Finite Group Databasebeta

← #34100662

Subgroups — #34100662

Order 20096.

344 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758039C211yesmaximal, characteristic, complemented
2#23757842C211yesmaximal, characteristic
3#23757279C211yesmaximal, characteristic, complemented
4#23757277C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757029C211yesmaximal, characteristic, complemented
7H(3,ℤ/4)×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×D628C2211yescharacteristic
10#19097779—22no—
11#19097777—22no—
12C2×#14578938C2211yescharacteristic
13M16×D157—22no—
14#19097768C412yescomplemented
15#19097759C412yescomplemented
16#19097746—22no—
17#19097741—22no—
18#19097730C2211yescharacteristic
19#19097693—22no—
20#19097691—22no—
21#19097685—22no—
22C2×M16×C157C2211yescharacteristic
23#117×C157C2211yescharacteristic
24#117×C157C2211yescharacteristic
25C23×D157—21no—
26#14578954—21no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30C2×Dic314C2×C411yescharacteristic
31C2×D628—24no—
32C2×D628—21no—
33C2×D628—21no—
34C2×D628—21no—
35C2×D628—21no—
36C2×D628—21no—
37C2×D628C2×C412yes—
38C2×D628C2×C411yescharacteristic
39C2×C22⋊D157D412yes—
40C2×C4×D157D412yes—
41#14578938—21no—
42#14578938—21no—
43#14578937—24no—
44C2×#11929434D412yescomplemented
45C8×D157—24no—
46C4⋊Dic157—21no—
47C22⋊Dic157—21no—
48#14578931—24no—
49C4×Dic157—21no—
50C22×C4×C157C2311yescharacteristic
51M16×C157—21no—
52M16×C157—21no—
53C2×C8×C157D412yescomplemented
54C4⋊C4×C157—21no—
55C4⋊C4×C157—21no—
56C22⋊C4×C157—21no—
57C22⋊C4×C157—21no—
58C42×C157—21no—
59C42×C157C2×C412yes—
60C22×D157—22no—
61C22×D157—41no—
62C22×D157—41no—
63C22×D157—82no—
64C22×D157C22⋊C412yes—
65Dic314—21no—
66Dic314—21no—
67D628—21no—
68D628—21no—
69D628—22no—
70D628—22no—
71D628—41no—
72D628—41no—
73D628—84no—
74C22⋊D157—22no—
75C22⋊D157—22no—
76C2×Dic157—41no—
77C2×Dic157C22⋊C412yes—
78C4×D157—22no—
79C4×D157—22no—
80#11929434—22no—
81C23×C157C2×D411yescharacteristic
82C2×C4×C157—22no—
83C2×C4×C157—41no—
84C2×C4×C157C22×C411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C2×C4×C157#4111yescharacteristic
89C2×C4×C157#4111yescharacteristic
90C8×C157—22no—
91C22×C157—41no—
92C22×C157C2×C22⋊C411yescharacteristic
93C22×C157#12111yescharacteristic
94C22×C157#12111yescharacteristic
95C2×D157—22no—
96C2×D157—42no—
97C2×D157—44no—
98C2×D157—42no—
99C2×D157—82no—
100C2×D157—82no—
101C4×C157—42no—
102C4×C157—81no—
103C4×C157C4×D411yescharacteristic
104C4×C157C4×D411yescharacteristic
105C4×C157#12011yescharacteristic
106C4×C157#12311yescharacteristic
107Dic157—22no—
108Dic157—81no—
109C2×C157—21no—
110C2×C157—21no—
111C2×C157—21no—
112C2×C157#38611yescharacteristic
113D157—42no—
114D157—82no—
115C157#191011yescharacteristic, complemented, Sylow
116#1910—1571nomaximal, Sylow
117#568—1571no—
118#535—1571no—
119#418—1571no—
120#418—1571no—
121#410—1571no—
