Group #22587094 — subgroups

Subgroups — #22587094

Order 9088.

184 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 81 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	#17986021	C₂	1	2	yes	maximal
2	C₄×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985750	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985749	C₂	1	2	yes	maximal, complemented
5	#334×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765848	—	2	2	no	—
7	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
8	C₂×C₈×D₇₁	C₄	1	1	yes	characteristic
9	C₄²×D₇₁	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	Dic₇₁⋊C₈	—	2	2	no	—
12	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₈×Dic₇₁	C₄	1	1	yes	characteristic
14	C₈×Dic₇₁	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
19	C₂×C₄×D₇₁	Q₈	1	1	yes	characteristic
20	#11765848	—	4	4	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₈×D₇₁	—	2	2	no	—
25	C₈×D₇₁	—	2	2	no	—
26	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₇₁	D₄	1	1	yes	characteristic
28	C₄×Dic₇₁	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₂²×D₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₇₁	—	2	1	no	—
41	C₄×D₇₁	—	2	1	no	—
42	C₄×D₇₁	—	2	2	no	—
43	C₄×D₇₁	SD₁₆	1	4	yes	—
44	C₈×C₇₁	—	2	1	no	—
45	C₈×C₇₁	—	4	2	no	—
46	C₈×C₇₁	M₁₆	1	2	yes	complemented
47	C₇₁⋊C₈	—	2	1	no	—
48	C₇₁⋊C₈	—	4	2	no	—
49	C₇₁⋊C₈	M₁₆	1	2	yes	complemented
50	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₇₁	—	2	1	no	—
52	C₂×D₇₁	—	2	1	no	—
53	C₂×D₇₁	#106	1	2	yes	—
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
56	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
58	Dic₇₁	—	2	1	no	—
59	Dic₇₁	#106	1	2	yes	—
60	Dic₇₁	#108	1	2	yes	—
61	C₂×C₇₁	#423	1	1	yes	characteristic
62	C₂×C₇₁	C₂×#106	1	1	yes	characteristic
63	C₂×C₇₁	C₂×#108	1	1	yes	characteristic
64	D₇₁	—	2	2	no	—
65	#1566	—	71	1	no	maximal, Sylow
66	C₇₁	#1566	1	1	yes	characteristic, complemented, Sylow
67	#423	—	71	2	no	—
68	C₂×C₄×C₈	—	71	1	no	—
69	#334	—	71	2	no	—
70	#334	—	71	1	no	—
71	#334	D₇₁	1	1	yes	characteristic, complemented
72	C₂×M₁₆	—	142	2	no	—
73	C₂²×C₈	—	71	1	no	—
74	C₂²×C₈	—	71	1	no	—
75	C₂×C₄²	—	71	1	no	—
76	C₄⋊C₈	—	71	2	no	—
77	C₄⋊C₈	—	142	2	no	—
78	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
79	C₄×C₈	—	71	1	no	—
80	C₄×C₈	—	71	1	no	—
81	C₄×C₈	—	71	1	no	—
82	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
83	C₂²×C₄	—	71	1	no	—
84	C₂²×C₄	—	71	1	no	—
85	C₂²×C₄	—	71	1	no	—
86	M₁₆	—	284	4	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	71	1	no	—
89	C₂×C₈	—	71	1	no	—
90	C₂×C₈	—	142	2	no	—
91	C₂×C₈	—	142	2	no	—
92	C₂×C₈	—	142	2	no	—
93	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
94	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
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	4	no	—
103	C₂×C₄	—	71	1	no	—
104	C₂×C₄	—	142	2	no	—
105	C₂×C₄	—	142	1	no	—
106	C₂×C₄	—	142	1	no	—
107	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
108	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	284	2	no	—
113	C₈	—	71	2	no	—
114	C₈	—	142	1	no	—
115	C₈	#11765848	1	2	yes	complemented
116	C₂²	—	71	2	no	—
117	C₂²	—	142	1	no	—
118	C₂²	—	142	1	no	—
119	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	71	2	no	—
122	C₄	—	71	2	no	—
123	C₄	—	142	1	no	—
124	C₄	M₁₆×D₇₁	1	1	yes	characteristic
125	C₄	C₂×#11765848	1	1	yes	characteristic
126	C₄	SD₁₆×D₇₁	1	2	yes	—
127	C₂	—	142	2	no	—
128	C₂	#17986021	1	1	yes	characteristic
129	C₂	#106×D₇₁	1	1	yes	characteristic
130	C₂	#108×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986021

C₂

yes

maximal

C₄×C₈×D₇₁

C₂

yes

maximal, characteristic

#17985750

C₂

yes

maximal, characteristic, complemented

#17985749

C₂

yes

maximal, complemented

#334×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765848

—

C₂×C₈×D₇₁

C₄

yes

characteristic

C₂×C₈×D₇₁

C₄

yes

characteristic

C₄²×D₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

Dic₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₁

C₄

yes

characteristic

C₈×Dic₇₁

C₄

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄×D₇₁

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₄×D₇₁

Q₈

yes

characteristic

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

characteristic

C₄×Dic₇₁

Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

C₄⋊C₄

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

SD₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

M₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

M₁₆

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

#106

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

—

Dic₇₁

—

Dic₇₁

#106

yes

—

Dic₇₁

#108

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1566

—

maximal, Sylow

C₇₁

#1566

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

—

#334

—

#334

—

#334

D₇₁

yes

characteristic, complemented

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

142

—

105

C₂×C₄

—

142

—

106

C₂×C₄

—

142

—

107

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

108

C₂×C₄

D₄×D₇₁

yes

characteristic

109

C₂×C₄

Q₈×D₇₁

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

284

—

113

C₈

—

114

C₈

—

142

—

115

C₈

#11765848

yes

complemented

116

C₂²

—

117

C₂²

—

142

—

118

C₂²

—

142

—

119

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

142

—

124

C₄

M₁₆×D₇₁

yes

characteristic

125

C₄

C₂×#11765848

yes

characteristic

126

C₄

SD₁₆×D₇₁

yes

—

127

C₂

—

142

—

128

C₂

#17986021

yes

characteristic

129

C₂

#106×D₇₁

yes

characteristic

130

C₂

#108×D₇₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#334×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	#334	D₇₁
hypocenter	C₇₁	#1566
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#334×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
3. #17985750	C₂	semidirect
4. #17985749	C₂	semidirect
5. #334×C₇₁	C₂	semidirect
46. C₈×C₇₁	M₁₆	semidirect
49. C₇₁⋊C₈	M₁₆	semidirect
66. C₇₁	#1566	semidirect
71. #334	D₇₁	semidirect
115. C₈	#11765848	semidirect

H ≅ G/N

type

3. #17985750

C₂

semidirect

4. #17985749

C₂

semidirect

5. #334×C₇₁

C₂

semidirect

46. C₈×C₇₁

M₁₆

semidirect

49. C₇₁⋊C₈

M₁₆

semidirect

66. C₇₁

#1566

semidirect

71. #334

D₇₁

semidirect

115. C₈

#11765848

semidirect

Subgroups — #22587094

Conjugacy classes of subgroups

Complements