Group #26028524 — subgroups

Subgroups — #26028524

Order 11392.

356 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 133 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	#19151540	C₂	1	4	yes	maximal
2	#19151537	C₂	1	2	yes	maximal
3	#19151519	C₂	1	4	yes	maximal
4	#19151518	C₂	1	2	yes	maximal
5	#19151464	C₂	1	2	yes	maximal
6	#382×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×#13083553	C₂²	1	4	yes	—
12	#15667919	C₂²	1	4	yes	—
13	#15667919	C₂²	1	4	yes	—
14	#15667919	C₂²	1	2	yes	—
15	#15667919	C₂²	1	4	yes	—
16	#15667919	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
20	#95×C₈₉	C₂²	1	4	yes	—
21	C₂×Dic₁₇₈	—	4	4	no	—
22	C₂×Dic₁₇₈	—	4	2	no	—
23	C₂²×Dic₈₉	C₂³	1	4	yes	—
24	C₂²×Dic₈₉	C₂³	1	4	yes	—
25	C₄⋊Dic₈₉	—	4	2	no	—
26	C₄⋊Dic₈₉	—	4	2	no	—
27	C₄⋊Dic₈₉	—	4	2	no	—
28	#13083553	—	4	4	no	—
29	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
31	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	—	4	1	no	—
34	C₄⋊C₄×C₈₉	—	4	1	no	—
35	C₄²×C₈₉	Q₈	1	4	yes	complemented
36	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₈₉	—	2	1	no	—
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	4	no	—
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	—	2	4	no	—
51	C₂×Dic₈₉	—	2	4	no	—
52	C₂×Dic₈₉	—	2	4	no	—
53	Dic₁₇₈	—	4	8	no	—
54	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
58	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
59	C₄×C₈₉	—	2	4	no	—
60	C₄×C₈₉	—	2	2	no	—
61	C₄×C₈₉	—	4	2	no	—
62	C₄×C₈₉	—	4	2	no	—
63	Dic₈₉	—	4	4	no	—
64	Dic₈₉	—	4	4	no	—
65	C₂×C₈₉	#544	1	2	yes	—
66	C₂×C₈₉	#544	1	1	yes	characteristic
67	C₂×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
68	C₂×C₈₉	#563	1	1	yes	characteristic
69	C₂×C₈₉	#563	1	2	yes	—
70	#2844	—	89	1	no	maximal, Sylow
71	C₈₉	#2844	1	1	yes	characteristic, complemented, Sylow
72	#400	—	89	2	no	—
73	#400	—	89	4	no	—
74	#395	—	89	2	no	—
75	#395	—	89	4	no	—
76	#382	—	89	2	no	—
77	#382	D₈₉	1	1	yes	characteristic
78	C₂²×Q₈	—	89	2	no	—
79	C₂×C₄⋊C₄	—	89	2	no	—
80	C₂×C₄⋊C₄	—	89	2	no	—
81	C₂×C₄⋊C₄	—	89	4	no	—
82	C₂×C₄⋊C₄	—	89	2	no	—
83	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
86	#95	—	89	4	no	—
87	#95	—	89	4	no	—
88	#95	—	89	2	no	—
89	#95	—	89	4	no	—
90	#95	—	89	2	no	—
91	#95	C₂×D₈₉	1	4	yes	—
92	C₂×Q₈	—	356	2	no	—
93	C₂×Q₈	—	356	4	no	—
94	C₂²×C₄	—	89	4	no	—
95	C₂²×C₄	—	89	4	no	—
96	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
98	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	356	4	no	—
103	C₄⋊C₄	—	356	2	no	—
104	C₄⋊C₄	—	356	2	no	—
105	C₄⋊C₄	—	356	2	no	—
106	C₄²	Dic₁₇₈	1	4	yes	complemented
107	C₂³	C₂³×D₈₉	1	1	yes	characteristic
108	Q₈	—	356	8	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	178	4	no	—
118	C₂×C₄	—	178	4	no	—
119	C₂×C₄	—	178	4	no	—
120	C₂×C₄	—	178	4	no	—
121	C₂×C₄	—	178	4	no	—
122	C₂×C₄	—	178	4	no	—
123	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
124	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
125	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
126	C₂²	#15668057	1	1	yes	characteristic
127	C₂²	#15668058	1	2	yes	—
128	C₂²	#15668059	1	2	yes	—
129	C₂²	#15668060	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	356	4	no	—
135	C₄	—	356	4	no	—
136	C₂	#19152226	1	2	yes	—
137	C₂	#19152229	1	1	yes	characteristic
138	C₂	#19152340	1	1	yes	characteristic
139	C₂	#19152348	1	1	yes	characteristic
140	C₂	#19152352	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151540

C₂

yes

maximal

#19151537

C₂

yes

maximal

#19151519

C₂

yes

maximal

#19151518

C₂

yes

maximal

#19151464

C₂

yes

maximal

#382×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

Q₈

yes

complemented

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×Q₈

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#544

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2844

—

maximal, Sylow

C₈₉

#2844

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

—

—

—

—

—

C₂×D₈₉

yes

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

356

—

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

356

—

105

C₄⋊C₄

—

356

—

106

C₄²

Dic₁₇₈

yes

complemented

107

C₂³

C₂³×D₈₉

yes

characteristic

108

Q₈

—

356

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

178

—

118

C₂×C₄

—

178

—

119

C₂×C₄

—

178

—

120

C₂×C₄

—

178

—

121

C₂×C₄

—

178

—

122

C₂×C₄

—

178

—

123

C₂×C₄

C₂×Dic₁₇₈

yes

—

124

C₂×C₄

C₂×Dic₁₇₈

yes

—

125

C₂²

C₂²×Dic₁₇₈

yes

characteristic

126

C₂²

#15668057

yes

characteristic

127

C₂²

#15668058

yes

—

128

C₂²

#15668059

yes

—

129

C₂²

#15668060

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

356

—

135

C₄

—

356

—

136

C₂

#19152226

yes

—

137

C₂

#19152229

yes

characteristic

138

C₂

#19152340

yes

characteristic

139

C₂

#19152348

yes

characteristic

140

C₂

#19152352

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#382×C₈₉	C₂
hypercenter	#382	D₈₉
hypocenter	C₈₉	#2844
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#382×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
35. C₄²×C₈₉	Q₈	semidirect
71. C₈₉	#2844	semidirect
106. C₄²	Dic₁₇₈	semidirect

H ≅ G/N

type

35. C₄²×C₈₉

Q₈

semidirect

71. C₈₉

#2844

semidirect

106. C₄²

Dic₁₇₈

semidirect

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

subgroup	index
#19151540	2
#19151537	2
#19151519	2
#19151518	2
#19151464	2
#382×C₈₉	2
#2844	89

subgroup

index

#382×C₈₉

Subgroups — #26028524

Conjugacy classes of subgroups

Complements

Maximal subgroups