Finite Group Databasebeta

← #34111650

Subgroups — #34111650

Order 20096.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097800C211yesmaximal, characteristic, complemented
2C22×#14578942C211yesmaximal, characteristic, complemented
3#23757936C211yesmaximal, characteristic, complemented
4#23757917C211yesmaximal, characteristic
5#23757664C211yesmaximal, characteristic, complemented
6#23757518C211yesmaximal, characteristic, complemented
7#23757512C211yesmaximal, characteristic, complemented
8#23757494C211yesmaximal, characteristic, complemented
9#23757476C211yesmaximal, characteristic, complemented
10#23757436C211yesmaximal, characteristic, complemented
11#23757431C211yesmaximal, characteristic, complemented
12#23757410C211yesmaximal, characteristic, complemented
13#23757370C211yesmaximal, characteristic, complemented
14#23757341C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic, complemented
19#19097800—24no—
20C2×Dic628C2211yescharacteristic, complemented
21C2×Dic628C2211yescharacteristic, complemented
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578942—24no—
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578942C2211yescharacteristic
27C2×#14578942C2211yescharacteristic
28C2×#14578937C2211yescharacteristic, complemented
29C22×#11929434C2211yescharacteristic
30#19097770—21no—
31#19097769C2211yescharacteristic, complemented
32#19097758C2211yescharacteristic
33#19097752—21no—
34#19097751C2211yescharacteristic, complemented
35#19097750C2211yescharacteristic
36#19097749—21no—
37#19097748—21no—
38#19097743—21no—
39#19097742C2211yescharacteristic
40#19097740—21no—
41#19097736C2211yescharacteristic, complemented
42D4×Dic157C2211yescharacteristic
43C4×C22⋊D157C2211yescharacteristic
44#19097731—21no—
45C2×C22⋊Dic157—21no—
46#19097704C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097696C2211yescharacteristic, complemented
50#19097695C2211yescharacteristic, complemented
51#19097694C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097681C2211yescharacteristic, complemented
54C22×D4×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C2×SD16×C157C2211yescharacteristic
57C22⋊Q8×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic
60C22⋊C8×C157C2211yescharacteristic
61#14578953—22no—
62#14578953—41no—
63C2×Dic314—21no—
64C2×Dic314—22no—
65C2×Dic314—21no—
66C2×Dic314—41no—
67C2×Dic314C2311yescharacteristic
68C2×Dic314C2311yescharacteristic
69C2×Dic314C2311yescharacteristic
70Q8×D157—22no—
71Q8×D157—41no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157D411yescharacteristic
75C22×Dic157—21no—
76C22×Dic157—21no—
77C22×Dic157—21no—
78C2×C4×D157—21no—
79C2×C4×D157C2311yescharacteristic
80Dic628—22no—
81Dic628—41no—
82#14578944—22no—
83#14578942—22no—
84#14578942—22no—
85#14578942—22no—
86#14578942D414yescomplemented
87#14578937—22no—
88C2×#11929434—22no—
89C2×#11929434C2311yescharacteristic
90C2×#11929434C2311yescharacteristic
91C4⋊Dic157—21no—
92C4⋊Dic157C2311yescharacteristic
93#14578933—21no—
94#14578933—21no—
95#14578933—21no—
96#14578933—21no—
97#14578933D411yescharacteristic
98C22⋊Dic157—21no—
99C22⋊Dic157—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157—21no—
102C22⋊Dic157—21no—
103C22⋊Dic157D411yescharacteristic
104#14578931—21no—
105#14578931D411yescharacteristic
106C4×Dic157—21no—