122H(3,ℤ/4)—1571no—
123H(3,ℤ/4)D15711yescharacteristic, complemented
124C2×#41—1571no—
125C22×D4—1571no—
126#131—3142no—
127#131—3142no—
128C2×M16—3142no—
129C2×M16—1571no—
130C2×M16C2×D15711yescharacteristic
131#127—1572no—
132#124—1572no—
133#121—3142no—
134#120—3142no—
135#117—1571no—
136#117C2×D15711yescharacteristic
137#117C2×D15711yescharacteristic
138#102—3142no—
139#102—3142no—
140#99—3142no—
141C24—3141no—
142#41—3141no—
143#41—3141no—
144#41—3141no—
145#41—3141no—
146C2×Q8—1571no—
147C2×D4—3144no—
148C2×D4—3141no—
149C2×D4—3141no—
150C2×D4—3141no—
151C2×D4—3141no—
152C2×D4—3141no—
153C2×D4—1572no—
154C2×D4—1572no—
155C2×D4—1571no—
156C22×C4—1572no—
157C22×C4C22×D15711yescharacteristic
158M16—21no—
159M16—21no—
160M16—3144no—
161M16—3141no—
162M16—3141no—
163C2×C8—3144no—
164C2×C8—1572no—
165C2×C8D62812yescomplemented
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3141no—
169C22⋊C4—21no—
170C22⋊C4—21no—
171C22⋊C4—3144no—
172C22⋊C4—3141no—
173C42—21no—
174C42—3141no—
175C42C4×D15712yes—
176C23—3142no—
177C23—6281no—
178C23—6281no—
179C23—1572no—
180C23—12562no—
181C23D4×D15711yescharacteristic
182Q8—3141no—
183Q8—3141no—
184D4—3142no—
185D4—3142no—
186D4—3141no—
187D4—3142no—
188D4—3141no—
189D4—3142no—
190D4—6281no—
191D4—6281no—
192D4—12564no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—3142no—
196C2×C4—3142no—
197C2×C4—6281no—
198C2×C4—1572no—
199C2×C4C2×C4×D15711yescharacteristic
200C2×C4D4×D15711yescharacteristic
201C2×C4D4×D15711yescharacteristic
202C2×C4C2×D62811yescharacteristic
203C2×C4#1457895411yescharacteristic
204C2×C4#1457895511yescharacteristic
205C8—22no—
206C8—3142no—
207C22—41no—
208C22—3142no—
209C22—6282no—
210C22—6284no—
211C22—6282no—
212C22—12562no—
213C22—12562no—
214C22C22⋊C4×D15711yescharacteristic
215C22#1909774411yescharacteristic
216C22#1909774611yescharacteristic
217C4—42no—
218C4—81no—
219C4—3142no—
220C4—12561no—
221C4#1909773311yescharacteristic
222C4C4×D62811yescharacteristic
223C4#1909774111yescharacteristic
224C4#1909775611yescharacteristic
225C2—21no—
226C2—21no—
227C2—21no—
228C2—6282no—
229C2—12562no—
230C2#2375715111yescharacteristic
subgroupG/H
centerC2#23757151
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingH(3,ℤ/4)×C157C2
socleC2×C157#386
prime radical#23757842C2
cosocleC22×C4×C157C23
hypercenterH(3,ℤ/4)D157
hypocenterC157#1910
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23758039C2semidirect
3. #23757279C2semidirect
4. #23757277C2semidirect
5. #23757246C2semidirect
6. #23757029C2semidirect
7. H(3,ℤ/4)×C157C2semidirect
14. #19097768C4semidirect
15. #19097759C4semidirect
44. C2×#11929434D4semidirect
53. C2×C8×C157D4semidirect
115. C157#1910semidirect
123. H(3,ℤ/4)D157semidirect
165. C2×C8D628semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237578422
#237572792
#237572772
#237572462
#237570292
H(3,ℤ/4)×C1572
#1910157