Group #36441862 — subgroups

Subgroups — #36441862

Order 22784.

170 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 39 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	#26020471	C₂	1	1	yes	maximal, characteristic
2	#26018178	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1321	C₂	1	1	yes	maximal, characteristic
4	#19152365	C₄	1	1	yes	characteristic
5	#19151646	—	2	1	no	—
6	#19151582	C₄	1	1	yes	characteristic
7	C₂²⋊#13083549	—	2	1	no	—
8	#19151363	—	2	1	no	—
9	#408×C₈₉	C₂²	1	1	yes	characteristic
10	#349×C₈₉	—	2	1	no	—
11	#15668019	—	4	2	no	—
12	C₂×#13083558	—	2	1	no	—
13	C₂×#13083558	—	2	1	no	—
14	C₂×#13083558	—	4	1	no	—
15	C₂×#13083549	—	4	1	no	—
16	#15667947	—	4	2	no	—
17	#15667931	C₈	1	2	yes	—
18	#15667929	—	2	1	no	—
19	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
22	M₃₂×C₈₉	—	4	1	no	—
23	C₂²⋊C₈×C₈₉	—	4	1	no	—
24	#13083558	—	2	2	no	—
25	#13083558	—	2	2	no	—
26	#13083558	—	4	1	no	—
27	#13083558	—	4	1	no	—
28	#13083558	—	4	2	no	—
29	C₂×#10623494	—	4	1	no	—
30	C₂×#10623494	—	4	1	no	—
31	C₂×#10623494	—	4	2	no	—
32	#13083549	—	8	1	no	—
33	#41×C₈₉	—	4	1	no	—
34	#41×C₈₉	—	4	1	no	—
35	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
36	C₂×D₄×C₈₉	—	2	1	no	—
37	C₂×D₄×C₈₉	M₁₆	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	—	2	1	no	—
39	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
40	M₁₆×C₈₉	—	4	1	no	—
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	—	4	1	no	—
44	C₁₆×C₈₉	—	8	1	no	—
45	C₂³×C₈₉	—	2	1	no	—
46	C₂³×C₈₉	#101	1	1	yes	characteristic
47	Q₈×C₈₉	—	4	1	no	—
48	D₄×C₈₉	—	4	1	no	—
49	D₄×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	—	2	1	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	4	1	no	—
54	C₂×C₄×C₈₉	—	4	1	no	—
55	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
57	C₈×C₈₉	—	4	1	no	—
58	C₈×C₈₉	—	4	1	no	—
59	#10623494	—	4	2	no	—
60	#10623494	—	4	2	no	—
61	C₂²×C₈₉	—	2	1	no	—
62	C₂²×C₈₉	—	4	1	no	—
63	C₂²×C₈₉	—	4	1	no	—
64	C₂²×C₈₉	—	8	1	no	—
65	C₂²×C₈₉	#324	1	1	yes	characteristic
66	C₄×C₈₉	—	2	1	no	—
67	C₄×C₈₉	—	4	1	no	—
68	C₄×C₈₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
69	C₄×C₈₉	#352	1	1	yes	characteristic
70	#7378	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	2	1	no	—
72	C₂×C₈₉	—	4	1	no	—
73	C₂×C₈₉	—	8	1	no	—
74	C₂×C₈₉	#1326	1	1	yes	characteristic
75	#1858	—	89	1	no	—
76	#1321	D₈₉	1	1	yes	characteristic
77	#1315	—	89	1	no	—
78	C₈₉	#7378	1	1	yes	characteristic, complemented, Sylow
79	#568	—	89	1	no	—
80	#430	—	178	1	no	—
81	#413	—	89	1	no	—
82	#408	C₂×D₈₉	1	1	yes	characteristic
83	#349	—	2	1	no	—
84	C₂²⋊C₁₆	—	178	1	no	—
85	#341	—	178	1	no	—
86	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
87	#131	—	356	2	no	—
88	C₂×M₁₆	—	356	1	no	—
89	C₂×M₁₆	—	178	1	no	—
90	C₂×M₁₆	—	178	1	no	—
91	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
92	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
93	M₃₂	—	4	1	no	—
94	C₂×C₁₆	—	356	1	no	—
95	#108	—	356	2	no	—
96	#101	—	89	2	no	—
97	#100	—	178	1	no	—
98	C₂²⋊C₈	—	4	1	no	—
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	#13083557	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	#13083551	1	1	yes	characteristic
106	M₁₆	—	4	1	no	—
107	M₁₆	—	356	1	no	—
108	M₁₆	—	356	1	no	—
109	M₁₆	—	356	2	no	—
110	M₁₆	—	178	2	no	—
111	M₁₆	—	178	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	356	1	no	—
116	C₂×C₈	—	356	1	no	—
117	C₂×C₈	—	356	2	no	—
118	C₁₆	—	8	1	no	—
119	C₁₆	—	712	1	no	—
120	C₂³	—	2	1	no	—
121	C₂³	#15667930	1	1	yes	characteristic
122	Q₈	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#15667924	1	1	yes	characteristic
131	C₂×C₄	#15667926	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	356	2	no	—
135	C₈	—	356	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#19151323	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	#19151384	1	1	yes	characteristic
144	C₄	#19151389	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	8	1	no	—
148	C₂	#26018199	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020471

