Group #26019009 — subgroups

Subgroups — #26019009

Order 11392.

184 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 81 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	#19151620	C₂	1	2	yes	maximal
2	C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151351	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151350	C₂	1	2	yes	maximal, complemented
5	#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083557	—	2	2	no	—
7	C₂×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂×C₈×D₈₉	C₄	1	1	yes	characteristic
9	C₄²×D₈₉	C₂²	1	1	yes	characteristic
10	C₄⋊#10623494	C₂²	1	2	yes	—
11	#15667941	—	2	2	no	—
12	C₄×#10623494	C₂²	1	1	yes	characteristic
13	C₈×Dic₈₉	C₄	1	1	yes	characteristic
14	C₈×Dic₈₉	C₄	1	1	yes	characteristic
15	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₈₉	Q₈	1	1	yes	characteristic
20	#13083557	—	4	4	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
23	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
24	C₈×D₈₉	—	2	2	no	—
25	C₈×D₈₉	—	2	2	no	—
26	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₈₉	D₄	1	1	yes	characteristic
28	C₄×Dic₈₉	Q₈	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×D₈₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
40	C₄×D₈₉	—	2	1	no	—
41	C₄×D₈₉	—	2	1	no	—
42	C₄×D₈₉	—	2	2	no	—
43	C₄×D₈₉	D₈	1	2	yes	—
44	C₄×D₈₉	Q₁₆	1	2	yes	—
45	C₈×C₈₉	—	2	1	no	—
46	C₈×C₈₉	—	4	2	no	—
47	C₈×C₈₉	M₁₆	1	2	yes	complemented
48	#10623494	—	2	1	no	—
49	#10623494	—	4	2	no	—
50	#10623494	M₁₆	1	2	yes	complemented
51	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×D₈₉	—	2	1	no	—
53	C₂×D₈₉	—	2	1	no	—
54	C₂×D₈₉	#107	1	2	yes	—
55	C₄×C₈₉	—	2	1	no	—
56	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
57	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
58	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
60	Dic₈₉	—	2	1	no	—
61	Dic₈₉	#107	1	2	yes	—
62	Dic₈₉	#108	1	2	yes	—
63	C₂×C₈₉	#423	1	1	yes	characteristic
64	C₂×C₈₉	C₂×#107	1	1	yes	characteristic
65	C₂×C₈₉	C₂×#108	1	1	yes	characteristic
66	D₈₉	—	2	2	no	—
67	#1567	—	89	1	no	maximal, Sylow
68	C₈₉	#1567	1	1	yes	characteristic, complemented, Sylow
69	#423	—	89	2	no	—
70	C₂×C₄×C₈	—	89	1	no	—
71	#335	—	89	2	no	—
72	#335	—	89	1	no	—
73	#335	D₈₉	1	1	yes	characteristic, complemented
74	C₂×M₁₆	—	178	2	no	—
75	C₂²×C₈	—	89	1	no	—
76	C₂²×C₈	—	89	1	no	—
77	C₂×C₄²	—	89	1	no	—
78	C₄⋊C₈	—	89	2	no	—
79	C₄⋊C₈	—	178	2	no	—
80	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
81	C₄×C₈	—	89	1	no	—
82	C₄×C₈	—	89	1	no	—
83	C₄×C₈	—	89	1	no	—
84	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
85	C₂²×C₄	—	89	1	no	—
86	C₂²×C₄	—	89	1	no	—
87	C₂²×C₄	—	89	1	no	—
88	M₁₆	—	356	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	89	1	no	—
91	C₂×C₈	—	89	1	no	—
92	C₂×C₈	—	178	2	no	—
93	C₂×C₈	—	178	2	no	—
94	C₂×C₈	—	178	2	no	—
95	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
96	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
97	C₄²	—	89	1	no	—
98	C₄²	—	89	1	no	—
99	C₄²	—	89	1	no	—
100	C₄²	C₂²×D₈₉	1	1	yes	characteristic
101	C₂³	—	89	1	no	—
102	C₂×C₄	—	89	1	no	—
103	C₂×C₄	—	89	1	no	—
104	C₂×C₄	—	89	2	no	—
105	C₂×C₄	—	89	2	no	—
106	C₂×C₄	—	89	1	no	—
107	C₂×C₄	—	178	2	no	—
108	C₂×C₄	—	178	1	no	—
109	C₂×C₄	—	178	1	no	—
110	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	4	2	no	—
115	C₈	—	89	2	no	—
116	C₈	—	356	2	no	—
117	C₈	—	178	1	no	—
118	C₈	#13083557	1	2	yes	complemented
119	C₂²	—	89	2	no	—
120	C₂²	—	178	1	no	—
121	C₂²	—	178	1	no	—
122	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	89	2	no	—
125	C₄	—	89	2	no	—
126	C₄	—	178	1	no	—
127	C₄	M₁₆×D₈₉	1	1	yes	characteristic
128	C₄	C₂×#13083557	1	1	yes	characteristic
129	C₄	D₈×D₈₉	1	1	yes	characteristic
130	C₄	Q₁₆×D₈₉	1	1	yes	characteristic
131	C₂	—	178	2	no	—
132	C₂	#19151620	1	1	yes	characteristic
133	C₂	#107×D₈₉	1	1	yes	characteristic
134	C₂	#108×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151620

