Group #36542233 — subgroups

Subgroups — #36542233

Order 22784.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#26030838	C₂	1	4	yes	maximal
2	#26030766	C₂	1	2	yes	maximal
3	#26019929	C₂	1	8	yes	maximal, complemented
4	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19152155	C₂²	1	4	yes	—
6	#19152003	C₂²	1	4	yes	—
7	#19152003	C₂²	1	4	yes	—
8	#19152000	—	2	4	no	—
9	#19151996	C₂²	1	4	yes	—
10	#19151978	C₂²	1	4	yes	—
11	#19151949	C₂²	1	8	yes	—
12	#19151839	—	2	8	no	—
13	C₄×Dic₃₅₆	—	2	8	no	—
14	#19151636	—	2	4	no	—
15	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
17	#322×C₈₉	C₂²	1	4	yes	—
18	C₂×Dic₃₅₆	—	2	8	no	—
19	#15668012	C₂³	1	8	yes	—
20	#15668004	—	2	4	no	—
21	#15667991	—	2	8	no	—
22	C₄×Dic₁₇₈	—	2	8	no	—
23	#15667971	—	2	4	no	—
24	#15667946	—	2	8	no	—
25	#15667946	—	2	8	no	—
26	#15667944	—	2	8	no	—
27	#15667937	—	2	16	no	—
28	C₂²×C₈×C₈₉	D₄	1	4	yes	—
29	C₂²×C₈×C₈₉	D₄	1	2	yes	—
30	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₈₉	D₄	1	4	yes	—
32	C₄×C₈×C₈₉	D₄	1	2	yes	—
33	C₄×C₈×C₈₉	C₂³	1	4	yes	—
34	C₄×C₈×C₈₉	C₂³	1	2	yes	—
35	C₂×Dic₁₇₈	—	2	8	no	—
36	Dic₃₅₆	—	4	16	no	—
37	C₄⋊Dic₈₉	—	2	8	no	—
38	C₄⋊Dic₈₉	—	2	8	no	—
39	#13083553	—	4	8	no	—
40	C₂²⋊Dic₈₉	—	4	4	no	—
41	C₄×Dic₈₉	—	4	4	no	—
42	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	—	4	4	no	—
45	C₂×C₈×C₈₉	Q₁₆	1	8	yes	complemented
46	C₂×C₈×C₈₉	C₂×D₄	1	8	yes	—
47	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
48	C₄²×C₈₉	C₂×D₄	1	2	yes	—
49	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
51	C₂³×C₈₉	#127	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	4	no	—
53	C₂×C₄×C₈₉	#127	1	2	yes	—
54	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
58	C₂×Dic₈₉	—	4	8	no	—
59	Dic₁₇₈	—	4	16	no	—
60	C₈×C₈₉	—	2	8	no	—
61	C₈×C₈₉	C₂×Q₁₆	1	8	yes	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	#494	1	2	yes	—
64	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	#494	1	2	yes	—
67	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
68	C₄×C₈₉	#575	1	4	yes	—
69	Dic₈₉	—	8	8	no	—
70	#19618	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	2	2	no	—
72	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
73	C₂×C₈₉	#3150	1	2	yes	—
74	#3170	—	89	4	no	—
75	#3160	—	89	2	no	—
76	#1723	—	89	8	no	—
77	C₂×#322	D₈₉	1	1	yes	characteristic
78	C₈₉	#19618	1	1	yes	characteristic, complemented, Sylow
79	#533	—	89	4	no	—
80	#500	—	89	4	no	—
81	#500	—	89	4	no	—
82	#499	—	178	4	no	—
83	#498	—	89	4	no	—
84	#494	—	89	4	no	—
85	#487	—	89	8	no	—
86	#467	—	178	8	no	—
87	C₄×Q₁₆	—	178	8	no	—
88	#427	—	178	4	no	—
89	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
91	#322	C₂×D₈₉	1	4	yes	—
92	C₂×Q₁₆	—	178	8	no	—
93	C₂²×C₈	D₃₅₆	1	4	yes	—
94	C₂²×C₈	D₃₅₆	1	2	yes	—
95	C₄⋊Q₈	—	89	8	no	—
96	#125	—	178	4	no	—
97	C₂²⋊Q₈	—	178	8	no	—
98	C₄×Q₈	—	178	8	no	—
99	#117	—	178	4	no	—
100	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
101	#107	—	178	8	no	—
102	#107	—	178	8	no	—
103	#106	—	178	8	no	—
104	#103	—	178	16	no	—
105	C₄×C₈	D₃₅₆	1	4	yes	—
106	C₄×C₈	D₃₅₆	1	2	yes	—
107	C₄×C₈	C₂²×D₈₉	1	4	yes	—
108	C₄×C₈	C₂²×D₈₉	1	2	yes	—
109	C₂×Q₈	—	178	8	no	—
110	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
111	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
112	Q₁₆	—	356	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₃₅₆	1	8	yes	complemented
115	C₂×C₈	C₂×D₃₅₆	1	8	yes	—
116	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
117	C₄⋊C₄	—	356	8	no	—
118	C₄⋊C₄	—	178	8	no	—
119	C₄⋊C₄	—	178	8	no	—
120	C₂²⋊C₄	—	356	4	no	—
121	C₄²	—	356	4	no	—
122	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
123	C₄²	C₂×D₃₅₆	1	2	yes	—
124	C₄²	C₂³×D₈₉	1	1	yes	characteristic
125	C₂³	#15668009	1	1	yes	characteristic
126	Q₈	—	356	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	356	8	no	—
129	C₂×C₄	#15668009	1	2	yes	—
130	C₂×C₄	#15668009	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₃₅₆	1	4	yes	—
132	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
133	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₃₅₆	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#19151978	1	2	yes	—
138	C₂²	C₂×#15668009	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	712	8	no	—
141	C₄	#19151978	1	2	yes	—
142	C₄	C₂²×Dic₃₅₆	1	2	yes	—
143	C₄	#19152398	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#19151978	1	1	yes	characteristic
146	C₂	#26030698	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26030838

