Finite Group Databasebeta

← #34110387

Subgroups — #34110387

Order 20096.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757881C211yesmaximal, characteristic, complemented
2#23757745C211yesmaximal, characteristic
3#23757616C211yesmaximal, characteristic, complemented
4#23757613C211yesmaximal, characteristic, complemented
5#23757576C211yesmaximal, characteristic, complemented
6#23757571C211yesmaximal, characteristic, complemented
7#23757477C211yesmaximal, characteristic, complemented
8#23757476C211yesmaximal, characteristic, complemented
9#23757425C211yesmaximal, characteristic
10#23757422C211yesmaximal, characteristic
11#23757375C211yesmaximal, characteristic, complemented
12C4×#14578941C211yesmaximal, characteristic, complemented
13C4×#14578942C211yesmaximal, characteristic, complemented
14#23757335C211yesmaximal, characteristic, complemented
15#440×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578943C2211yescharacteristic
18C2×#14578942C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20#19097769C2211yescharacteristic, complemented
21#19097766—21no—
22#19097765—21no—
23#19097763—21no—
24#19097762C2211yescharacteristic, complemented
25#19097761—21no—
26#19097751C2211yescharacteristic
27#19097747C2211yescharacteristic
28C4×Dic314C2211yescharacteristic, complemented
29C4×D628C2211yescharacteristic
30D4×Dic157C2211yescharacteristic
31#19097733C2211yescharacteristic
32C4×C22⋊D157—21no—
33#19097731—21no—
34#19097729C2211yescharacteristic
35#19097728C2211yescharacteristic
36C2×#14578931—21no—
37C22⋊C4×D157—21no—
38#19097705C2211yescharacteristic, complemented
39#19097702C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097700C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097694C2211yescharacteristic, complemented
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097683C2211yescharacteristic
51C4×#11929434C2211yescharacteristic, complemented
52C2×SD16×C157C2211yescharacteristic
53C4×Q8×C157C2211yescharacteristic, complemented
54C4×D4×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×Dic314C2311yescharacteristic
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63D4×D157—22no—
64D4×D157—22no—
65C22×Dic157—21no—
66C2×C4×D157—21no—
67C2×C4×D157—21no—
68C2×C4×D157C2311yescharacteristic
69#14578943—22no—
70#14578942—22no—
71#14578941—22no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C4⋊Dic157—21no—
75C4⋊Dic157—21no—
76C4⋊Dic157C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78#14578933—21no—
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84#14578933D411yescharacteristic
85C22⋊Dic157—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931D411yescharacteristic
92#14578931D411yescharacteristic
93C4×Dic157—21no—
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96C2×Q8×C157C2311yescharacteristic
97C2×D4×C157C2311yescharacteristic
98C22×C4×C157—21no—
99SD16×C157—22no—
100C2×C8×C157C2311yescharacteristic
101C2×C8×C157C2311yescharacteristic
102C4⋊C4×C157—21no—
103C4⋊C4×C157C2311yescharacteristic
104C4⋊C4×C157C2311yescharacteristic
105C22⋊C4×C157—21no—
106C42×C157—21no—
107C42×C157C2311yescharacteristic
108C22×D157—21no—
109C22×D157—24no—
110C22×D157—81no—
111C22×D157—81no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115D628—21no—
116D628#4112yes—
117C22⋊D157—24no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—81no—
122C2×Dic157C2×D411yescharacteristic
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C4×D157—21no—
126C4×D157—21no—
127C4×D157—81no—
128C4×D157—81no—
129#11929434—21no—
130#11929434—21no—
131C23×C157—21no—
132Q8×C157—21no—
133Q8×C157—21no—
134D4×C157—21no—
135D4×C157#4112yes—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—81no—
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2411yescharacteristic
143C8×C157—21no—
144C8×C157—21no—
145C22×C157—22no—
146C22×C157—41no—
147C22×C157—41no—
148C22×C157C22×D411yescharacteristic
149C2×D157—21no—
150C2×D157—22no—
151C2×D157—21no—
152C2×D157—21no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157—84no—
156C4×C157—21no—
157C4×C157—21no—
158C4×C157—41no—
159C4×C157—41no—
160C4×C157—41no—
161C4×C157C2×#4111yescharacteristic
162C4×C157C2×#4111yescharacteristic
163Dic157—21no—
164Dic157—21no—
165Dic157—21no—
166Dic157—41no—
167Dic157—41no—
168Dic157—41no—
169C2×C157—42no—
170C2×C157#52511yescharacteristic
171C2×C157#57711yescharacteristic
172C2×C157#57711yescharacteristic
173D157—41no—
174D157—42no—
175C157#312311yescharacteristic, complemented, Sylow
176#3123—1571nomaximal, Sylow
