Group #26028625 — subgroups

Subgroups — #26028625

Order 11392.

524 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 157 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	1	yes	maximal, characteristic
2	#19151503	C₂	1	2	yes	maximal, complemented
3	#19151499	C₂	1	2	yes	maximal
4	#19151488	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
9	#15668000	—	2	4	no	—
10	C₂×#13083551	C₂²	1	8	yes	—
11	C₂×#13083551	C₂²	1	4	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
14	#15667919	C₂²	1	8	yes	—
15	#125×C₈₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂³×D₈₉	C₂³	1	2	yes	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	—	2	4	no	—
22	C₂×D₃₅₆	—	2	2	no	—
23	C₂×D₃₅₆	—	2	4	no	—
24	C₂²×Dic₈₉	C₂³	1	2	yes	—
25	C₂²×Dic₈₉	C₂³	1	4	yes	—
26	#13083551	—	2	8	no	—
27	#13083551	—	2	16	no	—
28	C₄×Dic₈₉	—	2	4	no	—
29	C₄×Dic₈₉	—	2	8	no	—
30	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₈₉	—	2	8	no	—
34	C₄⋊C₄×C₈₉	—	2	4	no	—
35	C₄²×C₈₉	D₄	1	4	yes	—
36	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
37	C₂²×D₈₉	—	2	2	no	—
38	C₂²×D₈₉	—	2	8	no	—
39	C₂²×D₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	—	2	8	no	—
43	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	#41	1	4	yes	—
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	2	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	#41	1	8	yes	—
51	D₃₅₆	—	4	4	no	—
52	Dic₁₇₈	—	4	4	no	—
53	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₈₉	C₂×#41	1	2	yes	—
55	C₂²×C₈₉	C₂×#41	1	4	yes	—
56	C₂×D₈₉	—	4	2	no	—
57	C₂×D₈₉	—	4	8	no	—
58	C₂×D₈₉	—	4	4	no	—
59	C₄×C₈₉	—	2	4	no	—
60	C₄×C₈₉	—	4	4	no	—
61	C₄×C₈₉	#124	1	4	yes	—
62	Dic₈₉	—	2	8	no	—
63	Dic₈₉	—	4	2	no	—
64	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
65	C₂×C₈₉	#529	1	4	yes	—
66	C₂×C₈₉	#532	1	2	yes	—
67	D₈₉	—	8	2	no	—
68	#2861	—	89	1	no	maximal, Sylow
69	C₈₉	#2861	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	D₈₉	1	1	yes	characteristic, complemented
71	C₂×#124	—	89	1	no	—
72	#391	—	89	2	no	—
73	#390	—	89	2	no	—
74	#388	—	89	8	no	—
75	C₄³	—	89	1	no	—
76	C₂²×Q₈	—	89	1	no	—
77	C₂²×D₄	—	89	1	no	—
78	#125	—	2	4	no	—
79	#124	—	178	4	no	—
80	C₂×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
81	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
82	C₂×C₂²⋊C₄	—	89	4	no	—
83	C₂×C₂²⋊C₄	—	89	8	no	—
84	C₂×C₄²	—	89	4	no	—
85	C₂×C₄²	—	89	2	no	—
86	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
87	#95	—	89	8	no	—
88	C₂⁴	—	89	2	no	—
89	C₂×Q₈	—	178	2	no	—
90	C₂×Q₈	—	178	4	no	—
91	C₂×D₄	—	178	2	no	—
92	C₂×D₄	—	178	4	no	—
93	C₂²×C₄	—	89	4	no	—
94	C₂²×C₄	—	89	2	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	—	2	8	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₂²⋊C₄	—	178	16	no	—
101	C₂²⋊C₄	—	178	8	no	—
102	C₄²	—	178	8	no	—
103	C₄²	—	178	4	no	—
104	C₄²	C₂²⋊D₈₉	1	4	yes	—
105	C₂³	—	178	2	no	—
106	C₂³	—	178	8	no	—
107	C₂³	—	178	4	no	—
108	C₂³	C₂³×D₈₉	1	1	yes	characteristic
109	Q₈	—	356	4	no	—
110	D₄	—	356	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	89	8	no	—
115	C₂×C₄	—	178	2	no	—
116	C₂×C₄	—	178	4	no	—
117	C₂×C₄	—	178	4	no	—
118	C₂×C₄	—	178	4	no	—
119	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
121	C₂×C₄	#13083575	1	4	yes	—
122	C₂²	—	356	8	no	—
123	C₂²	—	356	4	no	—
124	C₂²	—	356	2	no	—
125	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂²	#41×D₈₉	1	4	yes	—
127	C₂²	C₂×#13083575	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	4	4	no	—
130	C₄	—	356	2	no	—
131	C₄	—	178	8	no	—
132	C₄	#15668001	1	4	yes	—
133	C₂	—	712	2	no	—
134	C₂	C₂×#15668001	1	1	yes	characteristic
135	C₂	#19152133	1	4	yes	—
136	C₂	#19152144	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668000

