Finite Group Databasebeta

← #34110690

Subgroups — #34110690

Order 20096.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757825C211yesmaximal, characteristic, complemented
2#23757818C211yesmaximal, characteristic
3#23757670C211yesmaximal, characteristic, complemented
4#23757644C211yesmaximal, characteristic, complemented
5#23757622C211yesmaximal, characteristic, complemented
6#23757618C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757500C211yesmaximal, characteristic, complemented
10#23757364C211yesmaximal, characteristic, complemented
11SD16×Dic157C211yesmaximal, characteristic, complemented
12#23757360C211yesmaximal, characteristic, complemented
13#23757358C211yesmaximal, characteristic, complemented
14C2×C8×Dic157C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×#14578943C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C22×#11929434C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097770C2211yescharacteristic
23#19097769C2211yescharacteristic
24#19097767C2211yescharacteristic
25#19097761—21no—
26#19097757—21no—
27#19097750C2211yescharacteristic
28#19097748—21no—
29#19097745C2211yescharacteristic
30#19097743—21no—
31Q8×Dic157C2211yescharacteristic
32#19097736—21no—
33#19097736C2211yescharacteristic
34D4×Dic157C2211yescharacteristic
35#19097733C2211yescharacteristic
36#19097731—21no—
37#19097727—21no—
38C2×C4×Dic157C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097702C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097694C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097691C2211yescharacteristic
48C8×Dic157C2211yescharacteristic
49C8×Dic157C2211yescharacteristic
50C8×Dic157C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×SD16×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54C22⋊Q8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#106×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578953—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—21no—
67C22×Dic157—21no—
68C22×Dic157D411yescharacteristic
69C22×Dic157D411yescharacteristic
70C2×C4×D157—21no—
71#14578943—22no—
72#14578942—22no—
73#14578941—22no—
74C2×#11929434—22no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C4⋊Dic157—21no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88#14578931—21no—
89#14578931—21no—
90C4×Dic157—21no—
91C4×Dic157—21no—
92C4×Dic157D411yescharacteristic
93C4×Dic157D411yescharacteristic
94C4×Dic157C2311yescharacteristic
95C4×Dic157C2311yescharacteristic
96C2×Q8×C157C2311yescharacteristic
97C2×D4×C157—21no—
98C2×D4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100SD16×C157—22no—
101C2×C8×C157—22no—
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C4⋊C4×C157—21no—
105C4⋊C4×C157C2311yescharacteristic
106C4⋊C4×C157C2311yescharacteristic
107C22⋊C4×C157—21no—
108C22⋊C4×C157—21no—
109C22×D157—21no—
110Dic314—21no—
111Dic314—21no—
112Dic314—42no—
113Dic314—42no—
114D628—21no—
115D628—21no—
116C22⋊D157—42no—
117C22⋊D157—42no—
118C22⋊D157—42no—
119C2×Dic157—22no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—42no—
124C2×Dic157SD1614yescomplemented
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—42no—
130#11929434—21no—
131#11929434#4112yescomplemented
132C23×C157—21no—
133C23×C157C2×D411yescharacteristic
134Q8×C157—21no—
135Q8×C157—21no—
136D4×C157—21no—
137D4×C157—21no—
138D4×C157—42no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—21no—
147C8×C157#4112yescomplemented
148C22×C157—21no—
149C22×C157—21no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157—41no—
153C22×C157C2×SD1612yes—
154C22×C157C22×D411yescharacteristic
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C4×C157—21no—
159C4×C157—41no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157C2×#4111yescharacteristic
163C4×C157C2×#4111yescharacteristic
164Dic157—21no—
165Dic157—41no—
166Dic157—41no—
167Dic157—41no—
168Dic157C2×SD1612yes—
169Dic157C2×SD1612yes—
170Dic157#13512yes—
171C2×C157—22no—
172C2×C157—81no—
173C2×C157#53211yescharacteristic
174C2×C157C22×SD1611yescharacteristic
