Finite Group Databasebeta

← #34111651

Subgroups — #34111651

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×#19097804C211yesmaximal, characteristic, complemented
2C22×#14578943C211yesmaximal, characteristic, complemented
3D4×Dic314C211yesmaximal, characteristic, complemented
4#23757919C211yesmaximal, characteristic
5#23757667C211yesmaximal, characteristic, complemented
6#23757519C211yesmaximal, characteristic, complemented
7#23757514C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757479C211yesmaximal, characteristic, complemented
10#23757437C211yesmaximal, characteristic, complemented
11#23757431C211yesmaximal, characteristic, complemented
12#23757426C211yesmaximal, characteristic, complemented
13#23757414C211yesmaximal, characteristic, complemented
14#23757375C211yesmaximal, characteristic, complemented
15#435×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C22×Dic314C2211yescharacteristic, complemented
19C22×D628C2211yescharacteristic
20#19097804—24no—
21C2×Dic628C2211yescharacteristic, complemented
22C2×Dic628C2211yescharacteristic, complemented
23C2×Dic628C2211yescharacteristic, complemented
24C2×#14578943—24no—
25C2×#14578943C2211yescharacteristic
26C2×#14578943C2211yescharacteristic
27C2×#14578943C2211yescharacteristic
28C2×#14578943C2211yescharacteristic
29C2×#14578943C2211yescharacteristic
30C2×#14578943C2211yescharacteristic
31#19097771C2211yescharacteristic, complemented
32#19097770—21no—
33#19097759C2211yescharacteristic
34#19097756—21no—
35#19097750C2211yescharacteristic
36#19097750C2211yescharacteristic
37#19097750C2211yescharacteristic, complemented
38#19097748—21no—
39#19097748—21no—
40#19097745C2211yescharacteristic
41#19097744—21no—
42#19097739—21no—
43C4×Dic314C2211yescharacteristic, complemented
44C4×D628C2211yescharacteristic
45D4×Dic157—21no—
46C22⋊C4×D157—21no—
47#19097705C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097696C2211yescharacteristic, complemented
50#19097696C2211yescharacteristic, complemented
51#19097696C2211yescharacteristic, complemented
52#19097693C2211yescharacteristic
53#19097693C2211yescharacteristic
54C2×M16×C157C2211yescharacteristic, complemented
55C22×C8×C157C2211yescharacteristic
56C4×D4×C157C2211yescharacteristic
57C4⋊C8×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic
59C22⋊C8×C157C2211yescharacteristic, complemented
60#97×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578954—22no—
63#14578954—41no—
64C2×Dic314—22no—
65C2×Dic314—22no—
66C2×Dic314—21no—
67C2×Dic314—21no—
68C2×Dic314—41no—
69C2×Dic314—41no—
70C2×Dic314C2311yescharacteristic
71C2×Dic314C2311yescharacteristic
72C2×Dic314C2311yescharacteristic
73C2×Dic314C2311yescharacteristic
74C2×D628—22no—
75C2×D628—21no—
76C2×D628—41no—
77C2×D628C2311yescharacteristic
78C2×D628C2311yescharacteristic
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81C22×Dic157—21no—
82C22×Dic157—21no—
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85Dic628—22no—
86Dic628—22no—
87Dic628—41no—
88#14578943—22no—
89#14578943—22no—
90#14578943—22no—
91#14578943—22no—
92#14578943—41no—
93#14578943D414yescomplemented
94C4⋊Dic157—21no—
95C4⋊Dic157—21no—
96C4⋊Dic157C2311yescharacteristic
97C4⋊Dic157C2311yescharacteristic
98#14578933—21no—
99#14578933—21no—
100#14578933—21no—
101#14578933—21no—
102C22⋊Dic157—21no—
103C22⋊Dic157—21no—
