Group #36468805 — subgroups

Subgroups — #36468805

Order 22784.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022527	C₂	1	1	yes	maximal, characteristic
2	#26022160	C₂	1	1	yes	maximal, characteristic
3	#26019647	C₂	1	2	yes	maximal
4	#26018308	C₂	1	2	yes	maximal
5	C₈₉×#1372	C₂	1	1	yes	maximal, characteristic
6	#19152003	C₂²	1	2	yes	—
7	#19151809	—	2	2	no	—
8	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	#19151424	—	2	1	no	—
10	#19151424	—	2	1	no	—
11	C₄⋊#13083549	C₂²	1	1	yes	characteristic
12	#19151416	—	2	1	no	—
13	#19151373	—	2	1	no	—
14	#19151345	—	2	2	no	—
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#335×C₈₉	C₂²	1	2	yes	—
17	C₂×Dic₃₅₆	—	4	2	no	—
18	#15668012	—	2	2	no	—
19	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
20	C₂×#13083549	—	2	1	no	—
21	#15667946	—	2	2	no	—
22	#15667946	—	2	2	no	—
23	C₄⋊#10623494	D₄	1	1	yes	characteristic
24	#15667941	—	2	1	no	—
25	#15667937	—	4	2	no	—
26	C₈×Dic₈₉	—	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	2	no	—
30	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₇₈	—	4	2	no	—
32	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
33	Dic₃₅₆	—	8	4	no	—
34	C₂×#10623494	—	2	1	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
38	#13083553	—	2	1	no	—
39	C₄×Dic₈₉	—	2	1	no	—
40	#13083549	—	4	1	no	—
41	C₂×C₈×C₈₉	—	4	2	no	—
42	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
44	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₁₆×C₈₉	—	2	1	no	—
46	C₁₆×C₈₉	Q₁₆	1	2	yes	complemented
47	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
50	C₂×Dic₈₉	—	2	1	no	—
51	C₂×Dic₈₉	—	2	1	no	—
52	C₂×Dic₈₉	—	8	2	no	—
53	Dic₁₇₈	—	2	2	no	—
54	Dic₁₇₈	—	8	2	no	—
55	Dic₁₇₈	—	8	2	no	—
56	C₈×C₈₉	—	2	1	no	—
57	C₈×C₈₉	—	8	2	no	—
58	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₈₉	#137	1	1	yes	characteristic
60	#10623494	—	4	1	no	—
61	C₂²×C₈₉	#466	1	1	yes	characteristic
62	C₄×C₈₉	—	2	1	no	—
63	C₄×C₈₉	#462	1	1	yes	characteristic
64	C₄×C₈₉	#473	1	1	yes	characteristic
65	C₄×C₈₉	#509	1	2	yes	—
66	Dic₈₉	—	4	1	no	—
67	Dic₈₉	—	4	1	no	—
68	Dic₈₉	—	16	2	no	—
69	#12460	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	#1674	1	1	yes	characteristic
71	C₂×C₈₉	#2218	1	1	yes	characteristic
72	C₂×C₈₉	#2222	1	1	yes	characteristic
73	#2252	—	89	1	no	—
74	#2183	—	89	1	no	—
75	#1674	—	89	2	no	—
76	#1372	D₈₉	1	1	yes	characteristic
77	#1364	—	89	2	no	—
78	C₈₉	#12460	1	1	yes	characteristic, complemented, Sylow
79	#500	—	89	2	no	—
80	#462	—	178	2	no	—
81	C₈×Q₈	—	89	1	no	—
82	#366	—	178	1	no	—
83	#366	—	178	1	no	—
84	C₄⋊C₁₆	—	89	1	no	—
85	C₄⋊C₁₆	—	178	1	no	—
86	#346	—	178	1	no	—
87	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	#335	C₂×D₈₉	1	2	yes	—
89	#332	—	178	2	no	—
90	C₂×Q₁₆	—	356	2	no	—
91	C₄⋊Q₈	—	178	2	no	—
92	C₄×Q₈	—	89	1	no	—
93	C₂×C₁₆	—	178	1	no	—
94	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
95	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
96	#107	—	178	2	no	—
97	#107	—	178	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	89	1	no	—
100	C₄⋊C₈	—	178	1	no	—
101	#103	—	356	2	no	—
102	C₄×C₈	—	178	1	no	—
103	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×Q₈	—	89	1	no	—
105	C₂×Q₈	—	356	2	no	—
106	Q₁₆	—	712	4	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	178	1	no	—
109	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
110	C₂×C₈	#13083575	1	1	yes	characteristic
111	C₄⋊C₄	—	89	1	no	—
112	C₄⋊C₄	—	356	2	no	—
113	C₄⋊C₄	—	356	2	no	—
114	C₄⋊C₄	—	178	1	no	—
115	C₄²	—	178	1	no	—
116	C₄²	D₄×D₈₉	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	356	1	no	—
119	C₁₆	Dic₃₅₆	1	2	yes	complemented
120	Q₈	—	178	2	no	—
121	Q₈	—	712	2	no	—
122	Q₈	—	712	2	no	—
123	C₂×C₄	—	178	1	no	—
124	C₂×C₄	—	178	1	no	—
125	C₂×C₄	—	712	2	no	—
126	C₂×C₄	#15667987	1	1	yes	characteristic
127	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
128	C₂×C₄	#15668035	1	1	yes	characteristic
129	C₈	—	2	1	no	—
130	C₈	—	8	2	no	—
131	C₈	—	356	1	no	—
132	C₈	C₂×Dic₃₅₆	1	1	yes	characteristic
133	C₈	#15668045	1	1	yes	characteristic
134	C₂²	#19151835	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	356	1	no	—
137	C₄	—	356	1	no	—
138	C₄	—	1424	2	no	—
139	C₄	#19151809	1	1	yes	characteristic
140	C₄	#19151874	1	1	yes	characteristic
141	C₄	#19152035	1	2	yes	—
142	C₂	#26019647	1	1	yes	characteristic
143	C₂	#26022356	1	1	yes	characteristic
144	C₂	#26022378	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic

