Group #22586641 — subgroups

Subgroups — #22586641

Order 9088.

178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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	#17986050	C₂	1	1	yes	maximal, characteristic
2	#17986050	C₂	1	1	yes	maximal, characteristic
3	#17986050	C₂	1	1	yes	maximal, characteristic
4	#17985721	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985720	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#322×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₇₁	C₂²	1	1	yes	characteristic
9	Dic₇₁⋊C₈	C₄	1	2	yes	—
10	Dic₇₁⋊C₈	C₄	1	2	yes	—
11	Dic₇₁⋊C₈	C₄	1	2	yes	—
12	(C₂×D₇₁)⋊C₈	C₄	1	2	yes	—
13	(C₂×D₇₁)⋊C₈	C₄	1	2	yes	—
14	(C₂×D₇₁)⋊C₈	C₄	1	2	yes	—
15	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₂²×D₇₁	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₇₁	C₄²	1	1	yes	characteristic
39	C₂×Dic₇₁	C₄²	1	1	yes	characteristic
40	C₂×Dic₇₁	C₄²	1	1	yes	characteristic
41	C₄×D₇₁	—	2	2	no	—
42	C₄×D₇₁	—	2	2	no	—
43	C₄×D₇₁	—	2	2	no	—
44	C₈×C₇₁	—	2	2	no	—
45	C₈×C₇₁	—	2	2	no	—
46	C₈×C₇₁	—	2	2	no	—
47	C₇₁⋊C₈	—	2	2	no	—
48	C₇₁⋊C₈	—	2	2	no	—
49	C₇₁⋊C₈	—	2	2	no	—
50	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
51	C₂×D₇₁	—	2	1	no	—
52	C₂×D₇₁	—	2	1	no	—
53	C₂×D₇₁	—	2	1	no	—
54	C₄×C₇₁	#131	1	2	yes	—
55	C₄×C₇₁	#131	1	2	yes	—
56	C₄×C₇₁	#131	1	2	yes	—
57	Dic₇₁	—	2	1	no	—
58	Dic₇₁	—	2	1	no	—
59	Dic₇₁	—	2	1	no	—
60	C₂×C₇₁	#405	1	1	yes	characteristic
61	C₂×C₇₁	#405	1	1	yes	characteristic
62	C₂×C₇₁	#405	1	1	yes	characteristic
63	D₇₁	—	4	1	no	—
64	#1454	—	71	1	no	maximal, Sylow
65	C₇₁	#1454	1	1	yes	characteristic, complemented, Sylow
66	#431	—	71	1	no	—
67	#431	—	71	1	no	—
68	#431	—	71	1	no	—
69	#322	—	71	1	no	—
70	#322	—	71	1	no	—
71	#322	D₇₁	1	1	yes	characteristic, complemented
72	C₈²	—	71	1	no	—
73	C₂×C₄²	—	71	1	no	—
74	C₄⋊C₈	—	71	2	no	—
75	C₄⋊C₈	—	71	2	no	—
76	C₄⋊C₈	—	71	2	no	—
77	C₂²⋊C₈	—	71	2	no	—
78	C₂²⋊C₈	—	71	2	no	—
79	C₂²⋊C₈	—	71	2	no	—
80	C₄×C₈	—	71	1	no	—
81	C₄×C₈	—	71	1	no	—
82	C₄×C₈	—	71	1	no	—
83	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
84	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
85	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
86	C₂²×C₄	—	71	1	no	—
87	C₂²×C₄	—	71	1	no	—
88	C₂²×C₄	—	71	1	no	—
89	C₂×C₈	—	71	2	no	—
90	C₂×C₈	—	71	2	no	—
91	C₂×C₈	—	71	2	no	—
92	C₂×C₈	C₄×D₇₁	1	2	yes	—
93	C₂×C₈	C₄×D₇₁	1	2	yes	—
94	C₂×C₈	C₄×D₇₁	1	2	yes	—
95	C₄²	—	71	1	no	—
96	C₄²	—	71	1	no	—
97	C₄²	—	71	1	no	—
98	C₄²	C₂²×D₇₁	1	1	yes	characteristic
99	C₂³	—	71	1	no	—
100	C₂×C₄	—	71	1	no	—
101	C₂×C₄	—	71	1	no	—
102	C₂×C₄	—	71	1	no	—
103	C₂×C₄	—	142	2	no	—
104	C₂×C₄	—	142	2	no	—
105	C₂×C₄	—	142	2	no	—
106	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	142	2	no	—
113	C₈	—	142	2	no	—
114	C₈	—	142	2	no	—
115	C₂²	—	142	1	no	—
116	C₂²	—	142	1	no	—
117	C₂²	—	142	1	no	—
118	C₂²	C₄²×D₇₁	1	1	yes	characteristic
119	C₄	—	142	1	no	—
120	C₄	—	142	1	no	—
121	C₄	—	142	1	no	—
122	C₄	#14556134	1	2	yes	—
123	C₄	#14556136	1	2	yes	—
124	C₄	#14556136	1	2	yes	—
125	C₂	—	284	1	no	—
126	C₂	#17985953	1	1	yes	characteristic
127	C₂	#17985956	1	1	yes	characteristic
128	C₂	#17985956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986050

