Finite Group Databasebeta

← #34110736

Subgroups — #34110736

Order 20096.

364 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757679C211yesmaximal, characteristic, complemented
3#23757665C211yesmaximal, characteristic, complemented
4#23757641C211yesmaximal, characteristic, complemented
5#23757635C211yesmaximal, characteristic, complemented
6#23757529C212yesmaximal, complemented
7#23757521C212yesmaximal
8C4×#14578944C212yesmaximal, complemented
9C4×#14578941C212yesmaximal, complemented
10C4×#14578938C211yesmaximal, characteristic, complemented
11#533×C157C211yesmaximal, characteristic, complemented
12C2×#14578944C2212yes—
13C2×#14578941C2212yes—
14C2×#14578938C2212yes—
15#19097765—21no—
16#19097763C2211yescharacteristic
17#19097759C2211yescharacteristic
18#19097754—21no—
19#19097750C2211yescharacteristic
20#19097745C2211yescharacteristic
21C4×Dic314C2211yescharacteristic
22C4×D628C2211yescharacteristic
23C4×C22⋊D157—21no—
24#19097728—21no—
25#19097704C2212yes—
26#19097702C2212yes—
27#19097695C2212yes—
28#19097695C2212yes—
29#19097694C2212yes—
30#19097691C2212yes—
31#19097682C2211yescharacteristic
32#19097682C2211yescharacteristic
33C4×#11929434C2212yes—
34C4⋊Q8×C157C2211yescharacteristic
35C4⋊Q8×C157C2211yescharacteristic
36#125×C157—21no—
37C22⋊Q8×C157—21no—
38C22⋊Q8×C157C2212yes—
39C4×Q8×C157—21no—
40C4×Q8×C157C2212yes—
41#117×C157—21no—
42C2×C42×C157C2211yescharacteristic
43C2×Dic314C2311yescharacteristic
44C2×D628C2311yescharacteristic
45C2×C22⋊D157—21no—
46C2×C4×D157—21no—
47#14578944—24no—
48#14578941—24no—
49#14578938—24no—
50C2×#11929434C2312yes—
51C2×#11929434C2312yes—
52C4⋊Dic157C2311yescharacteristic
53C4⋊Dic157C2311yescharacteristic
54#14578933—21no—
55#14578933—21no—
56#14578933—21no—
57#14578933—21no—
58C22⋊Dic157—21no—
59#14578931—21no—
60#14578931—21no—
61#14578931—21no—
62#14578931—21no—
63C4×Dic157—21no—
64C2×Q8×C157—21no—
65C2×Q8×C157C2312yes—
66C22×C4×C157D411yescharacteristic
67C22×C4×C157D411yescharacteristic
68C22×C4×C157C2311yescharacteristic
69C4⋊C4×C157—21no—
70C4⋊C4×C157—21no—
71C4⋊C4×C157—21no—
72C4⋊C4×C157—22no—
73C4⋊C4×C157—22no—
74C4⋊C4×C157C2312yes—
75C22⋊C4×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#4114yes—
91C23×C157C2×D411yescharacteristic
92Q8×C157—22no—
93Q8×C157—22no—
94Q8×C157—42no—
95C2×C4×C157—22no—
96C2×C4×C157—22no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—41no—
100C2×C4×C157C2×D411yescharacteristic
101C2×C4×C157C2×D411yescharacteristic
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157C2411yescharacteristic
106C22×C157—21no—
107C22×C157—21no—
108C22×C157—21no—
109C22×C157C22×D411yescharacteristic
110C2×D157—41no—
111C2×D157—41no—
112C2×D157—41no—
113C4×C157—21no—
114C4×C157—21no—
115C4×C157—21no—
116C4×C157—21no—
117C4×C157—42no—
118C4×C157—42no—
119C4×C157#13712yes—
120C4×C157C2×#4112yes—
121Dic157—41no—
122Dic157—41no—
123Dic157—41no—
124C2×C157—41no—
125C2×C157#53211yescharacteristic
126C2×C157C2×#13711yescharacteristic
127C2×C157#57511yescharacteristic
128D157—81no—
129C157#317211yescharacteristic, complemented, Sylow
130#3172—1571nomaximal, Sylow
