Group #22601967 — subgroups

Subgroups — #22601967

Order 9088.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	#17986715	C₂	1	9	yes	maximal, complemented
2	#17986671	C₂	1	6	yes	maximal, complemented
3	#17986550	C₂	1	9	yes	maximal, complemented
4	C₄×#11765866	C₂	1	6	yes	maximal, complemented
5	Q₈²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765866	C₂²	1	6	yes	—
7	C₂×#11765866	C₂²	1	6	yes	—
8	#14556125	C₂²	1	9	yes	—
9	#14556117	C₂²	1	18	yes	—
10	#14556103	C₂²	1	36	yes	—
11	Q₈×Dic₇₁	C₂²	1	2	yes	—
12	C₄×D₂₈₄	C₂²	1	18	yes	—
13	#14556090	C₂²	1	18	yes	—
14	#14556086	C₂²	1	18	yes	—
15	C₄²×D₇₁	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₇₁	C₂²	1	9	yes	—
17	C₄×Q₈×C₇₁	C₂²	1	6	yes	—
18	#11765866	—	2	24	no	—
19	#11765866	—	4	12	no	—
20	C₂×D₂₈₄	C₂³	1	18	yes	—
21	C₂×D₂₈₄	C₂³	1	18	yes	—
22	C₂×C₄×D₇₁	C₂³	1	9	yes	—
23	C₂×C₄×D₇₁	C₂³	1	18	yes	—
24	C₄⋊Dic₇₁	C₂³	1	6	yes	—
25	(C₂×D₇₁)⋊C₄	C₂³	1	36	yes	—
26	C₄×Dic₇₁	C₂³	1	6	yes	—
27	C₄×Dic₇₁	C₂³	1	9	yes	—
28	C₂×Q₈×C₇₁	C₂³	1	6	yes	—
29	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
30	C₄⋊C₄×C₇₁	C₂³	1	18	yes	—
31	C₄²×C₇₁	C₂³	1	9	yes	—
32	C₂²×D₇₁	C₂⁴	1	9	yes	—
33	Q₈×C₇₁	—	4	6	no	—
34	Q₈×C₇₁	#41	1	8	yes	complemented
35	C₂×C₄×C₇₁	C₂⁴	1	9	yes	—
36	C₂×C₄×C₇₁	C₂⁴	1	6	yes	—
37	C₂×Dic₇₁	C₂⁴	1	6	yes	—
38	C₂×Dic₇₁	C₂⁴	1	1	yes	characteristic
39	D₂₈₄	—	2	36	no	—
40	D₂₈₄	—	4	18	no	—
41	C₄×D₇₁	—	2	36	no	—
42	C₄×D₇₁	—	2	18	no	—
43	C₂²×C₇₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₇₁	—	2	18	no	—
45	C₂×D₇₁	—	2	9	no	—
46	C₄×C₇₁	—	2	9	no	—
47	C₄×C₇₁	C₂×#41	1	12	yes	—
48	Dic₇₁	—	2	6	no	—
49	Dic₇₁	H(5,𝔽₂)	1	2	yes	—
50	C₂×C₇₁	C₂²×#41	1	2	yes	—
51	C₂×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
52	D₇₁	—	4	9	no	—
53	#3483	—	71	1	no	maximal, Sylow
54	C₇₁	#3483	1	1	yes	characteristic, complemented, Sylow
55	Q₈²	D₇₁	1	1	yes	characteristic, complemented
56	#555	—	71	9	no	—
57	#550	—	71	6	no	—
58	#532	—	71	9	no	—
59	C₄×#41	—	71	6	no	—
60	C₂×#41	—	71	6	no	—
61	C₂×#41	—	71	6	no	—
62	C₄⋊Q₈	C₂×D₇₁	1	9	yes	—
63	#127	—	71	9	no	—
64	#124	—	71	18	no	—
65	#121	—	71	36	no	—
66	C₄×Q₈	—	71	2	no	—
67	C₄×Q₈	C₂×D₇₁	1	6	yes	—
68	C₄×D₄	—	71	18	no	—
69	C₄×D₄	—	71	18	no	—
70	#117	—	71	18	no	—
71	C₂×C₄²	—	71	9	no	—
72	#41	—	284	12	no	—
73	#41	—	142	24	no	—
74	C₂×Q₈	C₂²×D₇₁	1	6	yes	—
75	C₂×Q₈	C₂²×D₇₁	1	2	yes	—
76	C₂×D₄	—	71	18	no	—
77	C₂×D₄	—	71	18	no	—
78	C₂²×C₄	—	71	18	no	—
79	C₂²×C₄	—	71	9	no	—
80	C₄⋊C₄	—	71	6	no	—
81	C₄⋊C₄	C₂²×D₇₁	1	18	yes	—
82	C₂²⋊C₄	—	71	36	no	—
83	C₄²	—	71	6	no	—
84	C₄²	—	71	9	no	—
85	C₄²	C₂²×D₇₁	1	9	yes	—
86	C₂³	—	71	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#11765866	1	8	yes	complemented
89	D₄	—	284	18	no	—
90	D₄	—	142	36	no	—
91	C₂×C₄	—	71	1	no	—
92	C₂×C₄	—	71	6	no	—
93	C₂×C₄	—	142	18	no	—
94	C₂×C₄	—	142	36	no	—
95	C₂×C₄	C₂³×D₇₁	1	9	yes	—
96	C₂×C₄	C₂³×D₇₁	1	6	yes	—
97	C₂²	—	142	18	no	—
98	C₂²	—	142	9	no	—
99	C₂²	C₂⁴×D₇₁	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	71	2	no	—
102	C₄	—	142	6	no	—
103	C₄	C₂×#11765866	1	12	yes	—
104	C₂	—	284	9	no	—
105	C₂	C₂²×#11765866	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986715