104C22⋊Dic157—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931—21no—
109C4×Dic157—21no—
110C2×D4×C157D411yescharacteristic
111C22×C4×C157C2311yescharacteristic
112C22×C4×C157C2311yescharacteristic
113M16×C157—22no—
114C2×C8×C157—22no—
115C2×C8×C157C2311yescharacteristic
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C2×C8×C157C2311yescharacteristic
119C4⋊C4×C157D411yescharacteristic
120C22⋊C4×C157D411yescharacteristic
121C22⋊C4×C157D411yescharacteristic
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—42no—
126C22×D157—42no—
127C22×D157—81no—
128Dic314—21no—
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—21no—
133Dic314—21no—
134Dic314—21no—
135Dic314—42no—
136Dic314—42no—
137Dic314C2×D412yes—
138D628—21no—
139D628—21no—
140D628—21no—
141D628—42no—
142D628C2×D412yes—
143C22⋊D157—42no—
144C22⋊D157—81no—
145C2×Dic157—21no—
146C2×Dic157—21no—
147C2×Dic157—21no—
148C2×Dic157—21no—
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157—42no—
152C2×Dic157—42no—
153C4×D157—42no—
154C4×D157—81no—
155C23×C157C2×D411yescharacteristic
156C23×C157C2×D411yescharacteristic
157D4×C157—22no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157C2×D411yescharacteristic
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2411yescharacteristic
167C8×C157—21no—
168C8×C157—21no—
169C8×C157—21no—
170C8×C157C2×D412yes—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—21no—
175C22×C157—21no—
176C22×C157#13712yescomplemented
177C22×C157C22×D411yescharacteristic
178C2×D157—22no—
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—84no—
185C4×C157—21no—
186C4×C157—21no—
187C4×C157—21no—
188C4×C157—21no—
189C4×C157C22×D411yescharacteristic
190C4×C157H(5,𝔽2)11yescharacteristic
191Dic157—22no—
192Dic157—41no—
193Dic157—41no—
194Dic157—41no—
195Dic157—41no—
196Dic157—41no—
197C2×C157—22no—
198C2×C157—41no—
199C2×C157D4211yescharacteristic
200C2×C157C2×#13711yescharacteristic
201C2×C157#57711yescharacteristic
202D157—42no—
203D157—81no—
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#450—1571no—
218#448—1571no—
219#440—1571no—
220#435D15711yescharacteristic, complemented
221C2×#41—1571no—
222C22×Q8—1571no—
223C22×Q8—1571no—
224C22×D4—1571no—
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×SD16—1571no—
235C2×SD16—1571no—
236C2×M16C2×D15711yescharacteristic, complemented
237C22×C8C2×D15711yescharacteristic
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⋊Q8—1571no—
247#121—3141no—
248#121—1571no—
249#120—3141no—
250C4×Q8—1571no—
251C4×D4—3141no—
252C4×D4—1571no—
253C4×D4C2×D15711yescharacteristic
254C2×C22⋊C4—3141no—
255#107—1571no—
256C4⋊C8C2×D15711yescharacteristic, complemented
257#103—1571no—
258#103—1571no—
259#103—1571no—
260#103—1571no—
261#102—1571no—
262#102—1571no—
263C22⋊C8C2×D15711yescharacteristic
264C22⋊C8C2×D15711yescharacteristic, complemented
265#97C2×D15711yescharacteristic, complemented
266C24—3141no—
267#41—3142no—
268#41—6281no—
269C2×Q8—3142no—
270C2×Q8—3142no—
271C2×Q8—3141no—
272C2×Q8—3141no—
273C2×Q8—6281no—
274C2×Q8—6281no—
275C2×Q8—1571no—
276C2×Q8—1571no—
277C2×Q8—1571no—
278C2×Q8—1571no—
279C2×D4—3142no—
280C2×D4—3141no—
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—6281no—
284C2×D4—1571no—
285C2×D4—1571no—
286C2×D4D62811yescharacteristic
287C22×C4—3141no—
288C22×C4—3141no—
289C22×C4—3141no—
290C22×C4—3141no—
291C22×C4C22×D15711yescharacteristic