C₂

yes

maximal, characteristic

#26018178

C₂

yes

maximal, characteristic

C₈₉×#1321

C₂

yes

maximal, characteristic

#19152365

C₄

yes

characteristic

#19151646

—

#19151582

C₄

yes

characteristic

C₂²⋊#13083549

—

#19151363

—

#408×C₈₉

C₂²

yes

characteristic

#349×C₈₉

—

—

—

—

—

—

—

C₈

yes

—

#15667929

—

C₂×#41×C₈₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

D₄

yes

characteristic

M₃₂×C₈₉

—

C₂²⋊C₈×C₈₉

—

—

—

—

—

—

—

—

—

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₈

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

M₁₆

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₁₆×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#101

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

#99

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#324

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

AΣL(1,𝔽₁₆)

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1315

—

C₈₉

#7378

yes

characteristic, complemented, Sylow

#568

—

#430

—

178

—

#413

—

#408

C₂×D₈₉

yes

characteristic

#349

—

C₂²⋊C₁₆

—

178

—

#341

—

178

—

C₂×#41

C₄×D₈₉

yes

characteristic

#131

—

356

—

C₂×M₁₆

—

356

—

C₂×M₁₆

—

178

—

C₂×M₁₆

—

178

—

C₂×M₁₆

C₂²⋊D₈₉

yes

characteristic

C₂²×C₈

D₃₅₆

yes

characteristic

M₃₂

—

C₂×C₁₆

—

356

—

#108

—

356

—

#101

—

#100

—

178

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

C₈×D₈₉

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

#13083557

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

#13083551

yes

characteristic

106

M₁₆

—

107

M₁₆

—

356

—

108

M₁₆

—

356

—

109

M₁₆

—

356

—

110

M₁₆

—

178

—

111

M₁₆

—

178

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

356

—

116

C₂×C₈

—

356

—

117

C₂×C₈

—

356

—

118

C₁₆

—

119

C₁₆

—

712

—

120

C₂³

—

121

C₂³

#15667930

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#15667924

yes

characteristic

131

C₂×C₄

#15667926

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

356

—

135

C₈

—

356

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#19151323

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#19151384

yes

characteristic

144

C₄

#19151389

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#26018199

yes

characteristic

	subgroup	G/H
center	C₄	#19151384
commutator	C₂×Q₈×C₈₉	C₂×C₈
Frattini	#408	C₂×D₈₉
Fitting	C₈₉×#1321	C₂
socle	C₂×C₈₉	#1326
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	#408×C₈₉	C₂²
hypercenter	#1321	D₈₉
hypocenter	C₈₉	#7378
LCS	G ▷ C₂×Q₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

subgroup

G/H

center

C₄

#19151384

commutator

C₂×Q₈×C₈₉

C₂×C₈

Frattini

#408

C₂×D₈₉

Fitting

C₈₉×#1321

C₂

socle

C₂×C₈₉

#1326

prime radical

C₂×#41×C₈₉

C₂×C₄

cosocle

#408×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×Q₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1321

N	H ≅ G/N	type
78. C₈₉	#7378	semidirect

H ≅ G/N

type

78. C₈₉

#7378

semidirect

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

subgroup	index
#26020471	2
#26018178	2
C₈₉×#1321	2
#7378	89

subgroup

index

#26020471

#26018178

C₈₉×#1321

#7378

Subgroups — #36441862

Conjugacy classes of subgroups

Complements

Maximal subgroups