C₂

yes

maximal

#26030766

C₂

yes

maximal

#26019929

C₂

yes

maximal, complemented

C₂×#322×C₈₉

C₂

yes

maximal, characteristic

#19152155

C₂²

yes

—

#19152003

C₂²

yes

—

#19152003

C₂²

yes

—

#19152000

—

#19151996

C₂²

yes

—

#19151978

C₂²

yes

—

#19151949

C₂²

yes

—

#19151839

—

C₄×Dic₃₅₆

—

#19151636

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

—

#322×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

#15668012

C₂³

yes

—

#15668004

—

#15667991

—

C₄×Dic₁₇₈

—

—

—

—

—

—

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

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

Q₁₆

yes

complemented

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

characteristic

C₂³×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×D₄

yes

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

#494

yes

—

C₂²×C₈₉

C₂×#127

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#494

yes

—

C₄×C₈₉

C₂²×Q₁₆

yes

—

C₄×C₈₉

#575

yes

—

Dic₈₉

—

#19618

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#494

yes

characteristic

C₂×C₈₉

#3150

yes

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#19618

yes

characteristic, complemented, Sylow

—

—

—

—

178

—

—

—

—

—

178

—

C₄×Q₁₆

—

178

—

#427

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

—

#322

C₂×D₈₉

yes

—

C₂×Q₁₆

—

178

—

C₂²×C₈

D₃₅₆

yes

—

C₂²×C₈

D₃₅₆

yes

—

C₄⋊Q₈

—

#125

—

178

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

178

—

100

C₂×C₄²

C₂²×D₈₉

yes

characteristic

101

#107

—

178

—

102

#107

—

178

—

103

#106

—

178

—

104

#103

—

178

—

105

C₄×C₈

D₃₅₆

yes

—

106

C₄×C₈

D₃₅₆

yes

—

107

C₄×C₈

C₂²×D₈₉

yes

—

108

C₄×C₈

C₂²×D₈₉

yes

—

109

C₂×Q₈

—

178

—

110

C₂²×C₄

C₂×D₃₅₆

yes

—

111

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

112

Q₁₆

—

356

—

113

C₂×C₈

—

114

C₂×C₈

Dic₃₅₆

yes

complemented

115

C₂×C₈

C₂×D₃₅₆

yes

—

116

C₂×C₈

C₂×D₃₅₆

yes

—

117

C₄⋊C₄

—

356

—

118

C₄⋊C₄

—

178

—

119

C₄⋊C₄

—

178

—

120

C₂²⋊C₄

—

356

—

121

C₄²

—

356

—

122

C₄²

C₂×D₃₅₆

yes

characteristic

123

C₄²

C₂×D₃₅₆

yes

—

124

C₄²

C₂³×D₈₉

yes

characteristic

125

C₂³

#15668009

yes

characteristic

126

Q₈

—

356

—

127

C₂×C₄

—

128

C₂×C₄

—

356

—

129

C₂×C₄

#15668009

yes

—

130

C₂×C₄

#15668009

yes

characteristic

131

C₂×C₄

C₂×Dic₃₅₆

yes

—

132

C₂×C₄

C₂²×D₃₅₆

yes

—

133

C₂×C₄

C₂²×D₃₅₆

yes

characteristic

134

C₈

—

135

C₈

C₂×Dic₃₅₆

yes

—

136

C₂²

—

137

C₂²

#19151978

yes

—

138

C₂²

C₂×#15668009

yes

characteristic

139

C₄

—

140

C₄

—

712

—

141

C₄

#19151978

yes

—

142

C₄

C₂²×Dic₃₅₆

yes

—

143

C₄

#19152398

yes

—

144

C₂

—

145

C₂

C₂×#19151978

yes

characteristic

146

C₂

#26030698

yes

—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂³×C₈₉	#127
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#19618
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#15668009

commutator, cosocle

C₄²×C₈₉

C₂⁴

Frattini

C₄²

C₂³×D₈₉

Fitting

C₂×#322×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #26019929	C₂	semidirect
45. C₂×C₈×C₈₉	Q₁₆	semidirect
78. C₈₉	#19618	semidirect
114. C₂×C₈	Dic₃₅₆	semidirect

H ≅ G/N

type

3. #26019929

C₂

semidirect

45. C₂×C₈×C₈₉

Q₁₆

semidirect

78. C₈₉

#19618

semidirect

114. C₂×C₈

Dic₃₅₆

semidirect

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

subgroup	index
#26030838	2
#26030766	2
#26019929	2
C₂×#322×C₈₉	2
#19618	89

subgroup

index

C₂×#322×C₈₉

Subgroups — #36542233

Conjugacy classes of subgroups

Complements

Maximal subgroups