Group #22590083 — subgroups

Subgroups — C₂×#17985814 #22590083

Order 9088.

216 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556068	C₂	1	1	yes	maximal, characteristic
3	#17985814	C₂	1	4	yes	maximal, direct product factor
4	C₂×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₂	—	2	1	no	—
6	C₂×Dic₂₈₄	—	2	1	no	—
7	C₂×Dic₂₈₄	—	2	1	no	—
8	C₂×Dic₂₈₄	—	2	1	no	—
9	C₂×Dic₂₈₄	C₄	1	2	yes	—
10	C₂×Dic₂₈₄	C₄	1	2	yes	—
11	C₂×Dic₂₈₄	C₂²	1	2	yes	—
12	#14556068	—	2	1	no	—
13	#14556068	C₂²	1	2	yes	—
14	C₂×C₇₁⋊C₁₆	—	2	1	no	—
15	C₂×M₁₆×C₇₁	—	2	1	no	—
16	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#108×C₇₁	—	2	1	no	—
18	#108×C₇₁	C₂²	1	2	yes	—
19	C₂×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₁₄₂	—	2	2	no	—
21	C₂×Dic₁₄₂	—	2	2	no	—
22	C₂×Dic₁₄₂	—	4	1	no	—
23	C₂×Dic₁₄₂	—	4	1	no	—
24	C₂×Dic₁₄₂	—	4	1	no	—
25	C₂²×Dic₇₁	—	4	1	no	—
26	Dic₂₈₄	—	2	2	no	—
27	Dic₂₈₄	—	2	2	no	—
28	Dic₂₈₄	—	2	2	no	—
29	Dic₂₈₄	C₂×C₄	1	4	yes	—
30	C₇₁⋊C₁₆	—	2	2	no	—
31	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
32	M₁₆×C₇₁	—	2	2	no	—
33	M₁₆×C₇₁	—	4	1	no	—
34	C₂×C₈×C₇₁	—	4	1	no	—
35	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	C₂³×C₇₁	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
45	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₇₁	—	4	2	no	—
49	C₂×Dic₇₁	—	4	2	no	—
50	C₂×Dic₇₁	—	4	2	no	—
51	Dic₁₄₂	—	2	4	no	—
52	Dic₁₄₂	—	4	2	no	—
53	Dic₁₄₂	—	4	2	no	—
54	Dic₁₄₂	—	4	2	no	—
55	C₈×C₇₁	—	4	2	no	—
56	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₇₁	—	2	2	no	—
61	C₂²×C₇₁	#102	1	1	yes	characteristic
62	C₂²×C₇₁	#102	1	1	yes	characteristic
63	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₇₁	#102	1	1	yes	characteristic
65	C₄×C₇₁	#102	1	1	yes	characteristic
66	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
68	Dic₇₁	—	4	4	no	—
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	—	2	1	no	—
71	C₂×C₇₁	#362	1	2	yes	complemented
72	C₂×C₇₁	C₂×#102	1	1	yes	characteristic
73	C₂×#362	—	71	1	no	maximal, Sylow
74	C₇₁	C₂×#362	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₁₆	—	71	1	no	—
76	C₂×M₃₂	—	71	1	no	—
77	C₂×#108	D₇₁	1	1	yes	characteristic
78	#362	—	71	4	no	—
79	C₂²×Q₈	—	142	1	no	—
80	C₂×Q₁₆	—	71	2	no	—
81	C₂×Q₁₆	—	71	2	no	—
82	C₂×Q₁₆	—	71	2	no	—
83	C₂×Q₁₆	—	142	1	no	—
84	C₂×Q₁₆	—	142	1	no	—
85	C₂×Q₁₆	—	142	1	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
88	M₃₂	—	71	2	no	—
89	M₃₂	—	142	1	no	—
90	C₂×C₁₆	—	142	1	no	—
91	#108	—	2	1	no	—
92	#108	C₂×D₇₁	1	2	yes	—
93	C₂×Q₈	—	284	1	no	—
94	C₂×Q₈	—	284	1	no	—
95	C₂×Q₈	—	284	1	no	—
96	C₂×Q₈	—	142	2	no	—
97	C₂×Q₈	—	142	2	no	—
98	C₂×Q₈	—	142	2	no	—
99	C₂²×C₄	—	284	1	no	—
100	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
101	Q₁₆	—	71	4	no	—
102	Q₁₆	—	142	2	no	—
103	Q₁₆	—	142	2	no	—
104	Q₁₆	—	142	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	1	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
109	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
110	C₂×C₈	D₂₈₄	1	1	yes	characteristic
111	C₂×C₈	D₂₈₄	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₁₆	—	142	2	no	—
115	C₂³	#11765852	1	1	yes	characteristic
116	Q₈	—	284	2	no	—
117	Q₈	—	284	2	no	—
118	Q₈	—	284	2	no	—
119	Q₈	—	142	4	no	—
120	C₂×C₄	—	284	2	no	—
121	C₂×C₄	—	284	2	no	—
122	C₂×C₄	—	284	2	no	—
123	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#11765850	1	1	yes	characteristic
126	C₂×C₄	#11765850	1	1	yes	characteristic
127	C₂×C₄	#11765852	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
131	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
132	C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
133	C₈	C₂×D₂₈₄	1	1	yes	characteristic
134	C₂²	—	2	2	no	—
135	C₂²	#14556048	1	1	yes	characteristic
136	C₂²	#14556048	1	1	yes	characteristic
137	C₂²	C₂×#11765852	1	1	yes	characteristic
138	C₄	—	284	4	no	—
139	C₄	#14556048	1	1	yes	characteristic
140	C₄	#14556048	1	1	yes	characteristic
141	C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
142	C₄	C₂×#11765850	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	#17985814	1	2	yes	direct product factor
146	C₂	C₂×#14556048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₈₄

