Group #26028289 — subgroups

Subgroups — #26028289

Order 11392.

444 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#15667991	C₂	1	2	yes	maximal
2	#19151538	C₂	1	4	yes	maximal, complemented
3	#19151523	C₂	1	4	yes	maximal
4	#19151519	C₂	1	4	yes	maximal, complemented
5	#381×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	#15667991	—	2	8	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂×#13083553	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
12	#15667919	C₂²	1	8	yes	—
13	#15667919	C₂²	1	4	yes	—
14	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
16	#95×C₈₉	C₂²	1	4	yes	—
17	C₂×Dic₁₇₈	—	2	4	no	—
18	C₂×Dic₁₇₈	—	4	4	no	—
19	C₂²×Dic₈₉	C₂³	1	4	yes	—
20	C₂²×Dic₈₉	C₂³	1	4	yes	—
21	C₄⋊Dic₈₉	—	2	4	no	—
22	C₄⋊Dic₈₉	—	4	4	no	—
23	#13083553	—	2	8	no	—
24	C₂²⋊Dic₈₉	—	2	8	no	—
25	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	—	4	2	no	—
27	C₂²×C₄×C₈₉	D₄	1	4	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
30	C₂²⋊C₄×C₈₉	—	4	2	no	—
31	C₂³×C₈₉	—	2	2	no	—
32	C₂³×C₈₉	—	2	4	no	—
33	C₂³×C₈₉	C₂×D₄	1	2	yes	—
34	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	4	no	—
36	C₂×C₄×C₈₉	—	2	4	no	—
37	C₂×C₄×C₈₉	—	2	8	no	—
38	C₂×C₄×C₈₉	—	4	4	no	—
39	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
40	C₂×Dic₈₉	—	2	4	no	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	2	8	no	—
43	C₂×Dic₈₉	—	2	8	no	—
44	Dic₁₇₈	—	4	8	no	—
45	C₂²×C₈₉	—	2	2	no	—
46	C₂²×C₈₉	—	2	4	no	—
47	C₂²×C₈₉	—	4	4	no	—
48	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
50	C₂²×C₈₉	2₋¹⁺⁴	1	4	yes	—
51	C₄×C₈₉	—	2	4	no	—
52	C₄×C₈₉	—	4	4	no	—
53	Dic₈₉	—	4	4	no	—
54	Dic₈₉	—	4	4	no	—
55	C₂×C₈₉	—	4	2	no	—
56	C₂×C₈₉	#534	1	1	yes	characteristic
57	C₂×C₈₉	#536	1	2	yes	—
58	C₂×C₈₉	#563	1	4	yes	—
59	#2817	—	89	1	no	maximal, Sylow
60	C₈₉	#2817	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₂²⋊Q₈	—	89	2	no	—
62	#400	—	89	4	no	—
63	#397	—	89	4	no	—
64	#395	—	89	4	no	—
65	#381	D₈₉	1	1	yes	characteristic
66	C₂²×Q₈	—	89	2	no	—
67	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
68	C₂²⋊Q₈	—	178	8	no	—
69	C₂×C₄⋊C₄	—	89	4	no	—
70	C₂×C₄⋊C₄	—	89	4	no	—
71	C₂×C₄⋊C₄	—	89	2	no	—
72	C₂×C₂²⋊C₄	—	89	4	no	—
73	C₂×C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
74	#95	—	89	8	no	—
75	#95	—	89	4	no	—
76	#95	C₂×D₈₉	1	4	yes	—
77	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
78	C₂×Q₈	—	356	4	no	—
79	C₂×Q₈	—	178	4	no	—
80	C₂²×C₄	—	4	2	no	—
81	C₂²×C₄	—	89	4	no	—
82	C₂²×C₄	—	89	4	no	—
83	C₂²×C₄	D₃₅₆	1	4	yes	—
84	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
85	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
86	C₄⋊C₄	—	356	4	no	—
87	C₄⋊C₄	—	178	8	no	—
88	C₄⋊C₄	—	178	4	no	—
89	C₂²⋊C₄	—	4	2	no	—
90	C₂²⋊C₄	—	178	8	no	—
91	C₂³	—	2	2	no	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×D₃₅₆	1	2	yes	—
94	C₂³	C₂³×D₈₉	1	1	yes	characteristic
95	Q₈	—	356	8	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	178	4	no	—
101	C₂×C₄	—	178	4	no	—
102	C₂×C₄	—	178	8	no	—
103	C₂×C₄	—	178	8	no	—
104	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	4	4	no	—
108	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
109	C₂²	#15668057	1	2	yes	—
110	C₂²	#15668059	1	4	yes	—
111	C₄	—	2	4	no	—
112	C₄	—	4	4	no	—
113	C₄	—	356	4	no	—
114	C₄	—	356	4	no	—
115	C₂	—	4	2	no	—
116	C₂	#19152160	1	1	yes	characteristic
117	C₂	#19152174	1	2	yes	—
118	C₂	#19152351	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667991

C₂

yes

maximal

#19151538

C₂

yes

maximal, complemented

#19151523

C₂

yes

maximal

#19151519

C₂

yes

maximal, complemented

#381×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15667991

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#534

yes

characteristic

C₂×C₈₉

#536

yes

—

C₂×C₈₉

#563

yes

—

#2817

—

maximal, Sylow

C₈₉

#2817

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂²⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

#95

—

#95

—

#95

C₂×D₈₉

yes

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

C₂³

—

C₂³

—

C₂³

C₂×D₃₅₆

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

178

—

101

C₂×C₄

—

178

—

102

C₂×C₄

—

178

—

103

C₂×C₄

—

178

—

104

C₂×C₄

C₂×D₃₅₆

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²×D₃₅₆

yes

characteristic

109

C₂²

#15668057

yes

—

110

C₂²

#15668059

yes

—

111

C₄

—

112

C₄

—

113

C₄

—

356

—

114

C₄

—

356

—

115

C₂

—

116

C₂

#19152160

yes

characteristic

117

C₂

#19152174

yes

—

118

C₂

#19152351

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#381×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	#381	D₈₉
hypocenter	C₈₉	#2817
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #381

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#381×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #381

N	H ≅ G/N	type
2. #19151538	C₂	semidirect
4. #19151519	C₂	semidirect
60. C₈₉	#2817	semidirect

H ≅ G/N

type

2. #19151538

C₂

semidirect

4. #19151519

C₂

semidirect

60. C₈₉

#2817

semidirect

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

subgroup	index
C₂×#15667991	2
#19151538	2
#19151523	2
#19151519	2
#381×C₈₉	2
#2817	89

subgroup

index

#381×C₈₉

Subgroups — #26028289

Conjugacy classes of subgroups

Complements

Maximal subgroups