Group #36475315 — subgroups

Subgroups — #36475315

Order 22784.

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

normal subgroups only

This group has 164 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	#26018453	C₂	1	2	yes	maximal
3	C₄⋊#15667918	C₂	1	1	yes	maximal, characteristic
4	#26018433	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19152003	—	2	1	no	—
7	C₂×#15667918	C₂²	1	2	yes	—
8	#19151424	—	2	1	no	—
9	#19151424	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆×C₈₉	—	2	1	no	—
12	#366×C₈₉	C₂²	1	1	yes	characteristic
13	#358×C₈₉	—	2	1	no	—
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#15668012	—	4	1	no	—
16	#15667946	—	2	2	no	—
17	#15667946	—	4	1	no	—
18	#15667918	—	2	2	no	—
19	C₂×Q₁₆×C₈₉	—	2	1	no	—
20	C₄×Q₈×C₈₉	—	4	1	no	—
21	Q₃₂×C₈₉	—	2	1	no	—
22	Q₃₂×C₈₉	Q₈	1	2	yes	complemented
23	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
25	#107×C₈₉	—	2	1	no	—
26	#103×C₈₉	—	4	1	no	—
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂×Dic₁₇₈	—	8	1	no	—
29	C₄⋊Dic₈₉	—	4	2	no	—
30	C₄⋊Dic₈₉	—	8	1	no	—
31	C₂×Q₈×C₈₉	—	4	1	no	—
32	Q₁₆×C₈₉	—	2	2	no	—
33	Q₁₆×C₈₉	—	4	1	no	—
34	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	4	1	no	—
37	C₄⋊C₄×C₈₉	—	8	1	no	—
38	C₄²×C₈₉	—	8	1	no	—
39	C₄²×C₈₉	D₈	1	1	yes	characteristic
40	C₁₆×C₈₉	—	2	1	no	—
41	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
42	C₁₆×C₈₉	#41	1	1	yes	characteristic
43	Q₈×C₈₉	—	4	2	no	—
44	Q₈×C₈₉	—	8	1	no	—
45	C₂×C₄×C₈₉	—	8	1	no	—
46	C₂×C₄×C₈₉	—	8	1	no	—
47	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
48	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₈₉	—	8	2	no	—
51	Dic₁₇₈	—	16	2	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₈×C₈₉	#137	1	1	yes	characteristic
55	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	8	2	no	—
58	C₄×C₈₉	—	16	1	no	—
59	C₄×C₈₉	SD₆₄	1	2	yes	—
60	C₄×C₈₉	#474	1	1	yes	characteristic
61	C₄×C₈₉	#510	1	1	yes	characteristic
62	Dic₈₉	—	16	2	no	—
63	#13704	—	89	1	no	maximal, Sylow
64	C₂×C₈₉	#2224	1	1	yes	characteristic
65	C₂×C₈₉	C₂×SD₆₄	1	1	yes	characteristic
66	C₂×C₈₉	#2264	1	1	yes	characteristic
67	#2252	—	89	1	no	—
68	C₄×Q₃₂	D₈₉	1	1	yes	characteristic
69	#1424	—	89	2	no	—
70	C₄⋊C₃₂	—	89	1	no	—
71	#1416	—	89	2	no	—
72	C₈₉	#13704	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
74	#500	—	178	1	no	—
75	C₄×Q₁₆	—	2	1	no	—
76	C₂×C₃₂	—	89	2	no	—
77	#366	—	89	2	no	—
78	#366	—	178	1	no	—
79	#366	C₂×D₈₉	1	1	yes	characteristic
80	#358	—	2	1	no	—
81	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
82	C₂×Q₁₆	—	2	1	no	—
83	C₄⋊Q₈	—	356	1	no	—
84	C₄×Q₈	—	4	1	no	—
85	Q₃₂	—	2	1	no	—
86	Q₃₂	Dic₁₇₈	1	2	yes	complemented
87	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
88	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	#107	—	356	1	no	—
91	#107	—	178	2	no	—
92	#103	—	4	1	no	—
93	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
94	C₃₂	—	178	2	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	712	1	no	—
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	4	1	no	—
99	C₂×C₈	#13083559	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	356	2	no	—
104	C₄⋊C₄	—	712	1	no	—
105	C₄²	—	8	1	no	—
106	C₄²	#13083559	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
109	C₁₆	#13083574	1	1	yes	characteristic
110	Q₈	—	4	2	no	—
111	Q₈	—	8	1	no	—
112	Q₈	—	1424	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	712	2	no	—
116	C₂×C₄	#15667953	1	1	yes	characteristic
117	C₂×C₄	#15667953	1	1	yes	characteristic
118	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	#15667991	1	1	yes	characteristic
121	C₈	#15668042	1	1	yes	characteristic
122	C₂²	C₂×#15667953	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	8	2	no	—
125	C₄	—	16	1	no	—
126	C₄	—	1424	2	no	—
127	C₄	#19151435	1	2	yes	—
128	C₄	#19151878	1	1	yes	characteristic
129	C₄	#19152038	1	1	yes	characteristic
130	C₂	#26022389	1	1	yes	characteristic
131	C₂	C₂×#19151435	1	1	yes	characteristic
132	C₂	#26022569	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic

#26018453

C₂

yes

maximal

C₄⋊#15667918

C₂

yes

maximal, characteristic

#26018433

C₂

yes

maximal

C₄×Q₃₂×C₈₉

C₂

yes

maximal, characteristic

#19152003

—

C₂×#15667918

C₂²

yes

—

#19151424

—

#19151424

C₂²

yes

—

C₂×Q₃₂×C₈₉

C₂²

yes

characteristic

C₄×Q₁₆×C₈₉

—

#366×C₈₉

C₂²

yes

characteristic

#358×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

—

—

—

—

C₂×Q₁₆×C₈₉

—

C₄×Q₈×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

Q₈

yes

complemented

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#103×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×Dic₁₇₈

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₁₆×C₈₉

#41

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²⋊Q₈

yes

characteristic

C₈×C₈₉

#137

yes

characteristic

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

SD₆₄

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

C₂×SD₆₄

yes

C₂×C₈₉

yes

—

C₄×Q₃₂

D₈₉

yes

characteristic

#1424

—

C₄⋊C₃₂

—

#1416

—

C₈₉

#13704

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₈₉

yes

characteristic

#500

—

178

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#366

—

178

—

#366

C₂×D₈₉

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

356

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

Dic₁₇₈

yes

complemented

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

#107

—

356

—

#107

—

178

—

#103

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₃₂

—

178

—

C₂×Q₈

—

C₂×Q₈

—

712

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#13083559

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

712

—

105

C₄²

—

106

C₄²

#13083559

yes

characteristic

107

C₁₆

—

108

C₁₆

C₂×Dic₁₇₈

yes

characteristic

109

C₁₆

#13083574

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

1424

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

712

—

116

C₂×C₄

#15667953

yes

characteristic

117

C₂×C₄

#15667953

yes

characteristic

118

C₂×C₄

C₂×#13083559

yes

characteristic

119

C₈

—

120

C₈

#15667991

yes

characteristic

121

C₈

#15668042

yes

characteristic

122

C₂²

C₂×#15667953

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

1424

—

127

C₄

#19151435

yes

—

128

C₄

#19151878

yes

characteristic

129

C₄

#19152038

yes

characteristic

130

C₂

#26022389

yes

characteristic

131

C₂

C₂×#19151435

yes

characteristic

132

C₂

#26022569

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#15667953
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₄×Q₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	#13704
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#15667953

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₄×Q₃₂×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₈₉

hypocenter

C₈₉

#13704

LCS

G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
22. Q₃₂×C₈₉	Q₈	semidirect
72. C₈₉	#13704	semidirect
86. Q₃₂	Dic₁₇₈	semidirect

H ≅ G/N

type

22. Q₃₂×C₈₉

Q₈

semidirect

72. C₈₉

#13704

semidirect

86. Q₃₂

Dic₁₇₈

semidirect

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

subgroup	index
#26022527	2
#26018453	2
C₄⋊#15667918	2
#26018433	2
C₄×Q₃₂×C₈₉	2
#13704	89

subgroup

index

C₄×Q₃₂×C₈₉

Subgroups — #36475315

Conjugacy classes of subgroups

Complements

Maximal subgroups