C₂

yes

maximal, characteristic

C₂×#14556068

C₂

yes

maximal, characteristic

#17985814

C₂

yes

maximal, direct product factor

C₂×#108×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

C₄

yes

—

C₂×Dic₂₈₄

C₄

yes

—

C₂×Dic₂₈₄

C₂²

yes

—

#14556068

—

#14556068

C₂²

yes

—

C₂×C₇₁⋊C₁₆

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#108×C₇₁

—

#108×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

C₂×C₄

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂²×C₄

yes

characteristic

C₈×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

C₂×D₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×M₃₂

—

C₂×#108

D₇₁

yes

characteristic

#362

—

C₂²×Q₈

—

142

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

142

—

#108

—

#108

C₂×D₇₁

yes

—

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂²×C₄

—

284

—

100

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

101

Q₁₆

—

102

Q₁₆

—

142

—

103

Q₁₆

—

142

—

104

Q₁₆

—

142

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

C₄×D₇₁

yes

characteristic

109

C₂×C₈

C₄×D₇₁

yes

characteristic

110

C₂×C₈

D₂₈₄

yes

characteristic

111

C₂×C₈

D₂₈₄

yes

characteristic

112

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

113

C₂×C₈

C₂²×D₇₁

yes

characteristic

114

C₁₆

—

142

—

115

C₂³

#11765852

yes

characteristic

116

Q₈

—

284

—

117

Q₈

—

284

—

118

Q₈

—

284

—

119

Q₈

—

142

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

284

—

122

C₂×C₄

—

284

—

123

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

124

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

125

C₂×C₄

#11765850

yes

characteristic

126

C₂×C₄

#11765850

yes

characteristic

127

C₂×C₄

#11765852

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

129

C₈

—

130

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

131

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

132

C₈

C₂×C₄×D₇₁

yes

characteristic

133

C₈

C₂×D₂₈₄

yes

characteristic

134

C₂²

—

135

C₂²

#14556048

yes

characteristic

136

C₂²

#14556048

yes

characteristic

137

C₂²

C₂×#11765852

yes

characteristic

138

C₄

—

284

—

139

C₄

#14556048

yes

characteristic

140

C₄

#14556048

yes

characteristic

141

C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

142

C₄

C₂×#11765850

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

#17985814

yes

direct product factor

146

C₂

C₂×#14556048

yes

characteristic

	subgroup	G/H
center	C₂²	#14556048
commutator	C₈×C₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₂×#108×C₇₁	C₂
socle	C₂²×C₇₁	#102
hypercenter	C₂×#108	D₇₁
hypocenter	C₇₁	C₂×#362
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14556048

commutator

C₈×C₇₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₂×#108×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
3. #17985814	C₂	direct (N × H)
71. C₂×C₇₁	#362	semidirect
74. C₇₁	C₂×#362	semidirect
145. C₂	#17985814	direct (N × H)

H ≅ G/N

type

3. #17985814

C₂

direct (N × H)

71. C₂×C₇₁

#362

semidirect

74. C₇₁

C₂×#362

semidirect

145. C₂

#17985814

direct (N × H)

Subgroups — C₂×#17985814 #22590083

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17985814 #22590083

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17985814 #22590083