Finite Group Databasebeta

← #34109957

Subgroups — #34109957

Order 20096.

384 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19097762C211yesmaximal, characteristic, complemented
2#23757741C211yesmaximal, characteristic
3#23757583C212yesmaximal, complemented
4#23757579C212yesmaximal, complemented
5#23757576C212yesmaximal, complemented
6#23757572C212yesmaximal, complemented
7#23757316C211yesmaximal, characteristic, complemented
8#23757296C211yesmaximal, characteristic, complemented
9#23757275C211yesmaximal, characteristic
10#23757272C211yesmaximal, characteristic
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14#19097762—21no—
15#19097762—21no—
16#19097762C2212yes—
17#19097762C2212yes—
18#19097736C2211yescharacteristic
19#19097736C2211yescharacteristic
20#19097733C2211yescharacteristic
21#19097733C2211yescharacteristic
22#19097731—21no—
23#19097731—21no—
24#19097728—21no—
25C2×C4⋊Dic157—21no—
26C2×C4⋊Dic157C2211yescharacteristic
27C2×#14578933—21no—
28C2×C4×Dic157C2211yescharacteristic
29C42×D157—21no—
30#19097704C2212yes—
31#19097702C2212yes—
32#19097700C2212yes—
33#19097700C2212yes—
34#19097696C2212yes—
35#19097694C2212yes—
36#19097693C2212yes—
37#19097691C2212yes—
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yes—
42#106×C157C2212yes—
43#14578954—22no—
44#14578954Q814yes—
45C2×Dic314C2311yescharacteristic
46C2×D628C2311yescharacteristic
47C2×C22⋊D157—21no—
48C22×Dic157—21no—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51C2×C4×D157—21no—
52C2×C4×D157—21no—
53C2×C4×D157—21no—
54#14578938—22no—
55C2×#11929434C2312yes—
56C4⋊Dic157—22no—
57C4⋊Dic157—22no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2312yes—
60#14578933—22no—
61#14578933—22no—
62#14578933—21no—
63#14578933—21no—
64#14578931—21no—
65#14578931—21no—
66C4×Dic157—21no—
67C4×Dic157—21no—
68C4×Dic157D411yescharacteristic
69C4×Dic157D411yescharacteristic
70C4×Dic157C2311yescharacteristic
71C4×Dic157C2311yescharacteristic
72C22×C4×C157—21no—
73C22×C4×C157C2311yescharacteristic
74M16×C157—22no—
75C2×C8×C157C2312yes—
76C4⋊C4×C157—21no—
77C4⋊C4×C157C2311yescharacteristic
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C42×C157—21no—
82C22×D157—21no—
83Dic314—21no—
84Dic314C2×Q812yes—
85D628—21no—
86D628C2×Q812yes—
87C22⋊D157—24no—
88C2×Dic157—21no—
89C2×Dic157—22no—
90C2×Dic157—41no—
91C2×Dic157—41no—
92C2×Dic157—42no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C4×D157—24no—
98C4×D157—42no—
99C4×D157—42no—
100#11929434—22no—
101C23×C157C2×D411yescharacteristic
102C2×C4×C157—21no—
103C2×C4×C157—21no—
104C2×C4×C157—22no—
105C2×C4×C157—42no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×Q812yes—
108C2×C4×C157#4112yes—
109C2×C4×C157C2411yescharacteristic
110C8×C157—22no—
111C22×C157—21no—
112C22×C157—21no—
113C22×C157C22⋊Q812yes—
114C22×C157C22×D411yescharacteristic
115C2×D157—22no—
116C2×D157—41no—
117C2×D157—41no—
118C4×C157—42no—
119C4×C157—41no—
120C4×C157—41no—
121C4×C157C22⋊Q812yes—
122C4×C157C22×Q811yescharacteristic
123C4×C157C2×#4111yescharacteristic
124Dic157—22no—
125Dic157—21no—
126Dic157—21no—
127Dic157—21no—
128Dic157—21no—
129Dic157—42no—
130C2×C157—22no—
131C2×C157C2×C22⋊Q811yescharacteristic
132C2×C157#57511yescharacteristic
