Group #22590426 — subgroups

Subgroups — #22590426

Order 9088.

174 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 57 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×D₁₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986235	C₂	1	2	yes	maximal, complemented
3	#17985825	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985801	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₆₈	C₂²	1	2	yes	—
7	#14556102	—	2	2	no	—
8	D₁₁₃₆	—	2	2	no	—
9	#14556062	C₂²	1	2	yes	—
10	#14556050	—	2	2	no	—
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₁	—	2	2	no	—
13	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×D₂₈₄	—	2	2	no	—
16	C₂×C₂²⋊D₇₁	—	4	2	no	—
17	D₅₆₈	—	2	2	no	—
18	D₅₆₈	—	2	2	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	(C₂×D₇₁)⋊C₄	—	4	2	no	—
21	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	—	2	1	no	—
23	C₂×C₈×C₇₁	—	2	1	no	—
24	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₁₆×C₇₁	—	2	1	no	—
27	C₁₆×C₇₁	D₄	1	2	yes	complemented
28	C₂³×C₇₁	D₈	1	1	yes	characteristic
29	C₂²×D₇₁	—	4	2	no	—
30	C₂×C₄×C₇₁	—	2	1	no	—
31	C₂×C₄×C₇₁	—	2	1	no	—
32	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₇₁	—	8	4	no	—
35	C₂×Dic₇₁	—	4	2	no	—
36	D₂₈₄	—	4	2	no	—
37	D₂₈₄	—	4	2	no	—
38	C₈×C₇₁	—	2	1	no	—
39	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₇₁	#41	1	1	yes	characteristic
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	D₁₆	1	2	yes	complemented
44	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
45	C₂×D₇₁	—	8	2	no	—
46	C₂×D₇₁	—	8	2	no	—
47	C₂×D₇₁	—	8	2	no	—
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	#121	1	1	yes	characteristic
50	C₄×C₇₁	#135	1	1	yes	characteristic
51	Dic₇₁	—	8	2	no	—
52	C₂×C₇₁	—	2	2	no	—
53	C₂×C₇₁	#466	1	1	yes	characteristic
54	C₂×C₇₁	C₂×D₁₆	1	1	yes	characteristic
55	C₂×C₇₁	#508	1	1	yes	characteristic
56	D₇₁	—	16	2	no	—
57	#2215	—	71	1	no	maximal, Sylow
58	C₇₁	#2215	1	1	yes	characteristic, complemented, Sylow
59	C₂×D₁₆	—	71	1	no	—
60	C₂²×C₁₆	D₇₁	1	1	yes	characteristic, complemented
61	#466	—	71	2	no	—
62	#366	—	71	1	no	—
63	#357	—	71	2	no	—
64	C₂×D₈	—	71	2	no	—
65	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
66	#121	—	142	2	no	—
67	D₁₆	—	142	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
70	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
71	#107	—	71	2	no	—
72	#102	—	142	2	no	—
73	C₂×D₄	—	284	2	no	—
74	C₂×D₄	—	142	2	no	—
75	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
76	D₈	—	142	2	no	—
77	D₈	—	142	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	D₂₈₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
82	C₄⋊C₄	—	142	2	no	—
83	C₂²⋊C₄	—	284	2	no	—
84	C₁₆	—	2	1	no	—
85	C₁₆	C₂²⋊D₇₁	1	2	yes	complemented
86	C₂³	—	284	2	no	—
87	C₂³	D₅₆₈	1	1	yes	characteristic
88	D₄	—	284	2	no	—
89	D₄	—	284	2	no	—
90	D₄	—	568	4	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	284	2	no	—
94	C₂×C₄	D₅₆₈	1	1	yes	characteristic
95	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
98	C₈	#11765865	1	1	yes	characteristic
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	568	2	no	—
102	C₂²	—	568	2	no	—
103	C₂²	—	568	2	no	—
104	C₂²	D₁₁₃₆	1	2	yes	complemented
105	C₂²	C₂×D₅₆₈	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	568	2	no	—
108	C₄	#14556102	1	1	yes	characteristic
109	C₄	#14556150	1	1	yes	characteristic
110	C₂	—	2	2	no	—
111	C₂	—	1136	2	no	—
112	C₂	#17986235	1	1	yes	characteristic
113	C₂	C₂×D₁₁₃₆	1	1	yes	characteristic
114	C₂	#17986431	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₁₃₆

