Finite Group Databasebeta

← #34110307

Subgroups — #34110307

Order 20096.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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#23757807C211yesmaximal, characteristic
2#23757608C211yesmaximal, characteristic, complemented
3#23757591C211yesmaximal, characteristic
4#23757580C212yesmaximal, complemented
5#23757495C212yesmaximal, complemented
6#23757429C211yesmaximal, characteristic, complemented
7#23757412C211yesmaximal, characteristic
8#23757378C212yesmaximal, complemented
9#23757371C212yesmaximal, complemented
10#23757330C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C157C211yesmaximal, characteristic, complemented
12C2×#14578944C2212yes—
13C2×#14578941C2212yes—
14#19097765—21no—
15#19097763C2211yescharacteristic
16#19097759C2211yescharacteristic
17#19097754—21no—
18#19097750C2211yescharacteristic
19#19097745C2211yescharacteristic
20C4×Dic314C2211yescharacteristic
21C4×D628C2211yescharacteristic
22C4×C22⋊D157—21no—
23#19097728—21no—
24#19097704C2212yes—
25#19097702C2212yes—
26C4⋊#11929434C2212yes—
27#19097695C2212yes—
28#19097695C2212yes—
29#19097694C2212yes—
30#19097691C2212yes—
31C22⋊#11929434C2211yescharacteristic
32C22⋊#11929434C2211yescharacteristic
33#19097682C2212yes—
34C22×Q8×C157C2211yescharacteristic
35C4×Q8×C157—21no—
36C4×Q8×C157—21no—
37C4×Q8×C157C2212yes—
38C4×Q8×C157C2212yes—
39C2×C4⋊C4×C157—21no—
40C2×C4⋊C4×C157C2211yescharacteristic
41C2×C42×C157—21no—
42C2×C42×C157C2211yescharacteristic
43C2×Dic314C2311yescharacteristic
44C2×D628C2311yescharacteristic
45C2×C22⋊D157—21no—
46C2×C4×D157—21no—
47#14578944—24no—
48#14578941—24no—
49C2×#11929434C2312yes—
50C2×#11929434C2312yes—
51C4⋊Dic157C2311yescharacteristic
52C4⋊Dic157C2311yescharacteristic
53#14578933—21no—
54#14578933—21no—
55#14578933—21no—
56#14578933—21no—
57C22⋊Dic157—21no—
58#14578931—21no—
59#14578931—21no—
60#14578931—21no—
61#14578931—21no—
62C4×Dic157—21no—
63C2×Q8×C157—24no—
64C2×Q8×C157—21no—
65C2×Q8×C157C2312yes—
66C22×C4×C157—21no—
67C22×C4×C157—21no—
68C22×C4×C157D411yescharacteristic
69C22×C4×C157D411yescharacteristic
70C22×C4×C157C2311yescharacteristic
71C4⋊C4×C157—22no—
72C4⋊C4×C157—22no—
73C4⋊C4×C157—21no—
74C4⋊C4×C157C2312yes—
75C42×C157—22no—
76C42×C157—22no—
77C42×C157D411yescharacteristic
78C42×C157D411yescharacteristic
79C42×C157C2311yescharacteristic
80C42×C157C2311yescharacteristic
81C22×D157—21no—
82Dic314—22no—
83D628—22no—
84C22⋊D157—42no—
85C2×Dic157—21no—
86C2×Dic157—21no—
87C2×Dic157—21no—
88C4×D157—42no—
89#11929434—22no—
90#11929434—22no—
91C23×C157C2×D411yescharacteristic
92Q8×C157—24no—
93Q8×C157—22no—
94Q8×C157#4114yes—
95C2×C4×C157—24no—
96C2×C4×C157—22no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—41no—
101C2×C4×C157—42no—
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2411yescharacteristic
108C22×C157—21no—
109C22×C157—21no—
110C22×C157#13712yes—
111C22×C157C22×D411yescharacteristic
112C2×D157—41no—
113C2×D157—41no—
114C2×D157—41no—
115C4×C157—24no—
116C4×C157—21no—
117C4×C157—21no—
118C4×C157—21no—
119C4×C157—21no—
120C4×C157—21no—
121C4×C157—42no—
122C4×C157C2×#4112yes—
123Dic157—41no—
124Dic157—41no—
125Dic157—41no—
126C2×C157—22no—
127C2×C157#52511yescharacteristic
128C2×C157C2×#13711yescharacteristic
129C2×C157#57511yescharacteristic
