Group #36468707 — subgroups

Subgroups — #36468707

Order 22784.

212 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 71 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	#26022505	C₂	1	1	yes	maximal, characteristic
2	#26019622	C₂	1	2	yes	maximal
3	#26018326	C₂	1	1	yes	maximal, characteristic
4	#26018302	C₂	1	2	yes	maximal
5	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	—	2	1	no	—
7	C₂×Dic₇₁₂	—	2	1	no	—
8	#19151978	C₂²	1	2	yes	—
9	#19151883	—	2	2	no	—
10	#19151351	C₂²	1	2	yes	—
11	#19151345	—	2	2	no	—
12	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₉	—	2	1	no	—
14	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#346×C₈₉	—	2	1	no	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	—	2	2	no	—
18	C₂×Dic₃₅₆	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	Dic₇₁₂	—	4	2	no	—
21	Dic₇₁₂	—	4	2	no	—
22	#15667943	—	4	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	#15667936	—	4	2	no	—
25	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	—	2	1	no	—
27	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	—	2	1	no	—
30	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	—	2	1	no	—
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₇₈	—	4	2	no	—
34	Dic₃₅₆	—	4	2	no	—
35	Dic₃₅₆	—	4	2	no	—
36	Dic₃₅₆	—	8	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
46	C₄²×C₈₉	—	2	1	no	—
47	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₈₉	—	2	1	no	—
49	C₁₆×C₈₉	—	4	1	no	—
50	C₁₆×C₈₉	SD₁₆	1	2	yes	—
51	Q₈×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
57	C₂×Dic₈₉	—	8	2	no	—
58	Dic₁₇₈	—	8	2	no	—
59	Dic₁₇₈	—	8	2	no	—
60	C₈×C₈₉	—	2	1	no	—
61	C₈×C₈₉	—	4	1	no	—
62	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	#10623494	—	8	2	no	—
65	C₂²×C₈₉	#466	1	1	yes	characteristic
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	—	4	1	no	—
68	C₄×C₈₉	—	4	1	no	—
69	C₄×C₈₉	#460	1	1	yes	characteristic
70	C₄×C₈₉	#472	1	1	yes	characteristic
71	C₄×C₈₉	#510	1	2	yes	—
72	Dic₈₉	—	16	2	no	—
73	#12443	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	#1668	1	1	yes	characteristic
75	C₂×C₈₉	#2219	1	1	yes	characteristic
76	C₂×C₈₉	#2222	1	1	yes	characteristic
77	#2247	—	89	1	no	—
78	#2183	D₈₉	1	1	yes	characteristic
79	#1670	—	89	2	no	—
80	#1372	—	89	1	no	—
81	#1362	—	89	2	no	—
82	C₈₉	#12443	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	—	178	1	no	—
84	C₂×Q₃₂	—	178	1	no	—
85	#494	—	89	2	no	—
86	#475	—	178	2	no	—
87	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
88	C₄⋊C₁₆	—	2	1	no	—
89	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
90	#346	—	2	1	no	—
91	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
92	#335	—	89	2	no	—
93	#332	—	178	2	no	—
94	C₂×Q₁₆	—	178	2	no	—
95	C₂×Q₁₆	—	178	2	no	—
96	C₄⋊Q₈	—	178	2	no	—
97	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
98	Q₃₂	—	356	2	no	—
99	Q₃₂	—	356	2	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
103	#106	—	356	2	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	178	2	no	—
106	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
107	#103	—	356	2	no	—
108	C₄×C₈	—	2	1	no	—
109	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
110	C₂×Q₈	—	356	2	no	—
111	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
112	Q₁₆	—	356	2	no	—
113	Q₁₆	—	356	2	no	—
114	Q₁₆	—	712	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	356	2	no	—
117	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
118	C₂×C₈	#13083574	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	356	2	no	—
121	C₄⋊C₄	—	356	2	no	—
122	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	4	1	no	—
127	C₁₆	#13083561	1	2	yes	—
128	Q₈	—	2	2	no	—
129	Q₈	—	712	2	no	—
130	Q₈	—	712	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	712	2	no	—
134	C₂×C₄	#15667986	1	1	yes	characteristic
135	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
136	C₂×C₄	#15668034	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	712	2	no	—
140	C₈	C₂×#13083561	1	1	yes	characteristic
141	C₈	#15668037	1	1	yes	characteristic
142	C₂²	#19151834	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	1424	2	no	—
147	C₄	#19151798	1	1	yes	characteristic
148	C₄	#19151868	1	1	yes	characteristic
149	C₄	#19152041	1	2	yes	—
150	C₂	#26019612	1	1	yes	characteristic
151	C₂	#26022360	1	1	yes	characteristic
152	C₂	#26022377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022505

C₂

yes

maximal, characteristic

#26019622

C₂

yes

maximal

#26018326

C₂

yes

maximal, characteristic

#26018302

C₂

yes

maximal

C₈₉×#2183

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

—

#19151978

C₂²

yes

—

#19151883

—

#19151351

C₂²

yes

—

#19151345

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₄⋊C₁₆×C₈₉

C₂²

yes

characteristic

#346×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667943

—

C₄⋊#10623494

—

#15667936

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

SD₁₆

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×SD₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

—

—

—

C₈₉

#12443

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

C₂×Q₃₂

—

178

—

#494

—

#475

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#332

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

178

—

C₄×Q₈

D₃₅₆

yes

characteristic

Q₃₂

—

356

—

Q₃₂

—

356

—

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

103

#106

—

356

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

178

—

106

C₄⋊C₈

D₃₅₆

yes

characteristic

107

#103

—

356

—

108

C₄×C₈

—

109

C₄×C₈

C₂²×D₈₉

yes

characteristic

110

C₂×Q₈

—

356

—

111

C₂×Q₈

D₇₁₂

yes

characteristic

112

Q₁₆

—

356

—

113

Q₁₆

—

356

—

114

Q₁₆

—

712

—

115

C₂×C₈

—

116

C₂×C₈

—

356

—

117

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

118

C₂×C₈

#13083574

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

356

—

121

C₄⋊C₄

—

356

—

122

C₄⋊C₄

D₇₁₂

yes

characteristic

123

C₄²

—

124

C₄²

C₂×D₃₅₆

yes

characteristic

125

C₁₆

—

126

C₁₆

—

127

C₁₆

#13083561

yes

—

128

Q₈

—

129

Q₈

—

712

—

130

Q₈

—

712

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

712

—

134

C₂×C₄

#15667986

yes

characteristic

135

C₂×C₄

C₂×D₇₁₂

yes

characteristic

136

C₂×C₄

#15668034

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

712

—

140

C₈

C₂×#13083561

yes

characteristic

141

C₈

#15668037

yes

characteristic

142

C₂²

#19151834

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

1424

—

147

C₄

#19151798

yes

characteristic

148

C₄

#19151868

yes

characteristic

149

C₄

#19152041

yes

—

150

C₂

#26019612

yes

characteristic

151

C₂

#26022360

yes

characteristic

152

C₂

#26022377

yes

characteristic

	subgroup	G/H
center	C₂²	#19151834
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2183	C₂
socle, prime radical	C₂²×C₈₉	#466
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#12443
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#19151834

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2183

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
82. C₈₉	#12443	semidirect

H ≅ G/N

type

82. C₈₉

#12443

semidirect

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

subgroup	index
#26022505	2
#26019622	2
#26018326	2
#26018302	2
C₈₉×#2183	2
#12443	89

subgroup

index

C₈₉×#2183

Subgroups — #36468707

Conjugacy classes of subgroups

Complements

Maximal subgroups