177#541—1571no—
178#518—1571no—
179#485—1571no—
180#485—1571no—
181#478—1571no—
182#478—1571no—
183#461—1571no—
184#461—1571no—
185#450—1571no—
186#450—1571no—
187#440—1571no—
188#440D15711yescharacteristic, complemented
189C4×SD16—1571no—
190C4×SD16—1571no—
191#433—1571no—
192C22×D4—1571no—
193C2×SD16—1571no—
194C2×SD16—1571no—
195C2×SD16—1571no—
196C2×SD16C2×D15711yescharacteristic
197C4⋊Q8—1571no—
198#126—3141no—
199#126—3141no—
200#125—3141no—
201#125—3141no—
202#125—1571no—
203C22⋊Q8—1571no—
204C22⋊Q8—1571no—
205C4×Q8—1571no—
206C4×Q8C2×D15711yescharacteristic, complemented
207C4×D4—3141no—
208C4×D4—3141no—
209C4×D4—1571no—
210C4×D4—1571no—
211C4×D4—1571no—
212C4×D4C2×D15711yescharacteristic
213#117—1571no—
214#117—1571no—
215C2×C22⋊C4—3141no—
216C2×C22⋊C4—3141no—
217#107—1571no—
218#107C2×D15711yescharacteristic, complemented
219#106—1571no—
220#106—1571no—
221C4⋊C8—1571no—
222C4⋊C8—1571no—
223#103—1571no—
224#103—1571no—
225#103—1571no—
226#103C2×D15711yescharacteristic, complemented
227#102—1571no—
228#102—1571no—
229#102—1571no—
230#102C2×D15711yescharacteristic
231C22⋊C8—1571no—
232C22⋊C8—1571no—
233#97C2×D15711yescharacteristic, complemented
234C4×C8—1571no—
235C24—3141no—
236C2×Q8—1571no—
237C2×Q8C22×D15711yescharacteristic
238C2×D4—3142no—
239C2×D4—3142no—
240C2×D4—3141no—
241C2×D4—1571no—
242C2×D4C22×D15711yescharacteristic
243C22×C4—21no—
244C22×C4—3141no—
245C22×C4—3141no—
246C22×C4—3141no—
247C22×C4—1571no—
248SD16—22no—
249SD16—3142no—
250SD16—3142no—
251SD16—3142no—
252C2×C8—1571no—
253C2×C8—1571no—
254C2×C8C22×D15711yescharacteristic
255C2×C8C22×D15711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3141no—
258C4⋊C4—3141no—
259C4⋊C4—3141no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—3141no—
264C4⋊C4—1571no—
265C4⋊C4—1571no—
266C4⋊C4—1571no—
267C4⋊C4—1571no—
268C4⋊C4C22×D15711yescharacteristic
269C4⋊C4C22×D15711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3141no—
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—1571no—
278C22⋊C4—1571no—
279C42—21no—
280C42—3141no—
281C42—3141no—
282C42—1571no—
283C42C22×D15711yescharacteristic
284C23—21no—
285C23—3144no—
286C23—3141no—
287C23—1571no—
288C23—12561no—
289C23—12561no—
290Q8—21no—
291Q8—21no—
292Q8—3141no—
293Q8—3141no—
294D4—21no—
295D4—3144no—
296D4—3141no—
297D4—1572no—
298D4#1457895412yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3141no—
304C2×C4—3141no—
305C2×C4—3141no—
306C2×C4—3141no—
307C2×C4—3141no—
308C2×C4—1571no—
309C2×C4—1571no—
310C2×C4—1571no—
311C2×C4—12561no—
312C2×C4—12561no—
313C2×C4—12561no—
314C2×C4D4×D15711yescharacteristic
315C2×C4D4×D15711yescharacteristic
316C2×C4C23×D15711yescharacteristic
317C8—21no—
318C8—21no—
319C8—3141no—
320C8—3141no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3141no—
325C22—3141no—
326C22—3141no—
327C22—3142no—
328C22—6281no—
329C22—6281no—
330C22—12564no—
331C22C2×D4×D15711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3141no—
338C4—3141no—
339C4—3141no—
340C4—6281no—
341C4—6281no—
342C4—6281no—
343C4#41×D15711yescharacteristic
344C4C2×#1457895411yescharacteristic
345C2—42no—
346C2—6281no—
347C2—6282no—
348C2#2375778211yescharacteristic
349C2#2375808211yescharacteristic
350C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#440×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#440D157
hypocenterC157#3123
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #23757881C2semidirect
3. #23757616C2semidirect
4. #23757613C2semidirect
5. #23757576C2semidirect
6. #23757571C2semidirect
7. #23757477C2semidirect
8. #23757476C2semidirect
11. #23757375C2semidirect
12. C4×#14578941C2semidirect
13. C4×#14578942C2semidirect
14. #23757335C2semidirect
15. #440×C157C2semidirect
20. #19097769C22semidirect
24. #19097762C22semidirect
28. C4×Dic314C22semidirect
38. #19097705C22semidirect
39. #19097702C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097700C22semidirect
43. #19097696C22semidirect
44. #19097695C22semidirect
45. #19097694C22semidirect
51. C4×#11929434C22semidirect
53. C4×Q8×C157C22semidirect
55. #107×C157C22semidirect
56. #103×C157C22semidirect
58. #97×C157C22semidirect
175. C157#3123semidirect
188. #440D157semidirect
206. C4×Q8C2×D157semidirect
218. #107C2×D157semidirect
226. #103C2×D157semidirect
233. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578812
#237577452
#237576162
#237576132
#237575762
#237575712
#237574772
#237574762
#237574252
#237574222
#237573752
C4×#145789412
C4×#145789422
#237573352
#440×C1572
#3123157