Group #36441085 — subgroups

Subgroups — #36441085

Order 22784.

140 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#26018170	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1367	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₇₈	—	2	1	no	—
5	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
6	C₄×#13083549	—	2	1	no	—
7	#19151351	C₄	1	1	yes	characteristic
8	#19151330	—	2	1	no	—
9	C₄⋊C₁₆×C₈₉	—	2	1	no	—
10	C₈²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₅₆	C₈	1	1	yes	characteristic
12	C₄×Dic₁₇₈	—	2	1	no	—
13	C₂×#13083549	—	2	2	no	—
14	#15667946	C₈	1	1	yes	characteristic
15	C₄⋊#10623494	—	2	1	no	—
16	#15667941	—	4	1	no	—
17	#15667937	—	2	1	no	—
18	C₈×Dic₈₉	—	4	1	no	—
19	C₂×C₁₆×C₈₉	—	4	1	no	—
20	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
22	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	1	no	—
24	Dic₃₅₆	—	2	1	no	—
25	Dic₃₅₆	C₁₆	1	2	yes	complemented
26	C₂×#10623494	—	4	1	no	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	#13083553	—	4	1	no	—
29	C₄×Dic₈₉	—	4	1	no	—
30	#13083549	—	4	2	no	—
31	C₂×C₈×C₈₉	—	2	1	no	—
32	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
34	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	SD₁₆	1	1	yes	characteristic
36	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₁₆×C₈₉	—	8	1	no	—
38	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
41	C₂×Dic₈₉	—	4	1	no	—
42	C₂×Dic₈₉	—	4	1	no	—
43	Dic₁₇₈	—	2	2	no	—
44	Dic₁₇₈	—	4	1	no	—
45	C₈×C₈₉	—	2	2	no	—
46	C₈×C₈₉	—	2	1	no	—
47	C₈×C₈₉	—	2	1	no	—
48	C₈×C₈₉	—	2	1	no	—
49	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₈₉	M₃₂	1	1	yes	characteristic
51	#10623494	—	8	1	no	—
52	C₂²×C₈₉	D₄⋊C₈	1	1	yes	characteristic
53	C₄×C₈₉	—	2	1	no	—
54	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
55	C₄×C₈₉	#350	1	1	yes	characteristic
56	C₄×C₈₉	#360	1	1	yes	characteristic
57	C₄×C₈₉	#362	1	1	yes	characteristic
58	Dic₈₉	—	4	2	no	—
59	Dic₈₉	—	8	1	no	—
60	#7071	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	D₄⋊C₁₆	1	1	yes	characteristic
62	C₂×C₈₉	#1334	1	1	yes	characteristic
63	C₂×C₈₉	#1336	1	1	yes	characteristic
64	C₈×Q₁₆	—	89	1	no	—
65	#1367	D₈₉	1	1	yes	characteristic
66	#1312	—	89	1	no	—
67	C₈₉	#7071	1	1	yes	characteristic, complemented, Sylow
68	C₈×Q₈	—	178	1	no	—
69	C₄×Q₁₆	—	89	1	no	—
70	C₄⋊C₁₆	—	2	1	no	—
71	C₄×C₁₆	—	178	1	no	—
72	#335	—	89	1	no	—
73	Q₈⋊C₈	—	178	1	no	—
74	C₈²	C₂×D₈₉	1	1	yes	characteristic
75	C₂×Q₁₆	—	89	1	no	—
76	C₄×Q₈	—	178	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	178	2	no	—
79	#107	—	89	1	no	—
80	C₄⋊C₈	—	356	1	no	—
81	C₄⋊C₈	—	178	1	no	—
82	#103	—	178	1	no	—
83	C₄×C₈	—	356	1	no	—
84	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
85	C₄×C₈	D₃₅₆	1	1	yes	characteristic
86	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
87	C₂×Q₈	—	178	1	no	—
88	Q₁₆	—	89	2	no	—
89	Q₁₆	—	178	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	356	1	no	—
92	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
93	C₂×C₈	#13083557	1	1	yes	characteristic
94	C₂×C₈	#13083559	1	1	yes	characteristic
95	C₂×C₈	#13083561	1	1	yes	characteristic
96	C₄⋊C₄	—	356	1	no	—
97	C₄⋊C₄	—	178	1	no	—
98	C₄²	—	356	1	no	—
99	C₄²	#13083551	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	356	2	no	—
102	Q₈	—	356	1	no	—
103	Q₈	—	178	2	no	—
104	C₂×C₄	—	356	1	no	—
105	C₂×C₄	—	356	1	no	—
106	C₂×C₄	#15667924	1	1	yes	characteristic
107	C₂×C₄	#15667932	1	1	yes	characteristic
108	C₂×C₄	#15667938	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	712	1	no	—
114	C₈	C₁₆×D₈₉	1	1	yes	characteristic
115	C₈	#15667951	1	1	yes	characteristic
116	C₂²	#19151325	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	356	2	no	—
119	C₄	—	712	1	no	—
120	C₄	#19151377	1	1	yes	characteristic
121	C₄	#19151383	1	1	yes	characteristic
122	C₄	#19151407	1	1	yes	characteristic
123	C₄	#19151413	1	1	yes	characteristic
124	C₂	#26018206	1	1	yes	characteristic
125	C₂	#26018221	1	1	yes	characteristic
126	C₂	#26018227	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₃₅₆

