Group #36542101 — subgroups

Subgroups — C₂×#26019862 #36542101

Order 22784.

750 conjugacy classes of subgroups; only the 267 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 267 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×#19152003	C₂	1	2	yes	maximal
2	C₂×#19151949	C₂	1	4	yes	maximal
3	#26019862	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668012	C₂²	1	4	yes	—
6	#19152003	—	2	4	no	—
7	#19152003	C₂²	1	8	yes	—
8	#19151949	—	2	8	no	—
9	#19151949	C₂²	1	16	yes	—
10	C₂×#15667946	—	2	4	no	—
11	C₂×#15667937	—	2	8	no	—
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
14	C₈²×C₈₉	C₂²	1	4	yes	—
15	C₂²×Dic₁₇₈	—	2	4	no	—
16	#15668012	—	2	4	no	—
17	#15668012	—	2	4	no	—
18	#15668012	—	2	4	no	—
19	#15668012	C₂³	1	8	yes	—
20	C₂×C₄⋊Dic₈₉	—	2	4	no	—
21	C₂×C₄⋊Dic₈₉	—	2	4	no	—
22	#15667946	—	2	16	no	—
23	#15667937	—	2	32	no	—
24	C₂²×C₈×C₈₉	D₄	1	4	yes	—
25	C₂²×C₈×C₈₉	D₄	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	4	yes	—
28	C₄×C₈×C₈₉	D₄	1	2	yes	—
29	C₄×C₈×C₈₉	C₂³	1	4	yes	—
30	C₄×C₈×C₈₉	C₂³	1	2	yes	—
31	C₂×Dic₁₇₈	—	2	8	no	—
32	C₂×Dic₁₇₈	—	4	16	no	—
33	C₂×Dic₁₇₈	—	4	4	no	—
34	C₂²×Dic₈₉	—	4	4	no	—
35	C₄⋊Dic₈₉	—	2	8	no	—
36	C₄⋊Dic₈₉	—	2	8	no	—
37	C₄⋊Dic₈₉	—	4	4	no	—
38	C₄⋊Dic₈₉	—	4	4	no	—
39	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₈₉	—	2	8	no	—
42	C₂×C₈×C₈₉	Q₁₆	1	8	yes	—
43	C₂×C₈×C₈₉	C₂×D₄	1	8	yes	—
44	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
45	C₄²×C₈₉	C₂×D₄	1	2	yes	—
46	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
48	C₂³×C₈₉	#127	1	1	yes	characteristic
49	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#127	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
52	C₂×C₄×C₈₉	#135	1	8	yes	—
53	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
55	C₂×Dic₈₉	—	4	8	no	—
56	C₂×Dic₈₉	—	8	8	no	—
57	Dic₁₇₈	—	4	16	no	—
58	Dic₁₇₈	—	8	8	no	—
59	C₈×C₈₉	—	2	8	no	—
60	C₈×C₈₉	C₂×Q₁₆	1	8	yes	—
61	C₂²×C₈₉	#494	1	2	yes	—
62	C₂²×C₈₉	#495	1	4	yes	—
63	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
64	C₄×C₈₉	#494	1	2	yes	—
65	C₄×C₈₉	#495	1	4	yes	—
66	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
67	C₄×C₈₉	C₂×#135	1	4	yes	—
68	Dic₈₉	—	8	8	no	—
69	C₂×#1710	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	#1710	1	4	yes	complemented
