Finite Group Databasebeta

← #34100637

Subgroups — #34100637

Order 20096.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#23758071C211yesmaximal, characteristic
2#23757290C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic, complemented
4#23757246C211yesmaximal, characteristic, complemented
5#23757043C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7C4×#41×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097802—21no—
10#19097801C412yescomplemented
11#19097797—21no—
12#19097796C412yescomplemented
13#19097791—22no—
14C2×#14578938C2211yescharacteristic
15C2×#14578938C2211yescharacteristic
16#19097730C2211yescharacteristic
17#19097699—21no—
18#19097699—21no—
19#19097698—22no—
20#19097685—22no—
21C2×#41×C157C2211yescharacteristic
22C4×Q8×C157—21no—
23C4×D4×C157—21no—
24C4×D4×C157—22no—
25#117×C157—21no—
26#117×C157C2211yescharacteristic
27C2×C42×C157—21no—
28C2×C42×C157C2211yescharacteristic
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33C2×Dic314C2×C411yescharacteristic, complemented
34C2×D628C2×C411yescharacteristic, complemented
35C2×C22⋊D157—21no—
36C2×C4×D157—21no—
37#14578944—22no—
38#14578942—22no—
39#14578941—22no—
40#14578939—22no—
41#14578938—21no—
42#14578938—21no—
43#14578938—21no—
44#14578938C2×C412yescomplemented
45C2×#11929434—21no—
46C2×#11929434—21no—
47C4⋊Dic157—21no—
48C22⋊Dic157—21no—
49C4×Dic157—21no—
50#41×C157—21no—
51#41×C157—21no—
52#41×C157C2×C412yes—
53#41×C157D412yes—
54C2×Q8×C157D411yescharacteristic
55C2×D4×C157—21no—
56C2×D4×C157D411yescharacteristic
57C22×C4×C157—21no—
58C22×C4×C157—22no—
59C22×C4×C157—21no—
60C22×C4×C157C2311yescharacteristic
61C4⋊C4×C157—22no—
62C4⋊C4×C157—21no—
63C22⋊C4×C157—22no—
64C22⋊C4×C157D412yes—
65C42×C157—22no—
66C42×C157—21no—
67C42×C157—21no—
68C42×C157D411yescharacteristic
69C42×C157D411yescharacteristic
70C22×D157—21no—
71Dic314—21no—
72Dic314—21no—
73D628—21no—
74D628—21no—
75C22⋊D157—41no—
76C22⋊D157—41no—
77C2×Dic157—21no—
78C2×Dic157—41no—
79C4×D157—41no—
80C4×D157—41no—
81#11929434—22no—
82#11929434—41no—
83C23×C157—21no—
84C23×C157C2×D411yescharacteristic
85Q8×C157—21no—
86Q8×C157C22⋊C412yes—
87D4×C157—22no—
88D4×C157—22no—
89D4×C157—21no—
90D4×C157C22⋊C412yes—
91C2×C4×C157—21no—
92C2×C4×C157—21no—
93C2×C4×C157—22no—
94C2×C4×C157—22no—
95C2×C4×C157—22no—
96C2×C4×C157—21no—
97C2×C4×C157—42no—
98C2×C4×C157—44no—
99C2×C4×C157C22×C411yescharacteristic
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157#4111yescharacteristic
104C2×C4×C157#4111yescharacteristic
105C22×C157—22no—
106C22×C157—41no—
107C22×C157—42no—
108C22×C157C4×D411yescharacteristic
109C22×C157#12011yescharacteristic
110C22×C157#12111yescharacteristic
111C2×D157—41no—
112C2×D157—81no—
113C4×C157—22no—
114C4×C157—22no—
115C4×C157—42no—
116C4×C157—41no—
117C4×C157C2×C22⋊C411yescharacteristic
118C4×C157C4×D411yescharacteristic
119C4×C157#12111yescharacteristic
120C4×C157#12311yescharacteristic
121Dic157—41no—
122Dic157—81no—
123C2×C157—21no—
124C2×C157—21no—
125C2×C157—21no—
126C2×C157—42no—
127C2×C157#38611yescharacteristic
128D157—81no—
129C157#190611yescharacteristic, complemented, Sylow
