Finite Group Databasebeta

← #34112032

Subgroups — #34112032

Order 20096.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757932C211yesmaximal, characteristic, complemented
2#23757630C211yesmaximal, characteristic, complemented
3#23757614C211yesmaximal, characteristic, complemented
4#23757591C211yesmaximal, characteristic, complemented
5#23757577C211yesmaximal, characteristic, complemented
6#23757507C211yesmaximal, characteristic, complemented
7#23757500C211yesmaximal, characteristic, complemented
8#23757488C211yesmaximal, characteristic, complemented
9#23757416C211yesmaximal, characteristic, complemented
10#23757411C211yesmaximal, characteristic, complemented
11C4×#14578942C211yesmaximal, characteristic, complemented
12D4×#11929434C211yesmaximal, characteristic, complemented
13#23757267C211yesmaximal, characteristic, complemented
14C2×#19097694C211yesmaximal, characteristic, complemented
15#553×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21C22×#11929434C2211yescharacteristic
22C22×#11929434C2211yescharacteristic
23#19097759C2211yescharacteristic
24#19097757—21no—
25#19097750C2211yescharacteristic
26#19097749—21no—
27#19097745C2211yescharacteristic
28#19097743—21no—
29C4×Dic314C2211yescharacteristic
30#19097731—21no—
31#19097704C2211yescharacteristic
32#19097702C2211yescharacteristic
33#19097702C2211yescharacteristic
34C4⋊#11929434C2211yescharacteristic
35#19097694C2211yescharacteristic
36#19097694C2211yescharacteristic
37#19097694C2211yescharacteristic
38#19097694C2211yescharacteristic
39#19097694C2211yescharacteristic
40#19097694C2211yescharacteristic
41#19097692C2211yescharacteristic
42#19097691C2211yescharacteristic
43#19097691C2211yescharacteristic
44#19097691C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic
47C4×#11929434C2211yescharacteristic
48#126×C157—21no—
49#125×C157C2211yescharacteristic
50#123×C157—21no—
51#121×C157—21no—
52#121×C157C2211yescharacteristic
53#121×C157C2211yescharacteristic
54C4×D4×C157—21no—
55C4×D4×C157C2211yescharacteristic
56C4×D4×C157C2211yescharacteristic
57#117×C157C2211yescharacteristic
58C2×C4⋊C4×C157C2211yescharacteristic
59#14578954—22no—
60#14578954—41no—
61C2×Dic314—22no—
62C2×Dic314—21no—
63C2×Dic314—41no—
64C2×Dic314C2311yescharacteristic
65C2×Dic314C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C22×Dic157—21no—
70C2×C4×D157—21no—
71#14578942—22no—
72#14578942—41no—
73#14578939—41no—
74C2×#11929434—41no—
75C2×#11929434—41no—
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81#14578933—21no—
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87C4×Dic157—21no—
88C2×D4×C157—21no—
89C2×D4×C157—21no—
90C2×D4×C157D411yescharacteristic
91C2×D4×C157C2311yescharacteristic
92C22×C4×C157—21no—
93C22×C4×C157—21no—
94C22×C4×C157C2311yescharacteristic
95C22×C4×C157C2311yescharacteristic
96C4⋊C4×C157—21no—
97C4⋊C4×C157—21no—
98C4⋊C4×C157—21no—
99C4⋊C4×C157D411yescharacteristic
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C22⋊C4×C157—21no—
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C22⋊C4×C157D411yescharacteristic
108C22⋊C4×C157D411yescharacteristic
109C42×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112Dic314—21no—
113Dic314—21no—
114Dic314—21no—
115Dic314—42no—
116Dic314#4112yes—
117D628—21no—
118D628—21no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—42no—
125C4×D157—42no—
126#11929434—21no—
127#11929434—21no—
128#11929434—21no—
129#11929434—21no—
130C23×C157—21no—
131C23×C157C2×D411yescharacteristic
132C23×C157C2×D411yescharacteristic
133D4×C157—22no—
134D4×C157—21no—
135D4×C157—21no—
136D4×C157—42no—
137D4×C157—81no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—81no—
146C2×C4×C157—81no—
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2×D411yescharacteristic
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2411yescharacteristic
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—41no—
158C22×C157—41no—
159C22×C157—41no—
160C22×C157#13512yes—
161C22×C157C22×D411yescharacteristic
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C4×C157—21no—
166C4×C157—21no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157—41no—
170C4×C157—41no—
171C4×C157—41no—
172C4×C157C2×#4111yescharacteristic
