Finite Group Databasebeta

← #34101168

Subgroups — #34101168

Order 20096.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#19097804C211yesmaximal, characteristic, complemented
2#23757842C211yesmaximal, characteristic
3C2×#19097759C211yesmaximal, characteristic, complemented
4C2×#19097699C211yesmaximal, characteristic, complemented
5#23757281C211yesmaximal, characteristic, complemented
6C2×#19097687C211yesmaximal, characteristic, complemented
7#339×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314C2211yescharacteristic
10C22×D628C2211yescharacteristic
11#19097804—22no—
12#19097804—22no—
13C2×Dic628—21no—
14C2×#14578943—21no—
15C2×#14578938C2211yescharacteristic
16#19097768—21no—
17#19097759—22no—
18#19097759—21no—
19#19097759—22no—
20#19097746—22no—
21#19097741—22no—
22C2×#14578931—22no—
23#19097699—22no—
24#19097694—22no—
25#19097687—22no—
26C2×M16×C157C2211yescharacteristic
27#117×C157C2211yescharacteristic
28C2×C42×C157C2211yescharacteristic
29C23×D157D412yes—
30#14578954—21no—
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34C2×Dic314—21no—
35C2×Dic314—21no—
36C2×Dic314—21no—
37C2×Dic314—21no—
38C2×Dic314—21no—
39C2×Dic314D411yescharacteristic
40C2×Dic314D412yescomplemented
41C2×D628—24no—
42C2×D628—21no—
43C2×D628—21no—
44C2×D628—21no—
45C2×D628—21no—
46C2×D628—21no—
47C2×D628—21no—
48C2×D628D411yescharacteristic
49C2×C22⋊D157—21no—
50C22×Dic157—21no—
51C2×C4×D157—21no—
52Dic628—24no—
53#14578943—24no—
54#14578938—21no—
55#14578938—21no—
56C2×#11929434—21no—
57#14578931—28no—
58#14578931—24no—
59C22×C4×C157—21no—
60C22×C4×C157C2311yescharacteristic
61M16×C157—21no—
62M16×C157D412yescomplemented
63C2×C8×C157—21no—
64C4⋊C4×C157D412yes—
65C22⋊C4×C157—21no—
66C42×C157—21no—
67C42×C157—21no—
68C42×C157D412yescomplemented
69C22×D157—21no—
70C22×D157—22no—
71C22×D157—22no—
72C22×D157—22no—
73C22×D157—44no—
74Dic314—21no—
75Dic314—22no—
76Dic314—22no—
77Dic314—21no—
78Dic314—41no—
79Dic314—41no—
80D628—21no—
81D628—21no—
82D628—41no—
83D628—41no—
84D628—41no—
85D628—41no—
86D628—84no—
87C22⋊D157—41no—
88C22⋊D157—41no—
89C2×Dic157—21no—
90C2×Dic157—22no—
91C2×Dic157—41no—
92C2×Dic157—41no—
93C4×D157—41no—
94C4×D157—41no—
95#11929434—41no—
96C23×C157C2×D411yescharacteristic
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—42no—
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157#4111yescharacteristic
106C8×C157—22no—
107C22×C157—22no—
108C22×C157#12011yescharacteristic
109C22×C157#12011yescharacteristic
110C22×C157#12111yescharacteristic
111C2×D157—44no—
112C2×D157—41no—
113C2×D157—42no—
114C2×D157—42no—
115C2×D157—44no—
116C2×D157—82no—
117C4×C157—42no—
118C4×C157—42no—
119C4×C157#12011yescharacteristic
120C4×C157#12111yescharacteristic
121C4×C157#12111yescharacteristic
122C4×C157#12711yescharacteristic
123Dic157—41no—
124Dic157—42no—
125C2×C157—21no—
126C2×C157—21no—
127C2×C157#39211yescharacteristic
128C2×C157#45512yes—
129D157—81no—
130D157—82no—
131C157#200411yescharacteristic, complemented, Sylow
132#2004—1571nomaximal, Sylow
133C2×#137—1571no—
134#535—1571no—
135C2×#124—1571no—