131#533D15711yescharacteristic, complemented
132#529—1571no—
133#501—1571no—
134#497—1571no—
135#491—1571no—
136#489—1571no—
137#470—1572no—
138#468—1572no—
139C4×Q16—1572no—
140C4×SD16—1572no—
141C4×M16—1571no—
142C2×Q16—1572no—
143C2×SD16—1572no—
144C2×M16—1572no—
145C4⋊Q8C2×D15711yescharacteristic
146C4⋊Q8C2×D15711yescharacteristic
147#126—3141no—
148#125—21no—
149#125—1571no—
150#124—1571no—
151#123—3141no—
152C22⋊Q8—21no—
153C22⋊Q8—1571no—
154C22⋊Q8C2×D15712yes—
155#121—1571no—
156C4×Q8—21no—
157C4×Q8—1571no—
158C4×Q8C2×D15712yes—
159C4×D4—3141no—
160C4×D4—1571no—
161#117—21no—
162#117—3141no—
163C2×C42C2×D15711yescharacteristic
164#107—1572no—
165#106—1572no—
166#103—1572no—
167#103—1572no—
168#103—1572no—
169#102—1572no—
170#97—1571no—
171#97—1571no—
172C4×C8—1572no—
173C2×Q8—21no—
174C2×Q8—1571no—
175C2×Q8C22×D15712yes—
176C2×D4—3141no—
177C2×D4—1571no—
178C22×C4—3141no—
179C22×C4C22⋊D15711yescharacteristic
180C22×C4C22⋊D15711yescharacteristic
181C22×C4C22×D15711yescharacteristic
182Q16—3144no—
183SD16—3144no—
184M16—3144no—
185C2×C8—1572no—
186C2×C8—1572no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—3141no—
193C4⋊C4—3141no—
194C4⋊C4—3141no—
195C4⋊C4—3141no—
196C4⋊C4—1571no—
197C4⋊C4—1571no—
198C4⋊C4C22×D15712yes—
199C22⋊C4—22no—
200C22⋊C4—3141no—
201C22⋊C4—3141no—
202C22⋊C4—3141no—
203C22⋊C4—3141no—
204C22⋊C4—3141no—
205C42—22no—
206C42—3141no—
207C42C22⋊D15711yescharacteristic
208C42C22⋊D15711yescharacteristic
209C42C22×D15711yescharacteristic
210C42C22×D15711yescharacteristic
211C23—3141no—
212C23C2×C22⋊D15711yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—3142no—
217D4—3142no—
218D4—6282no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—3141no—
225C2×C4—3141no—
226C2×C4—3141no—
227C2×C4—6282no—
228C2×C4C2×C22⋊D15711yescharacteristic
229C2×C4C2×C22⋊D15711yescharacteristic
230C2×C4C2×C22⋊D15711yescharacteristic
231C2×C4C2×C22⋊D15711yescharacteristic
232C2×C4C2×C22⋊D15711yescharacteristic
233C2×C4C23×D15711yescharacteristic
234C8—3142no—
235C8—1574no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—6281no—
240C22—6281no—
241C22—6281no—
242C22C22×C22⋊D15711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—6281no—
250C4—6281no—
251C4—6281no—
252C4#1909780212yes—
253C4#41×D15712yes—
254C2—41no—
255C2—12561no—
256C2#2375781911yescharacteristic
257C2C2×#1909780211yescharacteristic
258C2#2375807111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#533×C157C2
socleC22×C157C22×D4
prime radical#19097745C22
hypercenter#533D157
hypocenterC157#3172
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #23757679C2semidirect
3. #23757665C2semidirect
4. #23757641C2semidirect
5. #23757635C2semidirect
6. #23757529C2semidirect
8. C4×#14578944C2semidirect
9. C4×#14578941C2semidirect
10. C4×#14578938C2semidirect
11. #533×C157C2semidirect
129. C157#3172semidirect
131. #533D157semidirect

Maximal subgroups

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

subgroupindex
#237578072
#237576792
#237576652
#237576412
#237576352
#237575292
#237575212
C4×#145789442
C4×#145789412
C4×#145789382
#533×C1572
#3172157