Group #22587963 — subgroups

Subgroups — #22587963

Order 9088.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986395	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986373	C₂	1	1	yes	maximal, characteristic
3	#17986372	C₂	1	1	yes	maximal, characteristic
4	#17986349	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986347	C₂	1	1	yes	maximal, characteristic
6	#17985720	C₂	1	1	yes	maximal, characteristic, complemented
7	#500×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765854	—	2	2	no	—
9	C₂×#11765852	—	2	2	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556125	C₂²	1	1	yes	characteristic
12	#14556059	—	2	2	no	—
13	#14556052	—	2	2	no	—
14	#14556048	—	2	2	no	—
15	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
19	#107×C₇₁	—	2	2	no	—
20	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×D₂₈₄	—	2	1	no	—
23	C₂×D₂₈₄	—	2	1	no	—
24	C₂×D₂₈₄	—	2	1	no	—
25	#11765854	—	4	4	no	—
26	#11765852	—	4	4	no	—
27	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₂×Q₈×C₇₁	—	2	1	no	—
32	C₂×Q₈×C₇₁	—	2	1	no	—
33	C₂×C₈×C₇₁	D₄	1	2	yes	—
34	C₄⋊C₄×C₇₁	—	2	1	no	—
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	C₂²×D₇₁	—	4	1	no	—
40	Q₈×C₇₁	—	4	2	no	—
41	Q₈×C₇₁	—	4	2	no	—
42	C₂×C₄×C₇₁	—	4	1	no	—
43	C₂×C₄×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₇₁	—	4	1	no	—
48	D₂₈₄	—	4	2	no	—
49	D₂₈₄	—	4	2	no	—
50	D₂₈₄	—	4	2	no	—
51	Dic₁₄₂	—	4	2	no	—
52	C₈×C₇₁	SD₁₆	1	4	yes	complemented
53	C₇₁⋊C₈	—	2	2	no	—
54	C₇₁⋊C₈	—	2	2	no	—
55	C₂²×C₇₁	#127	1	1	yes	characteristic
56	C₂×D₇₁	—	8	1	no	—
57	C₂×D₇₁	—	8	1	no	—
58	C₂×D₇₁	—	8	1	no	—
59	C₄×C₇₁	—	8	1	no	—
60	C₄×C₇₁	—	8	1	no	—
61	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
62	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₇₁	#137	1	2	yes	—
64	Dic₇₁	—	8	1	no	—
65	C₂×C₇₁	#492	1	1	yes	characteristic
66	C₂×C₇₁	#496	1	1	yes	characteristic
67	C₂×C₇₁	#497	1	1	yes	characteristic
68	D₇₁	—	16	1	no	—
69	#1714	—	71	1	no	maximal, Sylow
70	C₇₁	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	D₇₁	1	1	yes	characteristic, complemented
72	#498	—	71	1	no	—
73	#492	—	71	1	no	—
74	#492	—	71	1	no	—
75	#487	—	71	1	no	—
76	#486	—	71	1	no	—
77	#322	—	71	1	no	—
78	C₂×SD₁₆	—	142	2	no	—
79	C₂×SD₁₆	—	142	2	no	—
80	C₄⋊Q₈	—	71	1	no	—
81	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
83	#127	—	71	1	no	—
84	#107	—	2	2	no	—
85	#106	—	142	2	no	—
86	#103	—	142	2	no	—
87	#102	—	142	2	no	—
88	C₄×C₈	—	71	1	no	—
89	C₄×C₈	—	71	1	no	—
90	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	142	1	no	—
94	C₂×D₄	—	142	1	no	—
95	C₂×D₄	—	142	1	no	—
96	C₂×D₄	—	142	1	no	—
97	SD₁₆	—	284	4	no	—
98	SD₁₆	—	284	4	no	—
99	C₂×C₈	—	71	2	no	—
100	C₂×C₈	—	71	2	no	—
101	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	142	1	no	—
107	C₄⋊C₄	—	142	1	no	—
108	C₄²	C₂²×D₇₁	1	1	yes	characteristic
109	C₂³	—	284	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	284	2	no	—
113	D₄	—	284	2	no	—
114	D₄	—	284	2	no	—
115	D₄	—	284	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	284	1	no	—
119	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
120	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
122	C₈	—	142	2	no	—
123	C₈	—	142	2	no	—
124	C₈	#11765852	1	4	yes	complemented
125	C₂²	—	568	1	no	—
126	C₂²	—	568	1	no	—
127	C₂²	—	568	1	no	—
128	C₂²	#14556124	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	568	1	no	—
132	C₄	C₂×#11765852	1	2	yes	—
133	C₄	#14556152	1	2	yes	—
134	C₄	#14556160	1	2	yes	—
135	C₂	—	1136	1	no	—
136	C₂	#17986373	1	1	yes	characteristic
137	C₂	#17986387	1	1	yes	characteristic
138	C₂	#17986392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986395