173C4×C157H(5,𝔽2)11yescharacteristic
174Dic157—22no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—22no—
178C2×C157—41no—
179C2×C157—81no—
180C2×C157#54611yescharacteristic
181C2×C157C2×#13511yescharacteristic
182C2×C157#57711yescharacteristic
183D157—81no—
184C157#330111yescharacteristic, complemented, Sylow
185#3301—1571nomaximal, Sylow
186#553D15711yescharacteristic, complemented
187#548—1571no—
188#488—1571no—
189#485—1571no—
190#481—1571no—
191#479—1571no—
192#466—1571no—
193#465—1571no—
194#463—1571no—
195#449—1571no—
196#448—1571no—
197C4×SD16—1571no—
198C8×D4—1571no—
199#416—1571no—
200C2×#103—1571no—
201C2×#41—1571no—
202C22×Q8—1571no—
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×D8—1571no—
206C22×C8—1571no—
207C22×C8—1571no—
208#126—21no—
209#125C2×D15711yescharacteristic
210#124—3141no—
211#124—1571no—
212#123—21no—
213C22⋊Q8—3141no—
214C22⋊Q8—1571no—
215#121—21no—
216#121—3141no—
217#121—1571no—
218#121C2×D15711yescharacteristic
219#121C2×D15711yescharacteristic
220C4×Q8—1571no—
221C4×D4—21no—
222C4×D4—3141no—
223C4×D4C2×D15711yescharacteristic
224C4×D4C2×D15711yescharacteristic
225#117C2×D15711yescharacteristic
226C2×C4⋊C4C2×D15711yescharacteristic
227#107—1571no—
228#106—1571no—
229#106—1571no—
230C4⋊C8—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#102—1571no—
238#102—1571no—
239#102—1571no—
240#102—1571no—
241C22⋊C8—1571no—
242C22⋊C8—1571no—
243C4×C8—1571no—
244#41—3142no—
245#41—6281no—
246C2×Q8—3142no—
247C2×Q8—3141no—
248C2×Q8—6281no—
249C2×Q8—1571no—
250C2×Q8—1571no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—3141no—
254C2×D4—3141no—
255C2×D4—1571no—
256C2×D4C22⋊D15711yescharacteristic
257C2×D4C22×D15711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4C22×D15711yescharacteristic
263C22×C4C22×D15711yescharacteristic
264SD16—3142no—
265SD16—6281no—
266D8—6281no—
267C2×C8—6281no—
268C2×C8—6281no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8—1571no—
272C2×C8—1571no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—1571no—
279C4⋊C4C22⋊D15711yescharacteristic
280C4⋊C4C22×D15711yescharacteristic
281C4⋊C4C22×D15711yescharacteristic
282C4⋊C4C22×D15711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4C22⋊D15711yescharacteristic
292C22⋊C4C22⋊D15711yescharacteristic
293C42—21no—
294C42—3141no—
295C42C22×D15711yescharacteristic
296C23—21no—
297C23—3141no—
298C23C2×C22⋊D15711yescharacteristic
299C23C2×C22⋊D15711yescharacteristic
300Q8—3141no—
301Q8—3141no—
302Q8—3141no—
303Q8—6282no—
304Q8—1572no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—3141no—
311D4—3141no—
312D4—6282no—
313D4—6282no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4C2×C22⋊D15711yescharacteristic
329C2×C4C2×C22⋊D15711yescharacteristic
330C2×C4C2×C22⋊D15711yescharacteristic
331C2×C4C2×C22⋊D15711yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—3141no—
334C8—3141no—
335C8—3141no—
336C8—3141no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—6281no—
346C22—6281no—
347C22—6281no—
348C22#1909779112yes—
349C22C22×C22⋊D15711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3142no—
358C4—6281no—
359C4—6281no—
360C4#41×D15711yescharacteristic
361C4#1909781711yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—12561no—
366C2#2375791111yescharacteristic
367C2C2×#1909779111yescharacteristic
368C2#2375808411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#553×C157C2
socleC22×C157C22×D4
hypercenter#553D157
hypocenterC157#3301
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #23757932C2semidirect
2. #23757630C2semidirect
3. #23757614C2semidirect
4. #23757591C2semidirect
5. #23757577C2semidirect
6. #23757507C2semidirect
7. #23757500C2semidirect
8. #23757488C2semidirect
9. #23757416C2semidirect
10. #23757411C2semidirect
11. C4×#14578942C2semidirect
12. D4×#11929434C2semidirect
13. #23757267C2semidirect
14. C2×#19097694C2semidirect
15. #553×C157C2semidirect
184. C157#3301semidirect
186. #553D157semidirect

Maximal subgroups

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

subgroupindex
#237579322
#237576302
#237576142
#237575912
#237575772
#237575072
#237575002
#237574882
#237574162
#237574112
C4×#145789422
D4×#119294342
#237572672
C2×#190976942
#553×C1572
#3301157