Group #26028816 — subgroups

Subgroups — #26028816

Order 11392.

484 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 293 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	C₂²×#13083558	C₂	1	2	yes	maximal
2	#19151625	C₂	1	8	yes	maximal, complemented
3	#19151622	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13083558	—	2	8	no	—
6	C₂×#13083558	C₂²	1	4	yes	—
7	C₂×#13083558	C₂²	1	4	yes	—
8	C₂²×#10623494	C₂²	1	4	yes	—
9	C₄⋊#10623494	C₂²	1	16	yes	complemented
10	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
11	#117×C₈₉	C₄	1	4	yes	—
12	#117×C₈₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₈₉	C₄	1	2	yes	—
14	C₂×C₂²⋊C₄×C₈₉	C₄	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
16	#13083558	—	2	8	no	—
17	#13083558	—	2	8	no	—
18	C₂×#10623494	—	2	8	no	—
19	C₂×#10623494	C₂³	1	8	yes	—
20	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₈₉	D₄	1	4	yes	—
25	C₂²×C₄×C₈₉	Q₈	1	4	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
30	C₄²×C₈₉	C₂×C₄	1	4	yes	—
31	C₄²×C₈₉	C₂³	1	4	yes	—
32	C₂³×C₈₉	—	4	2	no	—
33	C₂³×C₈₉	C₄⋊C₄	1	4	yes	—
34	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
35	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	—	4	4	no	—
37	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₈₉	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
45	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
47	#10623494	—	2	8	no	—
48	C₂²×C₈₉	—	2	2	no	—
49	C₂²×C₈₉	—	2	4	no	—
50	C₂²×C₈₉	—	4	4	no	—
51	C₂²×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₈₉	—	2	8	no	—
55	C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
59	C₂×C₈₉	—	2	2	no	—
60	C₂×C₈₉	—	2	4	no	—
61	C₂×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈₉	#568	1	2	yes	—
63	#2905	—	89	1	no	maximal, Sylow
64	C₈₉	#2905	1	1	yes	characteristic, complemented, Sylow
65	C₂²×M₁₆	—	89	2	no	—
66	C₂×#117	D₈₉	1	1	yes	characteristic
67	#424	—	89	8	no	—
68	#423	—	89	4	no	—
69	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
70	C₂×M₁₆	—	89	4	no	—
71	C₂×M₁₆	—	89	4	no	—
72	C₂×M₁₆	—	178	8	no	—
73	C₂²×C₈	—	89	4	no	—
74	#117	Dic₈₉	1	4	yes	—
75	#117	C₂×D₈₉	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₈₉	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₈₉	1	2	yes	—
78	C₂×C₄²	C₂×D₈₉	1	2	yes	—
79	C₄⋊C₈	—	89	16	no	—
80	C₂⁴	C₂×Dic₈₉	1	1	yes	characteristic
81	C₂²×C₄	Dic₁₇₈	1	4	yes	—
82	C₂²×C₄	D₃₅₆	1	4	yes	—
83	C₂²×C₄	C₂×Dic₈₉	1	4	yes	—
84	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
86	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
88	M₁₆	—	178	8	no	—
89	M₁₆	—	178	8	no	—
90	C₂×C₈	—	89	8	no	—
91	C₂×C₈	—	178	8	no	—
92	C₄⋊C₄	C₂×Dic₈₉	1	8	yes	—
93	C₂²⋊C₄	C₂×Dic₈₉	1	8	yes	—
94	C₄²	C₂×Dic₈₉	1	4	yes	—
95	C₄²	C₂²×D₈₉	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊Dic₈₉	1	4	yes	—
98	C₂³	C₂²×Dic₈₉	1	2	yes	—
99	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊Dic₈₉	1	4	yes	—
102	C₂×C₄	C₄⋊Dic₈₉	1	8	yes	—
103	C₂×C₄	C₂²×Dic₈₉	1	8	yes	—
104	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
105	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
106	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
107	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
108	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
109	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
110	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
111	C₈	—	178	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊Dic₈₉	1	4	yes	—
116	C₂²	C₂×C₄⋊Dic₈₉	1	2	yes	—
117	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
120	C₄	C₂×C₄⋊Dic₈₉	1	4	yes	—
121	C₄	C₂²×D₃₅₆	1	1	yes	characteristic
122	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
126	C₂	#19152365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083558

C₂

yes

maximal

#19151625

C₂

yes

maximal, complemented

#19151622

C₂

yes

maximal, complemented

C₂×#117×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

—

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

complemented

C₂³×C₄×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₄

yes

—

#117×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₄

yes

—

C₂×C₂²⋊C₄×C₈₉

C₄

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

—

—

—

C₂³

yes

—

C₂⁴×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₈₉

#568

yes

—

#2905

—

maximal, Sylow

C₈₉

#2905

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×#117

D₈₉

yes

characteristic

#424

—

#423

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

#117

Dic₈₉

yes

—

#117

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

Dic₈₉

yes

—

C₂×C₂²⋊C₄

Dic₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

C₄⋊C₈

—

C₂⁴

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

Dic₁₇₈

yes

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₂²⋊C₄

C₂×Dic₈₉

yes

—

C₄²

C₂×Dic₈₉

yes

—

C₄²

C₂²×D₈₉

yes

—

C₂³

—

C₂³

C₄⋊Dic₈₉

yes

—

C₂³

C₂²×Dic₈₉

yes

—

C₂³

C₂²×Dic₈₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₈₉

yes

—

102

C₂×C₄

C₄⋊Dic₈₉

yes

—

103

C₂×C₄

C₂²×Dic₈₉

yes

—

104

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

105

C₂×C₄

C₂²×Dic₈₉

yes

—

106

C₂×C₄

C₂×D₃₅₆

yes

—

107

C₂×C₄

C₂×D₃₅₆

yes

—

108

C₂×C₄

C₂×Dic₁₇₈

yes

—

109

C₂×C₄

C₂×Dic₁₇₈

yes

—

110

C₂×C₄

C₂³×D₈₉

yes

characteristic

111

C₈

—

178

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊Dic₈₉

yes

—

116

C₂²

C₂×C₄⋊Dic₈₉

yes

—

117

C₂²

C₂³×Dic₈₉

yes

characteristic

118

C₄

—

119

C₄

C₂×C₄⋊Dic₈₉

yes

—

120

C₄

C₂×C₄⋊Dic₈₉

yes

—

121

C₄

C₂²×D₃₅₆

yes

characteristic

122

C₄

C₂²×Dic₁₇₈

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂²×C₄⋊Dic₈₉

yes

characteristic

126

C₂

#19152365

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	C₂×#117×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂×#117	D₈₉
hypocenter	C₈₉	#2905
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

C₂×#117×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #19151625	C₂	semidirect
3. #19151622	C₂	semidirect
9. C₄⋊#10623494	C₂²	semidirect
64. C₈₉	#2905	semidirect

H ≅ G/N

type

2. #19151625

C₂

semidirect

3. #19151622

C₂

semidirect

9. C₄⋊#10623494

C₂²

semidirect

64. C₈₉

#2905

semidirect

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

subgroup	index
C₂²×#13083558	2
#19151625	2
#19151622	2
C₂×#117×C₈₉	2
#2905	89

subgroup

index

C₂×#117×C₈₉

Subgroups — #26028816

Conjugacy classes of subgroups

Complements

Maximal subgroups