Group #26020300 — subgroups

Subgroups — #26020300

Order 11392.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15668019	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667947	C₂	1	2	yes	maximal, complemented
3	#19151582	C₂	1	1	yes	maximal, characteristic
4	#19151356	C₂	1	2	yes	maximal, complemented
5	#413×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668019	—	2	4	no	—
7	C₂×#13083558	C₄	1	2	yes	—
8	C₂×#13083558	C₂²	1	2	yes	—
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	C₂²×#10623494	C₄	1	2	yes	—
11	C₂²×#10623494	C₂²	1	1	yes	characteristic
12	#15667947	—	2	4	no	—
13	#15667931	—	2	1	no	—
14	#15667929	—	2	1	no	—
15	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
17	#101×C₈₉	—	2	1	no	—
18	#100×C₈₉	—	2	1	no	—
19	#13083558	—	2	2	no	—
20	#13083558	—	2	4	no	—
21	#13083558	—	2	2	no	—
22	#13083558	—	2	2	no	—
23	C₂×#10623494	—	2	4	no	—
24	C₂×#10623494	—	2	2	no	—
25	C₂×#10623494	C₂×C₄	1	2	yes	—
26	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
28	C₂×#10623494	D₄	1	4	yes	—
29	#41×C₈₉	—	2	2	no	—
30	#41×C₈₉	—	2	2	no	—
31	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂×D₄×C₈₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₈₉	—	2	2	no	—
37	M₁₆×C₈₉	—	2	2	no	—
38	C₂×C₈×C₈₉	—	2	2	no	—
39	C₂³×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₈₉	—	2	2	no	—
42	D₄×C₈₉	—	2	4	no	—
43	D₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#41	1	2	yes	—
51	C₈×C₈₉	—	4	2	no	—
52	#10623494	—	2	2	no	—
53	#10623494	—	4	2	no	—
54	#10623494	C₂²⋊C₄	1	2	yes	—
55	#10623494	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₈₉	—	2	2	no	—
57	C₂²×C₈₉	—	4	1	no	—
58	C₂²×C₈₉	—	4	2	no	—
59	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₈₉	#117	1	1	yes	characteristic
64	C₄×C₈₉	#121	1	2	yes	—
65	C₂×C₈₉	—	2	1	no	—
66	C₂×C₈₉	—	2	2	no	—
67	C₂×C₈₉	—	4	2	no	—
68	C₂×C₈₉	#380	1	1	yes	characteristic
69	#1813	—	89	1	no	maximal, Sylow
70	C₈₉	#1813	1	1	yes	characteristic, complemented, Sylow
71	C₂×#131	—	89	1	no	—
72	C₂×#108	—	89	2	no	—
73	#413	—	89	1	no	—
74	#413	D₈₉	1	1	yes	characteristic
75	#338	—	89	2	no	—
76	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
77	#131	—	178	4	no	—
78	C₂×M₁₆	—	89	2	no	—
79	C₂×M₁₆	—	89	1	no	—
80	C₂×M₁₆	—	89	2	no	—
81	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
82	C₂²×C₈	—	89	1	no	—
83	C₂²×C₈	—	89	2	no	—
84	#108	—	178	4	no	—
85	#101	—	2	1	no	—
86	#101	—	178	1	no	—
87	#100	—	2	1	no	—
88	#100	—	178	1	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
92	C₂×D₄	Dic₁₇₈	1	2	yes	—
93	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
94	C₂²×C₄	C₄×D₈₉	1	2	yes	—
95	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
96	M₁₆	—	2	2	no	—
97	M₁₆	—	2	2	no	—
98	M₁₆	—	178	4	no	—
99	M₁₆	—	178	2	no	—
100	M₁₆	—	178	2	no	—
101	M₁₆	—	178	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	89	1	no	—
104	C₂×C₈	—	89	1	no	—
105	C₂×C₈	—	89	4	no	—
106	C₂×C₈	—	89	2	no	—
107	C₂×C₈	—	178	4	no	—
108	C₂×C₈	—	178	2	no	—
109	C₂³	#13083553	1	2	yes	—
110	C₂³	C₂×Dic₁₇₈	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	D₄	—	2	4	no	—
113	D₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	#13083553	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
118	C₂×C₄	D₄×D₈₉	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂×C₄	#13083573	1	2	yes	—
121	C₈	—	4	2	no	—
122	C₈	—	89	2	no	—
123	C₈	—	89	2	no	—
124	C₈	—	356	2	no	—
125	C₈	—	178	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	4	1	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×#13083553	1	1	yes	characteristic
130	C₂²	#15667989	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
133	C₄	#15667968	1	1	yes	characteristic
134	C₄	#15667984	1	2	yes	—
135	C₂	—	2	1	no	—
136	C₂	—	2	2	no	—
137	C₂	—	4	2	no	—
138	C₂	#19151456	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668019

C₂

yes

maximal, characteristic

C₂×#15667947

C₂

yes

maximal, complemented

#19151582

C₂

yes

maximal, characteristic

#19151356

C₂

yes

maximal, complemented

#413×C₈₉

C₂

yes

maximal, characteristic

#15668019

—

C₂×#13083558

C₄

yes

—

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

characteristic

C₂²×#10623494

C₄

yes

—

C₂²×#10623494

C₂²

yes

—

—

—

C₂×#41×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

#101×C₈₉

—

#100×C₈₉

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

D₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×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

—

#10623494

C₂²⋊C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

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

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

#338

—

C₂×#41

C₂×D₈₉

yes

characteristic

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

178

—

#101

—

#101

—

178

—

#100

—

#100

—

178

—

#41

—

#41

—

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

C₂×D₄

Dic₁₇₈

yes

—

C₂×D₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

100

M₁₆

—

178

—

101

M₁₆

—

178

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

178

—

108

C₂×C₈

—

178

—

109

C₂³

#13083553

yes

—

110

C₂³

C₂×Dic₁₇₈

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#13083553

yes

—

117

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

118

C₂×C₄

D₄×D₈₉

yes

—

119

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

120

C₂×C₄

#13083573

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

356

—

125

C₈

—

178

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#13083553

yes

characteristic

130

C₂²

#15667989

yes

—

131

C₄

—

132

C₄

C₂²⋊C₄×D₈₉

yes

characteristic

133

C₄

#15667968

yes

characteristic

134

C₄

#15667984

yes

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#19151456

yes

characteristic

	subgroup	G/H
center	C₄	#15667968
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#413×C₈₉	C₂
socle	C₂×C₈₉	#380
prime radical	C₂×#41×C₈₉	C₂²
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#413	D₈₉
hypocenter	C₈₉	#1813
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

#15667968

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#413×C₈₉

C₂

socle

C₂×C₈₉

#380

prime radical

C₂×#41×C₈₉

C₂²

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
2. C₂×#15667947	C₂	semidirect
4. #19151356	C₂	semidirect
70. C₈₉	#1813	semidirect

H ≅ G/N

type

2. C₂×#15667947

C₂

semidirect

4. #19151356

C₂

semidirect

70. C₈₉

#1813

semidirect

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

subgroup	index
C₂×#15668019	2
C₂×#15667947	2
#19151582	2
#19151356	2
#413×C₈₉	2
#1813	89

subgroup

index

#413×C₈₉

Subgroups — #26020300

Conjugacy classes of subgroups

Complements

Maximal subgroups