107C4×Dic157C2311yescharacteristic
108C24×C157—21no—
109C2×Q8×C157C2311yescharacteristic
110C2×D4×C157—22no—
111C2×D4×C157—21no—
112C2×D4×C157—41no—
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115C22×C4×C157C2311yescharacteristic
116SD16×C157—22no—
117SD16×C157—41no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C4⋊C4×C157—21no—
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22×D157C2×D411yescharacteristic
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130Dic314—42no—
131Dic314C2×D412yes—
132C22⋊D157—22no—
133C22⋊D157—42no—
134C22⋊D157—81no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—42no—
140C2×Dic157—42no—
141C2×Dic157—81no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C4×D157—21no—
146C4×D157—21no—
147C4×D157—42no—
148#11929434—21no—
149#11929434C2×D412yes—
150C23×C157—21no—
151C23×C157—21no—
152C23×C157—42no—
153C23×C157—42no—
154C23×C157—81no—
155C23×C157C2×D411yescharacteristic
156Q8×C157—21no—
157Q8×C157—21no—
158D4×C157—21no—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157—42no—
162D4×C157C2×D412yes—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2411yescharacteristic
169C8×C157—21no—
170C8×C157—21no—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157—22no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157—41no—
178C22×C157—41no—
179C22×C157—84no—
180C22×C157#13712yescomplemented
181C22×C157C22×D411yescharacteristic
182C2×D157—21no—
183C2×D157—21no—
184C2×D157—21no—
185C4×C157—21no—
186C4×C157—41no—
187C4×C157—41no—
188C4×C157C22×D411yescharacteristic
189C4×C157H(5,𝔽2)11yescharacteristic
190Dic157—21no—
191Dic157—22no—
192Dic157—21no—
193Dic157—21no—
194Dic157—41no—
195Dic157—41no—
196Dic157—41no—
197C2×C157—22no—
198C2×C157—42no—
199C2×C157—81no—
200C2×C157D4211yescharacteristic
201C2×C157C2×#13711yescharacteristic
202C2×C157#57711yescharacteristic
203D157—41no—
204C157#326611yescharacteristic, complemented, Sylow
205#3266—1571nomaximal, Sylow
206C2×#137—1571no—
207C22×SD16—1571no—
208D4×Q8—1571no—
209#546—1571no—
210#497—1571no—
211#468—1571no—
212#467—1571no—
213#464—1571no—
214#461—1571no—
215#452—1571no—
216#451—1571no—
217#450D15711yescharacteristic, complemented
218#448—1571no—
219#440—1571no—
220#435—1571no—
221C2×#41—1571no—
222C22×Q8—1571no—
223C22×Q8—1571no—
224C22×D4C2×D15711yescharacteristic
225#137—3144no—
226C2×Q16—1571no—
227C2×Q16—1571no—
228C2×Q16—1571no—
229C2×SD16—3144no—
230C2×SD16—1571no—
231C2×SD16—1571no—
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16C2×D15711yescharacteristic
235C2×SD16C2×D15711yescharacteristic
236C2×M16—1571no—
237C22×C8—1571no—
238C4⋊Q8—3141no—
239C4⋊Q8—1571no—
240#124—1571no—
241#123—3141no—
242C22⋊Q8—3141no—
243C22⋊Q8—3141no—
244C22⋊Q8—1571no—
245C22⋊Q8—1571no—
246C22⋊Q8C2×D15711yescharacteristic
247#121—3141no—
248#121—1571no—
249#120—3141no—
250C4×Q8—1571no—
251C4×D4—3141no—
252C4×D4—1571no—
253C4×D4—1571no—
254C2×C22⋊C4—3141no—
255#107—1571no—
256C4⋊C8—1571no—
257#103—1571no—
258#103—1571no—
259#103—1571no—
260#103—1571no—
261#102C2×D15711yescharacteristic
262#102C2×D15711yescharacteristic
263C22⋊C8—1571no—
264C22⋊C8C2×D15711yescharacteristic
265#97—1571no—
266C24—21no—
267#41—3142no—
268#41—6281no—
269C2×Q8—3141no—
270C2×Q8—3142no—
271C2×Q8—3141no—
272C2×Q8—3142no—
273C2×Q8—6281no—
274C2×Q8—6281no—
275C2×Q8—1571no—
276C2×Q8—1571no—
