Group #36469082 — subgroups

Subgroups — #36469082

Order 22784.

306 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26022502	C₂	1	1	yes	maximal, characteristic
2	#26022477	C₂	1	2	yes	maximal, complemented
3	#26019845	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	—	2	2	no	—
7	#19151975	C₂²	1	2	yes	—
8	#19151972	—	2	2	no	—
9	#19151946	—	2	2	no	—
10	#19151398	—	2	4	no	—
11	C₄×#13083549	C₂²	1	1	yes	characteristic
12	C₈×#10623494	C₂²	1	1	yes	characteristic
13	#500×C₈₉	C₂²	1	2	yes	—
14	#366×C₈₉	—	2	2	no	—
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#13083561	—	4	4	no	—
17	C₂×D₇₁₂	—	2	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	#15668009	—	2	2	no	—
20	D₁₄₂₄	—	4	4	no	—
21	C₂×#13083549	D₄	1	2	yes	—
22	#15667932	—	4	4	no	—
23	C₄×#10623494	D₄	1	1	yes	characteristic
24	C₄×#10623494	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₈₉	—	2	2	no	—
26	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
27	#107×C₈₉	—	2	2	no	—
28	#107×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×D₃₅₆	—	4	2	no	—
31	C₂×D₃₅₆	—	4	2	no	—
32	C₂×D₃₅₆	—	4	2	no	—
33	#13083561	—	8	8	no	—
34	D₇₁₂	—	4	4	no	—
35	D₇₁₂	—	4	4	no	—
36	C₂×#10623494	D₈	1	2	yes	—
37	C₂×#10623494	D₈	1	2	yes	—
38	#13083549	—	2	2	no	—
39	C₂×Q₈×C₈₉	—	4	2	no	—
40	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₈₉	—	4	2	no	—
43	C₄⋊C₄×C₈₉	—	4	2	no	—
44	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₈₉	SD₁₆	1	4	yes	complemented
46	C₂²×D₈₉	—	8	2	no	—
47	Q₈×C₈₉	—	8	4	no	—
48	C₂×C₄×C₈₉	—	8	2	no	—
49	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
52	D₃₅₆	—	8	2	no	—
53	D₃₅₆	—	8	2	no	—
54	D₃₅₆	—	8	4	no	—
55	D₃₅₆	—	8	4	no	—
56	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
57	C₈×C₈₉	#135	1	2	yes	—
58	#10623494	—	2	2	no	—
59	#10623494	D₁₆	1	4	yes	complemented
60	C₂²×C₈₉	#493	1	1	yes	characteristic
61	C₂×D₈₉	—	16	2	no	—
62	C₂×D₈₉	—	16	2	no	—
63	C₂×D₈₉	—	16	2	no	—
64	C₄×C₈₉	—	16	2	no	—
65	C₄×C₈₉	#492	1	1	yes	characteristic
66	C₄×C₈₉	#495	1	1	yes	characteristic
67	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
68	C₄×C₈₉	#508	1	2	yes	—
69	#12503	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	#1706	1	1	yes	characteristic
71	C₂×C₈₉	#2246	1	1	yes	characteristic
72	C₂×C₈₉	#2249	1	1	yes	characteristic
73	D₈₉	—	32	2	no	—
74	#2252	D₈₉	1	1	yes	characteristic, complemented
75	#2246	—	89	1	no	—
76	#2240	—	89	2	no	—
77	#1706	—	89	2	no	—
78	C₈×C₁₆	—	89	1	no	—
79	C₈₉	#12503	1	1	yes	characteristic, complemented, Sylow
80	C₂×D₁₆	—	178	2	no	—
81	#500	C₂×D₈₉	1	2	yes	—
82	#493	—	89	2	no	—
83	#492	—	178	2	no	—
84	#486	—	178	2	no	—
85	#366	—	2	2	no	—
86	#357	—	178	4	no	—
87	C₄×C₁₆	—	89	1	no	—
88	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
89	C₈²	—	89	1	no	—
90	C₂×SD₁₆	—	356	4	no	—
91	C₂×D₈	—	178	2	no	—
92	C₂×D₈	—	178	2	no	—
93	C₄⋊Q₈	—	2	2	no	—
94	#127	—	178	2	no	—
95	D₁₆	—	356	4	no	—
96	C₂×C₁₆	—	89	2	no	—
97	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
98	#107	—	2	2	no	—
99	#107	—	2	2	no	—
100	#102	—	356	4	no	—
101	C₄×C₈	—	89	1	no	—
102	C₄×C₈	—	89	1	no	—
103	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×D₄	—	356	2	no	—
106	C₂×D₄	—	356	2	no	—
107	C₂×D₄	—	356	2	no	—
108	SD₁₆	—	712	8	no	—
109	D₈	—	356	4	no	—
110	D₈	—	356	4	no	—
111	C₂×C₈	—	89	2	no	—
112	C₂×C₈	—	89	2	no	—
113	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
114	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	D₄×D₈₉	1	1	yes	characteristic
118	C₁₆	—	178	2	no	—
119	C₁₆	#13083561	1	4	yes	complemented
120	C₂³	—	712	2	no	—
121	Q₈	—	8	4	no	—
122	D₄	—	712	2	no	—
123	D₄	—	712	2	no	—
124	D₄	—	712	4	no	—
125	D₄	—	712	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	#15668008	1	1	yes	characteristic
128	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
129	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
130	C₈	—	89	4	no	—
131	C₈	—	178	2	no	—
132	C₈	C₂×#13083561	1	2	yes	—
133	C₈	#15668035	1	2	yes	—
134	C₂²	—	1424	2	no	—
135	C₂²	—	1424	2	no	—
136	C₂²	—	1424	2	no	—
137	C₂²	#19151973	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	#19151972	1	1	yes	characteristic
140	C₄	#19151983	1	1	yes	characteristic
141	C₄	D₁₆×D₈₉	1	2	yes	—
142	C₄	#19152031	1	2	yes	—
143	C₂	—	2848	2	no	—
144	C₂	#26019845	1	1	yes	characteristic
145	C₂	#26022500	1	1	yes	characteristic
146	C₂	#26022514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022502

