Finite Group Databasebeta

← #34111726

Subgroups — #34111726

Order 20096.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097804C211yesmaximal, characteristic, complemented
2C2×#19097793C211yesmaximal, characteristic, complemented
3#23757944C211yesmaximal, characteristic, complemented
4#23757932C211yesmaximal, characteristic, complemented
5#23757662C211yesmaximal, characteristic, complemented
6#23757526C211yesmaximal, characteristic, complemented
7#23757502C211yesmaximal, characteristic, complemented
8#23757496C211yesmaximal, characteristic, complemented
9#23757474C211yesmaximal, characteristic, complemented
10#23757437C211yesmaximal, characteristic, complemented
11#23757420C211yesmaximal, characteristic, complemented
12#23757420C211yesmaximal, characteristic, complemented
13#23757414C211yesmaximal, characteristic, complemented
14#23757381C211yesmaximal, characteristic, complemented
15#435×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C2×#14578954C2211yescharacteristic
18C2×#14578954C2211yescharacteristic
19C22×Dic314C2211yescharacteristic
20#19097804—24no—
21#19097793—24no—
22C2×Dic628C2211yescharacteristic
23C2×Dic628C2211yescharacteristic
24C2×#14578943C2211yescharacteristic
25C2×#14578943C2211yescharacteristic
26C2×#14578943C2211yescharacteristic
27C2×#14578943C2211yescharacteristic
28C2×#14578943C2211yescharacteristic
29C2×D1256C2211yescharacteristic
30C2×D1256C2211yescharacteristic
31#19097768C2211yescharacteristic
32#19097767—21no—
33#19097759C2211yescharacteristic
34#19097757—21no—
35#19097750C2211yescharacteristic
36#19097749—21no—
37#19097745C2211yescharacteristic
38#19097745C2211yescharacteristic
39#19097745C2211yescharacteristic
40#19097744—21no—
41#19097744—21no—
42#19097743—21no—
43C4×Dic314C2211yescharacteristic
44C4×Dic314C2211yescharacteristic
45#19097733—21no—
46#19097731—21no—
47#19097703C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097696C2211yescharacteristic
50#19097696C2211yescharacteristic
51#19097693C2211yescharacteristic
52#19097693C2211yescharacteristic
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—22no—
63#14578954—22no—
64#14578954—41no—
65#14578954—41no—
66#14578954—41no—
67C2×Dic314—22no—
68C2×Dic314—21no—
69C2×Dic314—41no—
70C2×Dic314C2311yescharacteristic
71C2×Dic314C2311yescharacteristic
72C2×Dic314C2311yescharacteristic
73C2×D628—21no—
74C2×D628—21no—
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×D628C2311yescharacteristic
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81C2×C22⋊D157—21no—
82C2×C22⋊D157—21no—
83C2×C22⋊D157—21no—
84C22×Dic157—21no—
85C2×C4×D157—21no—
86C2×C4×D157—21no—
87C2×C4×D157—21no—
88Dic628—22no—
89Dic628D414yescomplemented
90#14578943—22no—
91#14578943—22no—
92#14578943—22no—
93#14578943—22no—
94#14578943—41no—
95D1256—22no—
96D1256—41no—
97C4⋊Dic157C2311yescharacteristic
98C4⋊Dic157C2311yescharacteristic
99#14578933—21no—
100#14578933—21no—
101#14578933—21no—
102C22⋊Dic157—21no—
103#14578931—21no—
104#14578931—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—
124C22×D157—21no—
125C22×D157—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—21no—
130Dic314—42no—
131Dic314C2×D412yes—
132Dic314C2×D412yes—
133D628—21no—
134D628—21no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628—21no—
139D628—42no—
140D628—81no—
141C22⋊D157—42no—
142C22⋊D157—42no—
143C22⋊D157—42no—
144C22⋊D157—42no—
145C22⋊D157—81no—
146C22⋊D157—81no—
147C2×Dic157—21no—
148C2×Dic157—21no—
149C2×Dic157—21no—
150C2×Dic157—21no—
151C2×Dic157—21no—
152C2×Dic157—42no—
153C4×D157—42no—
154C4×D157—42no—
155C4×D157—42no—
156C23×C157C2×D411yescharacteristic
157C23×C157C2×D411yescharacteristic
158D4×C157—41no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157C2×D411yescharacteristic
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2411yescharacteristic
168C8×C157—21no—
169C8×C157—21no—
170C8×C157—21no—
171C8×C157C2×D412yes—
172C22×C157—21no—
173C22×C157—21no—
174C22×C157—21no—
175C22×C157—21no—
176C22×C157—21no—
177C22×C157—21no—
178C22×C157C22×D411yescharacteristic
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C4×C157—21no—
189C4×C157—21no—
190C4×C157—21no—
191C4×C157—21no—
192C4×C157C22×D411yescharacteristic
193C4×C157H(5,𝔽2)11yescharacteristic
194Dic157—22no—
195Dic157—22no—
196Dic157—41no—
197Dic157—41no—
198Dic157—41no—
199C2×C157—41no—
200C2×C157—41no—
201C2×C157D4211yescharacteristic
202C2×C157#57511yescharacteristic
203C2×C157#57711yescharacteristic
204D157—81no—
