Group #36442003 — subgroups

Subgroups — #36442003

Order 22784.

136 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 41 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	#26022360	C₂	1	1	yes	maximal, characteristic
2	#26018392	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1385	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic
5	#19151839	—	2	1	no	—
6	#19151432	—	2	1	no	—
7	#19151426	C₄	1	1	yes	characteristic
8	#19151403	—	2	1	no	—
9	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×#108×C₈₉	—	2	1	no	—
11	C₂×Dic₃₅₆	—	2	1	no	—
12	#15667991	—	4	1	no	—
13	Dic₇₁₂	—	4	1	no	—
14	#15667946	—	2	1	no	—
15	#15667937	—	4	1	no	—
16	#15667918	—	4	1	no	—
17	C₂×M₁₆×C₈₉	—	4	1	no	—
18	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
19	M₃₂×C₈₉	—	4	1	no	—
20	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
22	#108×C₈₉	—	2	2	no	—
23	#108×C₈₉	—	4	1	no	—
24	C₂×Dic₁₇₈	—	4	1	no	—
25	Dic₃₅₆	—	4	1	no	—
26	Dic₃₅₆	—	4	1	no	—
27	C₄⋊Dic₈₉	—	4	1	no	—
28	#13083553	—	8	1	no	—
29	C₂²⋊Dic₈₉	—	8	1	no	—
30	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
31	M₁₆×C₈₉	—	4	2	no	—
32	M₁₆×C₈₉	—	8	1	no	—
33	C₂×C₈×C₈₉	—	2	1	no	—
34	C₂×C₈×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	—	8	1	no	—
36	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
38	C₁₆×C₈₉	—	2	1	no	—
39	C₁₆×C₈₉	—	2	1	no	—
40	C₂³×C₈₉	SD₃₂	1	1	yes	characteristic
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
44	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
45	C₂×Dic₈₉	—	8	1	no	—
46	C₂×Dic₈₉	—	8	1	no	—
47	Dic₁₇₈	—	8	1	no	—
48	Dic₁₇₈	—	8	1	no	—
49	C₈×C₈₉	—	2	1	no	—
50	C₈×C₈₉	—	8	2	no	—
51	C₈×C₈₉	#99	1	1	yes	characteristic
52	C₈×C₈₉	#104	1	1	yes	characteristic
53	C₂²×C₈₉	—	2	1	no	—
54	C₂²×C₈₉	—	4	1	no	—
55	C₂²×C₈₉	#357	1	1	yes	characteristic
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	#327	1	1	yes	characteristic
58	C₄×C₈₉	#360	1	1	yes	characteristic
59	Dic₈₉	—	16	1	no	—
60	Dic₈₉	—	16	1	no	—
61	#7425	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	2	1	no	—
63	C₂×C₈₉	—	4	1	no	—
64	C₂×C₈₉	#1347	1	1	yes	characteristic
65	#2219	—	89	1	no	—
66	#1400	—	89	1	no	—
67	#1385	D₈₉	1	1	yes	characteristic
68	C₈₉	#7425	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	89	1	no	—
70	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
71	#467	—	178	1	no	—
72	C₂×#108	—	2	1	no	—
73	M₆₄	—	178	1	no	—
74	#367	—	89	1	no	—
75	#358	—	178	1	no	—
76	C₂×Q₁₆	—	178	1	no	—
77	C₂×M₁₆	—	4	1	no	—
78	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
79	C₂²⋊Q₈	—	356	1	no	—
80	Q₃₂	—	356	1	no	—
81	M₃₂	—	4	1	no	—
82	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
83	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
84	#108	—	2	2	no	—
85	#108	—	4	1	no	—
86	#107	—	178	1	no	—
87	#103	—	356	1	no	—
88	C₃₂	—	356	1	no	—
89	C₂×Q₈	—	356	1	no	—
90	C₂²×C₄	#13083559	1	1	yes	characteristic
91	Q₁₆	—	356	1	no	—
92	Q₁₆	—	356	1	no	—
93	M₁₆	—	4	2	no	—
94	M₁₆	—	8	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	8	1	no	—
98	C₂×C₈	#13083551	1	1	yes	characteristic
99	C₂×C₈	#13083561	1	1	yes	characteristic
100	C₄⋊C₄	—	356	1	no	—
101	C₄⋊C₄	—	712	1	no	—
102	C₂²⋊C₄	—	712	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	2	1	no	—
105	C₂³	#15667955	1	1	yes	characteristic
106	Q₈	—	712	1	no	—
107	Q₈	—	712	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	712	1	no	—
111	C₂×C₄	—	712	1	no	—
112	C₂×C₄	#15667932	1	1	yes	characteristic
113	C₂×C₄	#15667953	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	8	2	no	—
116	C₈	#15667926	1	1	yes	characteristic
117	C₈	#15667940	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	4	1	no	—
120	C₂²	#19151398	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	1424	1	no	—
123	C₄	—	1424	1	no	—
124	C₄	#19151331	1	1	yes	characteristic
125	C₄	#19151407	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	1	no	—
128	C₂	#26018260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022360

