Group #22590382 — subgroups

Subgroups — #22590382

Order 9088.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14556071	C₂	1	2	yes	maximal
2	C₄×D₅₆₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985803	C₂	1	2	yes	maximal, complemented
5	#494×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
7	C₄×D₂₈₄	—	2	1	no	—
8	#14556071	—	2	4	no	—
9	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
10	#14556062	C₂²	1	1	yes	characteristic
11	#14556050	—	2	1	no	—
12	C₂×Q₁₆×C₇₁	—	2	1	no	—
13	C₂×Q₁₆×C₇₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₁	—	2	1	no	—
15	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×D₂₈₄	—	2	1	no	—
17	C₂×C₄×D₇₁	—	4	1	no	—
18	D₅₆₈	—	2	1	no	—
19	D₅₆₈	D₄	1	2	yes	—
20	C₄⋊Dic₇₁	—	2	1	no	—
21	(C₂×D₇₁)⋊C₄	—	4	1	no	—
22	C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×Q₈×C₇₁	—	2	2	no	—
24	Q₁₆×C₇₁	—	2	2	no	—
25	Q₁₆×C₇₁	—	2	2	no	—
26	Q₁₆×C₇₁	—	4	2	no	—
27	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	4	1	no	—
30	C₄⋊C₄×C₇₁	—	4	1	no	—
31	C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×D₇₁	—	4	1	no	—
33	Q₈×C₇₁	—	4	2	no	—
34	Q₈×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	—	4	2	no	—
36	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
37	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₁	—	4	1	no	—
40	D₂₈₄	—	2	2	no	—
41	D₂₈₄	—	4	1	no	—
42	C₄×D₇₁	—	8	2	no	—
43	C₈×C₇₁	—	2	1	no	—
44	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₇₁	#41	1	1	yes	characteristic
46	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
47	C₂×D₇₁	—	4	2	no	—
48	C₂×D₇₁	—	8	1	no	—
49	C₂×D₇₁	—	8	1	no	—
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	—	8	2	no	—
52	C₄×C₇₁	SD₃₂	1	2	yes	complemented
53	C₄×C₇₁	#121	1	1	yes	characteristic
54	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	Dic₇₁	—	8	1	no	—
56	C₂×C₇₁	#459	1	1	yes	characteristic
57	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
58	C₂×C₇₁	#510	1	1	yes	characteristic
59	D₇₁	—	8	2	no	—
60	#2208	—	71	1	no	maximal, Sylow
61	C₇₁	#2208	1	1	yes	characteristic, complemented, Sylow
62	C₂×SD₃₂	—	71	2	no	—
63	#494	D₇₁	1	1	yes	characteristic, complemented
64	C₄×D₈	—	71	1	no	—
65	C₄⋊C₁₆	—	71	1	no	—
66	#358	—	71	2	no	—
67	C₂×Q₁₆	—	2	1	no	—
68	C₂×Q₁₆	C₂×D₇₁	1	2	yes	—
69	C₂×D₈	—	71	1	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄×D₄	—	142	1	no	—
72	SD₃₂	—	142	4	no	—
73	C₂×C₁₆	—	71	2	no	—
74	#107	—	71	1	no	—
75	#102	—	142	1	no	—
76	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
77	C₂×Q₈	—	2	2	no	—
78	C₂×D₄	—	142	1	no	—
79	C₂²×C₄	—	284	1	no	—
80	Q₁₆	—	2	2	no	—
81	Q₁₆	—	2	2	no	—
82	Q₁₆	—	4	2	no	—
83	D₈	—	71	2	no	—
84	D₈	—	142	1	no	—
85	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	142	1	no	—
90	C₂²⋊C₄	—	284	1	no	—
91	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
92	C₁₆	—	142	2	no	—
93	C₂³	—	284	1	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	284	1	no	—
97	D₄	—	142	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	284	1	no	—
100	C₂×C₄	—	568	2	no	—
101	C₂×C₄	#11765850	1	1	yes	characteristic
102	C₂×C₄	#11765850	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	D₄×D₇₁	1	1	yes	characteristic
106	C₈	#11765864	1	1	yes	characteristic
107	C₂²	—	284	2	no	—
108	C₂²	—	568	1	no	—
109	C₂²	—	568	1	no	—
110	C₂²	C₂×#11765850	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	8	2	no	—
113	C₄	—	568	1	no	—
114	C₄	#14556071	1	2	yes	complemented
115	C₄	#14556099	1	1	yes	characteristic
116	C₄	#14556154	1	1	yes	characteristic
117	C₂	—	568	2	no	—
118	C₂	#17986191	1	1	yes	characteristic
119	C₂	C₂×#14556071	1	1	yes	characteristic
120	C₂	#17986440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556071

C₂

yes

maximal

C₄×D₅₆₈

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#17985803

C₂

yes

maximal, complemented

#494×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

characteristic

C₄×D₂₈₄

—

#14556071

—

C₂×C₇₁⋊C₁₆

C₂²

yes

—

#14556062

C₂²

yes

characteristic

#14556050

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×C₄×D₇₁

—

D₅₆₈

—

D₅₆₈

D₄

yes

—

C₄⋊Dic₇₁

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

SD₃₂

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

#459

yes

characteristic

C₂×C₇₁

C₂×SD₃₂

yes

characteristic

C₂×C₇₁

#510

yes

characteristic

D₇₁

—

#2208

—

maximal, Sylow

C₇₁

#2208

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

#494

D₇₁

yes

characteristic, complemented

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₇₁

yes

—

C₂×D₈

—

C₄⋊Q₈

—

C₄×D₄

—

142

—

SD₃₂

—

142

—

C₂×C₁₆

—

#107

—

#102

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

142

—

C₂²×C₄

—

284

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₄²

C₂²⋊D₇₁

yes

characteristic

C₁₆

—

142

—

C₂³

—

284

—

Q₈

—

Q₈

—

D₄

—

284

—

D₄

—

142

—

C₂×C₄

—

C₂×C₄

—

284

—

100

C₂×C₄

—

568

—

101

C₂×C₄

#11765850

yes

characteristic

102

C₂×C₄

#11765850

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

104

C₈

—

105

C₈

D₄×D₇₁

yes

characteristic

106

C₈

#11765864

yes

characteristic

107

C₂²

—

284

—

108

C₂²

—

568

—

109

C₂²

—

568

—

110

C₂²

C₂×#11765850

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

568

—

114

C₄

#14556071

yes

complemented

115

C₄

#14556099

yes

characteristic

116

C₄

#14556154

yes

characteristic

117

C₂

—

568

—

118

C₂

#17986191

yes

characteristic

119

C₂

C₂×#14556071

yes

characteristic

120

C₂

#17986440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#494×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#494	D₇₁
hypocenter	C₇₁	#2208
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

subgroup

G/H

center

C₂²

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#494×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

N	H ≅ G/N	type
3. C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
4. #17985803	C₂	semidirect
5. #494×C₇₁	C₂	semidirect
52. C₄×C₇₁	SD₃₂	semidirect
61. C₇₁	#2208	semidirect
63. #494	D₇₁	semidirect
114. C₄	#14556071	semidirect

H ≅ G/N

type

3. C₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

4. #17985803

C₂

semidirect

5. #494×C₇₁

C₂

semidirect

52. C₄×C₇₁

SD₃₂

semidirect

61. C₇₁

#2208

semidirect

63. #494

D₇₁

semidirect

114. C₄

#14556071

semidirect

Subgroups — #22590382

Conjugacy classes of subgroups

Complements