Group #22587480 — subgroups

Subgroups — #22587480

Order 9088.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986378	C₂	1	1	yes	maximal, characteristic
2	#17986294	C₂	1	2	yes	maximal
3	#17985748	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
5	Q₈²×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765855	—	2	2	no	—
7	#14556128	C₂²	1	1	yes	characteristic
8	#14556061	—	2	2	no	—
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	#14556052	—	2	2	no	—
11	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₇₁	—	2	2	no	—
13	C₄⋊Q₈×C₇₁	—	2	2	no	—
14	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₇₁	—	2	2	no	—
16	C₄×Q₈×C₇₁	C₂²	1	2	yes	—
17	C₂×Dic₁₄₂	—	2	1	no	—
18	#11765855	—	4	4	no	—
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₂×Q₈×C₇₁	—	2	2	no	—
23	C₂×Q₈×C₇₁	D₄	1	2	yes	—
24	C₂×Q₈×C₇₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₇₁	—	2	4	no	—
26	C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₄⋊C₄×C₇₁	D₄	1	2	yes	—
29	C₄²×C₇₁	—	2	2	no	—
30	C₄²×C₇₁	—	2	2	no	—
31	C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	Q₈×C₇₁	—	2	2	no	—
33	Q₈×C₇₁	—	4	2	no	—
34	Q₈×C₇₁	—	8	2	no	—
35	Q₈×C₇₁	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
41	C₂×Dic₇₁	—	4	1	no	—
42	Dic₁₄₂	—	4	2	no	—
43	C₇₁⋊C₈	—	2	2	no	—
44	C₇₁⋊C₈	—	4	2	no	—
45	C₂²×C₇₁	#120	1	1	yes	characteristic
46	C₄×C₇₁	—	2	4	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	—	4	2	no	—
49	C₄×C₇₁	—	4	2	no	—
50	C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
51	C₄×C₇₁	#137	1	2	yes	—
52	Dic₇₁	—	8	1	no	—
53	C₂×C₇₁	#451	1	2	yes	—
54	C₂×C₇₁	#453	1	1	yes	characteristic
55	#1633	—	71	1	no	maximal, Sylow
56	C₇₁	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	D₇₁	1	1	yes	characteristic
58	#494	—	71	1	no	—
59	#477	—	71	2	no	—
60	#333	—	71	1	no	—
61	Q₈⋊C₈	—	71	2	no	—
62	C₂×Q₁₆	—	142	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	71	1	no	—
66	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₇₁	1	2	yes	—
69	#107	—	142	2	no	—
70	C₄⋊C₈	—	71	2	no	—
71	#103	—	142	2	no	—
72	C₄×C₈	—	71	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	142	1	no	—
75	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
77	Q₁₆	—	284	4	no	—
78	C₂×C₈	—	142	2	no	—
79	C₂×C₈	—	142	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	142	2	no	—
84	C₄⋊C₄	C₂²⋊D₇₁	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₇₁	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	284	2	no	—
92	Q₈	#11765855	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	284	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
99	C₈	—	284	2	no	—
100	C₈	—	142	2	no	—
101	C₂²	#14556097	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	568	1	no	—
107	C₄	C₂×#11765855	1	2	yes	—
108	C₄	#14556159	1	2	yes	—
109	C₂	#17986155	1	2	yes	—
110	C₂	#17986167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986378

C₂

yes

maximal, characteristic

#17986294

C₂

yes

maximal

#17985748

C₂

yes

maximal, characteristic

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal

Q₈²×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765855

—

#14556128

C₂²

yes

characteristic

#14556061

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556052

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

#11765855

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

—

C₂×Q₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

Q₁₆

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#120

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

C₂×C₇₁

#451

yes

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

Q₈²

D₇₁

yes

—

—

—

Q₈⋊C₈

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

#107

—

142

—

C₄⋊C₈

—

#103

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

C₂²⋊D₇₁

yes

—

C₂×Q₈

C₂²⋊D₇₁

yes

—

Q₁₆

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

C₂²⋊D₇₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

284

—

Q₈

#11765855

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

C₈

—

284

—

100

C₈

—

142

—

101

C₂²

#14556097

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

568

—

107

C₄

C₂×#11765855

yes

—

108

C₄

#14556159

yes

—

109

C₂

#17986155

yes

—

110

C₂

#17986167

yes

characteristic

	subgroup	G/H
center	C₂²	#14556097
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	Q₈²×C₇₁	C₂
socle	C₂²×C₇₁	#120
hypercenter	Q₈²	D₇₁
hypocenter	C₇₁	#1633
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#14556097

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

Q₈²×C₇₁

C₂

socle

C₂²×C₇₁

#120

hypercenter

Q₈²

D₇₁

hypocenter

C₇₁

#1633

LCS

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₇₁	Q₁₆	semidirect
56. C₇₁	#1633	semidirect
92. Q₈	#11765855	semidirect

H ≅ G/N

type

35. Q₈×C₇₁

Q₁₆

semidirect

56. C₇₁

#1633

semidirect

92. Q₈

#11765855

semidirect

Subgroups — #22587480

Conjugacy classes of subgroups

Complements