C₂

yes

maximal, characteristic

#26022477

C₂

yes

maximal, complemented

#26019845

C₂

yes

maximal, complemented

C₁₆×#10623494

C₂

yes

maximal, characteristic, complemented

C₈₉×#2252

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

—

#19151975

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₈×#10623494

C₂²

yes

characteristic

#500×C₈₉

C₂²

yes

—

#366×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#13083561

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

D₁₄₂₄

—

C₂×#13083549

D₄

yes

—

#15667932

—

C₄×#10623494

D₄

yes

characteristic

C₄×#10623494

D₄

yes

characteristic

C₄⋊Q₈×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083561

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

D₈

yes

—

C₂×#10623494

D₈

yes

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

complemented

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#135

yes

—

#10623494

—

#10623494

D₁₆

yes

complemented

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

#508

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2252

D₈₉

yes

characteristic, complemented

#2246

—

#2240

—

#1706

—

C₈×C₁₆

—

C₈₉

#12503

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

178

—

#500

C₂×D₈₉

yes

—

#493

—

#492

—

178

—

#486

—

178

—

#366

—

#357

—

178

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

—

C₂×SD₁₆

—

356

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

178

—

D₁₆

—

356

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

—

#107

—

#107

—

100

#102

—

356

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂²×D₈₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×D₄

—

356

—

106

C₂×D₄

—

356

—

107

C₂×D₄

—

356

—

108

SD₁₆

—

712

—

109

D₈

—

356

—

110

D₈

—

356

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₄×D₈₉

yes

characteristic

114

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

D₄×D₈₉

yes

characteristic

118

C₁₆

—

178

—

119

C₁₆

#13083561

yes

complemented

120

C₂³

—

712

—

121

Q₈

—

122

D₄

—

712

—

123

D₄

—

712

—

124

D₄

—

712

—

125

D₄

—

712

—

126

C₂×C₄

—

127

C₂×C₄

#15668008

yes

characteristic

128

C₂×C₄

D₈×D₈₉

yes

characteristic

129

C₂×C₄

D₈×D₈₉

yes

characteristic

130

C₈

—

131

C₈

—

178

—

132

C₈

C₂×#13083561

yes

—

133

C₈

#15668035

yes

—

134

C₂²

—

1424

—

135

C₂²

—

1424

—

136

C₂²

—

1424

—

137

C₂²

#19151973

yes

characteristic

138

C₄

—

139

C₄

#19151972

yes

characteristic

140

C₄

#19151983

yes

characteristic

141

C₄

D₁₆×D₈₉

yes

—

142

C₄

#19152031

yes

—

143

C₂

—

2848

—

144

C₂

#26019845

yes

characteristic

145

C₂

#26022500

yes

characteristic

146

C₂

#26022514

yes

characteristic

	subgroup	G/H
center	C₂²	#19151973
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2252	C₂
socle	C₂²×C₈₉	#493
prime radical	#26022502	C₂
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#12503
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#19151973

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2252

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
2. #26022477	C₂	semidirect
3. #26019845	C₂	semidirect
4. C₁₆×#10623494	C₂	semidirect
5. C₈₉×#2252	C₂	semidirect
45. C₁₆×C₈₉	SD₁₆	semidirect
59. #10623494	D₁₆	semidirect
74. #2252	D₈₉	semidirect
79. C₈₉	#12503	semidirect
119. C₁₆	#13083561	semidirect

H ≅ G/N

type

2. #26022477

C₂

semidirect

3. #26019845

C₂

semidirect

4. C₁₆×#10623494

C₂

semidirect

5. C₈₉×#2252

C₂

semidirect

45. C₁₆×C₈₉

SD₁₆

semidirect

59. #10623494

D₁₆

semidirect

74. #2252

D₈₉

semidirect

79. C₈₉

#12503

semidirect

119. C₁₆

#13083561

semidirect

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

subgroup	index
#26022502	2
#26022477	2
#26019845	2
C₁₆×#10623494	2
C₈₉×#2252	2
#12503	89

subgroup

index

#26022502

#26022477

#26019845

C₁₆×#10623494

C₈₉×#2252

#12503

Subgroups — #36469082

Conjugacy classes of subgroups

Complements

Maximal subgroups