Group #36469087 — subgroups

Subgroups — #36469087

Order 22784.

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

normal subgroups only

This group has 262 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	#26022505	C₂	1	1	yes	maximal, characteristic
2	#26022481	C₂	1	2	yes	maximal
3	#26019850	C₂	1	2	yes	maximal
4	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	—	2	2	no	—
7	#19151994	—	2	2	no	—
8	#19151978	C₂²	1	2	yes	—
9	#19151948	—	2	2	no	—
10	#19151401	—	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×Dic₃₅₆	—	2	2	no	—
17	C₂×Dic₃₅₆	—	2	2	no	—
18	#15668012	—	2	2	no	—
19	Dic₇₁₂	—	4	4	no	—
20	C₂×#13083549	D₄	1	2	yes	—
21	#15667943	—	4	4	no	—
22	#15667936	—	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×Dic₁₇₈	—	4	2	no	—
31	Dic₃₅₆	—	4	4	no	—
32	Dic₃₅₆	—	4	4	no	—
33	C₂×#10623494	D₈	1	2	yes	—
34	C₂×#10623494	D₈	1	2	yes	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	#13083549	—	2	2	no	—
38	C₂×Q₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	—	4	2	no	—
42	C₄⋊C₄×C₈₉	—	4	2	no	—
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	SD₁₆	1	4	yes	—
45	Q₈×C₈₉	—	8	4	no	—
46	C₂×C₄×C₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₈₉	—	8	2	no	—
51	Dic₁₇₈	—	8	2	no	—
52	Dic₁₇₈	—	8	2	no	—
53	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
54	C₈×C₈₉	#135	1	2	yes	—
55	#10623494	—	2	2	no	—
56	#10623494	Q₃₂	1	4	yes	complemented
57	C₂²×C₈₉	#493	1	1	yes	characteristic
58	C₄×C₈₉	—	16	2	no	—
59	C₄×C₈₉	#492	1	1	yes	characteristic
60	C₄×C₈₉	#495	1	1	yes	characteristic
61	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
62	C₄×C₈₉	#508	1	2	yes	—
63	Dic₈₉	—	16	2	no	—
64	#12504	—	89	1	no	maximal, Sylow
65	C₂×C₈₉	#1706	1	1	yes	characteristic
66	C₂×C₈₉	#2247	1	1	yes	characteristic
67	C₂×C₈₉	#2249	1	1	yes	characteristic
68	#2252	D₈₉	1	1	yes	characteristic
69	#2247	—	89	1	no	—
70	#2241	—	89	2	no	—
71	#1707	—	89	2	no	—
72	C₈×C₁₆	—	89	1	no	—
73	C₈₉	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	178	2	no	—
75	#500	C₂×D₈₉	1	2	yes	—
76	#498	—	178	2	no	—
77	#494	—	89	2	no	—
78	#487	—	178	2	no	—
79	#366	—	2	2	no	—
80	#358	—	178	4	no	—
81	C₄×C₁₆	—	89	1	no	—
82	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
83	C₈²	—	89	1	no	—
84	C₂×Q₁₆	—	178	2	no	—
85	C₂×Q₁₆	—	178	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	178	2	no	—
88	Q₃₂	—	356	4	no	—
89	C₂×C₁₆	—	89	2	no	—
90	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#106	—	356	4	no	—
94	#103	—	356	4	no	—
95	C₄×C₈	—	89	1	no	—
96	C₄×C₈	—	89	1	no	—
97	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	356	2	no	—
100	Q₁₆	—	356	4	no	—
101	Q₁₆	—	356	4	no	—
102	C₂×C₈	—	89	2	no	—
103	C₂×C₈	—	89	2	no	—
104	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	356	2	no	—
109	C₄⋊C₄	—	356	2	no	—
110	C₄²	D₄×D₈₉	1	1	yes	characteristic
111	C₁₆	—	178	2	no	—
112	C₁₆	#13083561	1	4	yes	—
113	Q₈	—	8	4	no	—
114	Q₈	—	712	2	no	—
115	Q₈	—	712	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	712	2	no	—
118	C₂×C₄	#15668008	1	1	yes	characteristic
119	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
120	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
121	C₈	—	89	4	no	—
122	C₈	—	178	2	no	—
123	C₈	C₂×#13083561	1	2	yes	—
124	C₈	#15668035	1	2	yes	—
125	C₂²	#19151973	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	1424	2	no	—
128	C₄	#19151972	1	1	yes	characteristic
129	C₄	#19151983	1	1	yes	characteristic
130	C₄	Q₃₂×D₈₉	1	2	yes	—
131	C₄	#19152027	1	2	yes	—
132	C₂	#26019845	1	1	yes	characteristic
133	C₂	#26022503	1	1	yes	characteristic
134	C₂	#26022510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022505

