Group #22554036 — subgroups

Subgroups — #22554036

Order 8384.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16862241	C₂	1	4	yes	maximal, complemented
2	#16862235	C₂	1	2	yes	maximal
3	#16862226	C₂	1	2	yes	maximal, complemented
4	Q₈×Dic₁₃₁	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₂₄	C₂	1	1	yes	maximal, characteristic
7	#16862204	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₆₂	C₂²	1	1	yes	characteristic
10	C₂×D₅₂₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₃₁	C₂²	1	2	yes	—
12	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₃₁	C₂²	1	4	yes	—
15	Dic₁₃₁⋊C₄	C₂²	1	2	yes	—
16	(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
17	(C₂×D₁₃₁)⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₃₁)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₃₁	C₂²	1	2	yes	—
20	C₄×Dic₁₃₁	C₂²	1	4	yes	—
21	C₂×Q₈×C₁₃₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
24	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
25	C₂²×D₁₃₁	C₂³	1	2	yes	—
26	Dic₂₆₂	—	2	2	no	—
27	D₅₂₄	—	2	2	no	—
28	C₂×Dic₁₃₁	C₂³	1	2	yes	—
29	C₂×Dic₁₃₁	C₂³	1	2	yes	—
30	C₂×Dic₁₃₁	C₂³	1	2	yes	—
31	C₄×D₁₃₁	—	2	4	no	—
32	Q₈×C₁₃₁	—	2	4	no	—
33	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₃₁	C₂³	1	4	yes	—
35	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
37	C₂²×C₁₃₁	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₃₁	—	2	2	no	—
39	C₂×D₁₃₁	—	2	2	no	—
40	C₂×D₁₃₁	—	2	2	no	—
41	C₄×C₁₃₁	—	2	4	no	—
42	C₄×C₁₃₁	—	2	1	no	—
43	C₄×C₁₃₁	#41	1	2	yes	—
44	C₄×C₁₃₁	#41	1	2	yes	—
45	Dic₁₃₁	—	2	2	no	—
46	Dic₁₃₁	—	2	2	no	—
47	Dic₁₃₁	—	2	2	no	—
48	C₂×C₁₃₁	C₂×#41	1	1	yes	characteristic
49	C₂×C₁₃₁	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₃₁	—	4	2	no	—
52	C₁₃₁	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	131	1	no	maximal, Sylow
54	C₄⋊Q₈	D₁₃₁	1	1	yes	characteristic, complemented
55	#126	—	131	4	no	—
56	#124	—	131	2	no	—
57	C₂²⋊Q₈	—	131	2	no	—
58	C₄×Q₈	—	131	2	no	—
59	C₄×Q₈	—	131	1	no	—
60	C₄×D₄	—	131	1	no	—
61	#117	—	131	2	no	—
62	C₂×Q₈	—	131	1	no	—
63	C₂×Q₈	C₂×D₁₃₁	1	2	yes	—
64	C₂×D₄	—	131	1	no	—
65	C₂²×C₄	—	131	2	no	—
66	C₄⋊C₄	—	131	2	no	—
67	C₄⋊C₄	—	131	1	no	—
68	C₄⋊C₄	—	131	1	no	—
69	C₄⋊C₄	—	131	4	no	—
70	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
71	C₄⋊C₄	C₂×D₁₃₁	1	2	yes	—
72	C₂²⋊C₄	—	131	4	no	—
73	C₂²⋊C₄	—	131	2	no	—
74	C₂²⋊C₄	—	131	4	no	—
75	C₄²	—	131	4	no	—
76	C₄²	—	131	2	no	—
77	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
78	C₂³	—	131	2	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	262	2	no	—
81	D₄	—	262	2	no	—
82	C₂×C₄	—	262	4	no	—
83	C₂×C₄	—	131	2	no	—
84	C₂×C₄	—	131	2	no	—
85	C₂×C₄	—	131	2	no	—
86	C₂×C₄	C₂²×D₁₃₁	1	4	yes	—
87	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
90	C₂²	—	262	2	no	—
91	C₂²	—	262	2	no	—
92	C₂²	—	262	2	no	—
93	C₂²	C₂³×D₁₃₁	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	2	1	no	—
96	C₄	—	262	2	no	—
97	C₄	—	262	2	no	—
98	C₄	—	262	2	no	—
99	C₄	#14523512	1	2	yes	—
100	C₄	#14523514	1	2	yes	—
101	C₂	—	524	2	no	—
102	C₂	C₂×#14523512	1	1	yes	characteristic
103	C₂	C₂×#14523514	1	1	yes	characteristic
104	C₂	#16862294	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16862241