71	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
72	C₂×C₈₉	C₂×#495	1	2	yes	—
73	C₂×#500	—	89	2	no	—
74	C₂×#487	—	89	4	no	—
75	#1710	—	89	8	no	—
76	C₂×C₈²	D₈₉	1	1	yes	characteristic
77	C₈₉	C₂×#1710	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄⋊Q₈	—	89	4	no	—
79	#500	—	89	8	no	—
80	#500	—	178	4	no	—
81	#487	—	89	16	no	—
82	#487	—	178	8	no	—
83	C₂×#107	—	178	4	no	—
84	C₂×#103	—	178	8	no	—
85	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
87	C₈²	C₂×D₈₉	1	4	yes	—
88	C₂²×Q₈	—	178	4	no	—
89	C₂²×C₈	D₃₅₆	1	4	yes	—
90	C₂²×C₈	D₃₅₆	1	2	yes	—
91	C₄⋊Q₈	—	89	8	no	—
92	C₄⋊Q₈	—	178	4	no	—
93	C₄⋊Q₈	—	178	4	no	—
94	C₄⋊Q₈	—	178	4	no	—
95	C₂×C₄⋊C₄	—	178	4	no	—
96	C₂×C₄⋊C₄	—	178	4	no	—
97	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
98	#107	—	178	16	no	—
99	#103	—	178	32	no	—
100	C₄×C₈	D₃₅₆	1	4	yes	—
101	C₄×C₈	D₃₅₆	1	2	yes	—
102	C₄×C₈	C₂²×D₈₉	1	4	yes	—
103	C₄×C₈	C₂²×D₈₉	1	2	yes	—
104	C₂×Q₈	—	356	4	no	—
105	C₂×Q₈	—	356	16	no	—
106	C₂×Q₈	—	178	8	no	—
107	C₂²×C₄	—	356	4	no	—
108	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
109	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	Dic₃₅₆	1	8	yes	—
112	C₂×C₈	C₂×D₃₅₆	1	8	yes	—
113	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
114	C₄⋊C₄	—	356	4	no	—
115	C₄⋊C₄	—	356	4	no	—
116	C₄⋊C₄	—	178	8	no	—
117	C₄⋊C₄	—	178	8	no	—
118	C₄²	C₂×D₃₅₆	1	2	yes	—
119	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
120	C₄²	C₂³×D₈₉	1	1	yes	characteristic
121	C₂³	#15668009	1	1	yes	characteristic
122	Q₈	—	356	16	no	—
123	Q₈	—	712	8	no	—
124	C₂×C₄	—	356	8	no	—
125	C₂×C₄	—	712	8	no	—
126	C₂×C₄	#15668009	1	1	yes	characteristic
127	C₂×C₄	#15668009	1	2	yes	—
128	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
129	C₂×C₄	#15668034	1	8	yes	—
130	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
132	C₈	—	2	8	no	—
133	C₈	C₂×Dic₃₅₆	1	8	yes	—
134	C₂²	#19151978	1	2	yes	—
135	C₂²	#19151982	1	4	yes	—
136	C₂²	C₂×#15668009	1	1	yes	characteristic
137	C₄	—	712	8	no	—
138	C₄	#19151978	1	2	yes	—
139	C₄	#19151982	1	4	yes	—
140	C₄	C₂²×Dic₃₅₆	1	2	yes	—
141	C₄	C₂×#15668034	1	4	yes	—
142	C₂	#26019862	1	4	yes	direct product factor
143	C₂	C₂×#19151978	1	1	yes	characteristic
144	C₂	C₂×#19151982	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19152003