277C2×Q8—1571no—
278C2×Q8C22×D15711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—1571no—
285C2×D4C22×D15711yescharacteristic
286C2×D4C22×D15711yescharacteristic
287C22×C4—3141no—
288C22×C4—3141no—
289C22×C4—3141no—
290C22×C4—3141no—
291C22×C4—1571no—
292C22×C4C22×D15711yescharacteristic
293Q16—3142no—
294Q16—3142no—
295Q16—6281no—
296SD16—22no—
297SD16—41no—
298SD16—3142no—
299SD16—3142no—
300SD16—3142no—
301SD16—1574no—
302M16—3142no—
303C2×C8—3142no—
304C2×C8—1571no—
305C2×C8—1571no—
306C2×C8C22×D15711yescharacteristic
307C2×C8C22×D15711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3141no—
310C4⋊C4—3141no—
311C4⋊C4—3141no—
312C4⋊C4—3141no—
313C4⋊C4—3141no—
314C4⋊C4—1571no—
315C4⋊C4—1571no—
316C4⋊C4C22×D15711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—1571no—
325C22⋊C4—1571no—
326C42—3141no—
327C42—1571no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1571no—
334C23D4×D15711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—3141no—
338Q8—3141no—
339Q8—3141no—
340Q8—3141no—
341Q8—3141no—
342Q8—6282no—
343Q8—6282no—
344Q8—1572no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—3142no—
350D4—6282no—
351D4—12561no—
352D4D4×D15712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—3141no—
360C2×C4—3141no—
361C2×C4—3141no—
362C2×C4—3141no—
363C2×C4—6282no—
364C2×C4—6282no—
365C2×C4—6282no—
366C2×C4—1571no—
367C2×C4—1571no—
368C2×C4—1571no—
369C2×C4—12561no—
370C2×C4D4×D15711yescharacteristic
371C2×C4C23×D15711yescharacteristic
372C8—21no—
373C8—21no—
374C8—3141no—
375C8—1572no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3141no—
386C22—3141no—
387C22—3141no—
388C22#1909780012yescomplemented
389C22C2×D4×D15711yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—3141no—
394C4—3141no—
395C4—3142no—
396C4—3141no—
397C4—6281no—
398C4—6281no—
399C4—6281no—
400C4C2×D4×D15711yescharacteristic
401C4#1909781611yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—6281no—
406C2#2375790511yescharacteristic
407C2C2×#1909780011yescharacteristic
408C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radical#23757917C2
hypercenter#450D157
hypocenterC157#3266
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
2. C22×#14578942C2semidirect
3. #23757936C2semidirect
5. #23757664C2semidirect
6. #23757518C2semidirect
7. #23757512C2semidirect
8. #23757494C2semidirect
9. #23757476C2semidirect
10. #23757436C2semidirect
11. #23757431C2semidirect
12. #23757410C2semidirect
13. #23757370C2semidirect
14. #23757341C2semidirect
15. #450×C157C2semidirect
18. C2×Q8×D157C22semidirect
20. C2×Dic628C22semidirect
21. C2×Dic628C22semidirect
22. C2×#14578944C22semidirect
28. C2×#14578937C22semidirect
31. #19097769C22semidirect
34. #19097751C22semidirect
41. #19097736C22semidirect
46. #19097704C22semidirect
47. #19097700C22semidirect
48. #19097696C22semidirect
49. #19097696C22semidirect
50. #19097695C22semidirect
51. #19097694C22semidirect
52. #19097683C22semidirect
53. #19097681C22semidirect
86. #14578942D4semidirect
180. C22×C157#137semidirect
204. C157#3266semidirect
217. #450D157semidirect
388. C22#19097800semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C22×#145789422
#237579362
#237579172
#237576642
#237575182
#237575122
#237574942
#237574762
#237574362
#237574312
#237574102
#237573702
#237573412
#450×C1572
#3266157