C₂

yes

maximal, complemented

#17986671

C₂

yes

maximal, complemented

#17986550

C₂

yes

maximal, complemented

C₄×#11765866

C₂

yes

maximal, complemented

Q₈²×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765866

C₂²

yes

—

C₂×#11765866

C₂²

yes

—

#14556125

C₂²

yes

—

#14556117

C₂²

yes

—

#14556103

C₂²

yes

—

Q₈×Dic₇₁

C₂²

yes

—

C₄×D₂₈₄

C₂²

yes

—

#14556090

C₂²

yes

—

#14556086

C₂²

yes

—

C₄²×D₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

—

#11765866

—

#11765866

—

C₂×D₂₈₄

C₂³

yes

—

C₂×D₂₈₄

C₂³

yes

—

C₂×C₄×D₇₁

C₂³

yes

—

C₂×C₄×D₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

—

(C₂×D₇₁)⋊C₄

C₂³

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂²×D₇₁

C₂⁴

yes

—

Q₈×C₇₁

—

Q₈×C₇₁

#41

yes

complemented

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×Dic₇₁

C₂⁴

yes

—

C₂×Dic₇₁

C₂⁴

yes

characteristic

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₂²×C₇₁

C₂⁵

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#41

yes

—

Dic₇₁

—

Dic₇₁

H(5,𝔽₂)

yes

—

C₂×C₇₁

C₂²×#41

yes

—

C₂×C₇₁

C₂×H(5,𝔽₂)

yes

characteristic

D₇₁

—

#3483

—

maximal, Sylow

C₇₁

#3483

yes

characteristic, complemented, Sylow

Q₈²

D₇₁

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₇₁

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#41

—

284

—

#41

—

142

—

C₂×Q₈

C₂²×D₇₁

yes

—

C₂×Q₈

C₂²×D₇₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₇₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₁

yes

—

C₂³

—

Q₈

—

Q₈

#11765866

yes

complemented

D₄

—

284

—

D₄

—

142

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₂³×D₇₁

yes

—

C₂×C₄

C₂³×D₇₁

yes

—

C₂²

—

142

—

C₂²

—

142

—

C₂²

C₂⁴×D₇₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

142

—

103

C₄

C₂×#11765866

yes

—

104

C₂

—

284

—

105

C₂

C₂²×#11765866

yes

—

106

C₂

H(5,𝔽₂)×D₇₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₁
commutator, socle	C₂²×C₇₁	C₂⁵
Fitting	Q₈²×C₇₁	C₂
hypercenter	Q₈²	D₇₁
hypocenter	C₇₁	#3483
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₇₁

commutator, socle

C₂²×C₇₁

C₂⁵

Fitting

Q₈²×C₇₁

C₂

hypercenter

Q₈²

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. #17986715	C₂	semidirect
2. #17986671	C₂	semidirect
3. #17986550	C₂	semidirect
4. C₄×#11765866	C₂	semidirect
5. Q₈²×C₇₁	C₂	semidirect
34. Q₈×C₇₁	#41	semidirect
54. C₇₁	#3483	semidirect
55. Q₈²	D₇₁	semidirect
88. Q₈	#11765866	semidirect

H ≅ G/N

type

1. #17986715

C₂

semidirect

2. #17986671

C₂

semidirect

3. #17986550

C₂

semidirect

4. C₄×#11765866

C₂

semidirect

5. Q₈²×C₇₁

C₂

semidirect

34. Q₈×C₇₁

#41

semidirect

54. C₇₁

#3483

semidirect

55. Q₈²

D₇₁

semidirect

88. Q₈

#11765866

semidirect

Subgroups — #22601967

Conjugacy classes of subgroups

Complements