Group #33428 — subgroups

Subgroups — #33428

Order 256.

264 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₄×#131	C₂	1	1	yes	maximal, characteristic
2	#2166	C₂	1	4	yes	maximal, complemented
3	#2165	C₂	1	4	yes	maximal, complemented
4	C₂×#364	C₂	1	2	yes	maximal, complemented
5	#2115	C₂	1	2	yes	maximal
6	#2110	C₂	1	2	yes	maximal, complemented
7	C₂×#131	C₄	1	1	yes	characteristic
8	C₂×#131	C₂²	1	1	yes	characteristic
9	C₄×#41	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	4	yes	—
11	#405	C₄	1	1	yes	characteristic
12	#405	C₄	1	2	yes	—
13	#405	C₂²	1	2	yes	—
14	#405	C₂²	1	1	yes	characteristic
15	C₄×M₁₆	C₄	1	1	yes	characteristic
16	C₄×M₁₆	C₂²	1	2	yes	—
17	C₂×C₄×C₈	C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#364	C₂²	1	4	yes	—
20	#364	C₂²	1	4	yes	—
21	#349	C₂²	1	8	yes	—
22	#347	C₂²	1	8	yes	—
23	C₂×#41	C₂×C₄	1	1	yes	characteristic
24	#131	—	2	4	no	—
25	#131	—	2	2	no	—
26	#131	—	2	2	no	—
27	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆	C₂×C₄	1	2	yes	—
29	C₂×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆	C₂³	1	2	yes	—
31	C₂²×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	1	yes	characteristic
35	C₄×Q₈	—	2	1	no	—
36	C₄×D₄	—	2	2	no	—
37	C₄×D₄	—	2	1	no	—
38	#117	C₂×C₄	1	2	yes	—
39	#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	2	yes	—
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	M₃₂	—	2	4	no	—
43	M₃₂	C₂³	1	8	yes	—
44	C₂×C₁₆	—	2	4	no	—
45	#97	C₂×C₄	1	2	yes	—
46	#97	C₂×C₄	1	2	yes	—
47	#97	C₂³	1	2	yes	—
48	C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈	C₂×C₄	1	2	yes	—
50	C₄×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₈	C₂×C₄	1	2	yes	—
52	C₄×C₈	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈	C₂³	1	1	yes	characteristic
54	C₄×C₈	C₂³	1	1	yes	characteristic
55	#41	—	2	2	no	—
56	#41	—	2	2	no	—
57	C₂×Q₈	M₁₆	1	1	yes	characteristic
58	C₂×D₄	M₁₆	1	2	yes	—
59	C₂×D₄	M₁₆	1	1	yes	characteristic
60	C₂²×C₄	—	2	2	no	—
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	C₂²×C₄	1	2	yes	—
63	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	2	4	no	—
67	M₁₆	—	2	4	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	M₁₆	1	4	yes	—
72	C₂×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈	#41	1	2	yes	—
78	C₂×C₈	#41	1	2	yes	—
79	C₂×C₈	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	2	1	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	2	1	no	—
86	C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₄²	C₂²×C₄	1	2	yes	—
88	C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₁₆	—	2	8	no	—
90	C₂³	C₂×M₁₆	1	2	yes	—
91	C₂³	C₂×M₁₆	1	1	yes	characteristic
92	Q₈	—	2	2	no	—
93	D₄	—	2	4	no	—
94	D₄	—	2	2	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	#117	1	2	yes	—
103	C₂×C₄	C₂×M₁₆	1	2	yes	—
104	C₂×C₄	C₂×M₁₆	1	2	yes	—
105	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	2	no	—
109	C₈	#117	1	1	yes	characteristic
110	C₈	#117	1	1	yes	characteristic
111	C₈	C₂×M₁₆	1	2	yes	—
112	C₈	C₂×M₁₆	1	2	yes	—
113	C₈	C₂×#41	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	4	1	no	—
116	C₂²	—	4	2	no	—
117	C₂²	#432	1	2	yes	—
118	C₂²	C₂²×M₁₆	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	2	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	1	no	—
123	C₄	#432	1	2	yes	—
124	C₄	C₂×#117	1	1	yes	characteristic
125	C₄	C₂²×M₁₆	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	C₂×#432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#131

C₂

yes

maximal, characteristic

#2166

C₂

yes

maximal, complemented

#2165

C₂

yes

maximal, complemented

C₂×#364

C₂

yes

maximal, complemented

#2115

C₂

yes

maximal

#2110

C₂

yes

maximal, complemented

C₂×#131

C₄

yes

characteristic

C₂×#131

C₂²

yes

characteristic

C₄×#41

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

#405

C₄

yes

characteristic

#405

C₄

yes

—

#405

C₂²

yes

—

#405

C₂²

yes

characteristic

C₄×M₁₆

C₄

yes

characteristic

C₄×M₁₆

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#364

C₂²

yes

—

#364

C₂²

yes

—

#349

C₂²

yes

—

#347

C₂²

yes

—

C₂×#41

C₂×C₄

yes

—

—

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

characteristic

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

C₂×C₄

yes

—

#117

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

C₂³

yes

—

C₂×C₁₆

—

#97

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

#97

C₂³

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

#41

—

#41

—

C₂×Q₈

M₁₆

yes

characteristic

C₂×D₄

M₁₆

yes

—

C₂×D₄

M₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

characteristic

C₁₆

—

C₂³

C₂×M₁₆

yes

—

C₂³

C₂×M₁₆

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#117

yes

—

103

C₂×C₄

C₂×M₁₆

yes

—

104

C₂×C₄

C₂×M₁₆

yes

—

105

C₂×C₄

C₂×M₁₆

yes

characteristic

106

C₂×C₄

C₂³×C₄

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

#117

yes

characteristic

110

C₈

#117

yes

characteristic

111

C₈

C₂×M₁₆

yes

—

112

C₈

C₂×M₁₆

yes

—

113

C₈

C₂×#41

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#432

yes

—

118

C₂²

C₂²×M₁₆

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#432

yes

—

124

C₄

C₂×#117

yes

characteristic

125

C₄

C₂²×M₁₆

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

C₂×#432

yes

characteristic

	subgroup	G/H
center	C₈	#117
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂×#432
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#117

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂×C₈

C₂⁴

socle

C₂

C₂×#432

prime radical

C₂×#41

C₂×C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ #405 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2166	C₂	semidirect
3. #2165	C₂	semidirect
4. C₂×#364	C₂	semidirect
6. #2110	C₂	semidirect

H ≅ G/N

type

2. #2166

C₂

semidirect

3. #2165

C₂

semidirect

4. C₂×#364

C₂

semidirect

6. #2110

C₂

semidirect

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

subgroup	index
C₄×#131	2
#2166	2
#2165	2
C₂×#364	2
#2115	2
#2110	2

subgroup

index

Subgroups — #33428

Conjugacy classes of subgroups

Complements

Maximal subgroups