C₂

yes

maximal

C₄×C₈×D₈₉

C₂

yes

maximal, characteristic

#19151351

C₂

yes

maximal, characteristic, complemented

#19151350

C₂

yes

maximal, complemented

#335×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083557

—

C₂×C₈×D₈₉

C₄

yes

characteristic

C₂×C₈×D₈₉

C₄

yes

characteristic

C₄²×D₈₉

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

#15667941

—

C₄×#10623494

C₂²

yes

characteristic

C₈×Dic₈₉

C₄

yes

characteristic

C₈×Dic₈₉

C₄

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

Q₈

yes

—

—

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₈×D₈₉

—

C₈×D₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×Dic₈₉

C₄⋊C₄

yes

characteristic

C₂×Dic₈₉

C₄⋊C₄

yes

characteristic

C₂×Dic₈₉

C₄⋊C₄

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

D₈

yes

—

C₄×D₈₉

Q₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

M₁₆

yes

—

—

M₁₆

yes

complemented

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

#107

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₄×C₈₉

C₂×M₁₆

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

#107

yes

—

Dic₈₉

#108

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1567

—

maximal, Sylow

C₈₉

#1567

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

—

#335

—

#335

—

#335

D₈₉

yes

characteristic, complemented

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₄×D₈₉

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂²×D₈₉

yes

characteristic

101

C₂³

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

178

—

108

C₂×C₄

—

178

—

109

C₂×C₄

—

178

—

110

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

111

C₂×C₄

D₄×D₈₉

yes

characteristic

112

C₂×C₄

Q₈×D₈₉

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

356

—

117

C₈

—

178

—

118

C₈

#13083557

yes

complemented

119

C₂²

—

120

C₂²

—

178

—

121

C₂²

—

178

—

122

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

178

—

127

C₄

M₁₆×D₈₉

yes

characteristic

128

C₄

C₂×#13083557

yes

characteristic

129

C₄

D₈×D₈₉

yes

characteristic

130

C₄

Q₁₆×D₈₉

yes

characteristic

131

C₂

—

178

—

132

C₂

#19151620

yes

characteristic

133

C₂

#107×D₈₉

yes

characteristic

134

C₂

#108×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#335×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₄
prime radical	C₂²×D₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#335	D₈₉
hypocenter	C₈₉	#1567
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#335×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₄

prime radical

C₂²×D₈₉

C₄⋊C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
3. #19151351	C₂	semidirect
4. #19151350	C₂	semidirect
5. #335×C₈₉	C₂	semidirect
47. C₈×C₈₉	M₁₆	semidirect
50. #10623494	M₁₆	semidirect
68. C₈₉	#1567	semidirect
73. #335	D₈₉	semidirect
118. C₈	#13083557	semidirect

H ≅ G/N

type

3. #19151351

C₂

semidirect

4. #19151350

C₂

semidirect

5. #335×C₈₉

C₂

semidirect

47. C₈×C₈₉

M₁₆

semidirect

50. #10623494

M₁₆

semidirect

68. C₈₉

#1567

semidirect

73. #335

D₈₉

semidirect

118. C₈

#13083557

semidirect

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

subgroup	index
#19151620	2
C₄×C₈×D₈₉	2
#19151351	2
#19151350	2
#335×C₈₉	2
#1567	89

subgroup

index

C₄×C₈×D₈₉

#335×C₈₉

Subgroups — #26019009

Conjugacy classes of subgroups

Complements

Maximal subgroups