Finite Group Databasebeta

← #34111711

Subgroups — #34111711

Order 20096.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097804C211yesmaximal, characteristic, complemented
2C22×Dic628C211yesmaximal, characteristic, complemented
3D4×Dic314C211yesmaximal, characteristic, complemented
4#23757932C211yesmaximal, characteristic
5#23757667C211yesmaximal, characteristic, complemented
6#23757531C211yesmaximal, characteristic, complemented
7#23757502C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757484C211yesmaximal, characteristic, complemented
10#23757437C211yesmaximal, characteristic, complemented
11#23757431C211yesmaximal, characteristic, complemented
12#23757431C211yesmaximal, characteristic, complemented
13#23757414C211yesmaximal, characteristic, complemented
14#23757381C211yesmaximal, characteristic, complemented
15#435×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C22×Dic314C2211yescharacteristic
19C22×Dic314C2211yescharacteristic
20#19097804—24no—
21C2×Dic628—24no—
22C2×Dic628C2211yescharacteristic
23C2×Dic628C2211yescharacteristic
24C2×Dic628C2211yescharacteristic
25C2×Dic628C2211yescharacteristic
26C2×Dic628C2211yescharacteristic
27C2×Dic628C2211yescharacteristic
28C2×#14578943C2211yescharacteristic
29C2×#14578943C2211yescharacteristic
30C2×#14578943C2211yescharacteristic
31#19097771C2211yescharacteristic
32#19097770—21no—
33#19097759C2211yescharacteristic
34#19097757—21no—
35#19097750C2211yescharacteristic
36#19097750C2211yescharacteristic
37#19097750C2211yescharacteristic
38#19097749—21no—
39#19097748—21no—
40#19097748—21no—
41#19097745C2211yescharacteristic
42#19097743—21no—
43C4×Dic314C2211yescharacteristic
44C4×Dic314C2211yescharacteristic
45D4×Dic157—21no—
46#19097731—21no—
47#19097703C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097696C2211yescharacteristic
50#19097696C2211yescharacteristic
51#19097696C2211yescharacteristic
52#19097696C2211yescharacteristic
53#19097693C2211yescharacteristic
54C2×M16×C157C2211yescharacteristic
55C22×C8×C157C2211yescharacteristic
56C4×D4×C157C2211yescharacteristic
57C4⋊C8×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59C22⋊C8×C157C2211yescharacteristic
60#97×C157C2211yescharacteristic
61#14578954—22no—
62#14578954—41no—
63C2×Dic314—22no—
64C2×Dic314—22no—
65C2×Dic314—22no—
66C2×Dic314—21no—
67C2×Dic314—21no—
68C2×Dic314—21no—
69C2×Dic314—41no—
70C2×Dic314—41no—
71C2×Dic314—41no—
72C2×Dic314C2311yescharacteristic
73C2×Dic314C2311yescharacteristic
74C2×Dic314C2311yescharacteristic
75C2×Dic314C2311yescharacteristic
76C2×Dic314C2311yescharacteristic
77C2×D628C2311yescharacteristic
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C22×Dic157—21no—
81C22×Dic157—21no—
82C22×Dic157—21no—
83C2×C4×D157—21no—
84Dic628—22no—
85Dic628—22no—
86Dic628—22no—
87Dic628—22no—
88Dic628—41no—
89Dic628D414yescomplemented
90#14578943—22no—
91#14578943—22no—
92#14578943—41no—
93C4⋊Dic157—21no—
94C4⋊Dic157—21no—
95C4⋊Dic157C2311yescharacteristic
96C4⋊Dic157C2311yescharacteristic
97#14578933—21no—