C₂

yes

maximal, characteristic

#26018392

C₂

yes

maximal, characteristic

C₈₉×#1385

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₄

yes

—

—

C₄

yes

characteristic

#19151403

—

C₂×M₃₂×C₈₉

C₂²

yes

characteristic

C₂×#108×C₈₉

—

C₂×Dic₃₅₆

—

#15667991

—

Dic₇₁₂

—

#15667946

—

#15667937

—

#15667918

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

#108×C₈₉

—

#108×C₈₉

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

SD₃₂

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#102

yes

characteristic

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#357

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#7425

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

—

D₈₉

yes

characteristic

C₈₉

#7425

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×M₃₂

C₂×D₈₉

yes

characteristic

#467

—

178

—

C₂×#108

—

M₆₄

—

178

—

#367

—

#358

—

178

—

C₂×Q₁₆

—

178

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₈₉

yes

characteristic

C₂²⋊Q₈

—

356

—

Q₃₂

—

356

—

M₃₂

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

D₃₅₆

yes

—

—

—

178

—

#103

—

356

—

C₃₂

—

356

—

C₂×Q₈

—

356

—

C₂²×C₄

#13083559

yes

characteristic

Q₁₆

—

356

—

Q₁₆

—

356

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

100

C₄⋊C₄

—

356

—

101

C₄⋊C₄

—

712

—

102

C₂²⋊C₄

—

712

—

103

C₁₆

—

104

C₁₆

—

105

C₂³

#15667955

yes

characteristic

106

Q₈

—

712

—

107

Q₈

—

712

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

712

—

111

C₂×C₄

—

712

—

112

C₂×C₄

#15667932

yes

characteristic

113

C₂×C₄

#15667953

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

#15667926

yes

characteristic

117

C₈

#15667940

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

#19151398

yes

characteristic

121

C₄

—

122

C₄

—

1424

—

123

C₄

—

1424

—

124

C₄

#19151331

yes

characteristic

125

C₄

#19151407

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#26018260

yes

characteristic

	subgroup	G/H
center	C₂	#26018260
commutator	C₂×C₁₆×C₈₉	C₂×C₄
Frattini	C₂×M₃₂	C₂×D₈₉
Fitting	C₈₉×#1385	C₂
socle	C₂×C₈₉	#1347
prime radical	C₂³×C₈₉	SD₃₂
cosocle	C₂×M₃₂×C₈₉	C₂²
hypercenter	#1385	D₈₉
hypocenter	C₈₉	#7425
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #1385

subgroup

G/H

center

C₂

#26018260

commutator

C₂×C₁₆×C₈₉

C₂×C₄

Frattini

C₂×M₃₂

C₂×D₈₉

Fitting

C₈₉×#1385

C₂

socle

C₂×C₈₉

#1347

prime radical

C₂³×C₈₉

SD₃₂

cosocle

C₂×M₃₂×C₈₉

C₂²

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂ ◁ #1385

N	H ≅ G/N	type
68. C₈₉	#7425	semidirect

H ≅ G/N

type

68. C₈₉

#7425

semidirect

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

subgroup	index
#26022360	2
#26018392	2
C₈₉×#1385	2
#7425	89

subgroup

index

#26022360

#26018392

C₈₉×#1385

#7425

Subgroups — #36442003

Conjugacy classes of subgroups

Complements

Maximal subgroups