Group #22587986 — subgroups

Subgroups — #22587986

Order 9088.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986404	C₂	1	1	yes	maximal, characteristic
2	#17986397	C₂	1	1	yes	maximal, characteristic
3	#17986350	C₂	1	1	yes	maximal, characteristic
4	#17986350	C₂	1	1	yes	maximal, characteristic
5	#17986350	C₂	1	1	yes	maximal, characteristic
6	#17986350	C₂	1	1	yes	maximal, characteristic
7	#322×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556128	C₂²	1	1	yes	characteristic
12	#14556062	—	2	2	no	—
13	#14556060	—	2	2	no	—
14	#14556053	—	2	2	no	—
15	#14556053	—	2	2	no	—
16	#14556053	—	2	2	no	—
17	#14556053	—	2	2	no	—
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×Dic₁₄₂	—	2	1	no	—
22	C₂×Dic₁₄₂	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₄⋊Dic₇₁	—	2	1	no	—
26	C₄⋊Dic₇₁	—	2	1	no	—
27	C₄⋊Dic₇₁	—	2	1	no	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₂×C₈×C₇₁	D₄	1	2	yes	—
34	C₂×C₈×C₇₁	D₄	1	2	yes	—
35	C₂×C₈×C₇₁	D₄	1	2	yes	—
36	C₄²×C₇₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₇₁	—	4	1	no	—
41	C₂×Dic₇₁	—	4	1	no	—
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	—	4	1	no	—
44	Dic₁₄₂	—	4	2	no	—
45	Dic₁₄₂	—	4	2	no	—
46	Dic₁₄₂	—	4	2	no	—
47	Dic₁₄₂	—	4	2	no	—
48	C₈×C₇₁	—	2	2	no	—
49	C₈×C₇₁	—	2	2	no	—
50	C₈×C₇₁	—	2	2	no	—
51	C₂²×C₇₁	#127	1	1	yes	characteristic
52	C₄×C₇₁	#135	1	2	yes	—
53	C₄×C₇₁	#137	1	2	yes	—
54	C₄×C₇₁	#137	1	2	yes	—
55	Dic₇₁	—	8	1	no	—
56	Dic₇₁	—	8	1	no	—
57	Dic₇₁	—	8	1	no	—
58	Dic₇₁	—	8	1	no	—
59	C₂×C₇₁	#495	1	1	yes	characteristic
60	C₂×C₇₁	#497	1	1	yes	characteristic
61	C₂×C₇₁	#497	1	1	yes	characteristic
62	#1718	—	71	1	no	maximal, Sylow
63	C₇₁	#1718	1	1	yes	characteristic, complemented, Sylow
64	#500	—	71	1	no	—
65	#498	—	71	1	no	—
66	#487	—	71	1	no	—
67	#487	—	71	1	no	—
68	#487	—	71	1	no	—
69	#487	—	71	1	no	—
70	#322	D₇₁	1	1	yes	characteristic
71	C₄⋊Q₈	—	71	1	no	—
72	C₄⋊Q₈	—	71	1	no	—
73	C₄⋊Q₈	—	71	1	no	—
74	C₄⋊Q₈	—	71	1	no	—
75	#107	—	142	2	no	—
76	#106	—	142	2	no	—
77	#103	—	142	2	no	—
78	#103	—	142	2	no	—
79	#103	—	142	2	no	—
80	#103	—	142	2	no	—
81	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
82	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
83	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
84	C₂×Q₈	—	142	1	no	—
85	C₂×Q₈	—	142	1	no	—
86	C₂×Q₈	—	142	1	no	—
87	C₂×Q₈	—	142	1	no	—
88	C₂×C₈	D₂₈₄	1	2	yes	—
89	C₂×C₈	D₂₈₄	1	2	yes	—
90	C₂×C₈	D₂₈₄	1	2	yes	—
91	C₄⋊C₄	—	142	1	no	—
92	C₄⋊C₄	—	142	1	no	—
93	C₄⋊C₄	—	142	1	no	—
94	C₄⋊C₄	—	142	1	no	—
95	C₄⋊C₄	—	142	1	no	—
96	C₄⋊C₄	—	142	1	no	—
97	C₄⋊C₄	—	142	1	no	—
98	C₄⋊C₄	—	142	1	no	—
99	C₄²	C₂²×D₇₁	1	1	yes	characteristic
100	Q₈	—	284	2	no	—
101	Q₈	—	284	2	no	—
102	Q₈	—	284	2	no	—
103	Q₈	—	284	2	no	—
104	C₂×C₄	—	284	1	no	—
105	C₂×C₄	—	284	1	no	—
106	C₂×C₄	—	284	1	no	—
107	C₂×C₄	—	284	1	no	—
108	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
109	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₂²	#14556125	1	1	yes	characteristic
115	C₄	—	568	1	no	—
116	C₄	—	568	1	no	—
117	C₄	—	568	1	no	—
118	C₄	—	568	1	no	—
119	C₄	#14556150	1	2	yes	—
120	C₄	#14556161	1	2	yes	—
121	C₄	#14556161	1	2	yes	—
122	C₂	#17986383	1	1	yes	characteristic
123	C₂	#17986393	1	1	yes	characteristic
124	C₂	#17986393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986404

C₂

yes

maximal, characteristic

#17986397

C₂

yes

maximal, characteristic

#17986350

C₂

yes

maximal, characteristic

#17986350

C₂

yes

maximal, characteristic

#17986350

C₂

yes

maximal, characteristic

#17986350

C₂

yes

maximal, characteristic

#322×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

—

—

—

—

—

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

#127

yes

characteristic

C₄×C₇₁

#135

yes

—

C₄×C₇₁

#137

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#107

—

142

—

#106

—

142

—

#103

—

142

—

#103

—

142

—

#103

—

142

—

#103

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

D₂₈₄

yes

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

100

Q₈

—

284

—

101

Q₈

—

284

—

102

Q₈

—

284

—

103

Q₈

—

284

—

104

C₂×C₄

—

284

—

105

C₂×C₄

—

284

—

106

C₂×C₄

—

284

—

107

C₂×C₄

—

284

—

108

C₂×C₄

C₂×D₂₈₄

yes

characteristic

109

C₂×C₄

C₂×D₂₈₄

yes

characteristic

110

C₂×C₄

C₂×D₂₈₄

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

#14556125

yes

characteristic

115

C₄

—

568

—

116

C₄

—

568

—

117

C₄

—

568

—

118

C₄

—

568

—

119

C₄

#14556150

yes

—

120

C₄

#14556161

yes

—

121

C₄

#14556161

yes

—

122

C₂

#17986383

yes

characteristic

123

C₂

#17986393

yes

characteristic

124

C₂

#17986393

yes

characteristic

	subgroup	G/H
center	C₂²	#14556125
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#322×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#322	D₇₁
hypocenter	C₇₁	#1718
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#14556125

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#322×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
63. C₇₁	#1718	semidirect

H ≅ G/N

type

63. C₇₁

#1718

semidirect

Subgroups — #22587986

Conjugacy classes of subgroups

Complements