C₂

yes

maximal, characteristic, complemented

#17986235

C₂

yes

maximal, complemented

#17985825

C₂

yes

maximal, characteristic, complemented

#17985801

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

—

#14556102

—

D₁₁₃₆

—

#14556062

C₂²

yes

—

#14556050

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×C₂²⋊D₇₁

—

D₅₆₈

—

D₅₆₈

—

C₄⋊Dic₇₁

—

(C₂×D₇₁)⋊C₄

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

D₄

yes

complemented

C₂³×C₇₁

D₈

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

D₁₆

yes

complemented

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#466

yes

characteristic

C₂×C₇₁

C₂×D₁₆

yes

characteristic

C₂×C₇₁

#508

yes

characteristic

D₇₁

—

#2215

—

maximal, Sylow

C₇₁

#2215

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂²×C₁₆

D₇₁

yes

characteristic, complemented

#466

—

#366

—

#357

—

C₂×D₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

#121

—

142

—

D₁₆

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#102

—

142

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

C₂²×C₄

D₂₈₄

yes

characteristic

D₈

—

142

—

D₈

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₁₆

—

C₁₆

C₂²⋊D₇₁

yes

complemented

C₂³

—

284

—

C₂³

D₅₆₈

yes

characteristic

D₄

—

284

—

D₄

—

284

—

D₄

—

568

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

D₅₆₈

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₈

—

C₈

C₂×C₂²⋊D₇₁

yes

characteristic

C₈

#11765865

yes

characteristic

C₂²

—

100

C₂²

—

101

C₂²

—

568

—

102

C₂²

—

568

—

103

C₂²

—

568

—

104

C₂²

D₁₁₃₆

yes

complemented

105

C₂²

C₂×D₅₆₈

yes

characteristic

106

C₄

—

107

C₄

—

568

—

108

C₄

#14556102

yes

characteristic

109

C₄

#14556150

yes

characteristic

110

C₂

—

111

C₂

—

1136

—

112

C₂

#17986235

yes

characteristic

113

C₂

C₂×D₁₁₃₆

yes

characteristic

114

C₂

#17986431

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₂²×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	C₂²×C₁₆	D₇₁
hypocenter	C₇₁	#2215
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

C₂×D₅₆₈

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₂²×C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

hypercenter

C₂²×C₁₆

D₇₁

hypocenter

C₇₁

#2215

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
1. C₂×D₁₁₃₆	C₂	semidirect
2. #17986235	C₂	semidirect
3. #17985825	C₂	semidirect
4. #17985801	C₂	semidirect
5. C₂²×C₁₆×C₇₁	C₂	semidirect
27. C₁₆×C₇₁	D₄	semidirect
43. C₂²×C₇₁	D₁₆	semidirect
58. C₇₁	#2215	semidirect
60. C₂²×C₁₆	D₇₁	semidirect
85. C₁₆	C₂²⋊D₇₁	semidirect
104. C₂²	D₁₁₃₆	semidirect

H ≅ G/N

type

1. C₂×D₁₁₃₆

C₂

semidirect

2. #17986235

C₂

semidirect

3. #17985825

C₂

semidirect

4. #17985801

C₂

semidirect

5. C₂²×C₁₆×C₇₁

C₂

semidirect

27. C₁₆×C₇₁

D₄

semidirect

43. C₂²×C₇₁

D₁₆

semidirect

58. C₇₁

#2215

semidirect

60. C₂²×C₁₆

D₇₁

semidirect

85. C₁₆

C₂²⋊D₇₁

semidirect

104. C₂²

D₁₁₃₆

semidirect

Subgroups — #22590426

Conjugacy classes of subgroups

Complements