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—
108C4×Dic157—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—
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—42no—
133Dic314—42no—
134Dic314—42no—
135Dic314C2×D412yes—
136Dic314C2×D412yes—
137D628—21no—
138D628—21no—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C2×Dic157—21no—
142C2×Dic157—21no—
143C2×Dic157—21no—
144C2×Dic157—21no—
145C2×Dic157—21no—
146C2×Dic157—21no—
147C2×Dic157—21no—
148C2×Dic157—42no—
149C2×Dic157—42no—
150C2×Dic157—42no—
151C4×D157—42no—
152C23×C157C2×D411yescharacteristic
153C23×C157C2×D411yescharacteristic
154D4×C157—22no—
155C2×C4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157C2×D411yescharacteristic
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157C2×D411yescharacteristic
162C2×C4×C157C2×D411yescharacteristic
163C2×C4×C157C2411yescharacteristic
164C8×C157—21no—
165C8×C157—21no—
166C8×C157—21no—
167C8×C157C2×D412yes—
168C22×C157—21no—
169C22×C157—21no—
170C22×C157—21no—
171C22×C157—21no—
172C22×C157—21no—
173C22×C157#13712yescomplemented
174C22×C157C22×D411yescharacteristic
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—41no—
178C4×C157—21no—
179C4×C157—21no—
180C4×C157—21no—
181C4×C157—21no—
182C4×C157C22×D411yescharacteristic
183C4×C157H(5,𝔽2)11yescharacteristic
184Dic157—22no—
185Dic157—22no—
186Dic157—41no—
187Dic157—41no—
188Dic157—41no—
189Dic157—41no—
190Dic157—41no—
191C2×C157—22no—
192C2×C157—41no—
193C2×C157D4211yescharacteristic
194C2×C157C2×#13711yescharacteristic
195C2×C157#57811yescharacteristic
196D157—81no—
197C157#327011yescharacteristic, complemented, Sylow
198#3270—1571nomaximal, Sylow
199C2×#137—1571no—
200C22×Q16—1571no—
201D4×Q8—1571no—
202#548—1571no—
203#497—1571no—
204#470—1571no—
205#465—1571no—
206#464—1571no—
207#462—1571no—
208#452—1571no—
209#451—1571no—
210#451—1571no—
211#448—1571no—
212#441—1571no—
213#435D15711yescharacteristic, complemented
214C2×#41—1571no—
215C22×Q8—1571no—
216C22×Q8—1571no—
217C22×Q8—1571no—
218#137—3144no—
219C2×Q16—3144no—
220C2×Q16—1571no—
221C2×Q16—1571no—
222C2×Q16—1571no—
223C2×Q16—1571no—
224C2×Q16—1571no—
225C2×Q16—1571no—
226C2×SD16—1571no—
227C2×SD16—1571no—
228C2×SD16—1571no—
229C2×M16C2×D15711yescharacteristic
230C22×C8C2×D15711yescharacteristic
231C4⋊Q8—3141no—
232C4⋊Q8—1571no—
233#124—3141no—
234#124—1571no—
235C22⋊Q8—3141no—
236C22⋊Q8—3141no—
237C22⋊Q8—3141no—
238C22⋊Q8—1571no—
239C22⋊Q8—1571no—
240C22⋊Q8—1571no—
241#121—3141no—
242#121—1571no—
243C4×Q8—1571no—
244C4×Q8—1571no—
245C4×D4—3141no—
246C4×D4—3141no—
247C4×D4C2×D15711yescharacteristic
248#106—1571no—
249C4⋊C8C2×D15711yescharacteristic
250#103—1571no—
251#103—1571no—
252#103—1571no—
253#103—1571no—
254#103—1571no—
255#102—1571no—
256C22⋊C8C2×D15711yescharacteristic
257C22⋊C8C2×D15711yescharacteristic
258#97C2×D15711yescharacteristic
259#41—3142no—
260#41—6281no—
261C2×Q8—3142no—
262C2×Q8—3142no—
263C2×Q8—3142no—
264C2×Q8—3141no—
265C2×Q8—3141no—
266C2×Q8—3141no—
267C2×Q8—6281no—
268C2×Q8—6281no—
269C2×Q8—6281no—
270C2×Q8—1571no—
