Group #33825 — subgroups

Subgroups — #33825

Order 256.

256 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 97 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	#3176	C₂	1	2	yes	maximal, complemented
2	#2228	C₂	1	4	yes	maximal, complemented
3	#2218	C₂	1	2	yes	maximal, complemented
4	#2215	C₂	1	2	yes	maximal, complemented
5	#2206	C₂	1	4	yes	maximal, complemented
6	#2151	C₂	1	1	yes	maximal, characteristic, complemented
7	#540	—	2	2	no	—
8	C₂×D₁₆	C₂²	1	2	yes	—
9	C₂×D₁₆	C₂²	1	2	yes	—
10	C₂×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	1	yes	characteristic
12	#499	C₂²	1	2	yes	—
13	#493	C₂²	1	2	yes	—
14	#490	—	2	2	no	—
15	#469	—	2	2	no	—
16	#466	C₂²	1	4	yes	—
17	#442	—	2	2	no	—
18	C₄×D₈	C₂²	1	4	yes	—
19	#405	C₂²	1	1	yes	characteristic
20	#367	C₂²	1	2	yes	—
21	#366	C₂²	1	2	yes	—
22	C₄⋊C₁₆	C₂²	1	4	yes	—
23	#357	C₂²	1	4	yes	—
24	#357	C₂²	1	4	yes	—
25	C₂×D₈	—	2	2	no	—
26	C₂×D₈	C₂³	1	4	yes	—
27	C₂×M₁₆	D₄	1	1	yes	characteristic
28	C₂²×C₈	C₂³	1	1	yes	characteristic
29	#127	—	2	2	no	—
30	#125	—	2	2	no	—
31	#123	—	4	4	no	—
32	#121	—	2	4	no	—
33	#121	—	4	4	no	—
34	C₄×D₄	—	2	4	no	—
35	#117	D₄	1	1	yes	characteristic
36	D₁₆	—	4	2	no	—
37	D₁₆	—	4	2	no	—
38	M₃₂	—	4	1	no	—
39	C₂×C₁₆	—	4	1	no	—
40	C₂×C₁₆	C₂³	1	2	yes	—
41	C₂×C₁₆	C₂³	1	2	yes	—
42	#107	C₂³	1	4	yes	—
43	#106	—	2	2	no	—
44	#102	—	2	4	no	—
45	#102	—	2	4	no	—
46	#97	D₄	1	2	yes	—
47	C₄×C₈	C₂³	1	2	yes	—
48	C₂×D₄	—	2	4	no	—
49	C₂×D₄	—	4	4	no	—
50	C₂×D₄	—	4	4	no	—
51	C₂²×C₄	—	4	4	no	—
52	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
53	D₈	—	2	4	no	—
54	D₈	—	2	4	no	—
55	D₈	—	4	4	no	—
56	M₁₆	—	4	1	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄	—	2	4	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄⋊C₄	D₈	1	2	yes	—
65	C₂²⋊C₄	—	4	4	no	—
66	C₂²⋊C₄	—	4	4	no	—
67	C₂²⋊C₄	D₈	1	2	yes	—
68	C₄²	C₂×D₄	1	2	yes	—
69	C₁₆	—	2	2	no	—
70	C₁₆	—	2	2	no	—
71	C₂³	—	4	4	no	—
72	C₂³	C₂×D₈	1	1	yes	characteristic
73	D₄	—	4	4	no	—
74	D₄	—	4	4	no	—
75	D₄	—	16	4	no	—
76	D₄	—	16	4	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	4	4	no	—
80	C₂×C₄	—	16	4	no	—
81	C₂×C₄	C₂×D₈	1	4	yes	—
82	C₂×C₄	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	H(5,𝔽₂)	1	1	yes	characteristic
87	C₈	2₋¹⁺⁴	1	1	yes	characteristic
88	C₂²	—	2	2	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	8	4	no	—
91	C₂²	—	8	4	no	—
92	C₂²	—	8	4	no	—
93	C₂²	C₂²×D₈	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	2	1	no	—
96	C₄	—	8	4	no	—
97	C₄	#537	1	1	yes	characteristic
98	C₄	#575	1	1	yes	characteristic
99	C₂	—	4	1	no	—
100	C₂	—	16	4	no	—
101	C₂	#3245	1	1	yes	characteristic
102	C₂	#3415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3176

C₂

yes

maximal, complemented

#2228

C₂

yes

maximal, complemented

#2218

C₂

yes

maximal, complemented

#2215

C₂

yes

maximal, complemented

#2206

C₂

yes

maximal, complemented

#2151

C₂

yes

maximal, characteristic, complemented

#540

—

C₂×D₁₆

C₂²

yes

—

C₂×D₁₆

C₂²

yes

—

C₂×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

characteristic

#499

C₂²

yes

—

#493

C₂²

yes

—

#490

—

#469

—

#466

C₂²

yes

—

#442

—

C₄×D₈

C₂²

yes

—

#405

C₂²

yes

characteristic

#367

C₂²

yes

—

#366

C₂²

yes

—

C₄⋊C₁₆

C₂²

yes

—

#357

C₂²

yes

—

#357

C₂²

yes

—

C₂×D₈

—

C₂×D₈

C₂³

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

—

—

—

—

C₄×D₄

—

#117

D₄

yes

characteristic

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

#107

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

D₈

—

D₈

—

D₈

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₈

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₈

yes

—

C₄²

C₂×D₄

yes

—

C₁₆

—

C₁₆

—

C₂³

—

C₂³

C₂×D₈

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₈

yes

—

C₂×C₄

C₂×D₈

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₈

—

C₈

—

C₈

H(5,𝔽₂)

yes

characteristic

C₈

2₋¹⁺⁴

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×D₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

100

C₂

—

101

C₂

#3245

yes

characteristic

102

C₂

#3415

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ 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 ◁ #117 ◁ #405 ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #3176	C₂	semidirect
2. #2228	C₂	semidirect
3. #2218	C₂	semidirect
4. #2215	C₂	semidirect
5. #2206	C₂	semidirect
6. #2151	C₂	semidirect

H ≅ G/N

type

1. #3176

C₂

semidirect

2. #2228

C₂

semidirect

3. #2218

C₂

semidirect

4. #2215

C₂

semidirect

5. #2206

C₂

semidirect

6. #2151

C₂

semidirect

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

subgroup	index
#3176	2
#2228	2
#2218	2
#2215	2
#2206	2
#2151	2

subgroup

index

Subgroups — #33825

Conjugacy classes of subgroups

Complements

Maximal subgroups