Group #22587095 — subgroups

Subgroups — #22587095

Order 9088.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	#17986023	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#17985749	C₂	1	4	yes	maximal, complemented
4	#423×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	—	2	1	no	—
6	C₂²×C₇₁⋊C₈	C₄	1	2	yes	—
7	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
10	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
11	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
12	C₂×M₁₆×C₇₁	—	2	1	no	—
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	—	2	1	no	—
15	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	#11765849	—	4	2	no	—
17	C₂×C₇₁⋊C₈	—	2	4	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
24	M₁₆×C₇₁	—	4	2	no	—
25	C₂×C₈×C₇₁	—	2	2	no	—
26	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₇₁	D₄	1	1	yes	characteristic
28	C₄²×C₇₁	Q₈	1	1	yes	characteristic
29	C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
37	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₇₁	—	4	2	no	—
42	C₇₁⋊C₈	—	2	2	no	—
43	C₇₁⋊C₈	—	4	2	no	—
44	C₇₁⋊C₈	M₁₆	1	4	yes	complemented
45	C₂²×C₇₁	—	2	1	no	—
46	C₂²×C₇₁	—	2	1	no	—
47	C₂²×C₇₁	#106	1	2	yes	—
48	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	#106	1	2	yes	—
52	C₄×C₇₁	#108	1	2	yes	—
53	C₄×C₇₁	C₂×M₁₆	1	2	yes	—
54	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
55	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₇₁	—	2	2	no	—
57	C₂×C₇₁	#423	1	1	yes	characteristic
58	C₂×C₇₁	C₂×#106	1	1	yes	characteristic
59	C₂×C₇₁	C₂×#108	1	1	yes	characteristic
60	#1566	—	71	1	no	maximal, Sylow
61	C₇₁	#1566	1	1	yes	characteristic, complemented, Sylow
62	#423	—	71	1	no	—
63	#423	D₇₁	1	1	yes	characteristic
64	C₂×C₄×C₈	—	71	1	no	—
65	#334	—	71	4	no	—
66	C₂×M₁₆	—	2	1	no	—
67	C₂×M₁₆	—	142	1	no	—
68	C₂²×C₈	—	71	2	no	—
69	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	71	2	no	—
72	C₄⋊C₈	—	142	1	no	—
73	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
74	C₄×C₈	—	71	1	no	—
75	C₄×C₈	—	71	1	no	—
76	C₄×C₈	—	71	2	no	—
77	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	284	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	71	2	no	—
84	C₂×C₈	—	71	2	no	—
85	C₂×C₈	—	142	2	no	—
86	C₂×C₈	—	142	4	no	—
87	C₄²	Dic₁₄₂	1	1	yes	characteristic
88	C₄²	C₄×D₇₁	1	1	yes	characteristic
89	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
90	C₄²	C₂²×D₇₁	1	1	yes	characteristic
91	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#11765852	1	2	yes	—
98	C₂×C₄	#11765853	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	284	2	no	—
104	C₈	—	71	4	no	—
105	C₈	—	142	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#14556059	1	2	yes	—
109	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#14556059	1	2	yes	—
113	C₄	#14556063	1	2	yes	—
114	C₄	M₁₆×D₇₁	1	2	yes	—
115	C₄	C₂×#11765852	1	1	yes	characteristic
116	C₄	C₂×#11765853	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#17986022	1	1	yes	characteristic
119	C₂	C₂×#14556059	1	1	yes	characteristic
120	C₂	C₂×#14556063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986023

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, complemented

#423×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₄×C₇₁⋊C₈

C₄

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂×M₁₆×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

M₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#106

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#106

yes

—

C₄×C₇₁

#108

yes

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#423

—

#423

D₇₁

yes

characteristic

C₂×C₄×C₈

—

#334

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

C₄×D₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

#11765852

yes

—

C₂×C₄

#11765853

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

101

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

102

C₈

—

103

C₈

—

284

—

104

C₈

—

105

C₈

—

142

—

106

C₂²

—

107

C₂²

—

108

C₂²

#14556059

yes

—

109

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#14556059

yes

—

113

C₄

#14556063

yes

—

114

C₄

M₁₆×D₇₁

yes

—

115

C₄

C₂×#11765852

yes

characteristic

116

C₄

C₂×#11765853

yes

characteristic

117

C₂

—

118

C₂

#17986022

yes

characteristic

119

C₂

C₂×#14556059

yes

characteristic

120

C₂

C₂×#14556063

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#423×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	#423	D₇₁
hypocenter	C₇₁	#1566
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#423×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #17985749	C₂	semidirect
44. C₇₁⋊C₈	M₁₆	semidirect
61. C₇₁	#1566	semidirect

H ≅ G/N

type

3. #17985749

C₂

semidirect

44. C₇₁⋊C₈

M₁₆

semidirect

61. C₇₁

#1566

semidirect

Subgroups — #22587095

Conjugacy classes of subgroups

Complements