Group #22588493 — subgroups

Subgroups — #22588493

Order 9088.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₁⋊C₈	C₂	1	4	yes	maximal
3	#17985845	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
7	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
8	Dic₇₁⋊C₈	—	2	8	no	—
9	C₇₁⋊#95	C₄	1	2	yes	—
10	C₇₁⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₁	—	2	2	no	—
14	C₂²×Dic₇₁	C₂×C₄	1	4	yes	—
15	C₂×C₇₁⋊C₈	—	2	4	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₄×Dic₇₁	D₄	1	4	yes	—
18	C₄×Dic₇₁	Q₈	1	4	yes	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₇₁	—	2	2	no	—
23	C₂×C₈×C₇₁	—	2	4	no	—
24	C₄²×C₇₁	D₄	1	2	yes	—
25	C₄²×C₇₁	Q₈	1	2	yes	—
26	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	2	2	no	—
28	C₂×C₄×C₇₁	—	2	2	no	—
29	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₇₁	—	2	4	no	—
36	C₂×Dic₇₁	—	2	4	no	—
37	C₂×Dic₇₁	C₄⋊C₄	1	8	yes	—
38	C₈×C₇₁	—	4	2	no	—
39	C₇₁⋊C₈	—	4	2	no	—
40	C₂²×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₇₁	#131	1	2	yes	—
43	C₂²×C₇₁	#131	1	2	yes	—
44	C₄×C₇₁	—	2	2	no	—
45	C₄×C₇₁	—	2	2	no	—
46	C₄×C₇₁	#127	1	1	yes	characteristic
47	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
49	Dic₇₁	—	2	8	no	—
50	C₂×C₇₁	#384	1	1	yes	characteristic
51	C₂×C₇₁	#424	1	4	yes	—
52	C₂×C₇₁	#424	1	1	yes	characteristic
53	C₂×C₇₁	#424	1	1	yes	characteristic
54	#1843	—	71	1	no	maximal, Sylow
55	C₇₁	#1843	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊C₈	—	71	4	no	—
57	C₂×C₄⋊C₈	—	71	1	no	—
58	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
59	#376	—	71	1	no	—
60	C₂²×C₈	—	71	2	no	—
61	C₂²×C₈	C₂×D₇₁	1	2	yes	—
62	C₂×C₄²	—	71	2	no	—
63	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	142	8	no	—
66	C₄⋊C₈	—	142	2	no	—
67	#95	—	71	2	no	—
68	#95	—	71	2	no	—
69	C₂²×C₄	—	71	4	no	—
70	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	142	4	no	—
76	C₂×C₈	—	142	2	no	—
77	C₄²	—	71	4	no	—
78	C₄²	—	71	4	no	—
79	C₄²	Dic₁₄₂	1	2	yes	—
80	C₄²	C₂²⋊D₇₁	1	2	yes	—
81	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	71	8	no	—
85	C₂×C₄	—	142	4	no	—
86	C₂×C₄	—	142	4	no	—
87	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
89	C₂×C₄	D₄×D₇₁	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₇₁	1	2	yes	—
92	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	284	2	no	—
95	C₂²	C₄⋊C₄×D₇₁	1	2	yes	—
96	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
97	C₂²	#14556134	1	2	yes	—
98	C₂²	#14556136	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	142	8	no	—
102	C₄	#14556124	1	1	yes	characteristic
103	C₄	#14556126	1	1	yes	characteristic
104	C₄	#14556127	1	2	yes	—
105	C₂	#17985868	1	1	yes	characteristic
106	C₂	#17986024	1	4	yes	—
107	C₂	#17986025	1	1	yes	characteristic
108	C₂	#17986027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₇₁⋊C₈

C₂

yes

maximal

#17985845

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

—

Dic₇₁⋊C₈

—

C₇₁⋊#95

C₄

yes

—

C₇₁⋊#95

C₄

yes

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

D₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#131

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#127

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

#384

yes

characteristic

C₂×C₇₁

#424

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

#376

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

—

C₄²

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₂²⋊D₇₁

yes

—

C₂³

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

D₄×D₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

Q₈×D₇₁

yes

—

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₈

—

C₈

—

284

—

C₂²

C₄⋊C₄×D₇₁

yes

—

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

C₂²

#14556134

yes

—

C₂²

#14556136

yes

—

C₄

—

100

C₄

—

101

C₄

—

142

—

102

C₄

#14556124

yes

characteristic

103

C₄

#14556126

yes

characteristic

104

C₄

#14556127

yes

—

105

C₂

#17985868

yes

characteristic

106

C₂

#17986024

yes

—

107

C₂

#17986025

yes

characteristic

108

C₂

#17986027

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator, socle	C₂³×C₇₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1843
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator, socle

C₂³×C₇₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

hypercenter

C₂×C₄⋊C₈

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
55. C₇₁	#1843	semidirect

H ≅ G/N

type

55. C₇₁

#1843

semidirect

Subgroups — #22588493

Conjugacy classes of subgroups

Complements