Group #22589828 — subgroups

Subgroups — #22589828

Order 9088.

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

normal subgroups only

This group has 224 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	C₂×#14556062	C₂	1	1	yes	maximal, characteristic
2	#17985897	C₂	1	2	yes	maximal
3	#17985762	C₂	1	2	yes	maximal
4	#17985762	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₂×Dic₇₁⋊C₄	—	2	2	no	—
9	#14556062	—	2	2	no	—
10	C₇₁⋊#95	—	2	2	no	—
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	—	2	2	no	—
15	C₂²×Dic₇₁	—	2	2	no	—
16	C₂²×Dic₇₁	—	2	2	no	—
17	C₄⋊Dic₇₁	—	2	2	no	—
18	C₄⋊Dic₇₁	—	2	2	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	Q₈	1	4	yes	—
21	Dic₇₁⋊C₄	—	4	4	no	—
22	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₇₁	—	2	2	no	—
26	C₂×C₈×C₇₁	—	2	1	no	—
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	D₄	1	2	yes	—
29	C₄²×C₇₁	—	2	1	no	—
30	C₄²×C₇₁	—	2	1	no	—
31	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	1	no	—
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	—	2	2	no	—
36	C₂×C₄×C₇₁	D₈	1	2	yes	—
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₇₁	#41	1	2	yes	—
40	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
41	C₂×Dic₇₁	—	2	4	no	—
42	C₂×Dic₇₁	—	4	2	no	—
43	C₂×Dic₇₁	—	4	2	no	—
44	C₂×Dic₇₁	—	4	2	no	—
45	C₂×Dic₇₁	—	4	2	no	—
46	C₂×Dic₇₁	—	4	2	no	—
47	C₈×C₇₁	—	2	2	no	—
48	C₈×C₇₁	—	4	1	no	—
49	C₂²×C₇₁	#121	1	1	yes	characteristic
50	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₇₁	#135	1	1	yes	characteristic
53	C₂²×C₇₁	#137	1	2	yes	—
54	C₄×C₇₁	—	2	2	no	—
55	C₄×C₇₁	—	4	1	no	—
56	C₄×C₇₁	#123	1	1	yes	characteristic
57	C₄×C₇₁	#125	1	2	yes	—
58	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₇₁	—	4	4	no	—
60	Dic₇₁	—	8	2	no	—
61	C₂×C₇₁	#398	1	1	yes	characteristic
62	C₂×C₇₁	#466	1	1	yes	characteristic
63	C₂×C₇₁	#470	1	1	yes	characteristic
64	C₂×C₇₁	#474	1	2	yes	—
65	C₂×C₇₁	#479	1	2	yes	—
66	#2086	—	71	1	no	maximal, Sylow
67	C₇₁	#2086	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	71	1	no	—
69	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
70	#389	—	71	2	no	—
71	#340	—	71	1	no	—
72	#340	—	71	2	no	—
73	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	71	2	no	—
76	C₂×C₄⋊C₄	—	71	2	no	—
77	C₂×C₄⋊C₄	—	142	2	no	—
78	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
79	#107	—	142	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	142	2	no	—
82	C₂²×C₄	—	142	2	no	—
83	C₂²×C₄	—	142	2	no	—
84	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
85	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
91	C₄⋊C₄	—	284	4	no	—
92	C₄⋊C₄	—	71	4	no	—
93	C₄⋊C₄	—	142	2	no	—
94	C₄⋊C₄	—	142	2	no	—
95	C₄⋊C₄	—	142	2	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	284	2	no	—
104	C₂×C₄	—	284	2	no	—
105	C₂×C₄	—	284	2	no	—
106	C₂×C₄	—	284	2	no	—
107	C₂×C₄	—	284	2	no	—
108	C₂×C₄	—	142	4	no	—
109	C₂×C₄	D₅₆₈	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₇₁	1	2	yes	—
112	C₂×C₄	#11765864	1	2	yes	—
113	C₂×C₄	#11765865	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#14556102	1	1	yes	characteristic
117	C₂²	#14556108	1	2	yes	—
118	C₂²	C₂×D₅₆₈	1	1	yes	characteristic
119	C₂²	#14556150	1	1	yes	characteristic
120	C₂²	#14556161	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	284	4	no	—
124	C₄	—	568	2	no	—
125	C₄	#14556109	1	1	yes	characteristic
126	C₄	#14556118	1	2	yes	—
127	C₄	#14556126	1	1	yes	characteristic
128	C₂	#17985929	1	1	yes	characteristic
129	C₂	#17986235	1	1	yes	characteristic
130	C₂	#17986257	1	1	yes	characteristic
131	C₂	#17986280	1	2	yes	—
132	C₂	#17986307	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556062

C₂

yes

maximal, characteristic

#17985897

C₂

yes

maximal

#17985762

C₂

yes

maximal

#17985762

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

#14556062

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

Q₈

yes

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₂²×C₇₁

C₂×D₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#123

yes

characteristic

C₄×C₇₁

#125

yes

—

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#479

yes

—

#2086

—

maximal, Sylow

C₇₁

#2086

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₄⋊C₈

D₇₁

yes

—

—

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#107

—

142

—

C₄⋊C₈

—

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

C₄²

—

C₂³

C₂×D₂₈₄

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

284

—

104

C₂×C₄

—

284

—

105

C₂×C₄

—

284

—

106

C₂×C₄

—

284

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

142

—

109

C₂×C₄

D₅₆₈

yes

—

110

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

111

C₂×C₄

Q₈×D₇₁

yes

—

112

C₂×C₄

#11765864

yes

—

113

C₂×C₄

#11765865

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#14556102

yes

characteristic

117

C₂²

#14556108

yes

—

118

C₂²

C₂×D₅₆₈

yes

characteristic

119

C₂²

#14556150

yes

characteristic

120

C₂²

#14556161

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

284

—

124

C₄

—

568

—

125

C₄

#14556109

yes

characteristic

126

C₄

#14556118

yes

—

127

C₄

#14556126

yes

characteristic

128

C₂

#17985929

yes

characteristic

129

C₂

#17986235

yes

characteristic

130

C₂

#17986257

yes

characteristic

131

C₂

#17986280

yes

—

132

C₂

#17986307

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₈₄
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#2086
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₈₄

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₁

hypocenter

C₇₁

#2086

LCS

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
67. C₇₁	#2086	semidirect

H ≅ G/N

type

67. C₇₁

#2086

semidirect

Subgroups — #22589828

Conjugacy classes of subgroups

Complements