yes

—

#367

C₂²

yes

—

#367

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

—

#366

C₂²

yes

characteristic

#366

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

characteristic

#357

C₂²

yes

—

#357

C₂²

yes

—

#357

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

characteristic

C₂×D₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

—

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₄

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

D₁₆

—

M₃₂

—

C₂×C₁₆

C₂³

yes

characteristic

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

characteristic

#107

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#107

C₂³

yes

—

#106

—

#106

—

C₄⋊C₈

D₄

yes

—

—

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

#41

yes

—

C₂³

—

C₂³

C₂×D₈

yes

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×D₈

yes

—

111

C₂×C₄

C₂×D₈

yes

characteristic

112

C₂×C₄

C₂×D₈

yes

characteristic

113

C₂×C₄

C₂²×D₄

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

C₂×#41

yes

characteristic

117

C₈

2₋¹⁺⁴

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂²×D₈

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#547

yes

characteristic

132

C₄

#578

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#3334

yes

characteristic

136

C₂

#3414

yes

characteristic

137

C₂

#3415

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #3339	C₂	semidirect
2. #2253	C₂	semidirect
3. #2234	C₂	semidirect
4. #2232	C₂	semidirect
5. #2228	C₂	semidirect
6. #2218	C₂	semidirect
7. C₄×D₁₆	C₂	semidirect
8. #2169	C₂	semidirect
9. #2159	C₂	semidirect

H ≅ G/N

type

1. #3339

C₂

semidirect

2. #2253

C₂

semidirect

3. #2234

C₂

semidirect

4. #2232

C₂

semidirect

5. #2228

C₂

semidirect

6. #2218

C₂

semidirect

7. C₄×D₁₆

C₂

semidirect

8. #2169

C₂

semidirect

9. #2159

C₂

semidirect

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

subgroup	index
#3339	2
#2253	2
#2234	2
#2232	2
#2228	2
#2218	2
C₄×D₁₆	2
#2169	2
#2159	2

subgroup

index

C₄×D₁₆

Subgroups — #33922

Conjugacy classes of subgroups

Complements

Maximal subgroups