C₂

yes

maximal, complemented

#16862235

C₂

yes

maximal

#16862226

C₂

yes

maximal, complemented

Q₈×Dic₁₃₁

C₂

yes

maximal, complemented

C₄×Dic₂₆₂

C₂

yes

maximal, characteristic, complemented

C₄×D₅₂₄

C₂

yes

maximal, characteristic

#16862204

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₆₂

C₂²

yes

characteristic

C₂×D₅₂₄

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂²

yes

—

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

characteristic

C₄⋊Dic₁₃₁

C₂²

yes

—

Dic₁₃₁⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

(C₂×D₁₃₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₁

C₂²

yes

—

C₄×Dic₁₃₁

C₂²

yes

—

C₂×Q₈×C₁₃₁

C₂²

yes

—

C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₄²×C₁₃₁

C₂²

yes

characteristic

C₂²×D₁₃₁

C₂³

yes

—

Dic₂₆₂

—

D₅₂₄

—

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

—

C₂×Dic₁₃₁

C₂³

yes

—

C₄×D₁₃₁

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₄×C₁₃₁

C₂³

yes

characteristic

C₂²×C₁₃₁

C₂⁴

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#41

yes

—

C₄×C₁₃₁

#41

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

C₂×#41

yes

characteristic

C₂×C₁₃₁

C₂×#41

yes

characteristic

C₂×C₁₃₁

2₋¹⁺⁴

yes

characteristic

D₁₃₁

—

C₁₃₁

#556

yes

characteristic, complemented, Sylow

#556

—

131

maximal, Sylow

C₄⋊Q₈

D₁₃₁

yes

characteristic, complemented

#126

—

131

—

#124

—

131

—

C₂²⋊Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×D₄

—

131

—

#117

—

131

—

C₂×Q₈

—

131

—

C₂×Q₈

C₂×D₁₃₁

yes

—

C₂×D₄

—

131

—

C₂²×C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

—

131

—

C₄⋊C₄

C₂×D₁₃₁

yes

—

C₄⋊C₄

C₂×D₁₃₁

yes

—

C₂²⋊C₄

—

131

—

C₂²⋊C₄

—

131

—

C₂²⋊C₄

—

131

—

C₄²

—

131

—

C₄²

—

131

—

C₄²

C₂×D₁₃₁

yes

characteristic

C₂³

—

131

—

Q₈

—

Q₈

—

262

—

D₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₂²×D₁₃₁

yes

—

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₂×C₄

C₂²×D₁₃₁

yes

characteristic

C₂²

—

262

—

C₂²

—

262

—

C₂²

—

262

—

C₂²

C₂³×D₁₃₁

yes

characteristic

C₄

—

C₄

—

C₄

—

262

—

C₄

—

262

—

C₄

—

262

—

C₄

#14523512

yes

—

100

C₄

#14523514

yes

—

101

C₂

—

524

—

102

C₂

C₂×#14523512

yes

characteristic

103

C₂

C₂×#14523514

yes

characteristic

104

C₂

#16862294

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂⁴
Fitting	C₄⋊Q₈×C₁₃₁	C₂
hypercenter	C₄⋊Q₈	D₁₃₁
hypocenter	C₁₃₁	#556
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂⁴

Fitting

C₄⋊Q₈×C₁₃₁

C₂

hypercenter

C₄⋊Q₈

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #16862241	C₂	semidirect
3. #16862226	C₂	semidirect
4. Q₈×Dic₁₃₁	C₂	semidirect
5. C₄×Dic₂₆₂	C₂	semidirect
7. #16862204	C₂	semidirect
8. C₄⋊Q₈×C₁₃₁	C₂	semidirect
52. C₁₃₁	#556	semidirect
54. C₄⋊Q₈	D₁₃₁	semidirect

H ≅ G/N

type

1. #16862241

C₂

semidirect

3. #16862226

C₂

semidirect

4. Q₈×Dic₁₃₁

C₂

semidirect

5. C₄×Dic₂₆₂

C₂

semidirect

7. #16862204

C₂

semidirect

8. C₄⋊Q₈×C₁₃₁

C₂

semidirect

52. C₁₃₁

#556

semidirect

54. C₄⋊Q₈

D₁₃₁

semidirect

Subgroups — #22554036

Conjugacy classes of subgroups

Complements