205D157—81no—
206D157—81no—
207C157#327111yescharacteristic, complemented, Sylow
208#3271—1571nomaximal, Sylow
209C2×#137—1571no—
210C2×#135—1571no—
211#550—1571no—
212#548—1571no—
213#496—1571no—
214#469—1571no—
215#465—1571no—
216#464—1571no—
217#460—1571no—
218#452—1571no—
219#449—1571no—
220#449—1571no—
221#448—1571no—
222#441—1571no—
223#435D15711yescharacteristic, complemented
224C2×#41—1571no—
225C2×#41—1571no—
226C2×#41—1571no—
227C22×Q8—1571no—
228#137—3144no—
229#135—3144no—
230C2×Q16—1571no—
231C2×Q16—1571no—
232C2×SD16—1571no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×D8—1571no—
238C2×D8—1571no—
239C2×M16C2×D15711yescharacteristic
240C22×C8C2×D15711yescharacteristic
241#127—3141no—
242#127—1571no—
243#124—3141no—
244#124—1571no—
245C22⋊Q8—3141no—
246C22⋊Q8—1571no—
247#121—3141no—
248#121—3141no—
249#121—3141no—
250#121—1571no—
251#121—1571no—
252#121—1571no—
253C4×Q8—1571no—
254C4×Q8—1571no—
255C4×D4—3141no—
256C4×D4—3141no—
257C4×D4C2×D15711yescharacteristic
258#106—1571no—
259C4⋊C8C2×D15711yescharacteristic
260#103—1571no—
261#103—1571no—
262#103—1571no—
263#102—1571no—
264#102—1571no—
265#102—1571no—
266C22⋊C8C2×D15711yescharacteristic
267C22⋊C8C2×D15711yescharacteristic
268#97C2×D15711yescharacteristic
269#41—3142no—
270#41—3142no—
271#41—3142no—
272#41—6281no—
273#41—6281no—
274#41—6281no—
275C2×Q8—3142no—
276C2×Q8—3141no—
277C2×Q8—6281no—
278C2×Q8—1571no—
279C2×Q8—1571no—
280C2×Q8—1571no—
281C2×D4—3141no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4—1571no—
292C2×D4D62811yescharacteristic
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—3141no—
297C22×C4C22×D15711yescharacteristic
298C22×C4C22×D15711yescharacteristic
299Q16—3142no—
300Q16—1574no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—3142no—
305SD16—6281no—
306D8—3142no—
307D8—6281no—
308M16—22no—
309C2×C8—22no—
310C2×C8C22×D15711yescharacteristic
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C2×C8C22×D15711yescharacteristic
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—3141no—
317C4⋊C4—1571no—
318C4⋊C4—1571no—
319C4⋊C4D62811yescharacteristic
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4D62811yescharacteristic
327C22⋊C4D62811yescharacteristic
328C42—3141no—
329C42—3141no—
330C42C22×D15711yescharacteristic
331C23—3141no—
332C23—3141no—
333C23—3141no—
334C23C2×D62811yescharacteristic
335C23C2×D62811yescharacteristic
336Q8—3141no—
337Q8—3141no—
338Q8—3141no—
339Q8—3141no—
340Q8—6282no—
341Q8—1572no—
342Q8—1572no—
343D4—41no—
344D4—3141no—
345D4—3141no—
346D4—3141no—
347D4—3141no—
348D4—3141no—
349D4—3141no—
350D4—6282no—
351D4—6282no—
352D4—6282no—
353D4—6282no—
354D4—6282no—
355D4—12561no—
356D4—12561no—
357D4—12561no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—3141no—
363C2×C4—3141no—
364C2×C4—3141no—
365C2×C4—3141no—
366C2×C4—3141no—
367C2×C4—6282no—
368C2×C4—6282no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4C2×D62811yescharacteristic
372C2×C4C2×D62811yescharacteristic
373C2×C4C2×D62811yescharacteristic
374C2×C4C2×D62811yescharacteristic
375C2×C4C23×D15711yescharacteristic
376C8—21no—
377C8—21no—
378C8—21no—
379C8D4×D15712yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—6281no—
387C22—6281no—
388C22—6281no—
389C22—6281no—
390C22—6281no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22C22×D62811yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—3142no—
401C4—3142no—
402C4—6281no—
403C4—6281no—
404C4—6281no—
405C4C2×D4×D15711yescharacteristic
406C4#1909781711yescharacteristic
407C2—41no—
408C2—41no—
409C2—12561no—
410C2—12561no—
411C2—12561no—
412C2D4×D62811yescharacteristic
413C2#2375807311yescharacteristic
414C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#435×C157C2
socleC22×C157C22×D4
hypercenter#435D157
hypocenterC157#3271
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. C2×#19097793C2semidirect
3. #23757944C2semidirect
4. #23757932C2semidirect
5. #23757662C2semidirect
6. #23757526C2semidirect
7. #23757502C2semidirect
8. #23757496C2semidirect
9. #23757474C2semidirect
10. #23757437C2semidirect
11. #23757420C2semidirect
12. #23757420C2semidirect
13. #23757414C2semidirect
14. #23757381C2semidirect
15. #435×C157C2semidirect
89. Dic628D4semidirect
207. C157#3271semidirect
223. #435D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C2×#190977932
#237579442
#237579322
#237576622
#237575262
#237575022
#237574962
#237574742
#237574372
#237574202
#237574142
#237573812
#435×C1572
#3271157