Group #36519991 — subgroups

Subgroups — #36519991

Order 22784.

1034 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 251 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	#26018523	C₂	1	2	yes	maximal
2	#26018508	C₂	1	12	yes	maximal
3	C₈₉×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
5	C₂×#15667919	C₂²	1	12	yes	—
6	C₂×#15667919	C₂²	1	12	yes	—
7	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
8	C₂×#95×C₈₉	C₂²	1	6	yes	—
9	C₂³×Dic₈₉	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₈₉	—	4	12	no	—
11	#15667919	—	2	24	no	—
12	#15667919	—	4	12	no	—
13	C₂²×Q₈×C₈₉	—	4	6	no	—
14	C₂²×Q₈×C₈₉	Q₈	1	4	yes	—
15	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
16	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
17	#95×C₈₉	—	4	6	no	—
18	C₂²×Dic₈₉	—	2	24	no	—
19	C₂²×Dic₈₉	—	2	24	no	—
20	C₂²×Dic₈₉	—	2	8	no	—
21	C₄⋊Dic₈₉	—	4	16	no	—
22	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
23	C₂×Q₈×C₈₉	—	4	12	no	—
24	C₂×Q₈×C₈₉	—	4	16	no	—
25	C₂²×C₄×C₈₉	—	2	6	no	—
26	C₂²×C₄×C₈₉	—	2	12	no	—
27	C₂²×C₄×C₈₉	—	2	12	no	—
28	C₂²×C₄×C₈₉	—	2	12	no	—
29	C₂²×C₄×C₈₉	C₂×Q₈	1	6	yes	—
30	C₂²×C₄×C₈₉	#41	1	8	yes	—
31	C₂³×C₈₉	C₂²×Q₈	1	1	yes	characteristic
32	C₂³×C₈₉	C₂×#41	1	4	yes	—
33	C₂³×C₈₉	H(5,𝔽₂)	1	6	yes	—
34	C₂³×C₈₉	2₋¹⁺⁴	1	4	yes	—
35	Q₈×C₈₉	—	4	16	no	—
36	C₂×C₄×C₈₉	—	2	24	no	—
37	C₂×C₄×C₈₉	—	2	6	no	—
38	C₂×C₄×C₈₉	—	2	12	no	—
39	C₂×C₄×C₈₉	—	4	12	no	—
40	C₂×C₄×C₈₉	—	4	4	no	—
41	C₂×C₄×C₈₉	—	4	12	no	—
42	C₂×Dic₈₉	—	4	24	no	—
43	C₂×Dic₈₉	—	4	24	no	—
44	C₂×Dic₈₉	—	4	8	no	—
45	C₂²×C₈₉	#539	1	12	yes	—
46	C₂²×C₈₉	#543	1	3	yes	—
47	C₂²×C₈₉	#555	1	6	yes	—
48	C₂²×C₈₉	#557	1	4	yes	—
49	C₂²×C₈₉	H(3,𝔽₄)	1	4	yes	—
50	C₂²×C₈₉	#563	1	6	yes	—
51	C₄×C₈₉	—	4	8	no	—
52	C₄×C₈₉	—	4	6	no	—
53	Dic₈₉	—	8	8	no	—
54	#17308	—	89	1	no	maximal, Sylow
55	C₂×C₈₉	#2785	1	4	yes	—
56	C₂×C₈₉	#2809	1	6	yes	—
57	C₂×C₈₉	#2835	1	4	yes	—
58	C₂×C₈₉	#2840	1	1	yes	characteristic
59	#1446	—	89	2	no	—
60	#1446	D₈₉	1	1	yes	characteristic
61	#1442	—	89	12	no	—
62	C₈₉	#17308	1	1	yes	characteristic, complemented, Sylow
63	C₂³×Q₈	C₂×D₈₉	1	1	yes	characteristic
64	C₂²×C₄⋊C₄	—	89	4	no	—
65	C₂×#95	—	89	12	no	—
66	C₂×#95	—	89	12	no	—
67	C₂×#95	C₂×D₈₉	1	6	yes	—
68	C₂²×Q₈	—	4	6	no	—
69	C₂²×Q₈	Dic₁₇₈	1	4	yes	—
70	C₂³×C₄	—	89	8	no	—
71	C₂³×C₄	C₂²×D₈₉	1	3	yes	—
72	C₂³×C₄	C₂²×D₈₉	1	4	yes	—
73	C₂×C₄⋊C₄	—	356	12	no	—
74	#95	—	4	6	no	—
75	#95	—	356	12	no	—
76	#95	—	178	24	no	—
77	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
78	C₂×Q₈	—	4	12	no	—
79	C₂×Q₈	—	4	16	no	—
80	C₂²×C₄	—	2	12	no	—
81	C₂²×C₄	—	2	6	no	—
82	C₂²×C₄	—	2	12	no	—
83	C₂²×C₄	—	2	12	no	—
84	C₂²×C₄	—	178	24	no	—
85	C₂²×C₄	—	178	24	no	—
86	C₂²×C₄	—	178	8	no	—
87	C₂²×C₄	C₂×Dic₁₇₈	1	6	yes	—
88	C₂²×C₄	#13083575	1	8	yes	—
89	C₄⋊C₄	—	356	16	no	—
90	C₂³	C₂²×Dic₁₇₈	1	1	yes	characteristic
91	C₂³	C₂×#13083575	1	4	yes	—
92	C₂³	#15668057	1	6	yes	—
93	C₂³	#15668059	1	4	yes	—
94	Q₈	—	4	16	no	—
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	12	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	4	12	no	—
100	C₂×C₄	—	4	12	no	—
101	C₂×C₄	—	356	24	no	—
102	C₂×C₄	—	356	24	no	—
103	C₂×C₄	—	356	8	no	—
104	C₂²	#19152197	1	12	yes	—
105	C₂²	#19152224	1	3	yes	—
106	C₂²	#19152314	1	6	yes	—
107	C₂²	#19152327	1	4	yes	—
108	C₂²	#19152341	1	4	yes	—
109	C₂²	#19152348	1	6	yes	—
110	C₄	—	4	8	no	—
111	C₄	—	4	6	no	—
112	C₄	—	712	8	no	—
113	C₂	#26027960	1	4	yes	—
114	C₂	#26028240	1	6	yes	—
115	C₂	#26028443	1	4	yes	—
116	C₂	#26028505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26018523

