Group #22282 — subgroups

Subgroups — #22282

Order 256.

405 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 151 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	#3147	C₂	1	1	yes	maximal, characteristic
2	#2932	C₂	1	2	yes	maximal, complemented
3	#1963	C₂	1	2	yes	maximal, complemented
4	#1876	C₂	1	8	yes	maximal, complemented
5	#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135	C₂²	1	4	yes	—
8	C₂×#131	C₂²	1	4	yes	—
9	#532	C₂²	1	2	yes	—
10	#495	C₄	1	4	yes	complemented
11	#494	C₄	1	1	yes	characteristic, complemented
12	#493	C₄	1	1	yes	characteristic, complemented
13	#492	C₄	1	2	yes	complemented
14	#435	—	2	8	no	—
15	#423	C₂²	1	2	yes	—
16	C₂×#104	C₂²	1	8	yes	—
17	#408	C₂²	1	4	yes	—
18	C₂×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	#376	C₂²	1	1	yes	characteristic
21	#341	C₂²	1	4	yes	—
22	#336	C₂²	1	4	yes	—
23	C₂×#41	C₂³	1	4	yes	—
24	#135	—	4	8	no	—
25	C₂×Q₁₆	C₂×C₄	1	4	yes	—
26	C₂×SD₁₆	C₂×C₄	1	8	yes	—
27	C₂×D₈	C₂×C₄	1	4	yes	—
28	#131	—	4	8	no	—
29	C₂×M₁₆	—	2	4	no	—
30	C₂×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₈	—	2	4	no	—
32	C₂²×C₈	C₂³	1	2	yes	—
33	C₂²×C₈	C₂³	1	2	yes	—
34	C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	#127	C₂×C₄	1	2	yes	—
36	#124	C₂×C₄	1	4	yes	—
37	#121	—	2	4	no	—
38	C₄×D₄	—	2	4	no	—
39	C₂×C₄²	C₂³	1	2	yes	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	C₄⋊C₈	—	2	4	no	—
42	#104	—	4	8	no	—
43	C₂²⋊C₈	—	2	8	no	—
44	#97	D₄	1	4	yes	—
45	C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×C₄	1	2	yes	—
48	#95	—	2	2	no	—
49	#41	—	2	8	no	—
50	#41	—	2	8	no	—
51	C₂×Q₈	C₂²×C₄	1	4	yes	—
52	C₂×D₄	—	2	4	no	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	C₂²×C₄	1	4	yes	—
55	C₂²×C₄	—	2	4	no	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	#41	1	2	yes	—
58	C₂²×C₄	C₂⁴	1	1	yes	characteristic
59	Q₁₆	—	4	4	no	—
60	SD₁₆	—	4	8	no	—
61	D₈	—	4	4	no	—
62	M₁₆	—	2	8	no	—
63	M₁₆	—	4	8	no	—
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	4	8	no	—
68	C₂×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₈	C₂²×C₄	1	2	yes	—
70	C₂×C₈	C₂×D₄	1	4	yes	—
71	C₄⋊C₄	—	2	4	no	—
72	C₂²⋊C₄	—	2	8	no	—
73	C₄²	—	2	4	no	—
74	C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₄²	C₂²×C₄	1	1	yes	characteristic
76	C₄²	C₂×D₄	1	2	yes	—
77	C₂³	—	2	4	no	—
78	C₂³	C₂×#41	1	1	yes	characteristic
79	Q₈	—	2	8	no	—
80	D₄	—	2	8	no	—
81	D₄	—	4	8	no	—
82	D₄	—	8	4	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	4	no	—
86	C₂×C₄	—	4	4	no	—
87	C₂×C₄	—	4	4	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	C₄×D₄	1	2	yes	—
90	C₂×C₄	C₄×D₄	1	2	yes	—
91	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄	H(5,𝔽₂)	1	4	yes	—
94	C₈	—	2	2	no	—
95	C₈	—	2	2	no	—
96	C₈	—	2	4	no	—
97	C₈	—	4	4	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	4	4	no	—
100	C₂²	—	4	8	no	—
101	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²	#540	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	—	4	4	no	—
106	C₄	—	8	2	no	—
107	C₄	#518	1	2	yes	—
108	C₄	#535	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	—	8	4	no	—
111	C₂	#2328	1	1	yes	characteristic
112	C₂	#2957	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3147

C₂

yes

maximal, characteristic

#2932

C₂

yes

maximal, complemented

#1963

C₂

yes

maximal, complemented

#1876

C₂

yes

maximal, complemented

#1837

C₂

yes

maximal, characteristic, complemented

#1746

C₂

yes

maximal, characteristic, complemented

C₂×#135

C₂²

yes

—

C₂×#131

C₂²

yes

—

#532

C₂²

yes

—

#495

C₄

yes

complemented

#494

C₄

yes

characteristic, complemented

#493

C₄

yes

characteristic, complemented

#492

C₄

yes

complemented

#435

—

#423

C₂²

yes

—

C₂×#104

C₂²

yes

—

#408

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#376

C₂²

yes

characteristic

#341

C₂²

yes

—

#336

C₂²

yes

—

C₂×#41

C₂³

yes

—

#135

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×SD₁₆

C₂×C₄

yes

—

C₂×D₈

C₂×C₄

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

—

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

C₂×C₄

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

#104

—

C₂²⋊C₈

—

#97

D₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

#95

—

#41

—

#41

—

C₂×Q₈

C₂²×C₄

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#41

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

—

C₂³

—

C₂³

C₂×#41

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₂×C₄

C₂²×D₄

yes

characteristic

C₂×C₄

H(5,𝔽₂)

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

C₂×C₄×D₄

yes

characteristic

102

C₂²

#540

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#518

yes

—

108

C₄

#535

yes

—

109

C₂

—

110

C₂

—

111

C₂

#2328

yes

characteristic

112

C₂

#2957

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

socle

C₂²

C₂×C₄×D₄

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2932	C₂	semidirect
3. #1963	C₂	semidirect
4. #1876	C₂	semidirect
5. #1837	C₂	semidirect
6. #1746	C₂	semidirect
10. #495	C₄	semidirect
11. #494	C₄	semidirect
12. #493	C₄	semidirect
13. #492	C₄	semidirect

H ≅ G/N

type

2. #2932

C₂

semidirect

3. #1963

C₂

semidirect

4. #1876

C₂

semidirect

5. #1837

C₂

semidirect

6. #1746

C₂

semidirect

10. #495

C₄

semidirect

11. #494

C₄

semidirect

12. #493

C₄

semidirect

13. #492

C₄

semidirect

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

subgroup	index
#3147	2
#2932	2
#1963	2
#1876	2
#1837	2
#1746	2

subgroup

index

Subgroups — #22282

Conjugacy classes of subgroups

Complements

Maximal subgroups