C₂

yes

maximal, characteristic, complemented

#17986373

C₂

yes

maximal, characteristic

#17986372

C₂

yes

maximal, characteristic

#17986349

C₂

yes

maximal, characteristic, complemented

#17986347

C₂

yes

maximal, characteristic

#17985720

C₂

yes

maximal, characteristic, complemented

#500×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765854

—

C₂×#11765852

—

#14556128

C₂²

yes

characteristic

#14556125

C₂²

yes

—

—

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#107×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765854

—

#11765852

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#127

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1714

—

maximal, Sylow

C₇₁

#1714

yes

characteristic, complemented, Sylow

#500

D₇₁

yes

characteristic, complemented

—

—

—

—

—

—

C₂×SD₁₆

—

142

—

C₂×SD₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

C₂×D₇₁

yes

—

—

—

142

—

#103

—

142

—

#102

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

SD₁₆

—

284

—

SD₁₆

—

284

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₇₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

142

—

107

C₄⋊C₄

—

142

—

108

C₄²

C₂²×D₇₁

yes

characteristic

109

C₂³

—

284

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

284

—

113

D₄

—

284

—

114

D₄

—

284

—

115

D₄

—

284

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

284

—

119

C₂×C₄

D₄×D₇₁

yes

characteristic

120

C₂×C₄

D₄×D₇₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

122

C₈

—

142

—

123

C₈

—

142

—

124

C₈

#11765852

yes

complemented

125

C₂²

—

568

—

126

C₂²

—

568

—

127

C₂²

—

568

—

128

C₂²

#14556124

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

568

—

132

C₄

C₂×#11765852

yes

—

133

C₄

#14556152

yes

—

134

C₄

#14556160

yes

—

135

C₂

—

1136

—

136

C₂

#17986373

yes

characteristic

137

C₂

#17986387

yes

characteristic

138

C₂

#17986392

yes

characteristic

	subgroup	G/H
center	C₂²	#14556124
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#500×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#500	D₇₁
hypocenter	C₇₁	#1714
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14556124

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#500×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #17986395	C₂	semidirect
4. #17986349	C₂	semidirect
6. #17985720	C₂	semidirect
7. #500×C₇₁	C₂	semidirect
52. C₈×C₇₁	SD₁₆	semidirect
70. C₇₁	#1714	semidirect
71. #500	D₇₁	semidirect
124. C₈	#11765852	semidirect

H ≅ G/N

type

1. #17986395

C₂

semidirect

4. #17986349

C₂

semidirect

6. #17985720

C₂

semidirect

7. #500×C₇₁

C₂

semidirect

52. C₈×C₇₁

SD₁₆

semidirect

70. C₇₁

#1714

semidirect

71. #500

D₇₁

semidirect

124. C₈

#11765852

semidirect

Subgroups — #22587963

Conjugacy classes of subgroups

Complements