Group #36448135 — subgroups

Subgroups — #36448135

Order 22784.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#26019015	C₂	1	2	yes	maximal
2	#26018114	C₂	1	4	yes	maximal
3	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic
4	#19151655	C₂²	1	4	yes	—
5	#19151351	—	2	2	no	—
6	#19151349	—	2	2	no	—
7	#431×C₈₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	#322×C₈₉	C₄	1	2	yes	—
10	C₈²×C₈₉	C₄	1	2	yes	—
11	C₄⋊#10623494	—	2	4	no	—
12	C₂²⋊#10623494	—	2	4	no	—
13	#15667923	—	2	4	no	—
14	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₈₉	—	2	2	no	—
18	C₂²⋊C₈×C₈₉	—	2	2	no	—
19	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×#10623494	—	4	4	no	—
25	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	#41	1	2	yes	—
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	—	2	4	no	—
30	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₈₉	#41	1	2	yes	—
37	C₂³×C₈₉	#125	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#117	1	2	yes	—
43	C₂×C₄×C₈₉	#124	1	2	yes	—
44	C₂×C₄×C₈₉	#125	1	1	yes	characteristic
45	C₈×C₈₉	—	2	1	no	—
46	C₈×C₈₉	—	2	1	no	—
47	C₈×C₈₉	—	4	4	no	—
48	C₈×C₈₉	#108	1	2	yes	—
49	C₈×C₈₉	#108	1	2	yes	—
50	#10623494	—	8	4	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	—	2	1	no	—
53	C₂²×C₈₉	#382	1	1	yes	characteristic
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	—	2	2	no	—
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
58	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
59	#8796	—	89	1	no	maximal, Sylow
60	C₂×C₈₉	—	4	1	no	—
61	C₂×C₈₉	#1833	1	2	yes	—
62	C₂×C₈₉	#1837	1	1	yes	characteristic
63	#1568	—	89	2	no	—
64	#1450	D₈₉	1	1	yes	characteristic
65	#1287	—	89	4	no	—
66	C₈₉	#8796	1	1	yes	characteristic, complemented, Sylow
67	#432	—	89	4	no	—
68	#431	C₂×D₈₉	1	2	yes	—
69	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
70	#335	—	178	2	no	—
71	#334	—	178	2	no	—
72	#322	Dic₈₉	1	2	yes	—
73	C₈²	Dic₈₉	1	2	yes	—
74	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
75	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
76	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	178	4	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	178	4	no	—
81	#97	—	178	4	no	—
82	C₄×C₈	D₃₅₆	1	1	yes	characteristic
83	C₄×C₈	D₃₅₆	1	1	yes	characteristic
84	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
85	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
88	C₂²×C₄	#13083574	1	2	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	356	4	no	—
93	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
94	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
96	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
97	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
98	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
99	C₄²	#13083574	1	2	yes	—
100	C₂³	#15668004	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
105	C₂×C₄	#15667971	1	2	yes	—
106	C₂×C₄	#15668000	1	2	yes	—
107	C₂×C₄	#15668004	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	4	no	—
111	C₈	—	712	4	no	—
112	C₈	#15667948	1	2	yes	—
113	C₈	#15667948	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#19151464	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×#15667948	1	1	yes	characteristic
121	C₄	C₂×#15667948	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	#26020376	1	2	yes	—
124	C₂	#26020390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26019015

C₂

yes

maximal

#26018114

C₂

yes

maximal

C₈₉×#1450

C₂

yes

maximal, characteristic

#19151655

C₂²

yes

—

#19151351

—

#19151349

—

#431×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#322×C₈₉

C₄

yes

—

C₈²×C₈₉

C₄

yes

—

C₄⋊#10623494

—

C₂²⋊#10623494

—

#15667923

—

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂²×C₈×C₈₉

Q₈

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×#10623494

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

characteristic

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

#125

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#125

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#108

yes

—

C₈×C₈₉

#108

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#382

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

#1833

yes

—

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1287

—

C₈₉

#8796

yes

characteristic, complemented, Sylow

#432

—

#431

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#335

—

178

—

#334

—

178

—

#322

Dic₈₉

yes

—

C₈²

Dic₈₉

yes

—

C₂²×C₈

Dic₁₇₈

yes

characteristic

C₂²×C₈

Dic₁₇₈

yes

characteristic

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₂²⋊C₈

—

C₂²⋊C₈

—

178

—

#97

—

178

—

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

C₂×Dic₈₉

yes

—

C₄×C₈

C₂×Dic₈₉

yes

characteristic

C₄×C₈

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

C₂²×C₄

#13083574

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

C₂×C₈

C₄⋊Dic₈₉

yes

characteristic

C₄²

C₂²×Dic₈₉

yes

characteristic

C₄²

C₂×D₃₅₆

yes

characteristic

C₄²

#13083574

yes

—

100

C₂³

#15668004

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

105

C₂×C₄

#15667971

yes

—

106

C₂×C₄

#15668000

yes

—

107

C₂×C₄

#15668004

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

712

—

112

C₈

#15667948

yes

—

113

C₈

#15667948

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

#19151464

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#15667948

yes

characteristic

121

C₄

C₂×#15667948

yes

characteristic

122

C₂

—

123

C₂

#26020376

yes

—

124

C₂

#26020390

yes

characteristic

	subgroup	G/H
center	C₂²	#19151464
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	#382
prime radical	C₂³×C₈₉	#125
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1450	D₈₉
hypocenter	C₈₉	#8796
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#19151464

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1450

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
66. C₈₉	#8796	semidirect

H ≅ G/N

type

66. C₈₉

#8796

semidirect

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

subgroup	index
#26019015	2
#26018114	2
C₈₉×#1450	2
#8796	89

subgroup

index

#26019015

#26018114

C₈₉×#1450

#8796

Subgroups — #36448135

Conjugacy classes of subgroups

Complements

Maximal subgroups