C₂

yes

maximal, characteristic

#17986050

C₂

yes

maximal, characteristic

#17986050

C₂

yes

maximal, characteristic

#17985721

C₂

yes

maximal, characteristic, complemented

#17985720

C₂

yes

maximal, characteristic, complemented

C₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#322×C₇₁

C₂

yes

maximal, characteristic, complemented

C₄²×D₇₁

C₂²

yes

characteristic

Dic₇₁⋊C₈

C₄

yes

—

Dic₇₁⋊C₈

C₄

yes

—

Dic₇₁⋊C₈

C₄

yes

—

(C₂×D₇₁)⋊C₈

C₄

yes

—

(C₂×D₇₁)⋊C₈

C₄

yes

—

(C₂×D₇₁)⋊C₈

C₄

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

C₄²

yes

characteristic

C₂×Dic₇₁

C₄²

yes

characteristic

C₂×Dic₇₁

C₄²

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

#131

yes

—

C₄×C₇₁

#131

yes

—

C₄×C₇₁

#131

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1454

—

maximal, Sylow

C₇₁

#1454

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₈²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

C₄×D₇₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

142

—

104

C₂×C₄

—

142

—

105

C₂×C₄

—

142

—

106

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

107

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

108

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

142

—

113

C₈

—

142

—

114

C₈

—

142

—

115

C₂²

—

142

—

116

C₂²

—

142

—

117

C₂²

—

142

—

118

C₂²

C₄²×D₇₁

yes

characteristic

119

C₄

—

142

—

120

C₄

—

142

—

121

C₄

—

142

—

122

C₄

#14556134

yes

—

123

C₄

#14556136

yes

—

124

C₄

#14556136

yes

—

125

C₂

—

284

—

126

C₂

#17985953

yes

characteristic

127

C₂

#17985956

yes

characteristic

128

C₂

#17985956

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂×C₄²
Fitting	#322×C₇₁	C₂
hypercenter	#322	D₇₁
hypocenter	C₇₁	#1454
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂×C₄²

Fitting

#322×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. #17985721	C₂	semidirect
5. #17985720	C₂	semidirect
6. C₈×C₇₁⋊C₈	C₂	semidirect
7. #322×C₇₁	C₂	semidirect
65. C₇₁	#1454	semidirect
71. #322	D₇₁	semidirect

H ≅ G/N

type

4. #17985721

C₂

semidirect

5. #17985720

C₂

semidirect

6. C₈×C₇₁⋊C₈

C₂

semidirect

7. #322×C₇₁

C₂

semidirect

65. C₇₁

#1454

semidirect

71. #322

D₇₁

semidirect

Subgroups — #22586641

Conjugacy classes of subgroups

Complements