Group #25084402 — subgroups

Subgroups — #25084402

Order 11008.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#19145181	C₂	1	4	yes	maximal
2	#19145109	C₂	1	2	yes	maximal
3	#19134272	C₂	1	8	yes	maximal, complemented
4	C₂×C₄₃×#322	C₂	1	1	yes	maximal, characteristic
5	#14583456	C₂²	1	4	yes	—
6	#14583304	C₂²	1	4	yes	—
7	#14583304	C₂²	1	4	yes	—
8	#14583301	—	2	4	no	—
9	#14583297	C₂²	1	4	yes	—
10	#14583279	C₂²	1	4	yes	—
11	#14583250	C₂²	1	8	yes	—
12	#14583140	—	2	8	no	—
13	C₄×Dic₁₇₂	—	2	8	no	—
14	#14582937	—	2	4	no	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#322	C₂²	1	4	yes	—
18	C₂×Dic₁₇₂	—	2	8	no	—
19	#13063325	C₂³	1	8	yes	—
20	#13063317	—	2	4	no	—
21	#13063304	—	2	8	no	—
22	C₄×Dic₈₆	—	2	8	no	—
23	#13063284	—	2	4	no	—
24	#13063259	—	2	8	no	—
25	#13063259	—	2	8	no	—
26	#13063257	—	2	8	no	—
27	#13063250	—	2	16	no	—
28	C₂²×C₈×C₄₃	D₄	1	4	yes	—
29	C₂²×C₈×C₄₃	D₄	1	2	yes	—
30	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₃	D₄	1	2	yes	—
32	C₄×C₈×C₄₃	D₄	1	4	yes	—
33	C₄×C₈×C₄₃	C₂³	1	2	yes	—
34	C₄×C₈×C₄₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₈₆	—	2	8	no	—
38	C₂×C₈×C₄₃	—	4	4	no	—
39	C₂×C₈×C₄₃	Q₁₆	1	8	yes	complemented
40	C₂×C₈×C₄₃	C₂×D₄	1	8	yes	—
41	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
42	C₄²×C₄₃	C₂×D₄	1	2	yes	—
43	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
45	C₂²⋊Dic₄₃	—	4	4	no	—
46	C₄⋊Dic₄₃	—	2	8	no	—
47	C₄⋊Dic₄₃	—	2	8	no	—
48	Dic₄₃⋊C₄	—	4	8	no	—
49	C₄×Dic₄₃	—	4	4	no	—
50	Dic₁₇₂	—	4	16	no	—
51	C₂³×C₄₃	#127	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	4	no	—
53	C₂×C₄×C₄₃	#127	1	2	yes	—
54	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂²×D₄	1	2	yes	—
58	C₂×Dic₄₃	—	4	8	no	—
59	Dic₈₆	—	4	16	no	—
60	C₈×C₄₃	—	2	8	no	—
61	C₈×C₄₃	C₂×Q₁₆	1	8	yes	—
62	#19618	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	—	2	2	no	—
64	C₂²×C₄₃	#494	1	2	yes	—
65	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
66	C₄×C₄₃	—	2	2	no	—
67	C₄×C₄₃	#494	1	2	yes	—
68	C₄×C₄₃	C₂²×Q₁₆	1	2	yes	—
69	C₄×C₄₃	#575	1	4	yes	—
70	Dic₄₃	—	8	8	no	—
71	#3170	—	43	4	no	—
72	#3160	—	43	2	no	—
73	#1723	—	43	8	no	—
74	C₂×#322	D₄₃	1	1	yes	characteristic
75	C₂×C₄₃	—	2	2	no	—
76	C₂×C₄₃	C₂×#494	1	1	yes	characteristic
77	C₂×C₄₃	#3150	1	2	yes	—
78	#533	—	43	4	no	—
79	#500	—	43	4	no	—
80	#500	—	43	4	no	—
81	#499	—	86	4	no	—
82	#498	—	43	4	no	—
83	#494	—	43	4	no	—
84	#487	—	43	8	no	—
85	#467	—	86	8	no	—
86	C₄×Q₁₆	—	86	8	no	—
87	#427	—	86	4	no	—
88	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
90	#322	C₂×D₄₃	1	4	yes	—
91	C₄₃	#19618	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	86	8	no	—
93	C₂²×C₈	D₁₇₂	1	4	yes	—
94	C₂²×C₈	D₁₇₂	1	2	yes	—
95	C₄⋊Q₈	—	43	8	no	—
96	#125	—	86	4	no	—
97	C₂²⋊Q₈	—	86	8	no	—
98	C₄×Q₈	—	86	8	no	—
99	#117	—	86	4	no	—
100	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
101	#107	—	86	8	no	—
102	#107	—	86	8	no	—
103	#106	—	86	8	no	—
104	#103	—	86	16	no	—
105	C₄×C₈	D₁₇₂	1	4	yes	—
106	C₄×C₈	D₁₇₂	1	2	yes	—
107	C₄×C₈	C₂²×D₄₃	1	4	yes	—
108	C₄×C₈	C₂²×D₄₃	1	2	yes	—
109	C₂×Q₈	—	86	8	no	—
110	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
111	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
112	Q₁₆	—	172	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₁₇₂	1	8	yes	complemented
115	C₂×C₈	C₂×D₁₇₂	1	8	yes	—
116	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
117	C₄⋊C₄	—	86	8	no	—
118	C₄⋊C₄	—	86	8	no	—
119	C₄⋊C₄	—	172	8	no	—
120	C₂²⋊C₄	—	172	4	no	—
121	C₄²	—	172	4	no	—
122	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
123	C₄²	C₂×D₁₇₂	1	2	yes	—
124	C₄²	C₂³×D₄₃	1	1	yes	characteristic
125	C₂³	#13063322	1	1	yes	characteristic
126	Q₈	—	172	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	172	8	no	—
129	C₂×C₄	#13063322	1	2	yes	—
130	C₂×C₄	#13063322	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₁₇₂	1	4	yes	—
132	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₁₇₂	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₁₇₂	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#14583279	1	2	yes	—
138	C₂²	C₂×#13063322	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	344	8	no	—
141	C₄	#14583279	1	2	yes	—
142	C₄	C₂²×Dic₁₇₂	1	2	yes	—
143	C₄	#14583699	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#14583279	1	1	yes	characteristic
146	C₂	#19145041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145181

