Finite Group Databasebeta

← #34109661

Subgroups — #34109661

Order 20096.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic
2C2×Q16×D157C211yesmaximal, characteristic, complemented
3#23757848C211yesmaximal, characteristic
4#23757749C211yesmaximal, characteristic, complemented
5#23757498C211yesmaximal, characteristic, complemented
6#23757497C211yesmaximal, characteristic, complemented
7#23757496C211yesmaximal, characteristic, complemented
8#23757495C211yesmaximal, characteristic, complemented
9#23757482C211yesmaximal, characteristic, complemented
10#23757481C211yesmaximal, characteristic, complemented
11#23757480C211yesmaximal, characteristic, complemented
12#23757298C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757268C211yesmaximal, characteristic
15#462×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097803—24no—
20C2×Dic628C2211yescharacteristic
21C2×#14578944C2211yescharacteristic
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24Q16×D157—24no—
25C2×#14578943C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097769C2211yescharacteristic
30#19097760C2211yescharacteristic
31#19097759C2211yescharacteristic
32#19097756—21no—
33#19097756—21no—
34#19097749—21no—
35#19097749—21no—
36Q8×Dic157C2211yescharacteristic
37C4×Dic314—21no—
38C4×Dic314C2211yescharacteristic
39#19097736—21no—
40#19097736C2211yescharacteristic
41#19097729—21no—
42#19097729C2211yescharacteristic
43#19097728—21no—
44#19097728C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic
46#19097700C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097695C2211yescharacteristic
50#19097695C2211yescharacteristic
51#19097694C2211yescharacteristic
52#19097693C2211yescharacteristic
53#19097691C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic
56C4⋊Q8×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#103×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—22no—
63C2×Dic314—21no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—22no—
69Q8×D157—22no—
70Q8×D157D414yes—
71C2×D628—21no—
72C2×D628C2311yescharacteristic
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77Dic628—22no—
78#14578944—22no—
79#14578944—22no—
80#14578944—22no—
81#14578943—22no—
82#14578941—22no—
83#14578937—22no—
84C2×#11929434C2311yescharacteristic
85C2×#11929434C2311yescharacteristic
86C8×D157—22no—
87C4⋊Dic157—21no—
88C4⋊Dic157C2311yescharacteristic
89#14578933—21no—
90#14578933—21no—
91#14578933—21no—
92#14578933—21no—
93#14578933D411yescharacteristic
94#14578933D411yescharacteristic
95#14578931—21no—
96#14578931—21no—
97#14578931—21no—
98#14578931—21no—
99#14578931—21no—
100#14578931D411yescharacteristic
101#14578931D411yescharacteristic
102C4×Dic157—21no—
103C4×Dic157—21no—
104C4×Dic157—21no—
105C4×Dic157—21no—
106C4×Dic157C2311yescharacteristic
107C2×Q8×C157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C2×Q8×C157C2311yescharacteristic
110Q16×C157—22no—
111Q16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157—21no—
116C4⋊C4×C157—21no—
117C4⋊C4×C157C2311yescharacteristic
118C42×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157—21no—
121C22×D157C2×D411yescharacteristic
122Dic314—21no—
123Dic314—24no—
124Dic314—21no—
125Dic314—21no—
126Dic314—42no—
127Dic314C2×D412yes—
128D628—21no—
129D628—21no—
130D628—42no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157C2×D411yescharacteristic
135C2×Dic157C2×D411yescharacteristic
136C2×Dic157C2×D411yescharacteristic
137C4×D157—24no—
138C4×D157—42no—
139C4×D157—42no—
140C4×D157C2×D412yes—
141C4×D157#4112yes—
142#11929434—21no—
143#11929434—21no—
144Q8×C157—21no—
145Q8×C157—21no—
146Q8×C157—21no—
147Q8×C157—21no—
148Q8×C157—21no—
149Q8×C157C2×D412yes—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157—21no—
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2×D411yescharacteristic
156C2×C4×C157C2411yescharacteristic
157C8×C157—21no—
158C8×C157—21no—
159C22×C157C22×D411yescharacteristic
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157#12112yes—
166C4×C157—22no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157—41no—
170C4×C157—41no—
171C4×C157—41no—
172C4×C157C22×D411yescharacteristic
173C4×C157C2×#4111yescharacteristic
174Dic157—22no—
175Dic157—21no—
176Dic157—21no—
177Dic157—41no—
178Dic157—41no—