130D157—81no—
131C157#311711yescharacteristic, complemented, Sylow
132#3117—1571nomaximal, Sylow
133#529—1571no—
134C2×C4×Q8D15711yescharacteristic, complemented
135#484—1571no—
136#481—1571no—
137#479—1572no—
138#464—1572no—
139#451—1571no—
140#448—1571no—
141#441—1572no—
142#440—1572no—
143#432—1571no—
144C22×Q8C2×D15711yescharacteristic
145C2×Q16—1572no—
146C2×SD16—1572no—
147#126—3141no—
148#125—1571no—
149#124—1571no—
150#123—3141no—
151C22⋊Q8—1571no—
152#121—1571no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—1571no—
156C4×Q8C2×D15712yes—
157C4×Q8C2×D15712yes—
158C4×D4—3141no—
159C4×D4—1571no—
160#117—3141no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D15711yescharacteristic
163C2×C42—21no—
164C2×C42C2×D15711yescharacteristic
165#107—1572no—
166#106—1572no—
167C4⋊C8—1572no—
168#103—1572no—
169#103—1572no—
170#103—1572no—
171#102—1572no—
172C22⋊C8—1571no—
173C22⋊C8—1571no—
174#97—1572no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1571no—
178C2×Q8C22×D15712yes—
179C2×D4—3141no—
180C2×D4—1571no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—3141no—
184C22×C4C22⋊D15711yescharacteristic
185C22×C4C22⋊D15711yescharacteristic
186C22×C4C22×D15711yescharacteristic
187Q16—3144no—
188SD16—3144no—
189C2×C8—1572no—
190C2×C8—1572no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—3141no—
195C4⋊C4—3141no—
196C4⋊C4—3141no—
197C4⋊C4—3141no—
198C4⋊C4—1571no—
199C4⋊C4—1571no—
200C4⋊C4C22×D15712yes—
201C22⋊C4—3141no—
202C22⋊C4—3141no—
203C22⋊C4—3141no—
204C22⋊C4—3141no—
205C22⋊C4—3141no—
206C42—22no—
207C42—22no—
208C42—3141no—
209C42C22⋊D15711yescharacteristic
210C42C22⋊D15711yescharacteristic
211C42C22×D15711yescharacteristic
212C42C22×D15711yescharacteristic
213C23—3141no—
214C23C2×C22⋊D15711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—3142no—
218Q8#1457895414yes—
219D4—3142no—
220D4—6282no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—3141no—
229C2×C4—3141no—
230C2×C4—3141no—
231C2×C4—6282no—
232C2×C4C2×C22⋊D15711yescharacteristic
233C2×C4C2×C22⋊D15711yescharacteristic
234C2×C4C2×C22⋊D15711yescharacteristic
235C2×C4C2×C22⋊D15711yescharacteristic
236C2×C4C2×C22⋊D15711yescharacteristic
237C2×C4C23×D15711yescharacteristic
238C8—3142no—
239C8—3142no—
240C22—21no—
241C22—21no—
242C22—6281no—
243C22—6281no—
244C22—6281no—
245C22#1909780212yes—
246C22C22×C22⋊D15711yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—6281no—
255C4—6281no—
256C4—6281no—
257C4C2×#1457895412yes—
258C2—22no—
259C2—12561no—
260C2#2375778411yescharacteristic
261C2C2×#1909780211yescharacteristic
262C2#2375807111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×C4×Q8×C157C2
socleC22×C157C22×D4
prime radical#19097745C22
hypercenterC2×C4×Q8D157
hypocenterC157#3117
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #23757608C2semidirect
4. #23757580C2semidirect
5. #23757495C2semidirect
6. #23757429C2semidirect
8. #23757378C2semidirect
9. #23757371C2semidirect
10. #23757330C2semidirect
11. C2×C4×Q8×C157C2semidirect
131. C157#3117semidirect
134. C2×C4×Q8D157semidirect

Maximal subgroups

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

subgroupindex
#237578072
#237576082
#237575912
#237575802
#237574952
#237574292
#237574122
#237573782
#237573712
#237573302
C2×C4×Q8×C1572
#3117157