C₂

yes

maximal, characteristic

#19151503

C₂

yes

maximal, complemented

#19151499

C₂

yes

maximal

#19151488

C₂

yes

maximal, complemented

C₄²×Dic₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×D₃₅₆

C₂²

yes

characteristic

#15668000

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#125×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#41

yes

—

D₃₅₆

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

#529

yes

—

C₂×C₈₉

#532

yes

—

D₈₉

—

#2861

—

maximal, Sylow

C₈₉

#2861

yes

characteristic, complemented, Sylow

C₂×#125

D₈₉

yes

characteristic, complemented

—

—

—

—

C₄³

—

C₂²×Q₈

—

C₂²×D₄

—

#125

—

#124

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

C₂⁴

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

178

—

101

C₂²⋊C₄

—

178

—

102

C₄²

—

178

—

103

C₄²

—

178

—

104

C₄²

C₂²⋊D₈₉

yes

—

105

C₂³

—

178

—

106

C₂³

—

178

—

107

C₂³

—

178

—

108

C₂³

C₂³×D₈₉

yes

characteristic

109

Q₈

—

356

—

110

D₄

—

356

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

178

—

116

C₂×C₄

—

178

—

117

C₂×C₄

—

178

—

118

C₂×C₄

—

178

—

119

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

120

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

121

C₂×C₄

#13083575

yes

—

122

C₂²

—

356

—

123

C₂²

—

356

—

124

C₂²

—

356

—

125

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

126

C₂²

#41×D₈₉

yes

—

127

C₂²

C₂×#13083575

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

356

—

131

C₄

—

178

—

132

C₄

#15668001

yes

—

133

C₂

—

712

—

134

C₂

C₂×#15668001

yes

characteristic

135

C₂

#19152133

yes

—

136

C₂

#19152144

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#125×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₂×#125	D₈₉
hypocenter	C₈₉	#2861
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#125×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #19151503	C₂	semidirect
4. #19151488	C₂	semidirect
5. C₄²×Dic₈₉	C₂	semidirect
6. C₂×#125×C₈₉	C₂	semidirect
69. C₈₉	#2861	semidirect
70. C₂×#125	D₈₉	semidirect

H ≅ G/N

type

2. #19151503

C₂

semidirect

4. #19151488

C₂

semidirect

5. C₄²×Dic₈₉

C₂

semidirect

6. C₂×#125×C₈₉

C₂

semidirect

69. C₈₉

#2861

semidirect

70. C₂×#125

D₈₉

semidirect

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

subgroup	index
C₂×#15668000	2
#19151503	2
#19151499	2
#19151488	2
C₄²×Dic₈₉	2
C₂×#125×C₈₉	2
#2861	89

subgroup

index

C₄²×Dic₈₉

C₂×#125×C₈₉

#2861

Subgroups — #26028625

Conjugacy classes of subgroups

Complements

Maximal subgroups