Group #25011232 — subgroups

Subgroups — #25011232

Order 11008.

212 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19136748	C₂	1	2	yes	maximal
2	#19134213	C₂	1	1	yes	maximal, characteristic
3	#19133493	C₂	1	1	yes	maximal, characteristic
4	#19132652	C₂	1	2	yes	maximal
5	C₄₃×#1367	C₂	1	1	yes	maximal, characteristic
6	#14583304	—	2	1	no	—
7	#14583304	C₂²	1	2	yes	—
8	#14583279	—	2	1	no	—
9	#14583029	—	2	1	no	—
10	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	#14582725	—	2	2	no	—
12	#14582704	—	2	2	no	—
13	#14582652	C₂²	1	1	yes	characteristic
14	#14582631	—	2	1	no	—
15	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	4	2	no	—
18	C₂×Dic₁₇₂	D₄	1	1	yes	characteristic
19	#13063325	—	2	2	no	—
20	C₄×Dic₈₆	—	2	1	no	—
21	#13063259	—	2	2	no	—
22	#13063259	—	2	2	no	—
23	#13063259	—	4	1	no	—
24	#13063259	—	4	1	no	—
25	#13063259	D₄	1	1	yes	characteristic
26	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
27	Dic₄₃⋊C₈	—	4	1	no	—
28	#13063250	—	2	1	no	—
29	#13063235	—	4	1	no	—
30	C₂×C₁₆×C₄₃	—	2	2	no	—
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×Dic₈₆	—	2	1	no	—
35	C₂×Dic₈₆	—	4	2	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
41	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	2	1	no	—
43	C₄⋊Dic₄₃	—	4	2	no	—
44	C₄⋊Dic₄₃	—	4	2	no	—
45	Dic₄₃⋊C₄	—	4	1	no	—
46	C₄×Dic₄₃	—	4	1	no	—
47	C₂×C₄₃⋊C₈	—	4	1	no	—
48	Dic₁₇₂	—	2	2	no	—
49	Dic₁₇₂	—	8	4	no	—
50	C₁₆×C₄₃	—	4	2	no	—
51	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₂×Dic₄₃	—	4	1	no	—
55	C₂×Dic₄₃	—	4	1	no	—
56	C₂×Dic₄₃	—	8	2	no	—
57	Dic₈₆	—	4	1	no	—
58	Dic₈₆	—	4	1	no	—
59	Dic₈₆	—	8	4	no	—
60	C₈×C₄₃	—	2	2	no	—
61	C₈×C₄₃	—	2	1	no	—
62	C₈×C₄₃	—	2	1	no	—
63	C₈×C₄₃	Q₃₂	1	2	yes	—
64	C₈×C₄₃	#137	1	2	yes	—
65	C₄₃⋊C₈	—	8	1	no	—
66	#12500	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	#459	1	1	yes	characteristic
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	#470	1	1	yes	characteristic
70	C₄×C₄₃	#473	1	1	yes	characteristic
71	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
72	C₄×C₄₃	#510	1	1	yes	characteristic
73	Dic₄₃	—	8	1	no	—
74	Dic₄₃	—	8	1	no	—
75	Dic₄₃	—	16	2	no	—
76	#2227	—	43	2	no	—
77	#1713	—	43	1	no	—
78	#1591	—	43	1	no	—
79	#1367	D₄₃	1	1	yes	characteristic
80	#1365	—	43	2	no	—
81	C₂×C₄₃	#1689	1	1	yes	characteristic
82	C₂×C₄₃	#2209	1	1	yes	characteristic
83	C₂×C₄₃	#2213	1	1	yes	characteristic
84	#500	—	43	2	no	—
85	#500	—	86	1	no	—
86	#494	—	86	1	no	—
87	#446	—	86	1	no	—
88	C₄×Q₁₆	—	43	1	no	—
89	#366	—	86	2	no	—
90	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
91	#358	—	86	2	no	—
92	#335	—	43	1	no	—
93	Q₈⋊C₈	—	86	1	no	—
94	C₈²	C₂×D₄₃	1	1	yes	characteristic
95	C₄₃	#12500	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	43	1	no	—
97	C₂×Q₁₆	—	172	2	no	—
98	C₄⋊Q₈	—	86	2	no	—
99	C₄×Q₈	—	86	1	no	—
100	C₂×C₁₆	—	2	2	no	—
101	#107	—	43	1	no	—
102	#107	—	86	2	no	—
103	#107	—	86	2	no	—
104	#107	—	172	1	no	—
105	#107	—	172	1	no	—
106	C₄⋊C₈	—	86	1	no	—
107	C₄⋊C₈	—	172	1	no	—
108	#103	—	86	1	no	—
109	#97	—	172	1	no	—
110	C₄×C₈	D₁₇₂	1	1	yes	characteristic
111	C₄×C₈	D₁₇₂	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
113	C₂×Q₈	—	86	1	no	—
114	C₂×Q₈	—	172	2	no	—
115	Q₁₆	—	86	2	no	—
116	Q₁₆	—	344	4	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	172	1	no	—
119	C₂×C₈	D₃₄₄	1	1	yes	characteristic
120	C₂×C₈	D₃₄₄	1	1	yes	characteristic
121	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
122	C₂×C₈	#10621902	1	1	yes	characteristic
123	C₄⋊C₄	—	86	1	no	—
124	C₄⋊C₄	—	172	2	no	—
125	C₄⋊C₄	—	172	2	no	—
126	C₄⋊C₄	—	172	1	no	—
127	C₄²	—	172	1	no	—
128	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
129	C₁₆	—	4	2	no	—
130	Q₈	—	344	4	no	—
131	Q₈	—	172	1	no	—
132	Q₈	—	172	1	no	—
133	C₂×C₄	—	344	2	no	—
134	C₂×C₄	—	172	1	no	—
135	C₂×C₄	—	172	1	no	—
136	C₂×C₄	#13063300	1	1	yes	characteristic
137	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
138	C₂×C₄	#13063353	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	344	1	no	—
143	C₈	Dic₃₄₄	1	2	yes	—
144	C₈	#13063354	1	2	yes	—
145	C₂²	#14583095	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	344	1	no	—
148	C₄	—	344	1	no	—
149	C₄	—	688	2	no	—
150	C₄	#14583158	1	1	yes	characteristic
151	C₄	#14583175	1	1	yes	characteristic
152	C₄	C₂×Dic₃₄₄	1	1	yes	characteristic
153	C₄	#14583342	1	1	yes	characteristic
154	C₂	#19134085	1	1	yes	characteristic
155	C₂	#19136646	1	1	yes	characteristic
156	C₂	#19136672	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136748