179Dic157#12112yes—
180C2×C157C2×#12111yescharacteristic
181C2×C157#57811yescharacteristic
182C2×C157#57811yescharacteristic
183D157—22no—
184D157—81no—
185C157#304411yescharacteristic, complemented, Sylow
186#3044—1571nomaximal, Sylow
187C2×#137—1571no—
188C22×Q16—1571no—
189#536—1571no—
190#519—1571no—
191#464—1571no—
192#464—1571no—
193#464—1571no—
194#464—1571no—
195#462—1571no—
196#462—1571no—
197#462—1571no—
198#462D15711yescharacteristic, complemented
199#424—1571no—
200#419—1571no—
201#416—1571no—
202C2×#41—1571no—
203C22×Q8—1571no—
204C22×Q8—1571no—
205#137—3144no—
206C2×Q16—3144no—
207C2×Q16—1571no—
208C2×Q16—1571no—
209C2×Q16—1571no—
210C2×Q16—1571no—
211C2×Q16C2×D15711yescharacteristic
212C2×Q16C2×D15711yescharacteristic
213C2×SD16—1571no—
214C2×SD16—1571no—
215C2×M16—1571no—
216C22×C8—1571no—
217C4⋊Q8—1571no—
218C4⋊Q8C2×D15711yescharacteristic
219#124—1571no—
220#124—1571no—
221#123—3141no—
222#123—3141no—
223C22⋊Q8—3141no—
224C22⋊Q8—3141no—
225C4×Q8—3141no—
226C4×Q8—3141no—
227C4×Q8—1571no—
228C4×Q8—1571no—
229C4×Q8—1571no—
230C4×Q8C2×D15711yescharacteristic
231#117—3141no—
232#117—3141no—
233#117—1571no—
234#117—1571no—
235C4⋊C8—1571no—
236C4⋊C8—1571no—
237C4⋊C8—1571no—
238C4⋊C8C2×D15711yescharacteristic
239#103—1571no—
240#103—1571no—
241#103—1571no—
242#103—1571no—
243#103C2×D15711yescharacteristic
244#103C2×D15711yescharacteristic
245#102—1571no—
246#102—1571no—
247#41—3142no—
248#41—3142no—
249C2×Q8—3141no—
250C2×Q8—3141no—
251C2×Q8—3142no—
252C2×Q8—3142no—
253C2×Q8—3142no—
254C2×Q8—1574no—
255C2×Q8—1571no—
256C2×Q8—1571no—
257C2×Q8C22×D15711yescharacteristic
258C2×Q8C22×D15711yescharacteristic
259C2×Q8C22×D15711yescharacteristic
260C2×D4—3141no—
261C2×D4—1571no—
262C22×C4—3141no—
263C22×C4—3141no—
264C22×C4—3141no—
265C22×C4—1571no—
266Q16—22no—
267Q16—22no—
268Q16—3142no—
269Q16—3142no—
270Q16—3142no—
271Q16—3142no—
272SD16—3142no—
273SD16—3142no—
274M16—3142no—
275C2×C8—3142no—
276C2×C8—1571no—
277C2×C8—1571no—
278C2×C8C22×D15711yescharacteristic
279C2×C8C22×D15711yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3141no—
284C4⋊C4—3141no—
285C4⋊C4—3141no—
286C4⋊C4—3141no—
287C4⋊C4—3141no—
288C4⋊C4—1571no—
289C4⋊C4—1571no—
290C4⋊C4—1571no—
291C4⋊C4C22×D15711yescharacteristic
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—1571no—
298C22⋊C4—1571no—
299C42—21no—
300C42—3141no—
301C42—3141no—
302C42—3141no—
303C42—3141no—
304C42—1571no—
305C42C22×D15711yescharacteristic
306C23—3141no—
307C23—1571no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—3144no—
314Q8—3141no—
315Q8—3141no—
316Q8—3141no—
317Q8—6282no—
318Q8—1572no—
319Q8D4×D15712yes—
320D4—3141no—
321D4—3141no—
322D4—6282no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—3144no—
328C2×C4—3141no—
329C2×C4—3141no—
330C2×C4—3141no—
331C2×C4—6282no—
332C2×C4—6282no—
333C2×C4—1571no—
334C2×C4—1571no—
335C2×C4—1571no—
336C2×C4—1572no—
337C2×C4—1572no—
338C2×C4D4×D15711yescharacteristic
339C2×C4D4×D15711yescharacteristic
340C2×C4C23×D15711yescharacteristic
341C8—21no—
342C8—21no—
343C8—3141no—
344C8—3141no—
345C22—3141no—
346C22—3141no—
347C22—6281no—
348C22—6281no—
349C22—6281no—
350C22—1572no—
351C22C2×D4×D15711yescharacteristic
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3141no—
359C4—3141no—
360C4—3142no—
361C4—6281no—
362C4—6281no—
363C4—1572no—
364C4C2×D4×D15711yescharacteristic
365C4#41×D15711yescharacteristic
366C2—3142no—
367C2—12561no—
368C2#121×D15711yescharacteristic
369C2#2375808611yescharacteristic
370C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#462×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578955C22
hypercenter#462D157
hypocenterC157#3044
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D157C2semidirect
4. #23757749C2semidirect
5. #23757498C2semidirect
6. #23757497C2semidirect
7. #23757496C2semidirect
8. #23757495C2semidirect
9. #23757482C2semidirect
10. #23757481C2semidirect
11. #23757480C2semidirect
12. #23757298C2semidirect
13. #23757280C2semidirect
15. #462×C157C2semidirect
185. C157#3044semidirect
198. #462D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×Q16×D1572
#237578482
#237577492
#237574982
#237574972
#237574962
#237574952
#237574822
#237574812
#237574802
#237572982
#237572802
#237572682
#462×C1572
#3044157