C₂

yes

maximal

#26018508

C₂

yes

maximal

C₈₉×#1446

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#15667919

C₂²

yes

—

C₂×#15667919

C₂²

yes

—

C₂³×Q₈×C₈₉

C₂²

yes

characteristic

C₂×#95×C₈₉

C₂²

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂×C₄⋊Dic₈₉

—

#15667919

—

#15667919

—

C₂²×Q₈×C₈₉

—

C₂²×Q₈×C₈₉

Q₈

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

#95×C₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂²×C₄×C₈₉

#41

yes

—

C₂³×C₈₉

C₂²×Q₈

yes

characteristic

C₂³×C₈₉

C₂×#41

yes

—

C₂³×C₈₉

H(5,𝔽₂)

yes

—

C₂³×C₈₉

2₋¹⁺⁴

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

#539

yes

—

C₂²×C₈₉

#543

yes

—

C₂²×C₈₉

#555

yes

—

C₂²×C₈₉

#557

yes

—

C₂²×C₈₉

H(3,𝔽₄)

yes

—

C₂²×C₈₉

#563

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

#17308

—

maximal, Sylow

C₂×C₈₉

#2785

yes

—

C₂×C₈₉

#2809

yes

—

C₂×C₈₉

#2835

yes

—

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1442

—

C₈₉

#17308

yes

characteristic, complemented, Sylow

C₂³×Q₈

C₂×D₈₉

yes

characteristic

C₂²×C₄⋊C₄

—

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₈₉

yes

—

C₂²×Q₈

—

C₂²×Q₈

Dic₁₇₈

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂×C₄⋊C₄

—

356

—

#95

—

#95

—

356

—

#95

—

178

—

C₂⁴

C₂³×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂×Dic₁₇₈

yes

—

C₂²×C₄

#13083575

yes

—

C₄⋊C₄

—

356

—

C₂³

C₂²×Dic₁₇₈

yes

characteristic

C₂³

C₂×#13083575

yes

—

C₂³

#15668057

yes

—

C₂³

#15668059

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

356

—

102

C₂×C₄

—

356

—

103

C₂×C₄

—

356

—

104

C₂²

#19152197

yes

—

105

C₂²

#19152224

yes

—

106

C₂²

#19152314

yes

—

107

C₂²

#19152327

yes

—

108

C₂²

#19152341

yes

—

109

C₂²

#19152348

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

712

—

113

C₂

#26027960

yes

—

114

C₂

#26028240

yes

—

115

C₂

#26028443

yes

—

116

C₂

#26028505

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂⁴×C₈₉	C₂⁴
Fitting	C₈₉×#1446	C₂
hypercenter	#1446	D₈₉
hypocenter	C₈₉	#17308
LCS	G ▷ C₂⁴×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂⁴×C₈₉

C₂⁴

Fitting

C₈₉×#1446

C₂

D₈₉

C₈₉

G ▷ C₂⁴×C₈₉ ▷ C₈₉

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
62. C₈₉	#17308	semidirect

H ≅ G/N

type

62. C₈₉

#17308

semidirect

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

subgroup	index
#26018523	2
#26018508	2
C₈₉×#1446	2
#17308	89

subgroup

index

#26018523

#26018508

C₈₉×#1446

#17308

Subgroups — #36519991

Conjugacy classes of subgroups

Complements

Maximal subgroups