Group #26028298 — subgroups

Subgroups — #26028298

Order 11392.

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

normal subgroups only

This group has 580 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	C₂×#15668000	C₂	1	2	yes	maximal, complemented
2	#19151515	C₂	1	4	yes	maximal, complemented
3	#19151511	C₂	1	4	yes	maximal, complemented
4	#19151508	C₂	1	4	yes	maximal, complemented
5	#382×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂²×D₃₅₆	C₂²	1	2	yes	—
8	C₂²×D₃₅₆	C₂²	1	4	yes	—
9	#15668000	—	2	8	no	—
10	C₂×#13083551	C₂²	1	4	yes	—
11	C₂×#13083551	C₂²	1	4	yes	—
12	C₂×#13083551	C₂²	1	8	yes	—
13	#15667919	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	#95×C₈₉	C₂²	1	4	yes	—
17	C₂³×D₈₉	C₂³	1	4	yes	—
18	C₂×Dic₁₇₈	—	2	4	no	—
19	C₂×Dic₁₇₈	—	4	4	no	—
20	C₂×D₃₅₆	—	2	4	no	—
21	C₂×D₃₅₆	—	4	4	no	—
22	C₂×D₃₅₆	—	4	4	no	—
23	C₂×D₃₅₆	—	4	8	no	—
24	C₂²×Dic₈₉	C₂³	1	4	yes	—
25	#13083551	—	2	8	no	—
26	#13083551	—	2	8	no	—
27	#13083551	—	4	8	no	—
28	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₈₉	—	4	2	no	—
32	C₄²×C₈₉	D₄	1	4	yes	complemented
33	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
34	C₂²×D₈₉	—	2	4	no	—
35	C₂²×D₈₉	—	2	8	no	—
36	C₂²×D₈₉	—	2	4	no	—
37	C₂²×D₈₉	—	2	8	no	—
38	C₂²×D₈₉	—	2	4	no	—
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	1	no	—
42	C₂×C₄×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	2	8	no	—
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	—	2	8	no	—
48	D₃₅₆	—	4	8	no	—
49	D₃₅₆	—	4	16	no	—
50	Dic₁₇₈	—	4	8	no	—
51	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
52	C₂²×C₈₉	H(5,𝔽₂)	1	4	yes	—
53	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
54	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
55	C₂×D₈₉	—	4	4	no	—
56	C₂×D₈₉	—	4	4	no	—
57	C₂×D₈₉	—	4	4	no	—
58	C₂×D₈₉	—	4	8	no	—
59	C₂×D₈₉	—	4	8	no	—
60	C₄×C₈₉	—	2	4	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	—	4	4	no	—
63	Dic₈₉	—	4	4	no	—
64	C₂×C₈₉	#535	1	2	yes	—
65	C₂×C₈₉	#537	1	1	yes	characteristic
66	C₂×C₈₉	H(3,𝔽₄)	1	2	yes	—
67	C₂×C₈₉	#562	1	2	yes	—
68	D₈₉	—	8	4	no	—
69	#2819	—	89	1	no	maximal, Sylow
70	C₈₉	#2819	1	1	yes	characteristic, complemented, Sylow
71	C₂×#124	—	89	2	no	—
72	#394	—	89	4	no	—
73	#393	—	89	4	no	—
74	#392	—	89	4	no	—
75	#382	D₈₉	1	1	yes	characteristic, complemented
76	C₂²×Q₈	—	89	2	no	—
77	C₂²×D₄	—	89	4	no	—
78	C₂²×D₄	—	89	2	no	—
79	#124	—	178	8	no	—
80	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
81	C₂×C₂²⋊C₄	—	89	8	no	—
82	C₂×C₂²⋊C₄	—	89	4	no	—
83	C₂×C₂²⋊C₄	—	89	4	no	—
84	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
85	#95	—	89	4	no	—
86	#95	C₂×D₈₉	1	4	yes	—
87	C₂⁴	—	89	4	no	—
88	C₂×Q₈	—	356	4	no	—
89	C₂×Q₈	—	178	4	no	—
90	C₂×D₄	—	356	4	no	—
91	C₂×D₄	—	356	4	no	—
92	C₂×D₄	—	356	8	no	—
93	C₂×D₄	—	178	4	no	—
94	C₂²×C₄	—	89	4	no	—
95	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
96	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₂²⋊C₄	—	356	8	no	—
100	C₂²⋊C₄	—	178	8	no	—
101	C₂²⋊C₄	—	178	8	no	—
102	C₄²	D₃₅₆	1	4	yes	complemented
103	C₂³	—	178	4	no	—
104	C₂³	—	178	4	no	—
105	C₂³	—	178	8	no	—
106	C₂³	—	178	8	no	—
107	C₂³	—	178	4	no	—
108	C₂³	C₂³×D₈₉	1	1	yes	characteristic
109	Q₈	—	356	8	no	—
110	D₄	—	356	8	no	—
111	D₄	—	356	16	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	178	4	no	—
118	C₂×C₄	—	178	8	no	—
119	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
120	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
121	C₂²	—	356	4	no	—
122	C₂²	—	356	4	no	—
123	C₂²	—	356	4	no	—
124	C₂²	—	356	8	no	—
125	C₂²	—	356	8	no	—
126	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
127	C₂²	#15668057	1	1	yes	characteristic
128	C₂²	#15668058	1	4	yes	—
129	C₂²	#15668060	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	356	4	no	—
134	C₂	—	712	4	no	—
135	C₂	#19152167	1	2	yes	—
136	C₂	#19152179	1	1	yes	characteristic
137	C₂	#19152340	1	2	yes	—
138	C₂	#19152346	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668000

