Group #36475420 — subgroups

Subgroups — #36475420

Order 22784.

174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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	#26022505	C₂	1	1	yes	maximal, characteristic
2	#26018432	C₂	1	4	yes	maximal
3	#26018386	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2253	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₁₂	—	2	1	no	—
6	C₂×Dic₇₁₂	C₂²	1	2	yes	—
7	#19151978	—	2	1	no	—
8	C₂×#15667918	C₂²	1	2	yes	—
9	#499×C₈₉	—	2	1	no	—
10	#367×C₈₉	—	2	1	no	—
11	#366×C₈₉	C₂²	1	2	yes	—
12	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₅₆	—	2	2	no	—
14	C₂×Dic₃₅₆	—	4	1	no	—
15	#15668012	—	4	1	no	—
16	Dic₇₁₂	—	2	4	no	—
17	Dic₇₁₂	—	4	2	no	—
18	#15667918	—	2	2	no	—
19	#125×C₈₉	—	4	1	no	—
20	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
22	#107×C₈₉	—	2	2	no	—
23	#106×C₈₉	—	4	1	no	—
24	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₇₈	—	4	2	no	—
26	Dic₃₅₆	—	4	2	no	—
27	Dic₃₅₆	—	4	2	no	—
28	Dic₃₅₆	—	8	2	no	—
29	C₄⋊Dic₈₉	—	8	2	no	—
30	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₈₉	—	4	2	no	—
33	C₄⋊C₄×C₈₉	—	8	2	no	—
34	C₄²×C₈₉	D₈	1	1	yes	characteristic
35	C₁₆×C₈₉	SD₁₆	1	2	yes	—
36	C₁₆×C₈₉	#41	1	2	yes	—
37	C₂×C₄×C₈₉	—	8	2	no	—
38	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₈₉	—	8	2	no	—
41	Dic₁₇₈	—	8	2	no	—
42	Dic₁₇₈	—	8	2	no	—
43	C₈×C₈₉	SD₃₂	1	2	yes	—
44	C₈×C₈₉	#124	1	1	yes	characteristic
45	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
47	C₄×C₈₉	—	2	2	no	—
48	C₄×C₈₉	—	16	2	no	—
49	C₄×C₈₉	#486	1	1	yes	characteristic
50	C₄×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
51	Dic₈₉	—	16	2	no	—
52	#13724	—	89	1	no	maximal, Sylow
53	C₂×C₈₉	#2240	1	1	yes	characteristic
54	C₂×C₈₉	#2264	1	2	yes	—
55	#2253	D₈₉	1	1	yes	characteristic
56	#2247	—	89	1	no	—
57	#1416	—	89	4	no	—
58	#1397	—	89	1	no	—
59	C₈₉	#13724	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₃₂	—	89	2	no	—
61	C₂×Q₃₂	—	178	1	no	—
62	#499	—	2	1	no	—
63	#494	—	178	1	no	—
64	C₂×C₃₂	—	89	2	no	—
65	#367	—	2	1	no	—
66	#366	C₂×D₈₉	1	2	yes	—
67	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
68	C₂×Q₁₆	—	356	1	no	—
69	C₂×Q₁₆	—	178	2	no	—
70	C₄⋊Q₈	—	356	1	no	—
71	#125	—	4	1	no	—
72	Q₃₂	—	356	2	no	—
73	Q₃₂	—	178	4	no	—
74	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#106	—	4	1	no	—
78	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
79	C₃₂	—	178	2	no	—
80	C₂×Q₈	—	356	2	no	—
81	Q₁₆	—	356	2	no	—
82	Q₁₆	—	356	2	no	—
83	Q₁₆	—	712	2	no	—
84	C₂×C₈	#13083559	1	1	yes	characteristic
85	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	712	2	no	—
89	C₄²	#13083559	1	1	yes	characteristic
90	C₁₆	#13083561	1	2	yes	—
91	C₁₆	#13083575	1	2	yes	—
92	Q₈	—	712	2	no	—
93	Q₈	—	712	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	712	2	no	—
96	C₂×C₄	#15667953	1	2	yes	—
97	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
98	C₈	#15667955	1	2	yes	—
99	C₈	#15668001	1	1	yes	characteristic
100	C₈	C₂×#13083561	1	1	yes	characteristic
101	C₂²	C₂×#15667953	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	1424	2	no	—
105	C₄	#19151946	1	1	yes	characteristic
106	C₄	C₂×#15667955	1	1	yes	characteristic
107	C₂	#26022477	1	1	yes	characteristic
108	C₂	#26022569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022505

C₂

yes

maximal, characteristic

#26018432

C₂

yes

maximal

#26018386

C₂

yes

maximal, characteristic

C₈₉×#2253

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

C₂²

yes

—

#19151978

—

C₂×#15667918

C₂²

yes

—

#499×C₈₉

—

#367×C₈₉

—

#366×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667918

—

#125×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

#106×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

—

C₁₆×C₈₉

#41

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#124

yes

characteristic

C₈×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#486

yes

characteristic

C₄×C₈₉

C₂×SD₃₂

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2253

D₈₉

yes

—

—

—

C₈₉

#13724

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

178

—

#499

—

#494

—

178

—

C₂×C₃₂

—

#367

—

#366

C₂×D₈₉

yes

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

356

—

#125

—

Q₃₂

—

356

—

Q₃₂

—

178

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₃₂

—

178

—

C₂×Q₈

—

356

—

Q₁₆

—

356

—

Q₁₆

—

356

—

Q₁₆

—

712

—

C₂×C₈

#13083559

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

712

—

C₄²

#13083559

yes

characteristic

C₁₆

#13083561

yes

—

C₁₆

#13083575

yes

—

Q₈

—

712

—

Q₈

—

712

—

C₂×C₄

—

C₂×C₄

—

712

—

C₂×C₄

#15667953

yes

—

C₂×C₄

C₂×#13083559

yes

characteristic

C₈

#15667955

yes

—

C₈

#15668001

yes

characteristic

100

C₈

C₂×#13083561

yes

characteristic

101

C₂²

C₂×#15667953

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

1424

—

105

C₄

#19151946

yes

characteristic

106

C₄

C₂×#15667955

yes

characteristic

107

C₂

#26022477

yes

characteristic

108

C₂

#26022569

yes

—

	subgroup	G/H
center	C₂²	C₂×#15667953
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#2253	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	#2253	D₈₉
hypocenter	C₈₉	#13724
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

subgroup

G/H

center

C₂²

C₂×#15667953

commutator, cosocle

C₂×C₁₆×C₈₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₈₉×#2253

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₁₆

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

N	H ≅ G/N	type
59. C₈₉	#13724	semidirect

H ≅ G/N

type

59. C₈₉

#13724

semidirect

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

subgroup	index
#26022505	2
#26018432	2
#26018386	2
C₈₉×#2253	2
#13724	89

subgroup

index

C₈₉×#2253

Subgroups — #36475420

Conjugacy classes of subgroups

Complements

Maximal subgroups