Group #36469103 — subgroups

Subgroups — #36469103

Order 22784.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26022534	C₂	1	1	yes	maximal, characteristic
2	#26022505	C₂	1	1	yes	maximal, characteristic
3	#26022480	C₂	1	2	yes	maximal
4	#26019849	C₂	1	2	yes	maximal
5	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	—	2	2	no	—
7	#19152003	C₂²	1	2	yes	—
8	#19151996	—	2	2	no	—
9	#19151978	C₂²	1	2	yes	—
10	#19151949	—	2	2	no	—
11	#19151426	—	2	2	no	—
12	#19151403	—	2	4	no	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₈²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₆	—	2	2	no	—
17	C₂×Dic₃₅₆	—	2	2	no	—
18	#15668012	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	Dic₇₁₂	—	4	4	no	—
21	#15667946	—	2	2	no	—
22	#15667946	—	2	2	no	—
23	#15667944	—	4	4	no	—
24	#15667937	—	4	4	no	—
25	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
27	C₄×C₈×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	—	4	2	no	—
30	C₂×Dic₁₇₈	—	4	2	no	—
31	Dic₃₅₆	—	4	4	no	—
32	Dic₃₅₆	—	4	4	no	—
33	C₄⋊Dic₈₉	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	C₂×C₈×C₈₉	D₈	1	4	yes	—
38	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	—	2	2	no	—
42	C₁₆×C₈₉	SD₁₆	1	4	yes	—
43	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
45	C₂×Dic₈₉	—	8	2	no	—
46	C₂×Dic₈₉	—	8	2	no	—
47	Dic₁₇₈	—	8	2	no	—
48	Dic₁₇₈	—	8	2	no	—
49	Dic₁₇₈	—	8	4	no	—
50	C₈×C₈₉	—	2	4	no	—
51	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
52	C₈×C₈₉	#135	1	2	yes	—
53	C₂²×C₈₉	#493	1	1	yes	characteristic
54	C₄×C₈₉	#492	1	1	yes	characteristic
55	C₄×C₈₉	#495	1	1	yes	characteristic
56	C₄×C₈₉	#510	1	4	yes	—
57	Dic₈₉	—	16	2	no	—
58	Dic₈₉	—	16	2	no	—
59	#12507	—	89	1	no	maximal, Sylow
60	C₂×C₈₉	#1706	1	1	yes	characteristic
61	C₂×C₈₉	#2251	1	2	yes	—
62	#2254	—	89	1	no	—
63	#2247	—	89	1	no	—
64	#2241	—	89	2	no	—
65	#1707	—	89	2	no	—
66	#1311	D₈₉	1	1	yes	characteristic
67	C₈₉	#12507	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₃₂	—	178	2	no	—
69	#500	—	89	2	no	—
70	#498	—	178	2	no	—
71	#494	—	89	2	no	—
72	#487	—	178	2	no	—
73	#367	—	178	2	no	—
74	#358	—	178	4	no	—
75	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
76	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
77	C₈²	C₂×D₈₉	1	1	yes	characteristic
78	C₂×Q₁₆	—	178	2	no	—
79	C₂×Q₁₆	—	178	2	no	—
80	C₄⋊Q₈	—	178	2	no	—
81	C₄⋊Q₈	—	178	2	no	—
82	Q₃₂	—	356	4	no	—
83	C₂×C₁₆	D₃₅₆	1	2	yes	—
84	C₂×C₁₆	D₃₅₆	1	2	yes	—
85	#107	—	178	2	no	—
86	#107	—	178	2	no	—
87	#106	—	356	4	no	—
88	#103	—	356	4	no	—
89	C₄×C₈	D₃₅₆	1	2	yes	—
90	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
91	C₂×Q₈	—	356	2	no	—
92	C₂×Q₈	—	356	2	no	—
93	Q₁₆	—	356	4	no	—
94	Q₁₆	—	356	4	no	—
95	C₂×C₈	D₇₁₂	1	4	yes	—
96	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
97	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
98	C₄⋊C₄	—	356	2	no	—
99	C₄⋊C₄	—	356	2	no	—
100	C₄⋊C₄	—	356	2	no	—
101	C₄⋊C₄	—	356	2	no	—
102	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	#13083563	1	4	yes	—
105	Q₈	—	712	2	no	—
106	Q₈	—	712	2	no	—
107	Q₈	—	712	4	no	—
108	C₂×C₄	—	712	2	no	—
109	C₂×C₄	—	712	2	no	—
110	C₂×C₄	#15668009	1	1	yes	characteristic
111	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	C₂×#13083563	1	2	yes	—
114	C₈	#15668034	1	2	yes	—
115	C₂²	#19151975	1	1	yes	characteristic
116	C₄	—	1424	2	no	—
117	C₄	—	1424	2	no	—
118	C₄	#19151971	1	1	yes	characteristic
119	C₄	#19151982	1	1	yes	characteristic
120	C₄	#19152041	1	4	yes	—
121	C₂	#26019844	1	1	yes	characteristic
122	C₂	#26022523	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022534

C₂

yes

maximal, characteristic

#26022505

C₂

yes

maximal, characteristic

#26022480

C₂

yes

maximal

#26019849

C₂

yes

maximal

C₈₉×#1311

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

#19152003

C₂²

yes

—

#19151996

—

#19151978

C₂²

yes

—

#19151949

—

#19151426

—

#19151403

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

#15668012

—

Dic₇₁₂

—

—

—

—

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#12507

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

#500

—

#498

—

178

—

#494

—

#487

—

178

—

#367

—

178

—

#358

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

Q₃₂

—

356

—

C₂×C₁₆

D₃₅₆

yes

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

178

—

#107

—

178

—

#106

—

356

—

#103

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

Q₁₆

—

356

—

Q₁₆

—

356

—

C₂×C₈

D₇₁₂

yes

—

C₂×C₈

C₂×D₃₅₆

yes

characteristic

C₂×C₈

C₂×D₃₅₆

yes

characteristic

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

100

C₄⋊C₄

—

356

—

101

C₄⋊C₄

—

356

—

102

C₄²

C₂×D₃₅₆

yes

characteristic

103

C₁₆

—

104

C₁₆

#13083563

yes

—

105

Q₈

—

712

—

106

Q₈

—

712

—

107

Q₈

—

712

—

108

C₂×C₄

—

712

—

109

C₂×C₄

—

712

—

110

C₂×C₄

#15668009

yes

characteristic

111

C₂×C₄

C₂×D₇₁₂

yes

—

112

C₈

—

113

C₈

C₂×#13083563

yes

—

114

C₈

#15668034

yes

—

115

C₂²

#19151975

yes

characteristic

116

C₄

—

1424

—

117

C₄

—

1424

—

118

C₄

#19151971

yes

characteristic

119

C₄

#19151982

yes

characteristic

120

C₄

#19152041

yes

—

121

C₂

#26019844

yes

characteristic

122

C₂

#26022523

yes

—

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1311	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#12507
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1311

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
67. C₈₉	#12507	semidirect

H ≅ G/N

type

67. C₈₉

#12507

semidirect

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

subgroup	index
#26022534	2
#26022505	2
#26022480	2
#26019849	2
C₈₉×#1311	2
#12507	89

subgroup

index

C₈₉×#1311

Subgroups — #36469103

Conjugacy classes of subgroups

Complements

Maximal subgroups