C₂

yes

maximal, complemented

#19151515

C₂

yes

maximal, complemented

#19151511

C₂

yes

maximal, complemented

#19151508

C₂

yes

maximal, complemented

#382×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₇₈

C₂²

yes

—

C₂²×D₃₅₆

C₂²

yes

—

C₂²×D₃₅₆

C₂²

yes

—

#15668000

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

#13083551

—

#13083551

—

#13083551

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

complemented

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

#535

yes

—

C₂×C₈₉

#537

yes

characteristic

C₂×C₈₉

H(3,𝔽₄)

yes

—

C₂×C₈₉

#562

yes

—

D₈₉

—

#2819

—

maximal, Sylow

C₈₉

#2819

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

#124

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

#95

C₂×D₈₉

yes

—

C₂⁴

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

356

—

100

C₂²⋊C₄

—

178

—

101

C₂²⋊C₄

—

178

—

102

C₄²

D₃₅₆

yes

complemented

103

C₂³

—

178

—

104

C₂³

—

178

—

105

C₂³

—

178

—

106

C₂³

—

178

—

107

C₂³

—

178

—

108

C₂³

C₂³×D₈₉

yes

characteristic

109

Q₈

—

356

—

110

D₄

—

356

—

111

D₄

—

356

—

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

C₂×D₃₅₆

yes

—

120

C₂×C₄

C₂×D₃₅₆

yes

—

121

C₂²

—

356

—

122

C₂²

—

356

—

123

C₂²

—

356

—

124

C₂²

—

356

—

125

C₂²

—

356

—

126

C₂²

C₂²×D₃₅₆

yes

characteristic

127

C₂²

#15668057

yes

characteristic

128

C₂²

#15668058

yes

—

129

C₂²

#15668060

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

356

—

134

C₂

—

712

—

135

C₂

#19152167

yes

—

136

C₂

#19152179

yes

characteristic

137

C₂

#19152340

yes

—

138

C₂

#19152346

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#382×C₈₉	C₂
hypercenter	#382	D₈₉
hypocenter	C₈₉	#2819
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#382×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
1. C₂×#15668000	C₂	semidirect
2. #19151515	C₂	semidirect
3. #19151511	C₂	semidirect
4. #19151508	C₂	semidirect
5. #382×C₈₉	C₂	semidirect
32. C₄²×C₈₉	D₄	semidirect
70. C₈₉	#2819	semidirect
75. #382	D₈₉	semidirect
102. C₄²	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#15668000

C₂

semidirect

2. #19151515

C₂

semidirect

3. #19151511

C₂

semidirect

4. #19151508

C₂

semidirect

5. #382×C₈₉

C₂

semidirect

32. C₄²×C₈₉

D₄

semidirect

70. C₈₉

#2819

semidirect

75. #382

D₈₉

semidirect

102. C₄²

D₃₅₆

semidirect

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

subgroup	index
C₂×#15668000	2
#19151515	2
#19151511	2
#19151508	2
#382×C₈₉	2
#2819	89

subgroup

index

#382×C₈₉

Subgroups — #26028298

Conjugacy classes of subgroups

Complements

Maximal subgroups