Group #26028504 — subgroups

Subgroups — #26028504

Order 11392.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19151511	C₂	1	6	yes	maximal, complemented
2	#19151510	C₂	1	3	yes	maximal, complemented
3	#19151509	C₂	1	2	yes	maximal, complemented
4	#19151453	C₂	1	3	yes	maximal
5	#393×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₈₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₇₈	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₈₉	C₂²	1	3	yes	—
9	C₂×#13083551	C₂²	1	6	yes	—
10	C₂×#13083551	C₂²	1	6	yes	—
11	#15667919	C₂²	1	6	yes	complemented
12	#15667919	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
15	#95×C₈₉	C₂²	1	3	yes	complemented
16	C₂³×D₈₉	—	4	3	no	—
17	C₂³×D₈₉	—	4	3	no	—
18	C₂³×D₈₉	Q₈	1	4	yes	complemented
19	C₂³×D₈₉	C₂³	1	2	yes	—
20	C₂×Dic₁₇₈	—	4	3	no	—
21	C₂×Dic₁₇₈	—	4	6	no	—
22	C₂²×Dic₈₉	C₂³	1	6	yes	—
23	C₂²⋊Dic₈₉	—	4	3	no	—
24	#13083551	—	4	6	no	—
25	#13083551	—	4	6	no	—
26	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₈₉	—	4	3	no	—
28	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
29	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
30	C₂²⋊C₄×C₈₉	—	4	3	no	—
31	C₂³×C₈₉	—	2	3	no	—
32	C₂³×C₈₉	—	2	3	no	—
33	C₂³×C₈₉	C₂×Q₈	1	2	yes	—
34	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
35	C₂²×D₈₉	—	2	6	no	—
36	C₂²×D₈₉	—	2	6	no	—
37	C₂²×D₈₉	—	4	12	no	—
38	C₂²×D₈₉	—	4	12	no	—
39	C₂²×D₈₉	—	4	4	no	—
40	C₂²×D₈₉	C₂×Q₈	1	4	yes	—
41	Q₈×C₈₉	—	4	4	no	—
42	C₂×C₄×C₈₉	—	2	3	no	—
43	C₂×C₄×C₈₉	—	2	3	no	—
44	C₂×C₄×C₈₉	—	2	6	no	—
45	C₂×C₄×C₈₉	—	2	6	no	—
46	C₂×Dic₈₉	—	2	6	no	—
47	C₂×Dic₈₉	—	2	6	no	—
48	C₂×Dic₈₉	—	2	6	no	—
49	Dic₁₇₈	—	4	12	no	—
50	C₂²×C₈₉	—	2	6	no	—
51	C₂²×C₈₉	—	4	3	no	—
52	C₂²×C₈₉	—	4	1	no	—
53	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₈₉	—	2	12	no	—
57	C₂×D₈₉	—	4	16	no	—
58	C₂×D₈₉	—	4	6	no	—
59	C₂×D₈₉	—	4	2	no	—
60	C₄×C₈₉	—	4	3	no	—
61	C₄×C₈₉	—	4	3	no	—
62	Dic₈₉	—	4	6	no	—
63	C₂×C₈₉	—	4	2	no	—
64	C₂×C₈₉	#543	1	3	yes	—
65	C₂×C₈₉	H(3,𝔽₄)	1	3	yes	—
66	C₂×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
67	D₈₉	—	4	4	no	—
68	#2840	—	89	1	no	maximal, Sylow
69	C₈₉	#2840	1	1	yes	characteristic, complemented, Sylow
70	#393	—	89	2	no	—
71	#393	—	89	3	no	—
72	#393	—	89	6	no	—
73	#393	D₈₉	1	1	yes	characteristic, complemented
74	#379	—	89	3	no	—
75	C₂⁵	—	89	1	no	—
76	C₂²×Q₈	—	89	3	no	—
77	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	89	3	no	—
79	C₂×C₂²⋊C₄	—	89	6	no	—
80	C₂×C₂²⋊C₄	—	89	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
82	#95	—	89	6	no	—
83	#95	—	89	3	no	—
84	#95	C₂×D₈₉	1	3	yes	complemented
85	C₂⁴	—	89	4	no	—
86	C₂⁴	—	89	2	no	—
87	C₂⁴	—	356	3	no	—
88	C₂⁴	—	356	3	no	—
89	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	356	3	no	—
92	C₂×Q₈	—	356	6	no	—
93	C₂²×C₄	—	89	6	no	—
94	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
95	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	356	6	no	—
98	C₂²⋊C₄	—	356	6	no	—
99	C₂²⋊C₄	—	356	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	89	4	no	—
103	C₂³	—	356	12	no	—
104	C₂³	—	356	12	no	—
105	C₂³	—	356	4	no	—
106	C₂³	—	178	6	no	—
107	C₂³	—	178	6	no	—
108	C₂³	Q₈×D₈₉	1	2	yes	—
109	C₂³	C₂³×D₈₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	356	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	178	6	no	—
117	C₂×C₄	—	178	6	no	—
118	C₂×C₄	—	178	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	356	16	no	—
123	C₂²	—	356	6	no	—
124	C₂²	—	356	2	no	—
125	C₂²	—	178	12	no	—
126	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
127	C₂²	#15668057	1	3	yes	—
128	C₂²	#15668058	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	356	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	356	4	no	—
134	C₂	#19152223	1	3	yes	—
135	C₂	#19152340	1	3	yes	—
136	C₂	#19152341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151511