C₂

yes

maximal

C₂×#19151949

C₂

yes

maximal

#26019862

C₂

yes

maximal, direct product factor

C₂×C₈²×C₈₉

C₂

yes

maximal, characteristic

C₂×#15668012

C₂²

yes

—

#19152003

—

#19152003

C₂²

yes

—

#19151949

—

#19151949

C₂²

yes

—

C₂×#15667946

—

C₂×#15667937

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

—

C₂²×Dic₁₇₈

—

—

—

—

C₂³

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

#15667937

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂⁴

yes

C₂³×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

#494

yes

—

C₂²×C₈₉

#495

yes

—

C₂²×C₈₉

C₂×#127

yes

characteristic

C₄×C₈₉

#494

yes

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

C₂²×Q₁₆

yes

—

C₄×C₈₉

C₂×#135

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#500

—

C₂×#487

—

#1710

—

C₂×C₈²

D₈₉

yes

characteristic

C₈₉

C₂×#1710

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#500

—

#500

—

178

—

#487

—

#487

—

178

—

C₂×#107

—

178

—

C₂×#103

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₈²

C₂×D₈₉

yes

—

C₂²×Q₈

—

178

—

C₂²×C₈

D₃₅₆

yes

—

C₂²×C₈

D₃₅₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#107

—

178

—

#103

—

178

—

100

C₄×C₈

D₃₅₆

yes

—

101

C₄×C₈

D₃₅₆

yes

—

102

C₄×C₈

C₂²×D₈₉

yes

—

103

C₄×C₈

C₂²×D₈₉

yes

—

104

C₂×Q₈

—

356

—

105

C₂×Q₈

—

356

—

106

C₂×Q₈

—

178

—

107

C₂²×C₄

—

356

—

108

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

109

C₂²×C₄

C₂×D₃₅₆

yes

—

110

C₂×C₈

—

111

C₂×C₈

Dic₃₅₆

yes

—

112

C₂×C₈

C₂×D₃₅₆

yes

—

113

C₂×C₈

C₂×D₃₅₆

yes

—

114

C₄⋊C₄

—

356

—

115

C₄⋊C₄

—

356

—

116

C₄⋊C₄

—

178

—

117

C₄⋊C₄

—

178

—

118

C₄²

C₂×D₃₅₆

yes

—

119

C₄²

C₂×D₃₅₆

yes

characteristic

120

C₄²

C₂³×D₈₉

yes

characteristic

121

C₂³

#15668009

yes

characteristic

122

Q₈

—

356

—

123

Q₈

—

712

—

124

C₂×C₄

—

356

—

125

C₂×C₄

—

712

—

126

C₂×C₄

#15668009

yes

characteristic

127

C₂×C₄

#15668009

yes

—

128

C₂×C₄

C₂×Dic₃₅₆

yes

—

129

C₂×C₄

#15668034

yes

—

130

C₂×C₄

C₂²×D₃₅₆

yes

characteristic

131

C₂×C₄

C₂²×D₃₅₆

yes

—

132

C₈

—

133

C₈

C₂×Dic₃₅₆

yes

—

134

C₂²

#19151978

yes

—

135

C₂²

#19151982

yes

—

136

C₂²

C₂×#15668009

yes

characteristic

137

C₄

—

712

—

138

C₄

#19151978

yes

—

139

C₄

#19151982

yes

—

140

C₄

C₂²×Dic₃₅₆

yes

—

141

C₄

C₂×#15668034

yes

—

142

C₂

#26019862

yes

direct product factor

143

C₂

C₂×#19151978

yes

characteristic

144

C₂

C₂×#19151982

yes

—

	subgroup	G/H
center	C₂³	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	C₂×#1710
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#15668009

commutator, cosocle

C₄²×C₈₉

C₂⁴

Frattini

C₄²

C₂³×D₈₉

Fitting

C₂×C₈²×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

#127

hypercenter

C₂×C₈²

D₈₉

hypocenter

C₈₉

C₂×#1710

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
3. #26019862	C₂	direct (N × H)
70. C₂×C₈₉	#1710	semidirect
77. C₈₉	C₂×#1710	semidirect
142. C₂	#26019862	direct (N × H)

H ≅ G/N

type

3. #26019862

C₂

direct (N × H)

70. C₂×C₈₉

#1710

semidirect

77. C₈₉

C₂×#1710

semidirect

142. C₂

#26019862

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19152003	2
C₂×#19151949	2
#26019862	2
C₂×C₈²×C₈₉	2
C₂×#1710	89

subgroup

index

C₂×C₈²×C₈₉

Subgroups — C₂×#26019862 #36542101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26019862 #36542101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019862 #36542101