Finite Group Databasebeta

← #34100516

Subgroups — #34100516

Order 20096.

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

normal subgroups only

This group has 332 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#23757819C211yesmaximal, characteristic
3#23757290C211yesmaximal, characteristic, complemented
4#23757246C211yesmaximal, characteristic, complemented
5#23757034C211yesmaximal, characteristic, complemented
6#23757032C211yesmaximal, characteristic, complemented
7#421×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097779—22no—
10#19097777—22no—
11C2×#14578938C2211yescharacteristic
12M16×D157—22no—
13#19097769C411yescharacteristic, complemented
14#19097767C411yescharacteristic, complemented
15#19097760C411yescharacteristic, complemented
16#19097758C411yescharacteristic, complemented
17#19097743—22no—
18C4×C22⋊D157—22no—
19#19097730C2211yescharacteristic
20C2×C4×Dic157C2211yescharacteristic
21#19097698—22no—
22#19097685—22no—
23C2×#41×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26#104×C157—22no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31C2×Dic314C2×C411yescharacteristic
32C2×D628C2×C411yescharacteristic
33C2×C22⋊D157—22no—
34C2×C22⋊D157D412yes—
35C22×Dic157—21no—
36C2×C4×D157D412yes—
37#14578938—21no—
38#14578938D412yescomplemented
39#14578937—24no—
40C2×#11929434—21no—
41C8×D157—24no—
42C4⋊Dic157—21no—
43#14578933—22no—
44C22⋊Dic157—22no—
45C22⋊Dic157—21no—
46#14578931—22no—
47C4×Dic157—21no—
48C4×Dic157—21no—
49C4×Dic157C2×C411yescharacteristic
50C4×Dic157C2×C411yescharacteristic
51#41×C157—21no—
52#41×C157—21no—
53#41×C157—21no—
54#41×C157—21no—
55C2×Q8×C157C2×C411yescharacteristic
56C2×D4×C157—21no—
57C2×D4×C157C2×C411yescharacteristic
58C22×C4×C157—21no—
59C22×C4×C157C2311yescharacteristic
60M16×C157—21no—
61M16×C157D412yescomplemented
62C2×C8×C157—21no—
63C4⋊C4×C157—21no—
64C22⋊C4×C157—21no—
65C42×C157—21no—
66C22×D157C22⋊C412yes—
67Dic314—21no—
68Dic314—21no—
69D628—21no—
70D628—21no—
71C22⋊D157—22no—
72C22⋊D157—22no—
73C22⋊D157—44no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—42no—
77C2×Dic157—41no—
78C2×Dic157C22⋊C412yes—
79C4×D157—22no—
80C4×D157—22no—
81#11929434—22no—
82C23×C157—21no—
83C23×C157C2×D411yescharacteristic
84Q8×C157—21no—
85Q8×C157—21no—
86D4×C157—21no—
87D4×C157—21no—
88D4×C157—41no—
89D4×C157—41no—
90C2×C4×C157—21no—
91C2×C4×C157—41no—
92C2×C4×C157—41no—
93C2×C4×C157—41no—
94C2×C4×C157C22×C411yescharacteristic
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2×D411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157#4111yescharacteristic
99C2×C4×C157#4111yescharacteristic
100C8×C157—22no—
101C22×C157—41no—
102C22×C157—42no—
103C22×C157—41no—
104C22×C157C2×C22⋊C411yescharacteristic
105C22×C157#12111yescharacteristic
106C22×C157#12111yescharacteristic
107C2×D157—22no—
108C2×D157—42no—
109C4×C157—41no—
110C4×C157—81no—
111C4×C157C4×D411yescharacteristic
112C4×C157C4×D411yescharacteristic
113C4×C157#12011yescharacteristic
114C4×C157#12311yescharacteristic
115Dic157—22no—
116Dic157—22no—
117Dic157—41no—
118Dic157—81no—
119C2×C157—21no—
120C2×C157—21no—
121C2×C157—21no—
122C2×C157—81no—
123C2×C157#38611yescharacteristic