136C2×#104—1571no—
137#419—1571no—
138C2×#100—1571no—
139#339D15711yescharacteristic, complemented
140C2×#41—1571no—
141C22×Q8—1571no—
142C22×D4—1571no—
143#137—3142no—
144#137—3142no—
145C2×Q16—3141no—
146C2×SD16—3141no—
147C2×M16—1571no—
148C2×M16C2×D15711yescharacteristic
149#127—3141no—
150#124—3142no—
151#124—3141no—
152#124—3142no—
153#121—3142no—
154#120—3142no—
155#117C2×D15711yescharacteristic
156C2×C22⋊C4—3142no—
157C2×C42C2×D15711yescharacteristic
158#104—3142no—
159#103—3142no—
160#100—3142no—
161C24—1572no—
162#41—3141no—
163#41—3141no—
164#41—3141no—
165#41—3141no—
166C2×Q8—3141no—
167C2×Q8—3141no—
168C2×Q8—3141no—
169C2×Q8—3141no—
170C2×Q8—3141no—
171C2×Q8—1571no—
172C2×Q8—1572no—
173C2×D4—3144no—
174C2×D4—3141no—
175C2×D4—3141no—
176C2×D4—3141no—
177C2×D4—3141no—
178C2×D4—3141no—
179C2×D4—3141no—
180C2×D4—3141no—
181C2×D4—1571no—
182C22×C4—21no—
183C22×C4—3141no—
184C22×C4—3141no—
185C22×C4C22×D15711yescharacteristic
186Q16—3144no—
187SD16—3144no—
188M16—21no—
189M16—3141no—
190M16—3141no—
191M16D62812yescomplemented
192C2×C8—21no—
193C2×C8—3141no—
194C4⋊C4D62812yes—
195C22⋊C4—21no—
196C22⋊C4—3148no—
197C22⋊C4—3144no—
198C42—21no—
199C42—21no—
200C42D62812yescomplemented
201C23—3142no—
202C23—3142no—
203C23—3141no—
204C23—3142no—
205C23—6284no—
206C23D4×D15711yescharacteristic
207Q8—3142no—
208Q8—3141no—
209Q8—3142no—
210Q8—3141no—
211Q8—6281no—
212Q8—6281no—
213D4—3141no—
214D4—3141no—
215D4—6281no—
216D4—6281no—
217D4—6281no—
218D4—6281no—
219D4—6281no—
220D4—6281no—
221D4—12564no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—3142no—
226C2×C4—3141no—
227C2×C4—6281no—
228C2×C4—6281no—
229C2×C4—6281no—
230C2×C4—6281no—
231C2×C4D4×D15711yescharacteristic
232C2×C4D4×D15711yescharacteristic
233C2×C4C2×D62811yescharacteristic
234C2×C4C2×D62811yescharacteristic
235C2×C4C2×D62811yescharacteristic
236C2×C4#1457895511yescharacteristic
237C8—22no—
238C8—6281no—
239C22—22no—
240C22—6284no—
241C22—6281no—
242C22—6282no—
243C22—6282no—
244C22—6284no—
245C22—12562no—
246C22#1909774111yescharacteristic
247C22#1909774111yescharacteristic
248C22#1909774611yescharacteristic
249C4—42no—
250C4—42no—
251C4—6281no—
252C4—6282no—
253C4#1909774111yescharacteristic
254C4#1909774611yescharacteristic
255C4#1909774611yescharacteristic
256C4#1909776811yescharacteristic
257C2—21no—
258C2—21no—
259C2—12562no—
260C2—12561no—
261C2#2375718311yescharacteristic
262C2#2375745212yes—
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC22×C157#120
prime radical#23757842C2
hypercenter#339D157
hypocenterC157#2004
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#19097804C2semidirect
3. C2×#19097759C2semidirect
4. C2×#19097699C2semidirect
5. #23757281C2semidirect
6. C2×#19097687C2semidirect
7. #339×C157C2semidirect
40. C2×Dic314D4semidirect
62. M16×C157D4semidirect
68. C42×C157D4semidirect
131. C157#2004semidirect
139. #339D157semidirect
191. M16D628semidirect
200. C42D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
#237578422
C2×#190977592
C2×#190976992
#237572812
C2×#190976872
#339×C1572
#2004157