Group #36466314 — subgroups

Subgroups — #36466314

Order 22784.

246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 79 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	#26022129	C₂	1	1	yes	maximal, characteristic
2	#26019077	C₂	1	2	yes	maximal, complemented
3	#26018318	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018220	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	C₄	1	1	yes	characteristic
7	C₄×D₇₁₂	C₂²	1	2	yes	—
8	C₈×D₃₅₆	—	2	2	no	—
9	#19151426	C₄	1	1	yes	characteristic
10	#19151400	C₄	1	2	yes	—
11	#19151349	C₂²	1	2	yes	—
12	#19151327	—	2	2	no	—
13	#346×C₈₉	C₂²	1	1	yes	characteristic
14	#346×C₈₉	C₂²	1	1	yes	characteristic
15	#322×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₈₉	—	4	2	no	—
18	C₄×D₃₅₆	—	2	2	no	—
19	D₁₄₂₄	C₈	1	4	yes	complemented
20	#15667946	C₂×C₄	1	2	yes	—
21	C₄⋊#10623494	—	2	2	no	—
22	#15667934	—	2	2	no	—
23	#15667924	—	4	2	no	—
24	C₂×C₁₆×C₈₉	—	2	1	no	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×D₃₅₆	—	2	2	no	—
30	C₂×C₄×D₈₉	—	4	2	no	—
31	D₇₁₂	—	2	2	no	—
32	D₇₁₂	C₂×C₈	1	4	yes	complemented
33	C₂×#10623494	—	4	2	no	—
34	C₈×D₈₉	—	8	4	no	—
35	C₄⋊Dic₈₉	—	2	2	no	—
36	#13083551	—	4	2	no	—
37	C₂×C₈×C₈₉	—	2	2	no	—
38	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	—	2	1	no	—
42	C₁₆×C₈₉	—	4	1	no	—
43	C₁₆×C₈₉	C₂×C₈	1	2	yes	complemented
44	C₂²×D₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₈₉	—	4	2	no	—
48	D₃₅₆	—	2	4	no	—
49	D₃₅₆	—	4	2	no	—
50	C₄×D₈₉	—	8	2	no	—
51	C₄×D₈₉	—	8	2	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	—	4	2	no	—
54	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₈₉	#131	1	1	yes	characteristic
56	#10623494	—	8	2	no	—
57	C₂²×C₈₉	#442	1	1	yes	characteristic
58	C₂×D₈₉	—	4	4	no	—
59	C₂×D₈₉	—	8	2	no	—
60	C₂×D₈₉	—	8	2	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
63	C₄×C₈₉	#444	1	1	yes	characteristic
64	Dic₈₉	—	8	2	no	—
65	#12067	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	#1580	1	1	yes	characteristic
67	C₂×C₈₉	AGL(1,ℤ/16)	1	2	yes	—
68	D₈₉	—	8	4	no	—
69	#2177	—	89	1	no	—
70	#1580	—	89	2	no	—
71	#1368	—	89	1	no	—
72	#1333	—	89	2	no	—
73	#1313	D₈₉	1	1	yes	characteristic, complemented
74	C₈₉	#12067	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	89	1	no	—
76	C₄×D₈	—	89	2	no	—
77	C₈×D₄	—	178	2	no	—
78	#367	—	89	1	no	—
79	#357	—	89	2	no	—
80	#346	C₂×D₈₉	1	1	yes	characteristic
81	#346	C₂×D₈₉	1	1	yes	characteristic
82	#334	—	89	2	no	—
83	D₄⋊C₈	—	178	2	no	—
84	#322	C₂×D₈₉	1	1	yes	characteristic
85	C₂×D₈	—	89	2	no	—
86	C₂²×C₈	—	356	2	no	—
87	C₄×D₄	—	178	2	no	—
88	D₁₆	—	89	4	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
92	#107	—	89	2	no	—
93	C₄⋊C₈	—	178	2	no	—
94	#102	—	178	2	no	—
95	C₂²⋊C₈	—	356	2	no	—
96	C₄×C₈	D₃₅₆	1	2	yes	—
97	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	C₂×D₄	—	178	2	no	—
99	C₂²×C₄	—	356	2	no	—
100	D₈	—	89	4	no	—
101	D₈	—	178	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	356	2	no	—
104	C₂×C₈	—	712	4	no	—
105	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
106	C₂×C₈	#13083574	1	1	yes	characteristic
107	C₄⋊C₄	—	178	2	no	—
108	C₂²⋊C₄	—	356	2	no	—
109	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	C₈×D₈₉	1	2	yes	complemented
113	C₂³	—	356	2	no	—
114	D₄	—	356	2	no	—
115	D₄	—	178	4	no	—
116	C₂×C₄	—	356	2	no	—
117	C₂×C₄	—	712	2	no	—
118	C₂×C₄	—	712	2	no	—
119	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
120	C₂×C₄	#15668040	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	712	2	no	—
124	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
125	C₈	#15668018	1	1	yes	characteristic
126	C₂²	—	356	4	no	—
127	C₂²	—	712	2	no	—
128	C₂²	—	712	2	no	—
129	C₂²	#19151711	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	712	2	no	—
132	C₄	C₈×D₃₅₆	1	1	yes	characteristic
133	C₄	#19151721	1	1	yes	characteristic
134	C₂	—	712	4	no	—
135	C₂	#26019077	1	1	yes	characteristic
136	C₂	#26022152	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022129

