Group #26020288 — subgroups

Subgroups — #26020288

Order 11392.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×#10623494	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667947	C₂	1	2	yes	maximal, complemented
3	#19151562	C₂	1	1	yes	maximal, characteristic
4	#19151362	C₂	1	2	yes	maximal, complemented
5	#407×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂²×#10623494	—	2	4	no	—
8	C₂²×#10623494	C₄	1	4	yes	—
9	C₂²×#10623494	C₂²	1	1	yes	characteristic
10	C₂²×#10623494	C₂²	1	1	yes	characteristic
11	#15667947	—	2	4	no	—
12	C₂²⋊#10623494	—	2	2	no	—
13	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₈₉	—	2	2	no	—
16	#13083558	—	2	2	no	—
17	#13083558	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	—	2	8	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₂×#10623494	C₂×C₄	1	2	yes	—
22	C₂×#10623494	C₂×C₄	1	2	yes	—
23	C₂×#10623494	D₄	1	4	yes	—
24	C₂⁴×C₈₉	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	—	2	2	no	—
26	C₂²×C₄×C₈₉	—	2	2	no	—
27	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₈₉	D₄	1	2	yes	—
29	C₂²×C₄×C₈₉	Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₈₉	—	2	2	no	—
32	M₁₆×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂³×C₈₉	—	2	4	no	—
35	C₂³×C₈₉	—	2	2	no	—
36	C₂³×C₈₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#41	1	2	yes	—
47	C₈×C₈₉	—	4	2	no	—
48	#10623494	—	2	2	no	—
49	#10623494	—	4	2	no	—
50	#10623494	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₈₉	—	2	8	no	—
52	C₂²×C₈₉	—	2	2	no	—
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	#108	1	4	yes	complemented
56	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₈₉	#117	1	1	yes	characteristic
61	C₄×C₈₉	#121	1	2	yes	—
62	C₂×C₈₉	—	2	4	no	—
63	C₂×C₈₉	—	2	2	no	—
64	C₂×C₈₉	#380	1	1	yes	characteristic
65	C₂×C₈₉	C₂×#108	1	2	yes	—
66	#1810	—	89	1	no	maximal, Sylow
67	C₈₉	#1810	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₈	—	89	1	no	—
69	C₂×#108	—	89	2	no	—
70	#407	—	89	1	no	—
71	#407	D₈₉	1	1	yes	characteristic
72	#341	—	89	2	no	—
73	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
74	C₂×M₁₆	—	89	2	no	—
75	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
76	C₂²×C₈	—	89	1	no	—
77	C₂²×C₈	—	89	4	no	—
78	C₂²×C₈	—	89	1	no	—
79	C₂²×C₈	—	178	4	no	—
80	#108	—	178	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	178	2	no	—
83	C₂⁴	Dic₁₇₈	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₁₇₈	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₈₉	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	178	2	no	—
93	M₁₆	—	178	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	89	2	no	—
96	C₂×C₈	—	89	4	no	—
97	C₂×C₈	—	89	2	no	—
98	C₂×C₈	—	178	8	no	—
99	C₂×C₈	—	178	2	no	—
100	C₂×C₈	—	178	2	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	4	no	—
103	C₂³	#13083553	1	2	yes	—
104	C₂³	C₂×Dic₁₇₈	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#13083553	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₈₉	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
113	C₂×C₄	#13083573	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	89	4	no	—
116	C₈	—	356	2	no	—
117	C₈	—	178	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#15667947	1	4	yes	complemented
123	C₂²	C₂×#13083553	1	1	yes	characteristic
124	C₂²	#15667989	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
127	C₄	#15667968	1	1	yes	characteristic
128	C₄	#15667984	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	4	no	—
131	C₂	#19151456	1	1	yes	characteristic
132	C₂	C₂×#15667947	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#10623494

C₂

yes

maximal, characteristic

C₂×#15667947

C₂

yes

maximal, complemented

#19151562

C₂

yes

maximal, characteristic

#19151362

C₂

yes

maximal, complemented

#407×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

—

C₂²×#10623494

C₄

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15667947

—

C₂²⋊#10623494

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

D₄

yes

—

C₂⁴×C₈₉

Q₈

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

Q₈

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

#10623494

C₂²⋊C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#108

yes

complemented

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#380

yes

characteristic

C₂×C₈₉

C₂×#108

yes

—

#1810

—

maximal, Sylow

C₈₉

#1810

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₈₉

yes

characteristic

#341

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

178

—

#108

—

178

—

C₂²⋊C₈

—

C₂²⋊C₈

—

178

—

C₂⁴

Dic₁₇₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₇₈

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

100

C₂×C₈

—

178

—

101

C₂³

—

102

C₂³

—

103

C₂³

#13083553

yes

—

104

C₂³

C₂×Dic₁₇₈

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#13083553

yes

—

110

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

111

C₂×C₄

D₄×D₈₉

yes

—

112

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

113

C₂×C₄

#13083573

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

356

—

117

C₈

—

178

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#15667947

yes

complemented

123

C₂²

C₂×#13083553

yes

characteristic

124

C₂²

#15667989

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₈₉

yes

characteristic

127

C₄

#15667968

yes

characteristic

128

C₄

#15667984

yes

—

129

C₂

—

130

C₂

—

131

C₂

#19151456

yes

characteristic

132

C₂

C₂×#15667947

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#407×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₈₉	Q₈
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#407	D₈₉
hypocenter	C₈₉	#1810
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#407×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₄

prime radical

C₂⁴×C₈₉

Q₈

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#15667947	C₂	semidirect
4. #19151362	C₂	semidirect
55. C₂²×C₈₉	#108	semidirect
67. C₈₉	#1810	semidirect
122. C₂²	#15667947	semidirect

H ≅ G/N

type

2. C₂×#15667947

C₂

semidirect

4. #19151362

C₂

semidirect

55. C₂²×C₈₉

#108

semidirect

67. C₈₉

#1810

semidirect

122. C₂²

#15667947

semidirect

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

subgroup	index
C₂³×#10623494	2
C₂×#15667947	2
#19151562	2
#19151362	2
#407×C₈₉	2
#1810	89

subgroup

index

#407×C₈₉

Subgroups — #26020288

Conjugacy classes of subgroups

Complements

Maximal subgroups