Group #33400 — subgroups

Subgroups — C₂×#2153 #33400

Order 256.

349 conjugacy classes of subgroups; only the 263 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 263 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	C₂²×M₃₂	C₂	1	2	yes	maximal
2	C₂×#405	C₂	1	1	yes	maximal, characteristic
3	#2153	C₂	1	8	yes	maximal, direct product factor
4	C₂×#364	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	C₄	1	1	yes	characteristic
8	C₂×M₃₂	—	2	4	no	—
9	C₂×M₃₂	—	2	2	no	—
10	C₂×M₃₂	C₂²	1	4	yes	—
11	C₂×M₃₂	C₂²	1	8	yes	—
12	C₂²×C₁₆	—	2	2	no	—
13	#405	C₄	1	4	yes	—
14	#405	C₂²	1	4	yes	—
15	C₂×#97	C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	2	yes	—
17	#364	C₂²	1	16	yes	complemented
18	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆	C₈	1	4	yes	—
20	C₂×M₁₆	C₈	1	4	yes	—
21	C₂×M₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	D₄	1	2	yes	—
26	C₂²×C₈	D₄	1	2	yes	—
27	C₂²×C₈	Q₈	1	2	yes	—
28	C₂²×C₈	Q₈	1	2	yes	—
29	C₂²×C₈	C₂³	1	2	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	#117	—	2	2	no	—
32	#117	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²	C₂×C₄	1	2	yes	—
36	M₃₂	—	2	8	no	—
37	M₃₂	—	2	4	no	—
38	M₃₂	C₂³	1	8	yes	—
39	C₂×C₁₆	—	2	8	no	—
40	C₂×C₁₆	—	2	4	no	—
41	#97	C₂×C₄	1	8	yes	—
42	C₄×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₈	C₂³	1	4	yes	—
44	C₂⁴	M₁₆	1	1	yes	characteristic
45	C₂²×C₄	—	2	2	no	—
46	C₂²×C₄	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄	C₂×C₈	1	2	yes	—
49	C₂²×C₄	C₂×C₈	1	2	yes	—
50	C₂²×C₄	M₁₆	1	1	yes	characteristic
51	C₂²×C₄	M₁₆	1	2	yes	—
52	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂²×C₄	1	2	yes	—
54	M₁₆	C₂×C₈	1	16	yes	—
55	C₂×C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₈	C₄⋊C₄	1	4	yes	—
57	C₂×C₈	C₄⋊C₄	1	2	yes	—
58	C₂×C₈	C₄⋊C₄	1	4	yes	—
59	C₂×C₈	C₂×C₈	1	4	yes	—
60	C₂×C₈	C₂×C₈	1	4	yes	—
61	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₈	C₂×D₄	1	4	yes	—
65	C₂×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈	C₂×Q₈	1	4	yes	—
68	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄	—	2	4	no	—
72	C₂²⋊C₄	—	2	4	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂²×C₄	1	4	yes	—
75	C₁₆	—	2	8	no	—
76	C₂³	—	4	2	no	—
77	C₂³	C₄⋊C₈	1	2	yes	—
78	C₂³	C₄⋊C₈	1	2	yes	—
79	C₂³	C₂×M₁₆	1	1	yes	characteristic
80	C₂³	C₂×M₁₆	1	2	yes	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	C₄⋊C₈	1	2	yes	—
84	C₂×C₄	C₄⋊C₈	1	4	yes	—
85	C₂×C₄	C₄⋊C₈	1	4	yes	—
86	C₂×C₄	C₄⋊C₈	1	2	yes	—
87	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₈	1	4	yes	—
93	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄	C₂×M₁₆	1	2	yes	—
95	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
96	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₈	C₂×C₄⋊C₄	1	4	yes	—
98	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₈	C₂²×C₈	1	8	yes	—
100	C₈	C₂²×D₄	1	1	yes	characteristic
101	C₈	C₂²×Q₈	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	—	2	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₂²	C₂×C₄⋊C₈	1	4	yes	—
108	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
109	C₂²	C₂²×M₁₆	1	1	yes	characteristic
110	C₄	—	4	4	no	—
111	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄	C₂×C₄⋊C₈	1	4	yes	—
113	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄	C₂³×C₈	1	1	yes	characteristic
116	C₂	—	2	1	no	—
117	C₂	—	2	1	no	—
118	C₂	—	2	4	no	—
119	C₂	#2153	1	2	yes	direct product factor
120	C₂	C₂²×C₄⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₃₂

C₂

yes

maximal

C₂×#405

C₂

yes

maximal, characteristic

#2153

C₂

yes

maximal, direct product factor

C₂×#364

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₄

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

C₂²

yes

—

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

—

#405

C₄

yes

—

#405

C₂²

yes

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

—

#364

C₂²

yes

complemented

C₂³×C₄

C₂×C₄

yes

characteristic

C₂×M₁₆

C₈

yes

—

C₂×M₁₆

C₈

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

Q₈

yes

—

C₂²×C₈

Q₈

yes

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

—

#117

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×C₄

yes

—

M₃₂

—

M₃₂

—

M₃₂

C₂³

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

M₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

M₁₆

yes

characteristic

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

M₁₆

C₂×C₈

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

—

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₂³

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₄⋊C₈

yes

—

C₂³

C₂×M₁₆

yes

characteristic

C₂³

C₂×M₁₆

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₈

C₂×C₄⋊C₄

yes

characteristic

C₈

C₂×C₄⋊C₄

yes

—

C₈

C₂×C₄⋊C₄

yes

characteristic

C₈

C₂²×C₈

yes

—

100

C₈

C₂²×D₄

yes

characteristic

101

C₈

C₂²×Q₈

yes

characteristic

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂×C₄⋊C₈

yes

characteristic

107

C₂²

C₂×C₄⋊C₈

yes

—

108

C₂²

C₂×C₄⋊C₈

yes

characteristic

109

C₂²

C₂²×M₁₆

yes

characteristic

110

C₄

—

111

C₄

C₂×C₄⋊C₈

yes

characteristic

112

C₄

C₂×C₄⋊C₈

yes

—

113

C₄

C₂×C₄⋊C₈

yes

characteristic

114

C₄

C₂²×C₄⋊C₄

yes

characteristic

115

C₄

C₂³×C₈

yes

characteristic

116

C₂

—

117

C₂

—

118

C₂

—

119

C₂

#2153

yes

direct product factor

120

C₂

C₂²×C₄⋊C₈

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂⁴	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₄

commutator

C₄

C₂³×C₈

Frattini, cosocle

C₂×C₈

C₂⁴

socle

C₂²

C₂×C₄⋊C₈

prime radical

C₂⁴

M₁₆

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #2153	C₂	direct (N × H)
4. C₂×#364	C₂	semidirect
17. #364	C₂²	semidirect
119. C₂	#2153	direct (N × H)

H ≅ G/N

type

3. #2153

C₂

direct (N × H)

4. C₂×#364

C₂

semidirect

17. #364

C₂²

semidirect

119. C₂

#2153

direct (N × H)

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

subgroup	index
C₂²×M₃₂	2
C₂×#405	2
#2153	2
C₂×#364	2

subgroup

index

C₂²×M₃₂

C₂×#405

#2153

C₂×#364

Subgroups — C₂×#2153 #33400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#2153 #33400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2153 #33400