292C22×C4C22×D15711yescharacteristic
293Q16—3142no—
294Q16—3142no—
295Q16—6281no—
296SD16—3142no—
297SD16—3142no—
298SD16—3142no—
299SD16—3142no—
300SD16—6281no—
301SD16—1574no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D15711yescharacteristic
305C2×C8C22×D15711yescharacteristic
306C2×C8C22×D15711yescharacteristic
307C2×C8C22×D15711yescharacteristic
308C4⋊C4—3141no—
309C4⋊C4—3141no—
310C4⋊C4—3141no—
311C4⋊C4—3141no—
312C4⋊C4—3141no—
313C4⋊C4—3141no—
314C4⋊C4—1571no—
315C4⋊C4—1571no—
316C4⋊C4D62811yescharacteristic
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4D62811yescharacteristic
325C22⋊C4D62811yescharacteristic
326C42—3141no—
327C42C22×D15711yescharacteristic
328C23—3141no—
329C23—3141no—
330C23—6282no—
331C23—6282no—
332C23—12561no—
333C23C2×D62811yescharacteristic
334C23C2×D62811yescharacteristic
335Q8—3141no—
336Q8—3141no—
337Q8—3141no—
338Q8—3141no—
339Q8—3141no—
340Q8—3141no—
341Q8—3141no—
342Q8—6282no—
343Q8—6282no—
344Q8—1572no—
345D4—22no—
346D4—3141no—
347D4—3141no—
348D4—3141no—
349D4—6282no—
350D4—6282no—
351D4—1572no—
352D4—12561no—
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—12561no—
367C2×C4C2×D62811yescharacteristic
368C2×C4C2×D62811yescharacteristic
369C2×C4C2×D62811yescharacteristic
370C2×C4C2×D62811yescharacteristic
371C2×C4C23×D15711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D15712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—3142no—
382C22—6281no—
383C22—6281no—
384C22—6281no—
385C22—6281no—
386C22—6281no—
387C22—12564no—
388C22#1909780412yescomplemented
389C22C22×D62811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3142no—
395C4—6281no—
396C4—6281no—
397C4—6281no—
398C4—6281no—
399C4—6281no—
400C4C2×D4×D15711yescharacteristic
401C4#1909781711yescharacteristic
402C2—22no—
403C2—41no—
404C2—6282no—
405C2—12561no—
406C2D4×D62811yescharacteristic
407C2C2×#1909780411yescharacteristic
408C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#435×C157C2
socleC22×C157C22×D4
prime radical#23757919C2
hypercenter#435D157
hypocenterC157#3266
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19097804C2semidirect
2. C22×#14578943C2semidirect
3. D4×Dic314C2semidirect
5. #23757667C2semidirect
6. #23757519C2semidirect
7. #23757514C2semidirect
8. #23757496C2semidirect
9. #23757479C2semidirect
10. #23757437C2semidirect
11. #23757431C2semidirect
12. #23757426C2semidirect
13. #23757414C2semidirect
14. #23757375C2semidirect
15. #435×C157C2semidirect
18. C22×Dic314C22semidirect
21. C2×Dic628C22semidirect
22. C2×Dic628C22semidirect
23. C2×Dic628C22semidirect
31. #19097771C22semidirect
37. #19097750C22semidirect
43. C4×Dic314C22semidirect
47. #19097705C22semidirect
48. #19097696C22semidirect
49. #19097696C22semidirect
50. #19097696C22semidirect
51. #19097696C22semidirect
54. C2×M16×C157C22semidirect
57. C4⋊C8×C157C22semidirect
59. C22⋊C8×C157C22semidirect
60. #97×C157C22semidirect
93. #14578943D4semidirect
176. C22×C157#137semidirect
204. C157#3266semidirect
220. #435D157semidirect
236. C2×M16C2×D157semidirect
256. C4⋊C8C2×D157semidirect
264. C22⋊C8C2×D157semidirect
265. #97C2×D157semidirect
388. C22#19097804semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C22×#145789432
D4×Dic3142
#237579192
#237576672
#237575192
#237575142
#237574962
#237574792
#237574372
#237574312
#237574262
#237574142
#237573752
#435×C1572
#3266157