Group #33494 — subgroups

Subgroups — #33494

Order 256.

278 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 177 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	#2168	C₂	1	8	yes	maximal, complemented
3	#2163	C₂	1	2	yes	maximal, complemented
4	#2113	C₂	1	4	yes	maximal, complemented
5	C₂×#131	C₂²	1	2	yes	—
6	#532	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	4	yes	—
8	C₂²×C₁₆	C₂²	1	4	yes	—
9	#436	C₄	1	4	yes	—
10	C₈×D₄	C₂²	1	4	yes	—
11	#423	C₄	1	1	yes	characteristic
12	#408	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆	C₂²	1	4	yes	—
15	C₂²⋊C₁₆	C₂²	1	4	yes	—
16	C₂²⋊C₁₆	C₂²	1	8	yes	—
17	#346	C₂²	1	4	yes	—
18	C₂×#41	C₂×C₄	1	2	yes	—
19	#131	—	4	2	no	—
20	#131	—	4	2	no	—
21	C₂×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₈	C₂³	1	4	yes	—
26	C₄⋊Q₈	C₈	1	1	yes	characteristic
27	#127	C₈	1	1	yes	characteristic
28	#124	C₈	1	2	yes	—
29	#121	C₈	1	4	yes	—
30	C₄×D₄	C₂×C₄	1	4	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	M₃₂	—	4	4	no	—
33	C₂×C₁₆	—	4	4	no	—
34	C₂×C₁₆	C₂³	1	8	yes	—
35	C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	C₂³	1	2	yes	—
39	#41	—	4	2	no	—
40	#41	—	4	2	no	—
41	C₂×Q₈	C₂×C₈	1	2	yes	—
42	C₂×D₄	C₂×C₈	1	4	yes	—
43	C₂×D₄	C₂×C₈	1	4	yes	—
44	C₂×D₄	C₂×C₈	1	2	yes	—
45	C₂²×C₄	C₂×C₈	1	2	yes	—
46	C₂²×C₄	C₂²×C₄	1	4	yes	—
47	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
48	M₁₆	—	4	2	no	—
49	M₁₆	—	4	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	4	1	no	—
54	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈	C₂²×C₄	1	2	yes	—
56	C₂×C₈	C₂²×C₄	1	4	yes	—
57	C₂×C₈	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄	C₂×C₈	1	4	yes	—
59	C₂²⋊C₄	C₂×C₈	1	8	yes	—
60	C₄²	C₂×C₈	1	1	yes	characteristic
61	C₄²	C₂×C₈	1	1	yes	characteristic
62	C₄²	C₂²×C₄	1	2	yes	—
63	C₁₆	—	2	8	no	—
64	C₂³	C₂²×C₈	1	4	yes	—
65	C₂³	C₂²×C₈	1	1	yes	characteristic
66	Q₈	—	4	2	no	—
67	D₄	—	4	4	no	—
68	D₄	—	4	2	no	—
69	D₄	—	4	4	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×C₈	1	2	yes	—
77	C₂×C₄	C₂²×C₈	1	2	yes	—
78	C₂×C₄	C₂²×C₈	1	4	yes	—
79	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
80	C₈	—	2	4	no	—
81	C₈	—	2	1	no	—
82	C₈	—	2	2	no	—
83	C₈	H(5,𝔽₂)	1	2	yes	—
84	C₂²	—	2	2	no	—
85	C₂²	—	2	8	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	4	no	—
88	C₂²	C₂³×C₈	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	—	2	4	no	—
93	C₄	#518	1	1	yes	characteristic
94	C₄	#568	1	1	yes	characteristic
95	C₂	—	4	1	no	—
96	C₂	—	4	4	no	—
97	C₂	#2972	1	1	yes	characteristic
98	C₂	#3407	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

#2168

C₂

yes

maximal, complemented

#2163

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

—

C₂²⋊C₁₆

C₂²

yes

—

C₂²⋊C₁₆

C₂²

yes

—

#346

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

—

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

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

—

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

H(5,𝔽₂)

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×C₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

#2972

yes

characteristic

C₂

#3407

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. #2168	C₂	semidirect
3. #2163	C₂	semidirect
4. #2113	C₂	semidirect

H ≅ G/N

type

2. #2168

C₂

semidirect

3. #2163

C₂

semidirect

4. #2113

C₂

semidirect

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

subgroup	index
#2931	2
#2168	2
#2163	2
#2113	2

subgroup

index

Subgroups — #33494

Conjugacy classes of subgroups

Complements

Maximal subgroups