Group #26025494 — subgroups

Subgroups — #26025494

Order 11392.

380 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 165 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	#19151537	C₂	1	4	yes	maximal
2	#19151497	C₂	1	4	yes	maximal
3	#19151463	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₈₉	C₂	1	2	yes	maximal
5	#384×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
9	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
10	#15667919	C₂²	1	4	yes	—
11	#15667919	C₂²	1	4	yes	—
12	#15667919	C₂²	1	8	yes	—
13	C₂×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₈₉	C₂³	1	4	yes	—
17	C₂²×Dic₈₉	C₂³	1	4	yes	—
18	C₄⋊Dic₈₉	—	2	4	no	—
19	C₄⋊Dic₈₉	—	2	4	no	—
20	C₄⋊Dic₈₉	—	4	4	no	—
21	C₄×Dic₈₉	—	2	8	no	—
22	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
23	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
25	C₄⋊C₄×C₈₉	—	2	8	no	—
26	C₄⋊C₄×C₈₉	—	4	2	no	—
27	C₄²×C₈₉	Q₈	1	4	yes	—
28	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
29	C₂×C₄×C₈₉	—	2	1	no	—
30	C₂×C₄×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈₉	—	2	8	no	—
32	C₂×C₄×C₈₉	—	2	4	no	—
33	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₈₉	#41	1	4	yes	—
36	C₂×C₄×C₈₉	#41	1	4	yes	—
37	C₂×Dic₈₉	—	2	4	no	—
38	C₂×Dic₈₉	—	2	4	no	—
39	C₂×Dic₈₉	—	2	4	no	—
40	C₂×Dic₈₉	—	2	4	no	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	2	4	no	—
43	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
44	C₂²×C₈₉	C₂×#41	1	2	yes	—
45	C₂²×C₈₉	C₂×#41	1	2	yes	—
46	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
47	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄×C₈₉	—	2	2	no	—
49	C₄×C₈₉	—	4	4	no	—
50	C₄×C₈₉	#125	1	8	yes	—
51	Dic₈₉	—	4	4	no	—
52	Dic₈₉	—	4	4	no	—
53	C₂×C₈₉	C₂×#125	1	2	yes	—
54	C₂×C₈₉	#544	1	1	yes	characteristic
55	C₂×C₈₉	#555	1	1	yes	characteristic
56	C₂×C₈₉	#556	1	2	yes	—
57	C₂×C₈₉	#559	1	1	yes	characteristic
58	#2571	—	89	1	no	maximal, Sylow
59	C₈₉	#2571	1	1	yes	characteristic, complemented, Sylow
60	#400	—	89	4	no	—
61	#389	—	89	4	no	—
62	#384	D₈₉	1	1	yes	characteristic
63	#382	—	89	4	no	—
64	C₄×C₄⋊C₄	—	89	2	no	—
65	C₂×C₄⋊C₄	—	89	4	no	—
66	C₂×C₄⋊C₄	—	89	2	no	—
67	C₂×C₄⋊C₄	—	89	2	no	—
68	C₂×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
69	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
70	C₂×C₄²	—	89	4	no	—
71	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
72	#95	—	89	4	no	—
73	#95	—	89	4	no	—
74	#95	—	89	8	no	—
75	C₂²×C₄	—	89	4	no	—
76	C₂²×C₄	—	89	4	no	—
77	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
78	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	4	2	no	—
82	C₄⋊C₄	—	356	4	no	—
83	C₄⋊C₄	—	178	4	no	—
84	C₄⋊C₄	—	178	4	no	—
85	C₄²	—	178	8	no	—
86	C₄²	Dic₁₇₈	1	4	yes	—
87	C₂³	C₂³×D₈₉	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	178	4	no	—
93	C₂×C₄	—	178	4	no	—
94	C₂×C₄	—	178	4	no	—
95	C₂×C₄	—	178	4	no	—
96	C₂×C₄	—	178	4	no	—
97	C₂×C₄	—	178	4	no	—
98	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
99	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
100	C₂×C₄	#13083573	1	4	yes	—
101	C₂×C₄	#13083575	1	4	yes	—
102	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
103	C₂²	C₂×#13083573	1	2	yes	—
104	C₂²	C₂×#13083575	1	2	yes	—
105	C₂²	#15668057	1	1	yes	characteristic
106	C₂²	#15668060	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	356	4	no	—
110	C₄	—	356	4	no	—
111	C₄	#15668003	1	8	yes	—
112	C₂	C₂×#15668003	1	2	yes	—
113	C₂	#19152229	1	1	yes	characteristic
114	C₂	#19152314	1	1	yes	characteristic
115	C₂	#19152322	1	2	yes	—
116	C₂	#19152334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151537

C₂

yes

maximal

#19151497

C₂

yes

maximal

#19151463

C₂

yes

maximal

C₄×C₄⋊Dic₈₉

C₂

yes

maximal

#384×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#125

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#125

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#400

—

#389

—

#384

D₈₉

yes

characteristic

#382

—

C₄×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

Dic₁₇₈

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

100

C₂×C₄

#13083573

yes

—

101

C₂×C₄

#13083575

yes

—

102

C₂²

C₂²×Dic₁₇₈

yes

characteristic

103

C₂²

C₂×#13083573

yes

—

104

C₂²

C₂×#13083575

yes

—

105

C₂²

#15668057

yes

characteristic

106

C₂²

#15668060

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

356

—

110

C₄

—

356

—

111

C₄

#15668003

yes

—

112

C₂

C₂×#15668003

yes

—

113

C₂

#19152229

yes

characteristic

114

C₂

#19152314

yes

characteristic

115

C₂

#19152322

yes

—

116

C₂

#19152334

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#384×C₈₉	C₂
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2571
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#384×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
59. C₈₉	#2571	semidirect

H ≅ G/N

type

59. C₈₉

#2571

semidirect

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

subgroup	index
#19151537	2
#19151497	2
#19151463	2
C₄×C₄⋊Dic₈₉	2
#384×C₈₉	2
#2571	89

subgroup

index

#19151537

#19151497

#19151463

C₄×C₄⋊Dic₈₉

#384×C₈₉

#2571

Subgroups — #26025494

Conjugacy classes of subgroups

Complements

Maximal subgroups