130#1906—1571nomaximal, Sylow
131#575—1571no—
132C4×#41D15711yescharacteristic, complemented
133#421—1571no—
134C2×#104—1571no—
135#410—1571no—
136#344—1571no—
137#339—1571no—
138C2×#41—1571no—
139C2×#41C2×D15711yescharacteristic
140#137—3141no—
141#137—1572no—
142AGL(1,ℤ/8)—3141no—
143AGL(1,ℤ/8)—1572no—
144#135—3142no—
145C2×M16—1571no—
146C2×M16—1571no—
147C4×Q8—21no—
148C4×D4—21no—
149C4×D4—22no—
150#117—21no—
151#117—1571no—
152#117C2×D15711yescharacteristic
153C2×C42—21no—
154C2×C42C2×D15711yescharacteristic
155#104—3142no—
156#104—3141no—
157#104—3141no—
158#99—3142no—
159#41—21no—
160#41—21no—
161#41—3141no—
162#41—3141no—
163#41—3141no—
164#41—3141no—
165#41C4×D15712yes—
166#41D62812yes—
167C2×Q8—1571no—
168C2×Q8C22⋊D15711yescharacteristic
169C2×D4—21no—
170C2×D4—3141no—
171C2×D4—1571no—
172C2×D4C22⋊D15711yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—3141no—
177C22×C4C22×D15711yescharacteristic
178Q16—3142no—
179SD16—3142no—
180SD16—3142no—
181D8—3142no—
182M16—3141no—
183M16—3141no—
184M16—3141no—
185M16—1572no—
186C2×C8—3141no—
187C2×C8—3141no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—3141no—
191C22⋊C4—22no—
192C22⋊C4—3141no—
193C22⋊C4C22⋊D15712yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—3141no—
198C42C22⋊D15711yescharacteristic
199C42C22⋊D15711yescharacteristic
200C23—21no—
201C23—3141no—
202C23C2×C22⋊D15711yescharacteristic
203Q8—21no—
204Q8—3141no—
205Q8—3141no—
206Q8#1457893112yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—3141no—
211D4—3141no—
212D4—6281no—
213D4—6281no—
214D4#1457893112yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—3141no—
224C2×C4—6281no—
225C2×C4—6281no—
226C2×C4—6281no—
227C2×C4C2×C4×D15711yescharacteristic
228C2×C4C2×C22⋊D15711yescharacteristic
229C2×C4C2×C22⋊D15711yescharacteristic
230C2×C4C2×D62811yescharacteristic
231C2×C4#1457895411yescharacteristic
232C2×C4#1457895411yescharacteristic
233C8—3142no—
234C8—6281no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—6281no—
239C22—12561no—
240C22C4×C22⋊D15711yescharacteristic
241C22#1909774011yescharacteristic
242C22#1909774511yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—6281no—
248C4—12561no—
249C4C2×#1457893111yescharacteristic
250C4C4×C22⋊D15711yescharacteristic
251C4#1909774511yescharacteristic
252C4#1909775411yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—12561no—
258C2#2375715211yescharacteristic
subgroupG/H
centerC4#19097745
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC4×#41×C157C2
socleC2×C157#386
prime radical#23758071C2
cosocleC22×C4×C157C23
hypercenterC4×#41D157
hypocenterC157#1906
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #23757290C2semidirect
3. C2×#19097699C2semidirect
4. #23757246C2semidirect
5. #23757043C2semidirect
6. #23757033C2semidirect
7. C4×#41×C157C2semidirect
10. #19097801C4semidirect
12. #19097796C4semidirect
33. C2×Dic314C2×C4semidirect
34. C2×D628C2×C4semidirect
44. #14578938C2×C4semidirect
129. C157#1906semidirect
132. C4×#41D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237572902
C2×#190976992
#237572462
#237570432
#237570332
C4×#41×C1572
#1906157