C₂

yes

maximal, characteristic

#26018170

C₂

yes

maximal, characteristic

C₈₉×#1367

C₂

yes

maximal, characteristic

C₈×Dic₁₇₈

—

C₄×Dic₃₅₆

C₄

yes

characteristic

C₄×#13083549

—

#19151351

C₄

yes

characteristic

#19151330

—

C₄⋊C₁₆×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

C₈

yes

characteristic

C₄×Dic₁₇₈

—

C₂×#13083549

—

#15667946

C₈

yes

—

—

—

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

C₁₆

yes

complemented

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

M₁₆

yes

characteristic

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

SD₁₆

yes

characteristic

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₁₆×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×C₁₆

yes

characteristic

C₈×C₈₉

M₃₂

yes

characteristic

#10623494

—

C₂²×C₈₉

D₄⋊C₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊C₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

#7071

—

maximal, Sylow

C₂×C₈₉

D₄⋊C₁₆

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₈×Q₁₆

—

#1367

D₈₉

yes

characteristic

#1312

—

C₈₉

#7071

yes

characteristic, complemented, Sylow

C₈×Q₈

—

178

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

178

—

#335

—

Q₈⋊C₈

—

178

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

#107

—

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

#103

—

178

—

C₄×C₈

—

356

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×Q₈

—

178

—

Q₁₆

—

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₈×D₈₉

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄²

—

356

—

C₄²

#13083551

yes

characteristic

100

C₁₆

—

101

C₁₆

—

356

—

102

Q₈

—

356

—

103

Q₈

—

178

—

104

C₂×C₄

—

356

—

105

C₂×C₄

—

356

—

106

C₂×C₄

#15667924

yes

characteristic

107

C₂×C₄

#15667932

yes

characteristic

108

C₂×C₄

#15667938

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

712

—

114

C₈

C₁₆×D₈₉

yes

characteristic

115

C₈

#15667951

yes

characteristic

116

C₂²

#19151325

yes

characteristic

117

C₄

—

118

C₄

—

356

—

119

C₄

—

712

—

120

C₄

#19151377

yes

characteristic

121

C₄

#19151383

yes

characteristic

122

C₄

#19151407

yes

characteristic

123

C₄

#19151413

yes

characteristic

124

C₂

#26018206

yes

characteristic

125

C₂

#26018221

yes

characteristic

126

C₂

#26018227

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#15667932
commutator	C₈×C₈₉	C₂×C₁₆
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1367	C₂
socle, prime radical	C₂²×C₈₉	D₄⋊C₈
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1367	D₈₉
hypocenter	C₈₉	#7071
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂×C₄

#15667932

commutator

C₈×C₈₉

C₂×C₁₆

Frattini

C₈²

C₂×D₈₉

Fitting

C₈₉×#1367

C₂

socle, prime radical

C₂²×C₈₉

D₄⋊C₈

cosocle

C₈²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

N	H ≅ G/N	type
25. Dic₃₅₆	C₁₆	semidirect
67. C₈₉	#7071	semidirect

H ≅ G/N

type

25. Dic₃₅₆

C₁₆

semidirect

67. C₈₉

#7071

semidirect

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

subgroup	index
C₈×Dic₃₅₆	2
#26018170	2
C₈₉×#1367	2
#7071	89

subgroup

index

C₈×Dic₃₅₆

#26018170

C₈₉×#1367

#7071

Subgroups — #36441085

Conjugacy classes of subgroups

Complements

Maximal subgroups