Finite Group Databasebeta

← #34109598

Subgroups — #34109598

Order 20096.

548 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×#19097798C211yesmaximal, characteristic, complemented
2C2×#19097795C211yesmaximal, characteristic, complemented
3#23757822C211yesmaximal, characteristic, complemented
4C4×D4×D157C211yesmaximal, characteristic, complemented
5#23757489C211yesmaximal, characteristic, complemented
6#23757488C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757477C211yesmaximal, characteristic, complemented
9#23757476C211yesmaximal, characteristic, complemented
10#23757467C211yesmaximal, characteristic, complemented
11#23757466C211yesmaximal, characteristic, complemented
12#23757295C211yesmaximal, characteristic, complemented
13#23757273C211yesmaximal, characteristic, complemented
14#23757271C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19C22×C4×D157—21no—
20#19097798—24no—
21#19097795—24no—
22C2×#14578943C2211yescharacteristic
23C2×#14578942C2211yescharacteristic
24C2×#14578941C2211yescharacteristic
25C2×D1256C2211yescharacteristic
26C2×#14578939C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C2×#14578937C2211yescharacteristic
29C2×#14578937C2211yescharacteristic
30#19097769C2211yescharacteristic, complemented
31#19097767C2211yescharacteristic, complemented
32#19097759C2211yescharacteristic, complemented
33#19097744—21no—
34#19097744—21no—
35C4×D628C2211yescharacteristic
36D4×Dic157C2211yescharacteristic
37#19097733C2211yescharacteristic
38C4×C22⋊D157—21no—
39#19097731—21no—
40#19097731—21no—
41#19097731—21no—
42C4⋊C4×D157C2211yescharacteristic
43C22⋊C4×D157—21no—
44C42×D157C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097700C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097694C2211yescharacteristic, complemented
51#19097693C2211yescharacteristic, complemented
52#19097692C2211yescharacteristic, complemented
53#19097691C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic
55C2×D8×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62#14578955—22no—
63#14578955—22no—
64#14578953—22no—
65#14578953—22no—
66C2×Dic314C2311yescharacteristic
67C2×D628—21no—
68C2×D628C2311yescharacteristic
69C2×D628C2311yescharacteristic
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74D4×D157—22no—
75D4×D157D414yescomplemented
76C22×Dic157—21no—
77C22×Dic157—21no—
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157—44no—
83C2×C4×D157D411yescharacteristic
84C2×C4×D157D411yescharacteristic
85C2×C4×D157C2311yescharacteristic
86#14578943—22no—
87#14578942—22no—
88#14578941—22no—
89D1256—22no—
90#14578939—22no—
91#14578939—22no—
92#14578937—22no—
93#14578937—22no—
94C2×#11929434C2311yescharacteristic
95C2×#11929434C2311yescharacteristic
96C4⋊Dic157C2311yescharacteristic
97#14578933—21no—
98#14578933—21no—
99#14578933—21no—
100C22⋊Dic157—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105C4×Dic157D411yescharacteristic
106C4×Dic157D411yescharacteristic
107C4×Dic157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110C2×D4×C157C2311yescharacteristic
111C22×C4×C157—21no—
112SD16×C157—22no—
113D8×C157—22no—
114C2×C8×C157C2311yescharacteristic
115C2×C8×C157C2311yescharacteristic
116C4⋊C4×C157C2311yescharacteristic
117C22⋊C4×C157—21no—
118C22⋊C4×C157—21no—
119C22⋊C4×C157—21no—
120C42×C157C2311yescharacteristic
121C22×D157—21no—
122C22×D157—24no—
