Finite Group Databasebeta

← #34110386

Subgroups — #34110386

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#23757884C211yesmaximal, characteristic, complemented
2#23757745C211yesmaximal, characteristic
3#23757613C211yesmaximal, characteristic, complemented
4#23757612C211yesmaximal, characteristic, complemented
5#23757574C211yesmaximal, characteristic, complemented
6#23757573C211yesmaximal, characteristic, complemented
7#23757475C211yesmaximal, characteristic, complemented
8#23757422C211yesmaximal, characteristic
9#23757422C211yesmaximal, characteristic
10#23757376C211yesmaximal, characteristic, complemented
11#23757374C211yesmaximal, characteristic, complemented
12SD16×Dic157C211yesmaximal, characteristic, complemented
13#23757358C211yesmaximal, characteristic, complemented
14#23757333C211yesmaximal, characteristic, complemented
15#461×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578941C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19#19097766—21no—
20#19097766—21no—
21#19097763C2211yescharacteristic, complemented
22#19097761—21no—
23#19097761—21no—
24#19097747C2211yescharacteristic
25#19097747C2211yescharacteristic
26Q8×Dic157C2211yescharacteristic, complemented
27Q8×Dic157C2211yescharacteristic, complemented
28C4×D628C2211yescharacteristic
29D4×Dic157C2211yescharacteristic
30#19097733C2211yescharacteristic
31C4×C22⋊D157—21no—
32#19097731—21no—
33#19097729C2211yescharacteristic
34#19097728C2211yescharacteristic
35C2×#14578931—21no—
36C22⋊C4×D157—21no—
37#19097705C2211yescharacteristic, complemented
38#19097704C2211yescharacteristic, complemented
39#19097703C2211yescharacteristic, complemented
40#19097702C2211yescharacteristic, complemented
41C4⋊#11929434C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097695C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097683C2211yescharacteristic
49#19097683C2211yescharacteristic
50#19097681C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C2×SD16×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C4⋊Q8×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56C4⋊C8×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58#103×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62D4×D157—22no—
63D4×D157—22no—
64C22×Dic157—21no—
65C2×C4×D157—21no—
66C2×C4×D157—21no—
67C2×C4×D157C2311yescharacteristic
68#14578941—22no—
69#14578941—22no—
70C2×#11929434C2311yescharacteristic
71C2×#11929434C2311yescharacteristic
72C4⋊Dic157—21no—
73C4⋊Dic157—21no—
74C4⋊Dic157C2311yescharacteristic
75C4⋊Dic157C2311yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77#14578933—21no—
78#14578933—21no—
79#14578933—21no—
80#14578933—21no—
81#14578933D411yescharacteristic
82#14578933D411yescharacteristic
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931D411yescharacteristic
90#14578931D411yescharacteristic
91C4×Dic157—21no—
92C4×Dic157—21no—
93C4×Dic157—21no—
94C4×Dic157C2311yescharacteristic
95C2×Q8×C157C2311yescharacteristic
96C2×Q8×C157C2311yescharacteristic
97C2×D4×C157C2311yescharacteristic
98C22×C4×C157—21no—
99SD16×C157—22no—
100SD16×C157—22no—
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C4⋊C4×C157—21no—
104C4⋊C4×C157—21no—
105C4⋊C4×C157C2311yescharacteristic
106C22⋊C4×C157—21no—
107C42×C157C2311yescharacteristic
108C22×D157—21no—
109C22×D157—24no—
110C22×D157—81no—
111C22×D157—81no—
112C22×D157C2×D411yescharacteristic
113D628—21no—
114D628#4112yes—
115C22⋊D157—24no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—81no—
120C2×Dic157C2×D411yescharacteristic
121C2×Dic157C2×D411yescharacteristic
122C2×Dic157C2×D411yescharacteristic
123C4×D157—21no—
124C4×D157—21no—
125C4×D157—81no—
126C4×D157—81no—
127#11929434—21no—
128#11929434—21no—
129C23×C157—21no—
130Q8×C157—21no—
131Q8×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#461D15711yescharacteristic, complemented
185#450—1571no—
186#450—1571no—
187#440—1571no—
188#440—1571no—
189C4×SD16—1571no—
190C4×SD16—1571no—
191#433—1571no—
192C22×D4—1571no—
193C2×SD16—1571no—
194C2×SD16—1571no—
195C2×SD16C2×D15711yescharacteristic
196C2×SD16C2×D15711yescharacteristic
197C4⋊Q8C2×D15711yescharacteristic, complemented
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×Q8—1571no—
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#107—1571no—
219#106—1571no—
220#106—1571no—
221C4⋊C8—1571no—
222C4⋊C8C2×D15711yescharacteristic, complemented
223#103—1571no—
224#103—1571no—
225#103C2×D15711yescharacteristic, complemented
226#103C2×D15711yescharacteristic, complemented
227#102—1571no—
228#102—1571no—
229#102—1571no—
230#102—1571no—
231C22⋊C8—1571no—
232C22⋊C8—1571no—
233#97—1571no—
234C4×C8—1571no—
235C24—3141no—
236C2×Q8C22×D15711yescharacteristic
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—22no—
250SD16—3142no—
251SD16—3142no—
252C2×C8—1571no—
253C2×C8—1571no—
254C2×C8C22×D15711yescharacteristic
255C2×C8C22×D15711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
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⋊C4—1571no—
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—3141no—
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—21no—
293Q8—21no—
294D4—21no—
295D4—3144no—
296D4—3141no—
297D4—1572no—
298D4#1457895312yes—
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—3142no—
327C22—3141no—
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×#1457895311yescharacteristic
345C2—42no—
346C2—6281no—
347C2—6282no—
348C2#2375778311yescharacteristic
349C2#2375807911yescharacteristic
350C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#461×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#461D157
hypocenterC157#3123
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #23757884C2semidirect
3. #23757613C2semidirect
4. #23757612C2semidirect
5. #23757574C2semidirect
6. #23757573C2semidirect
7. #23757475C2semidirect
10. #23757376C2semidirect
11. #23757374C2semidirect
12. SD16×Dic157C2semidirect
13. #23757358C2semidirect
14. #23757333C2semidirect
15. #461×C157C2semidirect
21. #19097763C22semidirect
26. Q8×Dic157C22semidirect
27. Q8×Dic157C22semidirect
37. #19097705C22semidirect
38. #19097704C22semidirect
39. #19097703C22semidirect
40. #19097702C22semidirect
41. C4⋊#11929434C22semidirect
42. #19097695C22semidirect
43. #19097695C22semidirect
50. #19097681C22semidirect
51. C8×Dic157C22semidirect
54. C4⋊Q8×C157C22semidirect
56. C4⋊C8×C157C22semidirect
57. #103×C157C22semidirect
58. #103×C157C22semidirect
175. C157#3123semidirect
184. #461D157semidirect
197. C4⋊Q8C2×D157semidirect
222. C4⋊C8C2×D157semidirect
225. #103C2×D157semidirect
226. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578842
#237577452
#237576132
#237576122
#237575742
#237575732
#237574752
#237574222
#237573762
#237573742
SD16×Dic1572
#237573582
#237573332
#461×C1572
#3123157