271C2×Q8—1571no—
272C2×Q8—1571no—
273C2×Q8—1571no—
274C2×Q8—1571no—
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—1571no—
278C2×D4D62811yescharacteristic
279C22×C4—3141no—
280C22×C4—3141no—
281C22×C4—3141no—
282C22×C4—3141no—
283C22×C4C22×D15711yescharacteristic
284C22×C4C22×D15711yescharacteristic
285Q16—3142no—
286Q16—3142no—
287Q16—3142no—
288Q16—3142no—
289Q16—6281no—
290Q16—1574no—
291SD16—3142no—
292SD16—3142no—
293SD16—6281no—
294M16—22no—
295C2×C8—22no—
296C2×C8C22×D15711yescharacteristic
297C2×C8C22×D15711yescharacteristic
298C2×C8C22×D15711yescharacteristic
299C2×C8C22×D15711yescharacteristic
300C4⋊C4—3141no—
301C4⋊C4—3141no—
302C4⋊C4—3141no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—3141no—
306C4⋊C4—3141no—
307C4⋊C4—1571no—
308C4⋊C4—1571no—
309C4⋊C4D62811yescharacteristic
310C22⋊C4—3141no—
311C22⋊C4—3141no—
312C22⋊C4—3141no—
313C22⋊C4—3141no—
314C22⋊C4—3141no—
315C22⋊C4—3141no—
316C22⋊C4D62811yescharacteristic
317C22⋊C4D62811yescharacteristic
318C42—3141no—
319C42—3141no—
320C42C22×D15711yescharacteristic
321C23—3141no—
322C23C2×D62811yescharacteristic
323C23C2×D62811yescharacteristic
324Q8—3141no—
325Q8—3141no—
326Q8—3141no—
327Q8—3141no—
328Q8—3141no—
329Q8—3141no—
330Q8—3141no—
331Q8—3141no—
332Q8—6282no—
333Q8—6282no—
334Q8—6282no—
335Q8—1572no—
336Q8—1572no—
337D4—22no—
338D4—3141no—
339D4—3141no—
340D4—6282no—
341D4—6282no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—6282no—
354C2×C4—6282no—
355C2×C4—6282no—
356C2×C4—6282no—
357C2×C4C2×D62811yescharacteristic
358C2×C4C2×D62811yescharacteristic
359C2×C4C2×D62811yescharacteristic
360C2×C4C2×D62811yescharacteristic
361C2×C4C23×D15711yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8D4×D15712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—6281no—
372C22—6281no—
373C22—6281no—
374C22#1909780412yescomplemented
375C22C22×D62811yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—3142no—
381C4—3142no—
382C4—6281no—
383C4—6281no—
384C4—6281no—
385C4—6281no—
386C4—6281no—
387C4C2×D4×D15711yescharacteristic
388C4#1909781711yescharacteristic
389C2—22no—
390C2—41no—
391C2—12561no—
392C2D4×D62811yescharacteristic
393C2C2×#1909780411yescharacteristic
394C2#2375809011yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#435×C157C2
socleC22×C157C22×D4
prime radical#23757932C2
hypercenter#435D157
hypocenterC157#3270
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×Dic628C2semidirect
3. D4×Dic314C2semidirect
5. #23757667C2semidirect
6. #23757531C2semidirect
7. #23757502C2semidirect
8. #23757496C2semidirect
9. #23757484C2semidirect
10. #23757437C2semidirect
11. #23757431C2semidirect
12. #23757431C2semidirect
13. #23757414C2semidirect
14. #23757381C2semidirect
15. #435×C157C2semidirect
89. Dic628D4semidirect
173. C22×C157#137semidirect
197. C157#3270semidirect
213. #435D157semidirect
374. 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×Dic6282
D4×Dic3142
#237579322
#237576672
#237575312
#237575022
#237574962
#237574842
#237574372
#237574312
#237574142
#237573812
#435×C1572
#3270157