123C22×D157—21no—
124C22×D157—42no—
125C22×D157—42no—
126C22×D157C2×D411yescharacteristic
127Dic314—21no—
128Dic314—21no—
129D628—21no—
130D628—21no—
131D628—21no—
132D628—42no—
133D628C2×D412yes—
134C22⋊D157—24no—
135C22⋊D157—42no—
136C22⋊D157—42no—
137C22⋊D157—42no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—42no—
141C2×Dic157—42no—
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C4×D157—22no—
146C4×D157—41no—
147C4×D157—42no—
148C4×D157—42no—
149C4×D157—42no—
150C4×D157—42no—
151C4×D157C2×D412yes—
152C4×D157#4112yes—
153#11929434—21no—
154#11929434—21no—
155C23×C157—21no—
156C23×C157—21no—
157Q8×C157—21no—
158Q8×C157—21no—
159D4×C157—21no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157C2×D412yes—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—42no—
166C2×C4×C157C2×D411yescharacteristic
167C2×C4×C157C2×D411yescharacteristic
168C2×C4×C157C2411yescharacteristic
169C8×C157—21no—
170C8×C157—21no—
171C22×C157—22no—
172C22×C157—41no—
173C22×C157—41no—
174C22×C157—41no—
175C22×C157—41no—
176C22×C157—41no—
177C22×C157C22×D411yescharacteristic
178C2×D157—21no—
179C2×D157—21no—
180C2×D157—22no—
181C2×D157—48no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157#12112yes—
188C4×C157—21no—
189C4×C157—21no—
190C4×C157—41no—
191C4×C157—41no—
192C4×C157C22×D411yescharacteristic
193C4×C157C2×#4111yescharacteristic
194Dic157—21no—
195Dic157—41no—
196Dic157—41no—
197Dic157#12112yes—
198Dic157AGL(1,ℤ/8)12yescomplemented
199C2×C157—42no—
200C2×C157—81no—
201C2×C157C2×#12111yescharacteristic
202C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
203C2×C157#57511yescharacteristic
204D157—22no—
205D157—42no—
206D157—81no—
207C157#303611yescharacteristic, complemented, Sylow
208#3036—1571nomaximal, Sylow
209C2×AGL(1,ℤ/8)—1571no—
210C2×AGL(1,ℤ/8)—1571no—
211#532—1571no—
212C2×C4×D4—1571no—
213#463—1571no—
214#463—1571no—
215#463—1571no—
216#463D15711yescharacteristic, complemented
217#461—1571no—
218#461—1571no—
219#459—1571no—
220#459—1571no—
221#423—1571no—
222#417—1571no—
223#417—1571no—
224C2×#41—1571no—
225C2×#41—1571no—
226C22×D4—1571no—
227C23×C4—3141no—
228AGL(1,ℤ/8)—3144no—
229AGL(1,ℤ/8)—3144no—
230C2×SD16—1571no—
231C2×SD16—1571no—
232C2×SD16—1571no—
233C2×SD16C2×D15711yescharacteristic
234C2×D8—1571no—
235C2×D8—1571no—
236C2×D8—1571no—
237C2×D8C2×D15711yescharacteristic
238C2×M16—1571no—
239C2×M16—1571no—
240C4⋊Q8—1571no—
241#127—1571no—
242#124—1571no—
243#124C2×D15711yescharacteristic, complemented
244#121—3141no—
245#121—3141no—
246C4×D4—3141no—
247C4×D4—3141no—
248C4×D4—3141no—
249C4×D4—3141no—
250C4×D4—1571no—
251C4×D4—1571no—
252C4×D4—1571no—
253C4×D4C2×D15711yescharacteristic
254C2×C4⋊C4—1571no—
255C2×C22⋊C4—3141no—
256C2×C42—1571no—
257C4⋊C8—1571no—
258C4⋊C8—1571no—
259C4⋊C8—1571no—
260C4⋊C8C2×D15711yescharacteristic, complemented
261#103—1571no—
262#103—1571no—
263#103—1571no—
264#103C2×D15711yescharacteristic, complemented
265#102—1571no—
266#102—1571no—
267#102—1571no—
268#102C2×D15711yescharacteristic, complemented
269C24—3141no—
270#41—3142no—
271#41—3142no—
272#41—3142no—
273#41—3142no—
274C2×Q8—1571no—
275C2×Q8C22×D15711yescharacteristic
276C2×D4—3141no—
277C2×D4—3141no—
278C2×D4—3142no—
279C2×D4—3141no—
280C2×D4—3141no—
281C2×D4—3141no—
282C2×D4—1574no—
283C2×D4—1571no—
284C2×D4—1571no—
285C2×D4C22×D15711yescharacteristic
286C2×D4C22×D15711yescharacteristic
