Group #22588197 — subgroups

Subgroups — #22588197

Order 9088.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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	2	yes	maximal
2	C₂×#14556039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#17985754	C₂	1	2	yes	maximal
5	#376×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊(C₇₁⋊C₈)	C₄	1	8	yes	complemented
9	#14556039	—	2	2	no	—
10	C₄×C₇₁⋊C₈	—	2	2	no	—
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	#95×C₇₁	C₄	1	4	yes	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
25	C₄²×C₇₁	—	2	2	no	—
26	C₄²×C₇₁	C₂×C₄	1	4	yes	—
27	C₄²×C₇₁	D₄	1	2	yes	—
28	C₄²×C₇₁	Q₈	1	2	yes	—
29	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	4	no	—
31	C₂×C₄×C₇₁	—	2	4	no	—
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	—	2	4	no	—
34	C₂×C₄×C₇₁	C₄²	1	4	yes	—
35	C₂×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₇₁	#41	1	2	yes	—
40	C₇₁⋊C₈	—	2	4	no	—
41	C₇₁⋊C₈	—	2	4	no	—
42	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₇₁	#117	1	1	yes	characteristic
45	C₂²×C₇₁	#131	1	4	yes	—
46	C₄×C₇₁	—	2	4	no	—
47	C₄×C₇₁	—	2	4	no	—
48	C₄×C₇₁	—	4	2	no	—
49	C₄×C₇₁	C₄×D₄	1	2	yes	—
50	C₄×C₇₁	C₄×Q₈	1	2	yes	—
51	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₇₁	#405	1	2	yes	—
53	C₂×C₇₁	#424	1	2	yes	—
54	C₂×C₇₁	#433	1	2	yes	—
55	#1768	—	71	1	no	maximal, Sylow
56	C₇₁	#1768	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄⋊C₈	—	71	2	no	—
58	C₂×#97	—	71	1	no	—
59	C₂×C₄×C₈	—	71	1	no	—
60	#376	D₇₁	1	1	yes	characteristic
61	#336	—	71	2	no	—
62	C₂²×C₈	—	71	2	no	—
63	C₂²×C₈	—	71	2	no	—
64	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
67	C₄⋊C₈	—	71	8	no	—
68	#97	—	142	2	no	—
69	C₄×C₈	—	142	2	no	—
70	#95	Dic₇₁	1	4	yes	—
71	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
72	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
73	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
75	C₂×C₈	—	71	4	no	—
76	C₂×C₈	—	71	4	no	—
77	C₂×C₈	—	142	2	no	—
78	C₂×C₈	—	142	2	no	—
79	C₂×C₈	—	142	2	no	—
80	C₂×C₈	—	142	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₁₄₂	1	2	yes	—
83	C₄²	C₄×D₇₁	1	4	yes	—
84	C₄²	D₂₈₄	1	2	yes	—
85	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₇₁	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₇₁	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
93	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
95	C₂×C₄	#11765865	1	2	yes	—
96	C₈	—	142	4	no	—
97	C₈	—	142	4	no	—
98	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₂²	#14556087	1	1	yes	characteristic
101	C₂²	#14556135	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₂₈₄	1	2	yes	—
106	C₄	C₄×Dic₁₄₂	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
108	C₂	#17985955	1	2	yes	—
109	C₂	#17986026	1	2	yes	—
110	C₂	#17986060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal

C₂×#14556039

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal

#376×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

#14556039

—

C₄×C₇₁⋊C₈

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₂×C₄⋊C₄

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₄

yes

—

C₄×C₇₁

C₄×Q₈

yes

—

C₂×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₁

#405

yes

—

C₂×C₇₁

#424

yes

—

C₂×C₇₁

#433

yes

—

#1768

—

maximal, Sylow

C₇₁

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

D₇₁

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

#97

—

142

—

C₄×C₈

—

142

—

#95

Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₄×D₇₁

yes

—

C₄²

D₂₈₄

yes

—

C₂³

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₇₁

yes

—

C₂×C₄

C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

—

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

#11765865

yes

—

C₈

—

142

—

C₈

—

142

—

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

100

C₂²

#14556087

yes

characteristic

101

C₂²

#14556135

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₂₈₄

yes

—

106

C₄

C₄×Dic₁₄₂

yes

—

107

C₂

C₄×C₄⋊Dic₇₁

yes

characteristic

108

C₂

#17985955

yes

—

109

C₂

#17986026

yes

—

110

C₂

#17986060

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄²
Fitting	#376×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	#376	D₇₁
hypocenter	C₇₁	#1768
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄²

Fitting

#376×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
56. C₇₁	#1768	semidirect

H ≅ G/N

type

8. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

56. C₇₁

#1768

semidirect

Subgroups — #22588197

Conjugacy classes of subgroups

Complements