175C2×C157C2×#13511yescharacteristic
176D157—81no—
177C157#316711yescharacteristic, complemented, Sylow
178#3167—1571nomaximal, Sylow
179#533—1571no—
180#532—1571no—
181#498—1571no—
182#492—1571no—
183#487—1571no—
184#486—1571no—
185#465—1571no—
186#465—1571no—
187#465—1571no—
188#465D15711yescharacteristic, complemented
189C4×SD16—1571no—
190C4×SD16—1571no—
191C4×SD16—1571no—
192C4×SD16—1571no—
193C2×C4×C8—1571no—
194C2×#41—3141no—
195C2×SD16—1571no—
196C2×SD16—1571no—
197C2×SD16—1571no—
198C2×SD16C2×D15711yescharacteristic
199C22×C8—1571no—
200C22×C8C2×D15711yescharacteristic
201C4⋊Q8—1571no—
202C4⋊Q8—1571no—
203C4⋊Q8—1571no—
204#127—1571no—
205#125—3141no—
206#124—3141no—
207C22⋊Q8—3141no—
208C22⋊Q8—1571no—
209C22⋊Q8C2×D15711yescharacteristic
210#121—3141no—
211#121—1571no—
212#121C2×D15711yescharacteristic
213C4×Q8—3141no—
214C4×Q8—1571no—
215C4×Q8—1571no—
216C4×D4—3141no—
217C4×D4—1571no—
218C4×D4—1571no—
219#117—3141no—
220C2×C42—1571no—
221#106—1571no—
222#106—1571no—
223#106—1571no—
224#106C2×D15711yescharacteristic
225#103—1571no—
226#103—1571no—
227#103—1571no—
228#103C2×D15711yescharacteristic
229#102—1571no—
230#102—1571no—
231#102—1571no—
232#102C2×D15711yescharacteristic
233C4×C8—1571no—
234C4×C8—1571no—
235C4×C8—1571no—
236C4×C8—1571no—
237#41—6284no—
238C2×Q8—3141no—
239C2×Q8—3141no—
240C2×Q8—1571no—
241C2×Q8C22×D15711yescharacteristic
242C2×D4—21no—
243C2×D4—3141no—
244C2×D4—3141no—
245C2×D4—3141no—
246C2×D4—1571no—
247C2×D4C22×D15711yescharacteristic
248C22×C4—3141no—
249C22×C4—3141no—
250C22×C4—1571no—
251C22×C4—1571no—
252C22×C4C22×D15711yescharacteristic
253SD16—22no—
254SD16—3142no—
255SD16—3142no—
256SD16—3142no—
257C2×C8—22no—
258C2×C8—3142no—
259C2×C8—1571no—
260C2×C8—1571no—
261C2×C8C22×D15711yescharacteristic
262C2×C8C22×D15711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—1571no—
272C4⋊C4—1571no—
273C4⋊C4C22×D15711yescharacteristic
274C4⋊C4C22×D15711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C42—3141no—
282C42—3141no—
283C42—1571no—
284C42—1571no—
285C42—1571no—
286C42—1571no—
287C23—21no—
288C23—3141no—
289C23D4×D15711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3141no—
293Q8—3141no—
294Q8—6282no—
295Q8—6282no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—3141no—
300D4—3141no—
301D4—6282no—
302D4—6282no—
303D4—6282no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3142no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—6282no—
314C2×C4—6282no—
315C2×C4—1571no—
316C2×C4—1574no—
317C2×C4—1571no—
318C2×C4—1571no—
319C2×C4—1571no—
320C2×C4D4×D15711yescharacteristic
321C2×C4C23×D15711yescharacteristic
322C8—21no—
323C8—3141no—
324C8—1572no—
325C8#1457895312yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—6281no—
332C22—6281no—
333C22—6281no—
334C22SD16×D15712yes—
335C22C2×D4×D15711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—3141no—
341C4—6281no—
342C4—6281no—
343C4—6281no—
344C4—1572no—
345C4—1572no—
346C4—1572no—
347C4#41×D15711yescharacteristic
348C4C2×#1457895311yescharacteristic
349C2—22no—
350C2—81no—
351C2—12561no—
352C2#2375781811yescharacteristic
353C2C2×SD16×D15711yescharacteristic
354C2#135×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
prime radical#23757818C2
hypercenter#465D157
hypocenterC157#3167
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23757825C2semidirect
3. #23757670C2semidirect
4. #23757644C2semidirect
5. #23757622C2semidirect
6. #23757618C2semidirect
7. #23757504C2semidirect
8. #23757502C2semidirect
9. #23757500C2semidirect
10. #23757364C2semidirect
11. SD16×Dic157C2semidirect
12. #23757360C2semidirect
13. #23757358C2semidirect
14. C2×C8×Dic157C2semidirect
15. #465×C157C2semidirect
124. C2×Dic157SD16semidirect
131. #11929434#41semidirect
147. C8×C157#41semidirect
177. C157#3167semidirect
188. #465D157semidirect
325. C8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237578252
#237578182
#237576702
#237576442
#237576222
#237576182
#237575042
#237575022
#237575002
#237573642
SD16×Dic1572
#237573602
#237573582
C2×C8×Dic1572
#465×C1572
#3167157