C₂

yes

maximal, characteristic

#26019077

C₂

yes

maximal, complemented

#26018318

C₂

yes

maximal, characteristic, complemented

#26018220

C₂

yes

maximal, complemented

C₈₉×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

C₄

yes

characteristic

C₄×D₇₁₂

C₂²

yes

—

C₈×D₃₅₆

—

#19151426

C₄

yes

characteristic

#19151400

C₄

yes

—

#19151349

C₂²

yes

—

#19151327

—

#346×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

characteristic

#322×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

C₂×C₄

yes

—

C₂×C₈×D₈₉

—

C₄×D₃₅₆

—

D₁₄₂₄

C₈

yes

complemented

#15667946

C₂×C₄

yes

—

C₄⋊#10623494

—

#15667934

—

#15667924

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

C₂×C₈

yes

complemented

C₂×#10623494

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×C₈

yes

complemented

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×C₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#442

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

#444

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#12067

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

178

—

#367

—

#357

—

#346

C₂×D₈₉

yes

characteristic

#346

C₂×D₈₉

yes

characteristic

#334

—

D₄⋊C₈

—

178

—

#322

C₂×D₈₉

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

356

—

C₄×D₄

—

178

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

—

178

—

#102

—

178

—

C₂²⋊C₈

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×D₄

—

178

—

C₂²×C₄

—

356

—

100

D₈

—

101

D₈

—

178

—

102

C₂×C₈

—

103

C₂×C₈

—

356

—

104

C₂×C₈

—

712

—

105

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

106

C₂×C₈

#13083574

yes

characteristic

107

C₄⋊C₄

—

178

—

108

C₂²⋊C₄

—

356

—

109

C₄²

C₂×D₃₅₆

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

C₈×D₈₉

yes

complemented

113

C₂³

—

356

—

114

D₄

—

356

—

115

D₄

—

178

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

712

—

118

C₂×C₄

—

712

—

119

C₂×C₄

C₄×D₃₅₆

yes

characteristic

120

C₂×C₄

#15668040

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

712

—

124

C₈

C₂×C₈×D₈₉

yes

characteristic

125

C₈

#15668018

yes

characteristic

126

C₂²

—

356

—

127

C₂²

—

712

—

128

C₂²

—

712

—

129

C₂²

#19151711

yes

characteristic

130

C₄

—

131

C₄

—

712

—

132

C₄

C₈×D₃₅₆

yes

characteristic

133

C₄

#19151721

yes

characteristic

134

C₂

—

712

—

135

C₂

#26019077

yes

characteristic

136

C₂

#26022152

yes

—

	subgroup	G/H
center	C₂²	#19151711
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1313	C₂
socle	C₂²×C₈₉	#442
prime radical	C₂×D₁₄₂₄	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1313	D₈₉
hypocenter	C₈₉	#12067
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#19151711

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1313

C₂

socle

C₂²×C₈₉

#442

prime radical

C₂×D₁₄₂₄

C₄

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
2. #26019077	C₂	semidirect
3. #26018318	C₂	semidirect
4. #26018220	C₂	semidirect
5. C₈₉×#1313	C₂	semidirect
19. D₁₄₂₄	C₈	semidirect
32. D₇₁₂	C₂×C₈	semidirect
43. C₁₆×C₈₉	C₂×C₈	semidirect
73. #1313	D₈₉	semidirect
74. C₈₉	#12067	semidirect
112. C₁₆	C₈×D₈₉	semidirect

H ≅ G/N

type

2. #26019077

C₂

semidirect

3. #26018318

C₂

semidirect

4. #26018220

C₂

semidirect

5. C₈₉×#1313

C₂

semidirect

19. D₁₄₂₄

C₈

semidirect

32. D₇₁₂

C₂×C₈

semidirect

43. C₁₆×C₈₉

C₂×C₈

semidirect

73. #1313

D₈₉

semidirect

74. C₈₉

#12067

semidirect

112. C₁₆

C₈×D₈₉

semidirect

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

subgroup	index
#26022129	2
#26019077	2
#26018318	2
#26018220	2
C₈₉×#1313	2
#12067	89

subgroup

index

C₈₉×#1313

Subgroups — #36466314

Conjugacy classes of subgroups

Complements

Maximal subgroups