C₂

yes

maximal

#19145109

C₂

yes

maximal

#19134272

C₂

yes

maximal, complemented

C₂×C₄₃×#322

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583304

C₂²

yes

—

#14583304

C₂²

yes

—

#14583301

—

#14583297

C₂²

yes

—

#14583279

C₂²

yes

—

#14583250

C₂²

yes

—

#14583140

—

C₄×Dic₁₇₂

—

#14582937

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄₃×#322

C₂²

yes

—

C₂×Dic₁₇₂

—

#13063325

C₂³

yes

—

#13063317

—

#13063304

—

C₄×Dic₈₆

—

—

—

—

—

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

C₂³

yes

—

C₄×C₈×C₄₃

C₂³

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

Q₁₆

yes

complemented

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂⁴

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

C₂³×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#127

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂×Q₁₆

yes

—

#19618

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#494

yes

—

C₂²×C₄₃

C₂×#127

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

#494

yes

—

C₄×C₄₃

C₂²×Q₁₆

yes

—

C₄×C₄₃

#575

yes

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

C₂×#494

yes

characteristic

C₂×C₄₃

#3150

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#427

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₂×C₄×C₈

C₂×D₄₃

yes

—

#322

C₂×D₄₃

yes

—

C₄₃

#19618

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂²×C₈

D₁₇₂

yes

—

C₂²×C₈

D₁₇₂

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

100

C₂×C₄²

C₂²×D₄₃

yes

characteristic

101

#107

—

102

#107

—

103

#106

—

104

#103

—

105

C₄×C₈

D₁₇₂

yes

—

106

C₄×C₈

D₁₇₂

yes

—

107

C₄×C₈

C₂²×D₄₃

yes

—

108

C₄×C₈

C₂²×D₄₃

yes

—

109

C₂×Q₈

—

110

C₂²×C₄

C₂×D₁₇₂

yes

—

111

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

112

Q₁₆

—

172

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₇₂

yes

complemented

115

C₂×C₈

C₂×D₁₇₂

yes

—

116

C₂×C₈

C₂×D₁₇₂

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

172

—

120

C₂²⋊C₄

—

172

—

121

C₄²

—

172

—

122

C₄²

C₂×D₁₇₂

yes

characteristic

123

C₄²

C₂×D₁₇₂

yes

—

124

C₄²

C₂³×D₄₃

yes

characteristic

125

C₂³

#13063322

yes

characteristic

126

Q₈

—

172

—

127

C₂×C₄

—

128

C₂×C₄

—

172

—

129

C₂×C₄

#13063322

yes

—

130

C₂×C₄

#13063322

yes

characteristic

131

C₂×C₄

C₂×Dic₁₇₂

yes

—

132

C₂×C₄

C₂²×D₁₇₂

yes

characteristic

133

C₂×C₄

C₂²×D₁₇₂

yes

—

134

C₈

—

135

C₈

C₂×Dic₁₇₂

yes

—

136

C₂²

—

137

C₂²

#14583279

yes

—

138

C₂²

C₂×#13063322

yes

characteristic

139

C₄

—

140

C₄

—

344

—

141

C₄

#14583279

yes

—

142

C₄

C₂²×Dic₁₇₂

yes

—

143

C₄

#14583699

yes

—

144

C₂

—

145

C₂

C₂×#14583279

yes

characteristic

146

C₂

#19145041

yes

—

	subgroup	G/H
center	C₂×C₄	#13063322
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#322	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	C₂³×C₄₃	#127
hypercenter	C₂×#322	D₄₃
hypocenter	C₄₃	#19618
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13063322

commutator, cosocle

C₄²×C₄₃

C₂⁴

Frattini

C₄²

C₂³×D₄₃

Fitting

C₂×C₄₃×#322

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #19134272	C₂	semidirect
39. C₂×C₈×C₄₃	Q₁₆	semidirect
91. C₄₃	#19618	semidirect
114. C₂×C₈	Dic₁₇₂	semidirect

H ≅ G/N

type

3. #19134272

C₂

semidirect

39. C₂×C₈×C₄₃

Q₁₆

semidirect

91. C₄₃

#19618

semidirect

114. C₂×C₈

Dic₁₇₂

semidirect

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

subgroup	index
#19145181	2
#19145109	2
#19134272	2
C₂×C₄₃×#322	2
#19618	43

subgroup

index

C₂×C₄₃×#322

Subgroups — #25084402

Conjugacy classes of subgroups

Complements

Maximal subgroups