133C2×C157#57511yescharacteristic
134D157—42no—
135C157#308011yescharacteristic, complemented, Sylow
136#3080—1571nomaximal, Sylow
137C2×#125—1571no—
138C4×#41—1571no—
139#479—1572no—
140#479—1572no—
141#478—1572no—
142#478—1572no—
143#428—1571no—
144#428D15711yescharacteristic, complemented
145#423—1571no—
146#417—1571no—
147#417—1571no—
148C2×#41—1571no—
149C2×M16—1571no—
150C2×M16C2×D15711yescharacteristic
151#125—3141no—
152#125—3141no—
153#125—1572no—
154#125—1572no—
155C4×Q8—1571no—
156C4×Q8—1571no—
157C4×D4—3141no—
158C4×D4—3141no—
159C4×D4—1571no—
160C4×D4—1571no—
161#117—3141no—
162#117C2×D15711yescharacteristic
163C2×C4⋊C4—3141no—
164C2×C4⋊C4—3141no—
165C2×C4⋊C4—1571no—
166C2×C4⋊C4C2×D15711yescharacteristic
167C2×C42—3141no—
168C2×C42—1571no—
169#107—1572no—
170#107C2×D15712yes—
171#106—1572no—
172#106C2×D15712yes—
173C4⋊C8—1572no—
174C4⋊C8—1572no—
175#103—1572no—
176#103—1572no—
177#102—1572no—
178#102—1572no—
179#41—3142no—
180#41—1574no—
181C2×Q8—1571no—
182C2×D4—3141no—
183C2×D4—1571no—
184C22×C4—21no—
185C22×C4—3141no—
186C22×C4—3141no—
187C22×C4—3141no—
188C22×C4—3141no—
189C22×C4—1571no—
190C22×C4—1571no—
191C22×C4C22×D15711yescharacteristic
192M16—22no—
193M16—3142no—
194C2×C8—1572no—
195C2×C8C22×D15712yes—
196C4⋊C4—21no—
197C4⋊C4—3142no—
198C4⋊C4—3142no—
199C4⋊C4—3142no—
200C4⋊C4—3142no—
201C4⋊C4—3141no—
202C4⋊C4—3141no—
203C4⋊C4—3141no—
204C4⋊C4—1572no—
205C4⋊C4C22×D15712yes—
206C4⋊C4C22×D15711yescharacteristic
207C4⋊C4C22×D15711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—3141no—
210C22⋊C4—3141no—
211C42—21no—
212C42—3141no—
213C42—3141no—
214C42—1571no—
215C42—1571no—
216C42—1571no—
217C42—1571no—
218C23—3141no—
219C23D4×D15711yescharacteristic
220Q8—3141no—
221Q8—1572no—
222D4—3144no—
223D4—3141no—
224D4—1572no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—3141no—
230C2×C4—3144no—
231C2×C4—3142no—
232C2×C4—6282no—
233C2×C4—6282no—
234C2×C4—6281no—
235C2×C4—6281no—
236C2×C4—6282no—
237C2×C4—1571no—
238C2×C4—1571no—
239C2×C4—1571no—
240C2×C4—1571no—
241C2×C4D4×D15711yescharacteristic
242C2×C4Q8×D15712yes—
243C2×C4#1457895412yes—
244C2×C4C23×D15711yescharacteristic
245C8—22no—
246C8—3142no—
247C22—21no—
248C22—21no—
249C22—3142no—
250C22—6281no—
251C22—6281no—
252C22#1909774912yes—
253C22C2×D4×D15711yescharacteristic
254C4—42no—
255C4—41no—
256C4—41no—
257C4—3142no—
258C4—3141no—
259C4—3141no—
260C4—3141no—
261C4—3141no—
262C4—6282no—
263C4#1909774912yes—
264C4C2×Q8×D15711yescharacteristic
265C4C2×#1457895411yescharacteristic
266C2—22no—
267C2—6282no—
268C2C2×#1909774911yescharacteristic
269C2#2375806911yescharacteristic
270C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578954C22
hypercenter#428D157
hypocenterC157#3080
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19097762C2semidirect
3. #23757583C2semidirect
4. #23757579C2semidirect
5. #23757576C2semidirect
6. #23757572C2semidirect
7. #23757316C2semidirect
8. #23757296C2semidirect
11. #428×C157C2semidirect
135. C157#3080semidirect
144. #428D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977622
#237577412
#237575832
#237575792
#237575762
#237575722
#237573162
#237572962
#237572752
#237572722
#428×C1572
#3080157