#26022160

C₂

yes

maximal, characteristic

#26019647

C₂

yes

maximal

#26018308

C₂

yes

maximal

C₈₉×#1372

C₂

yes

maximal, characteristic

#19152003

C₂²

yes

—

#19151809

—

C₈×Dic₁₇₈

C₂²

yes

—

—

C₂²

yes

—

—

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#335×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

—

#15668012

—

C₄×Dic₁₇₈

D₄

yes

—

—

—

D₄

yes

characteristic

#15667941

—

#15667937

—

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₈

yes

characteristic

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₈

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

D₈₉

yes

characteristic

#1364

—

C₈₉

#12460

yes

characteristic, complemented, Sylow

#500

—

#462

—

178

—

C₈×Q₈

—

#366

—

178

—

#366

—

178

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

#346

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

C₂×D₈₉

yes

—

#332

—

178

—

C₂×Q₁₆

—

356

—

C₄⋊Q₈

—

178

—

C₄×Q₈

—

C₂×C₁₆

—

178

—

C₂×C₁₆

D₃₅₆

yes

characteristic

C₂×C₁₆

D₃₅₆

yes

characteristic

#107

—

178

—

#107

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

—

178

—

101

#103

—

356

—

102

C₄×C₈

—

178

—

103

C₄×C₈

C₂²×D₈₉

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

356

—

106

Q₁₆

—

712

—

107

C₂×C₈

—

108

C₂×C₈

—

178

—

109

C₂×C₈

C₂×D₃₅₆

yes

characteristic

110

C₂×C₈

#13083575

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

356

—

113

C₄⋊C₄

—

356

—

114

C₄⋊C₄

—

178

—

115

C₄²

—

178

—

116

C₄²

D₄×D₈₉

yes

characteristic

117

C₁₆

—

118

C₁₆

—

356

—

119

C₁₆

Dic₃₅₆

yes

complemented

120

Q₈

—

178

—

121

Q₈

—

712

—

122

Q₈

—

712

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

—

712

—

126

C₂×C₄

#15667987

yes

characteristic

127

C₂×C₄

D₈×D₈₉

yes

characteristic

128

C₂×C₄

#15668035

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

356

—

132

C₈

C₂×Dic₃₅₆

yes

characteristic

133

C₈

#15668045

yes

characteristic

134

C₂²

#19151835

yes

characteristic

135

C₄

—

136

C₄

—

356

—

137

C₄

—

356

—

138

C₄

—

1424

—

139

C₄

#19151809

yes

characteristic

140

C₄

#19151874

yes

characteristic

141

C₄

#19152035

yes

—

142

C₂

#26019647

yes

characteristic

143

C₂

#26022356

yes

characteristic

144

C₂

#26022378

yes

characteristic

	subgroup	G/H
center	C₂²	#19151835
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1372	C₂
socle, prime radical	C₂²×C₈₉	#466
hypercenter	#1372	D₈₉
hypocenter	C₈₉	#12460
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

subgroup

G/H

center

C₂²

#19151835

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1372

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1372

N	H ≅ G/N	type
46. C₁₆×C₈₉	Q₁₆	semidirect
78. C₈₉	#12460	semidirect
119. C₁₆	Dic₃₅₆	semidirect

H ≅ G/N

type

46. C₁₆×C₈₉

Q₁₆

semidirect

78. C₈₉

#12460

semidirect

119. C₁₆

Dic₃₅₆

semidirect

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

subgroup	index
#26022527	2
#26022160	2
#26019647	2
#26018308	2
C₈₉×#1372	2
#12460	89

subgroup

index

C₈₉×#1372

Subgroups — #36468805

Conjugacy classes of subgroups

Complements

Maximal subgroups