124D157—42no—
125C157#188411yescharacteristic, complemented, Sylow
126#1884—1571nomaximal, Sylow
127#568—1571no—
128#532—1571no—
129#421—1571no—
130#421D15711yescharacteristic, complemented
131#410—1571no—
132#339—1571no—
133#339—1571no—
134C2×#41—1571no—
135C2×#41C2×D15711yescharacteristic
136#131—3142no—
137#131—3142no—
138C2×M16—3142no—
139C2×M16—1571no—
140C2×M16C2×D15711yescharacteristic
141C4⋊Q8—1571no—
142#127—1571no—
143#124—1571no—
144#124—1571no—
145#121—3142no—
146C4×D4—3142no—
147#117—1571no—
148#117C2×D15711yescharacteristic
149C2×C42—1571no—
150#104—22no—
151#104—3142no—
152#99—3142no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—3141no—
158#41—3141no—
159#41—3141no—
160#41—3141no—
161C2×Q8—1571no—
162C2×Q8C4×D15711yescharacteristic
163C2×D4—21no—
164C2×D4—3142no—
165C2×D4—1572no—
166C2×D4—1571no—
167C2×D4C4×D15711yescharacteristic
168C22×C4—21no—
169C22×C4—3141no—
170C22×C4—1572no—
171C22×C4C22×D15711yescharacteristic
172M16—21no—
173M16—3144no—
174M16—3141no—
175M16—1572no—
176M16C22⋊D15712yescomplemented
177C2×C8—21no—
178C2×C8—3144no—
179C2×C8—3141no—
180C4⋊C4—21no—
181C4⋊C4—3142no—
182C4⋊C4—3141no—
183C22⋊C4—21no—
184C22⋊C4—3142no—
185C22⋊C4—3142no—
186C22⋊C4—3141no—
187C42—21no—
188C42—3141no—
189C42—3141no—
190C42—1571no—
191C42—1571no—
192C23—21no—
193C23—1572no—
194C23D4×D15711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—3141no—
198Q8—3141no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—3142no—
204D4—3142no—
205D4—3141no—
206D4—3141no—
207D4—6284no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—3141no—
213C2×C4—3142no—
214C2×C4—3142no—
215C2×C4—3141no—
216C2×C4—6282no—
217C2×C4—6281no—
218C2×C4—1572no—
219C2×C4C2×C4×D15711yescharacteristic
220C2×C4D4×D15711yescharacteristic
221C2×C4D4×D15711yescharacteristic
222C2×C4C2×C22⋊D15711yescharacteristic
223C2×C4#1457895311yescharacteristic
224C2×C4#1457895411yescharacteristic
225C8—22no—
226C8—3142no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—3142no—
231C22—6282no—
232C22C22⋊C4×D15711yescharacteristic
233C22#1909774211yescharacteristic
234C22#1909774411yescharacteristic
235C4—41no—
236C4—81no—
237C4—3142no—
238C4—3142no—
239C4—6281no—
240C4—12561no—
241C4#1909773111yescharacteristic
242C4C4×C22⋊D15711yescharacteristic
243C4#1909773911yescharacteristic
244C4#1909775311yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—81no—
249C2—6282no—
250C2#2375714911yescharacteristic
subgroupG/H
centerC4#19097739
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#421×C157C2
socleC2×C157#386
prime radical#23757819C2
cosocleC22×C4×C157C23
hypercenter#421D157
hypocenterC157#1884
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. #23758039C2semidirect
3. #23757290C2semidirect
4. #23757246C2semidirect
5. #23757034C2semidirect
6. #23757032C2semidirect
7. #421×C157C2semidirect
13. #19097769C4semidirect
14. #19097767C4semidirect
15. #19097760C4semidirect
16. #19097758C4semidirect
38. #14578938D4semidirect
61. M16×C157D4semidirect
125. C157#1884semidirect
130. #421D157semidirect
176. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237578192
#237572902
#237572462
#237570342
#237570322
#421×C1572
#1884157