Group #22590171 — subgroups

Subgroups — #22590171

Order 9088.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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×#14556067	C₂	1	2	yes	maximal, complemented
2	C₈×D₂₈₄	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₇₁)⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇₁	C₂²	1	2	yes	—
7	C₄×D₂₈₄	C₄	1	1	yes	characteristic
8	#14556067	—	2	4	no	—
9	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
12	(C₂×D₇₁)⋊C₈	C₄	1	2	yes	—
13	C₂×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×D₂₈₄	C₈	1	1	yes	characteristic
16	C₂×C₄×D₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
18	C₈×D₇₁	—	2	2	no	—
19	C₈×D₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	C₈	1	1	yes	characteristic
21	(C₂×D₇₁)⋊C₄	C₈	1	2	yes	—
22	C₇₁⋊C₁₆	—	2	1	no	—
23	C₇₁⋊C₁₆	D₄	1	2	yes	complemented
24	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₇₁	—	2	2	no	—
28	C₂²×D₇₁	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
33	D₂₈₄	—	2	2	no	—
34	C₄×D₇₁	—	2	2	no	—
35	C₄×D₇₁	—	2	2	no	—
36	C₈×C₇₁	—	2	1	no	—
37	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₇₁	#41	1	1	yes	characteristic
39	C₇₁⋊C₈	—	2	2	no	—
40	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
41	C₂×D₇₁	—	2	2	no	—
42	C₂×D₇₁	—	2	2	no	—
43	C₂×D₇₁	—	2	2	no	—
44	C₄×C₇₁	—	2	1	no	—
45	C₄×C₇₁	M₃₂	1	2	yes	complemented
46	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
47	C₄×C₇₁	#131	1	1	yes	characteristic
48	Dic₇₁	—	2	2	no	—
49	C₂×C₇₁	C₈×D₄	1	1	yes	characteristic
50	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
51	C₂×C₇₁	#504	1	1	yes	characteristic
52	D₇₁	—	4	2	no	—
53	#2169	—	71	1	no	maximal, Sylow
54	C₇₁	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	71	2	no	—
56	C₈×D₄	—	71	1	no	—
57	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic, complemented
58	C₂²⋊C₁₆	—	71	2	no	—
59	C₄×C₁₆	—	71	1	no	—
60	C₂²×C₈	—	71	2	no	—
61	C₄×D₄	—	71	1	no	—
62	M₃₂	—	142	4	no	—
63	C₂×C₁₆	—	71	1	no	—
64	C₂×C₁₆	—	71	1	no	—
65	C₂×C₁₆	C₂×D₇₁	1	2	yes	—
66	C₄⋊C₈	—	71	1	no	—
67	C₂²⋊C₈	—	71	2	no	—
68	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
69	C₂×D₄	—	71	1	no	—
70	C₂²×C₄	—	71	2	no	—
71	C₂×C₈	—	71	2	no	—
72	C₂×C₈	—	142	2	no	—
73	C₂×C₈	—	142	2	no	—
74	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
76	C₄⋊C₄	—	71	1	no	—
77	C₂²⋊C₄	—	71	2	no	—
78	C₄²	C₄×D₇₁	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	C₁₆	—	71	2	no	—
81	C₁₆	—	142	1	no	—
82	C₂³	—	71	2	no	—
83	D₄	—	142	2	no	—
84	C₂×C₄	—	71	2	no	—
85	C₂×C₄	—	142	2	no	—
86	C₂×C₄	—	142	2	no	—
87	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
88	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	142	2	no	—
92	C₈	D₄×D₇₁	1	1	yes	characteristic
93	C₈	#11765866	1	1	yes	characteristic
94	C₂²	—	142	2	no	—
95	C₂²	—	142	2	no	—
96	C₂²	—	142	2	no	—
97	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	142	2	no	—
100	C₄	#14556067	1	2	yes	complemented
101	C₄	#14556090	1	1	yes	characteristic
102	C₄	#14556136	1	1	yes	characteristic
103	C₂	—	284	2	no	—
104	C₂	#17986064	1	1	yes	characteristic
105	C₂	C₂×#14556067	1	1	yes	characteristic
106	C₂	#17986417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556067

C₂

yes

maximal, complemented

C₈×D₂₈₄

C₂

yes

maximal, characteristic

(C₂×D₇₁)⋊C₁₆

C₂

yes

maximal, complemented

C₄×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₁

C₂²

yes

—

C₄×D₂₈₄

C₄

yes

characteristic

#14556067

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₄

yes

characteristic

(C₂×D₇₁)⋊C₈

C₄

yes

—

C₂×C₁₆×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

C₈

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

C₈

yes

characteristic

(C₂×D₇₁)⋊C₄

C₈

yes

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

D₄

yes

complemented

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

M₃₂

yes

complemented

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

#131

yes

characteristic

Dic₇₁

—

C₂×C₇₁

C₈×D₄

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

characteristic

C₂×C₇₁

#504

yes

characteristic

D₇₁

—

#2169

—

maximal, Sylow

C₇₁

#2169

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

D₇₁

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₄×D₄

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₄×D₇₁

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

142

—

C₂³

—

D₄

—

142

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₈×D₇₁

yes

characteristic

C₂×C₄

C₈×D₇₁

yes

characteristic

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₈

—

C₈

—

142

—

C₈

D₄×D₇₁

yes

characteristic

C₈

#11765866

yes

characteristic

C₂²

—

142

—

C₂²

—

142

—

C₂²

—

142

—

C₂²

C₂×C₈×D₇₁

yes

characteristic

C₄

—

C₄

—

142

—

100

C₄

#14556067

yes

complemented

101

C₄

#14556090

yes

characteristic

102

C₄

#14556136

yes

characteristic

103

C₂

—

284

—

104

C₂

#17986064

yes

characteristic

105

C₂

C₂×#14556067

yes

characteristic

106

C₂

#17986417

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₈
Fitting	C₄⋊C₁₆×C₇₁	C₂
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#2169
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₈

Fitting

C₄⋊C₁₆×C₇₁

C₂

hypercenter

C₄⋊C₁₆

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×#14556067	C₂	semidirect
3. (C₂×D₇₁)⋊C₁₆	C₂	semidirect
4. C₄×C₇₁⋊C₁₆	C₂	semidirect
5. C₄⋊C₁₆×C₇₁	C₂	semidirect
23. C₇₁⋊C₁₆	D₄	semidirect
45. C₄×C₇₁	M₃₂	semidirect
54. C₇₁	#2169	semidirect
57. C₄⋊C₁₆	D₇₁	semidirect
100. C₄	#14556067	semidirect

H ≅ G/N

type

1. C₂×#14556067

C₂

semidirect

3. (C₂×D₇₁)⋊C₁₆

C₂

semidirect

4. C₄×C₇₁⋊C₁₆

C₂

semidirect

5. C₄⋊C₁₆×C₇₁

C₂

semidirect

23. C₇₁⋊C₁₆

D₄

semidirect

45. C₄×C₇₁

M₃₂

semidirect

54. C₇₁

#2169

semidirect

57. C₄⋊C₁₆

D₇₁

semidirect

100. C₄

#14556067

semidirect

Subgroups — #22590171

Conjugacy classes of subgroups

Complements