C₂

yes

maximal, characteristic

#26022481

C₂

yes

maximal

#26019850

C₂

yes

maximal

C₁₆×#10623494

C₂

yes

maximal, characteristic

C₈₉×#2252

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

#19151994

—

#19151978

C₂²

yes

—

#19151948

—

#19151401

—

C₄×#13083549

C₂²

yes

characteristic

C₈×#10623494

C₂²

yes

characteristic

#500×C₈₉

C₂²

yes

—

#366×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

Dic₇₁₂

—

C₂×#13083549

D₄

yes

—

#15667943

—

#15667936

—

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×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₂×#10623494

D₈

yes

—

C₂×#10623494

D₈

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#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

—

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

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#135

yes

—

#10623494

—

#10623494

Q₃₂

yes

complemented

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

—

C₄×C₈₉

#508

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

—

—

—

C₈×C₁₆

—

C₈₉

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

#500

C₂×D₈₉

yes

—

#498

—

178

—

#494

—

#487

—

178

—

#366

—

#358

—

178

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

Q₃₂

—

356

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

—

#107

—

#107

—

#106

—

356

—

#103

—

356

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

100

Q₁₆

—

356

—

101

Q₁₆

—

356

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₈₉

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

356

—

109

C₄⋊C₄

—

356

—

110

C₄²

D₄×D₈₉

yes

characteristic

111

C₁₆

—

178

—

112

C₁₆

#13083561

yes

—

113

Q₈

—

114

Q₈

—

712

—

115

Q₈

—

712

—

116

C₂×C₄

—

117

C₂×C₄

—

712

—

118

C₂×C₄

#15668008

yes

characteristic

119

C₂×C₄

D₈×D₈₉

yes

characteristic

120

C₂×C₄

D₈×D₈₉

yes

characteristic

121

C₈

—

122

C₈

—

178

—

123

C₈

C₂×#13083561

yes

—

124

C₈

#15668035

yes

—

125

C₂²

#19151973

yes

characteristic

126

C₄

—

127

C₄

—

1424

—

128

C₄

#19151972

yes

characteristic

129

C₄

#19151983

yes

characteristic

130

C₄

Q₃₂×D₈₉

yes

—

131

C₄

#19152027

yes

—

132

C₂

#26019845

yes

characteristic

133

C₂

#26022503

yes

characteristic

134

C₂

#26022510

yes

characteristic

	subgroup	G/H
center	C₂²	#19151973
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2252	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#12504
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

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
56. #10623494	Q₃₂	semidirect
73. C₈₉	#12504	semidirect

H ≅ G/N

type

56. #10623494

Q₃₂

semidirect

73. C₈₉

#12504

semidirect

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

subgroup	index
#26022505	2
#26022481	2
#26019850	2
C₁₆×#10623494	2
C₈₉×#2252	2
#12504	89

subgroup

index

#26022505

#26022481

#26019850

C₁₆×#10623494

C₈₉×#2252

#12504

Subgroups — #36469087

Conjugacy classes of subgroups

Complements

Maximal subgroups