C₂

yes

maximal, complemented

#19151510

C₂

yes

maximal, complemented

#19151509

C₂

yes

maximal, complemented

#19151453

C₂

yes

maximal

#393×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₈₉

C₂²

yes

characteristic

C₂²×Dic₁₇₈

C₂²

yes

complemented

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

complemented

#15667919

C₂²

yes

complemented

C₂²×Q₈×C₈₉

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

complemented

C₂³×D₈₉

—

C₂³×D₈₉

—

C₂³×D₈₉

Q₈

yes

complemented

C₂³×D₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²⋊Dic₈₉

—

#13083551

—

#13083551

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂×Q₈

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#543

yes

—

C₂×C₈₉

H(3,𝔽₄)

yes

—

C₂×C₈₉

H(3,𝔽₄)

yes

characteristic

D₈₉

—

#2840

—

maximal, Sylow

C₈₉

#2840

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

#379

—

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

#95

—

#95

—

#95

C₂×D₈₉

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

—

356

—

C₂⁴

—

356

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

356

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

356

—

104

C₂³

—

356

—

105

C₂³

—

356

—

106

C₂³

—

178

—

107

C₂³

—

178

—

108

C₂³

Q₈×D₈₉

yes

—

109

C₂³

C₂³×D₈₉

yes

characteristic

110

Q₈

—

111

Q₈

—

356

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

178

—

117

C₂×C₄

—

178

—

118

C₂×C₄

—

178

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

356

—

123

C₂²

—

356

—

124

C₂²

—

356

—

125

C₂²

—

178

—

126

C₂²

C₂×Q₈×D₈₉

yes

characteristic

127

C₂²

#15668057

yes

—

128

C₂²

#15668058

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

356

—

132

C₂

—

133

C₂

—

356

—

134

C₂

#19152223

yes

—

135

C₂

#19152340

yes

—

136

C₂

#19152341

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#393×C₈₉	C₂
prime radical	C₂⁴×D₈₉	C₂²
hypercenter	#393	D₈₉
hypocenter	C₈₉	#2840
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#393×C₈₉

C₂

prime radical

C₂⁴×D₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #19151511	C₂	semidirect
2. #19151510	C₂	semidirect
3. #19151509	C₂	semidirect
5. #393×C₈₉	C₂	semidirect
7. C₂²×Dic₁₇₈	C₂²	semidirect
11. #15667919	C₂²	semidirect
12. #15667919	C₂²	semidirect
13. C₂²×Q₈×C₈₉	C₂²	semidirect
15. #95×C₈₉	C₂²	semidirect
18. C₂³×D₈₉	Q₈	semidirect
69. C₈₉	#2840	semidirect
73. #393	D₈₉	semidirect
77. C₂²×Q₈	C₂×D₈₉	semidirect
84. #95	C₂×D₈₉	semidirect

H ≅ G/N

type

1. #19151511

C₂

semidirect

2. #19151510

C₂

semidirect

3. #19151509

C₂

semidirect

5. #393×C₈₉

C₂

semidirect

7. C₂²×Dic₁₇₈

C₂²

semidirect

11. #15667919

C₂²

semidirect

12. #15667919

C₂²

semidirect

13. C₂²×Q₈×C₈₉

C₂²

semidirect

15. #95×C₈₉

C₂²

semidirect

18. C₂³×D₈₉

Q₈

semidirect

69. C₈₉

#2840

semidirect

73. #393

D₈₉

semidirect

77. C₂²×Q₈

C₂×D₈₉

semidirect

84. #95

C₂×D₈₉

semidirect

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

subgroup	index
#19151511	2
#19151510	2
#19151509	2
#19151453	2
#393×C₈₉	2
#2840	89

subgroup

index

#393×C₈₉

Subgroups — #26028504

Conjugacy classes of subgroups

Complements

Maximal subgroups