C₂

yes

maximal

#19134213

C₂

yes

maximal, characteristic

#19133493

C₂

yes

maximal, characteristic

#19132652

C₂

yes

maximal

C₄₃×#1367

C₂

yes

maximal, characteristic

#14583304

—

#14583304

C₂²

yes

—

#14583279

—

#14583029

—

C₄×Dic₁₇₂

C₂²

yes

—

—

C₂²

yes

characteristic

#14582631

—

C₄₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

D₄

yes

characteristic

#13063325

—

C₄×Dic₈₆

—

—

—

—

—

D₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063250

—

#13063235

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

Q₃₂

yes

—

C₈×C₄₃

#137

yes

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#510

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic

#1365

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

C₂×D₄₃

yes

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12500

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

172

—

C₄⋊Q₈

—

C₄×Q₈

—

100

C₂×C₁₆

—

101

#107

—

102

#107

—

103

#107

—

104

#107

—

172

—

105

#107

—

172

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

172

—

108

#103

—

109

#97

—

172

—

110

C₄×C₈

D₁₇₂

yes

characteristic

111

C₄×C₈

D₁₇₂

yes

characteristic

112

C₄×C₈

C₂²×D₄₃

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

172

—

115

Q₁₆

—

116

Q₁₆

—

344

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₂×C₈

D₃₄₄

yes

characteristic

120

C₂×C₈

D₃₄₄

yes

characteristic

121

C₂×C₈

D₄×D₄₃

yes

characteristic

122

C₂×C₈

#10621902

yes

characteristic

123

C₄⋊C₄

—

124

C₄⋊C₄

—

172

—

125

C₄⋊C₄

—

172

—

126

C₄⋊C₄

—

172

—

127

C₄²

—

172

—

128

C₄²

C₂×D₁₇₂

yes

characteristic

129

C₁₆

—

130

Q₈

—

344

—

131

Q₈

—

172

—

132

Q₈

—

172

—

133

C₂×C₄

—

344

—

134

C₂×C₄

—

172

—

135

C₂×C₄

—

172

—

136

C₂×C₄

#13063300

yes

characteristic

137

C₂×C₄

C₂×D₃₄₄

yes

characteristic

138

C₂×C₄

#13063353

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

344

—

143

C₈

Dic₃₄₄

yes

—

144

C₈

#13063354

yes

—

145

C₂²

#14583095

yes

characteristic

146

C₄

—

147

C₄

—

344

—

148

C₄

—

344

—

149

C₄

—

688

—

150

C₄

#14583158

yes

characteristic

151

C₄

#14583175

yes

characteristic

152

C₄

C₂×Dic₃₄₄

yes

characteristic

153

C₄

#14583342

yes

characteristic

154

C₂

#19134085

yes

characteristic

155

C₂

#19136646

yes

characteristic

156

C₂

#19136672

yes

characteristic

	subgroup	G/H
center	C₂²	#14583095
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1367	C₂
socle, prime radical	C₂²×C₄₃	#459
hypercenter	#1367	D₄₃
hypocenter	C₄₃	#12500
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂²

#14583095

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1367

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1367

N	H ≅ G/N	type
95. C₄₃	#12500	semidirect

H ≅ G/N

type

95. C₄₃

#12500

semidirect

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

subgroup	index
#19136748	2
#19134213	2
#19133493	2
#19132652	2
C₄₃×#1367	2
#12500	43

subgroup

index

C₄₃×#1367

Subgroups — #25011232

Conjugacy classes of subgroups

Complements

Maximal subgroups