Group #33493 — subgroups

Subgroups — #33493

Order 256.

286 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 181 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	#2931	C₂	1	1	yes	maximal, characteristic
2	#2169	C₂	1	4	yes	maximal, complemented
3	D₄×C₁₆	C₂	1	4	yes	maximal, complemented
4	#2162	C₂	1	2	yes	maximal, complemented
5	#2113	C₂	1	4	yes	maximal, complemented
6	C₂×#131	C₂²	1	2	yes	—
7	#532	C₄	1	1	yes	characteristic
8	C₂×M₃₂	C₂²	1	4	yes	—
9	C₂²×C₁₆	C₂²	1	4	yes	—
10	#436	C₄	1	4	yes	—
11	C₈×D₄	C₂²	1	4	yes	—
12	#423	C₄	1	1	yes	characteristic
13	#408	C₄	1	2	yes	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂²⋊C₁₆	C₂²	1	4	yes	—
18	C₂²⋊C₁₆	C₂²	1	8	yes	—
19	C₄×C₁₆	C₂²	1	4	yes	—
20	C₂×#41	C₂×C₄	1	2	yes	—
21	#131	—	4	2	no	—
22	#131	—	4	2	no	—
23	C₂×M₁₆	C₂×C₄	1	2	yes	—
24	C₂×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₈	C₂³	1	4	yes	—
28	C₄⋊Q₈	C₈	1	1	yes	characteristic
29	#127	C₈	1	1	yes	characteristic
30	#124	C₈	1	2	yes	—
31	#121	C₈	1	4	yes	—
32	C₄×D₄	C₂×C₄	1	4	yes	—
33	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
34	M₃₂	—	4	4	no	—
35	C₂×C₁₆	—	4	4	no	—
36	C₂×C₁₆	C₂³	1	4	yes	—
37	C₂×C₁₆	C₂³	1	4	yes	—
38	C₄⋊C₈	C₂×C₄	1	4	yes	—
39	C₂²⋊C₈	C₂×C₄	1	8	yes	—
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	C₂³	1	2	yes	—
42	#41	—	4	2	no	—
43	#41	—	4	2	no	—
44	C₂×Q₈	C₂×C₈	1	2	yes	—
45	C₂×D₄	C₂×C₈	1	4	yes	—
46	C₂×D₄	C₂×C₈	1	4	yes	—
47	C₂×D₄	C₂×C₈	1	2	yes	—
48	C₂²×C₄	C₂×C₈	1	2	yes	—
49	C₂²×C₄	C₂²×C₄	1	4	yes	—
50	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
51	M₁₆	—	4	2	no	—
52	M₁₆	—	4	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	4	1	no	—
58	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₈	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄	C₂×C₈	1	4	yes	—
63	C₂²⋊C₄	C₂×C₈	1	8	yes	—
64	C₄²	C₂×C₈	1	1	yes	characteristic
65	C₄²	C₂×C₈	1	1	yes	characteristic
66	C₄²	C₂²×C₄	1	2	yes	—
67	C₁₆	—	2	4	no	—
68	C₁₆	—	2	4	no	—
69	C₂³	C₂²×C₈	1	4	yes	—
70	C₂³	C₂²×C₈	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	D₄	—	2	8	no	—
73	D₄	—	4	2	no	—
74	D₄	—	4	4	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄	C₂²×C₈	1	2	yes	—
82	C₂×C₄	C₂²×C₈	1	4	yes	—
83	C₂×C₄	C₂²×C₈	1	2	yes	—
84	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
85	C₈	—	2	4	no	—
86	C₈	—	2	1	no	—
87	C₈	—	2	2	no	—
88	C₈	H(5,𝔽₂)	1	1	yes	characteristic
89	C₈	H(5,𝔽₂)	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	—	2	8	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	4	no	—
94	C₂²	C₂³×C₈	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	4	no	—
98	C₄	#504	1	4	yes	—
99	C₄	#518	1	1	yes	characteristic
100	C₄	#568	1	1	yes	characteristic
101	C₂	—	4	1	no	—
102	C₂	—	4	4	no	—
103	C₂	#2972	1	1	yes	characteristic
104	C₂	C₂×#504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

#2169

C₂

yes

maximal, complemented

D₄×C₁₆

C₂

yes

maximal, complemented

#2162

C₂

yes

maximal, complemented

#2113

C₂

yes

maximal, complemented

C₂×#131

C₂²

yes

—

#532

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

—

#436

C₄

yes

—

C₈×D₄

C₂²

yes

—

#423

C₄

yes

characteristic

#408

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

C₄×C₁₆

C₂²

yes

—

C₂×#41

C₂×C₄

yes

—

#131

—

#131

—

C₂×M₁₆

C₂×C₄

yes

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

C₈

yes

characteristic

#127

C₈

yes

characteristic

#124

C₈

yes

—

#121

C₈

yes

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

#41

—

#41

—

C₂×Q₈

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×C₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#504

yes

—

C₄

#518

yes

characteristic

100

C₄

#568

yes

characteristic

101

C₂

—

102

C₂

—

103

C₂

#2972

yes

characteristic

104

C₂

C₂×#504

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₈

C₂⁴

commutator, socle

C₂²

C₂³×C₈

C₄

G ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ G

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2169	C₂	semidirect
3. D₄×C₁₆	C₂	semidirect
4. #2162	C₂	semidirect
5. #2113	C₂	semidirect

H ≅ G/N

type

2. #2169

C₂

semidirect

3. D₄×C₁₆

C₂

semidirect

4. #2162

C₂

semidirect

5. #2113

C₂

semidirect

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

subgroup	index
#2931	2
#2169	2
D₄×C₁₆	2
#2162	2
#2113	2

subgroup

index

D₄×C₁₆

Subgroups — #33493

Conjugacy classes of subgroups

Complements

Maximal subgroups