Group #36469119 — subgroups

Subgroups — #36469119

Order 22784.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#26022481	C₂	1	4	yes	maximal
2	#26019870	C₂	1	1	yes	maximal, characteristic
3	C₈×#13083549	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic
5	#19152003	—	2	1	no	—
6	#19152003	—	2	1	no	—
7	#19151978	C₂²	1	2	yes	—
8	#19151401	—	2	8	no	—
9	C₄×#13083549	C₂²	1	2	yes	—
10	#500×C₈₉	—	2	1	no	—
11	#500×C₈₉	C₂²	1	2	yes	—
12	#494×C₈₉	—	2	1	no	—
13	C₈²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₆	—	2	4	no	—
15	#15668012	—	2	2	no	—
16	C₂×#13083549	D₄	1	4	yes	—
17	#15667946	—	4	2	no	—
18	#15667946	—	4	2	no	—
19	C₂×Q₁₆×C₈₉	—	4	2	no	—
20	C₄⋊Q₈×C₈₉	—	2	2	no	—
21	#107×C₈₉	—	2	4	no	—
22	#107×C₈₉	—	4	2	no	—
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₇₈	—	4	2	no	—
27	Dic₃₅₆	—	4	8	no	—
28	C₄⋊Dic₈₉	—	4	2	no	—
29	C₄⋊Dic₈₉	—	4	2	no	—
30	#13083549	—	2	4	no	—
31	C₂×Q₈×C₈₉	—	4	2	no	—
32	Q₁₆×C₈₉	—	8	4	no	—
33	C₂×C₈×C₈₉	D₈	1	2	yes	—
34	C₂×C₈×C₈₉	D₈	1	2	yes	—
35	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
36	C₄⋊C₄×C₈₉	—	4	2	no	—
37	C₄⋊C₄×C₈₉	—	4	2	no	—
38	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₈₉	—	8	4	no	—
40	C₂×C₄×C₈₉	—	8	2	no	—
41	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₈₉	—	8	2	no	—
45	Dic₁₇₈	—	8	4	no	—
46	C₈×C₈₉	SD₃₂	1	4	yes	—
47	C₈×C₈₉	Q₃₂	1	4	yes	—
48	C₈×C₈₉	#135	1	4	yes	—
49	C₂²×C₈₉	#493	1	1	yes	characteristic
50	C₄×C₈₉	—	16	2	no	—
51	C₄×C₈₉	#495	1	2	yes	—
52	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
53	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
54	Dic₈₉	—	16	2	no	—
55	#12510	—	89	1	no	maximal, Sylow
56	C₂×C₈₉	#1709	1	1	yes	characteristic
57	C₂×C₈₉	#2247	1	1	yes	characteristic
58	C₂×C₈₉	#2248	1	1	yes	characteristic
59	#2241	—	89	4	no	—
60	#1713	—	89	1	no	—
61	#1713	D₈₉	1	1	yes	characteristic
62	C₈×C₁₆	—	89	1	no	—
63	C₈₉	#12510	1	1	yes	characteristic, complemented, Sylow
64	#500	—	2	1	no	—
65	#500	—	178	1	no	—
66	#500	—	178	1	no	—
67	#500	C₂×D₈₉	1	2	yes	—
68	#494	—	2	1	no	—
69	#494	—	89	2	no	—
70	#358	—	178	8	no	—
71	C₄×C₁₆	—	89	2	no	—
72	C₈²	C₂×D₈₉	1	1	yes	characteristic
73	C₂×Q₁₆	—	4	2	no	—
74	C₂×Q₁₆	—	178	4	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	178	2	no	—
77	C₂×C₁₆	—	89	4	no	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	356	2	no	—
81	#107	—	356	2	no	—
82	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	356	2	no	—
87	Q₁₆	—	8	4	no	—
88	Q₁₆	—	356	8	no	—
89	C₂×C₈	#13083559	1	2	yes	—
90	C₂×C₈	#13083559	1	2	yes	—
91	C₂×C₈	D₄×D₈₉	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	356	2	no	—
95	C₄⋊C₄	—	356	2	no	—
96	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
97	C₁₆	—	178	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	712	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	712	2	no	—
102	C₂×C₄	#15668008	1	1	yes	characteristic
103	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
104	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
105	C₈	#15667956	1	4	yes	—
106	C₈	#15667958	1	4	yes	—
107	C₈	#15668035	1	4	yes	—
108	C₂²	#19151974	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	1424	2	no	—
111	C₄	#19151983	1	2	yes	—
112	C₄	C₂×#15667956	1	2	yes	—
113	C₄	C₂×#15667958	1	2	yes	—
114	C₂	#26019859	1	1	yes	characteristic
115	C₂	#26022504	1	1	yes	characteristic
116	C₂	#26022507	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022481

C₂

yes

maximal

#26019870

C₂

yes

maximal, characteristic

C₈×#13083549

C₂

yes

maximal, characteristic

C₈₉×#1713

C₂

yes

maximal, characteristic

#19152003

—

#19152003

—

#19151978

C₂²

yes

—

#19151401

—

C₄×#13083549

C₂²

yes

—

#500×C₈₉

—

#500×C₈₉

C₂²

yes

—

#494×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

#15668012

—

C₂×#13083549

D₄

yes

—

#15667946

—

#15667946

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

Q₃₂

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

C₄×C₈₉

C₂×Q₃₂

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

D₈₉

yes

characteristic

C₈×C₁₆

—

C₈₉

#12510

yes

characteristic, complemented, Sylow

#500

—

#500

—

178

—

#500

—

178

—

#500

C₂×D₈₉

yes

—

#494

—

#494

—

#358

—

178

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

356

—

#107

—

356

—

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₄×C₈

C₂²⋊D₈₉

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

Q₁₆

—

Q₁₆

—

356

—

C₂×C₈

#13083559

yes

—

C₂×C₈

#13083559

yes

—

C₂×C₈

D₄×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

C₁₆

—

178

—

Q₈

—

Q₈

—

712

—

100

C₂×C₄

—

101

C₂×C₄

—

712

—

102

C₂×C₄

#15668008

yes

characteristic

103

C₂×C₄

C₂×#13083559

yes

characteristic

104

C₂×C₄

C₂×#13083559

yes

characteristic

105

C₈

#15667956

yes

—

106

C₈

#15667958

yes

—

107

C₈

#15668035

yes

—

108

C₂²

#19151974

yes

characteristic

109

C₄

—

110

C₄

—

1424

—

111

C₄

#19151983

yes

—

112

C₄

C₂×#15667956

yes

—

113

C₄

C₂×#15667958

yes

—

114

C₂

#26019859

yes

characteristic

115

C₂

#26022504

yes

characteristic

116

C₂

#26022507

yes

characteristic

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1713	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#12510
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1713

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
63. C₈₉	#12510	semidirect

H ≅ G/N

type

63. C₈₉

#12510

semidirect

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

subgroup	index
#26022481	2
#26019870	2
C₈×#13083549	2
C₈₉×#1713	2
#12510	89

subgroup

index

C₈₉×#1713

Subgroups — #36469119

Conjugacy classes of subgroups

Complements

Maximal subgroups