287C22×C4—21no—
288C22×C4—3141no—
289C22×C4—3141no—
290C22×C4—3141no—
291C22×C4—3141no—
292C22×C4—3141no—
293C22×C4—3141no—
294C22×C4—6284no—
295C22×C4—1571no—
296C22×C4—1571no—
297C22×C4—1571no—
298SD16—22no—
299SD16—3142no—
300SD16—3142no—
301SD16—3142no—
302D8—22no—
303D8—3142no—
304D8—3142no—
305D8—3142no—
306M16—3142no—
307M16—3142no—
308C2×C8—1571no—
309C2×C8—1571no—
310C2×C8C22×D15711yescharacteristic
311C2×C8C22×D15711yescharacteristic
312C4⋊C4—3141no—
313C4⋊C4—3141no—
314C4⋊C4—3141no—
315C4⋊C4—1571no—
316C4⋊C4C22×D15711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C42—1571no—
326C42—1571no—
327C42—1571no—
328C42C22×D15711yescharacteristic
329C23—21no—
330C23—21no—
331C23—3144no—
332C23—3141no—
333C23—3141no—
334C23—6282no—
335C23—6282no—
336C23—1571no—
337Q8—21no—
338Q8—21no—
339Q8—3141no—
340Q8—3141no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—3144no—
345D4—3141no—
346D4—3141no—
347D4—3141no—
348D4—6282no—
349D4—6282no—
350D4—6282no—
351D4—6282no—
352D4—1572no—
353D4D4×D15712yes—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—3142no—
358C2×C4—3141no—
359C2×C4—3141no—
360C2×C4—6281no—
361C2×C4—6282no—
362C2×C4—6282no—
363C2×C4—6282no—
364C2×C4—6282no—
365C2×C4—6282no—
366C2×C4—6282no—
367C2×C4—1571no—
368C2×C4—1571no—
369C2×C4—1571no—
370C2×C4—1572no—
371C2×C4—1572no—
372C2×C4D4×D15711yescharacteristic
373C2×C4D4×D15711yescharacteristic
374C2×C4C23×D15711yescharacteristic
375C8—21no—
376C8—21no—
377C8—3141no—
378C8—3141no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—3141no—
386C22—3141no—
387C22—3142no—
388C22—6288no—
389C22—6281no—
390C22—6281no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—1572no—
395C22C2×D4×D15711yescharacteristic
396C4—21no—
397C4—21no—
398C4—41no—
399C4—41no—
400C4—3141no—
401C4—6281no—
402C4—6281no—
403C4—1572no—
404C4—1572no—
405C4C2×D4×D15711yescharacteristic
406C4#41×D15711yescharacteristic
407C2—42no—
408C2—81no—
409C2—3142no—
410C2—6282no—
411C2—12561no—
412C2#121×D15711yescharacteristic
413C2AGL(1,ℤ/8)×D15711yescharacteristic
414C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
hypercenter#463D157
hypocenterC157#3036
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097795C2semidirect
3. #23757822C2semidirect
4. C4×D4×D157C2semidirect
5. #23757489C2semidirect
6. #23757488C2semidirect
7. #23757485C2semidirect
8. #23757477C2semidirect
9. #23757476C2semidirect
10. #23757467C2semidirect
11. #23757466C2semidirect
12. #23757295C2semidirect
13. #23757273C2semidirect
14. #23757271C2semidirect
15. #463×C157C2semidirect
30. #19097769C22semidirect
31. #19097767C22semidirect
32. #19097759C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097700C22semidirect
47. #19097700C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097694C22semidirect
51. #19097693C22semidirect
52. #19097692C22semidirect
53. #19097691C22semidirect
56. #124×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
75. D4×D157D4semidirect
198. Dic157AGL(1,ℤ/8)semidirect
207. C157#3036semidirect
216. #463D157semidirect
243. #124C2×D157semidirect
260. C4⋊C8C2×D157semidirect
264. #103C2×D157semidirect
268. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C2×#190977952
#237578222
C4×D4×D1572
#237574892
#237574882
#237574852
#237574772
#237574762
#237574672
#237574662
#237572952
#237572732
#237572712
#463×C1572
#3036157