Finite Group Databasebeta

← #34109664

Subgroups — #34109664

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×#19097800C211yesmaximal, characteristic
2C2×Q16×D157C211yesmaximal, characteristic, complemented
3#23757843C211yesmaximal, characteristic
4#23757749C211yesmaximal, characteristic, complemented
5#23757494C211yesmaximal, characteristic, complemented
6#23757493C211yesmaximal, characteristic, complemented
7#23757492C211yesmaximal, characteristic, complemented
8#23757484C211yesmaximal, characteristic, complemented
9#23757483C211yesmaximal, characteristic, complemented
10#23757482C211yesmaximal, characteristic, complemented
11#23757481C211yesmaximal, characteristic, complemented
12#23757298C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic
15#464×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097800—24no—
20C2×Dic628C2211yescharacteristic
21C2×Dic628C2211yescharacteristic
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578944C2211yescharacteristic
25Q16×D157—24no—
26C2×#14578942C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097771C2211yescharacteristic
30#19097769C2211yescharacteristic
31#19097758C2211yescharacteristic
32#19097753—21no—
33#19097753—21no—
34#19097748—21no—
35#19097748—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#19097696C2211yescharacteristic
50#19097695C2211yescharacteristic
51#19097694C2211yescharacteristic
52#19097694C2211yescharacteristic
53#19097692C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#102×C157C2211yescharacteristic
61#14578953—22no—
62#14578953—22no—
63C2×Dic314—21no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68Q8×D157—22no—
69Q8×D157—22no—
70Q8×D157—22no—
71Q8×D157D414yes—
72C2×C22⋊D157—21no—
73C22×Dic157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77Dic628—22no—
78Dic628—22no—
79#14578944—22no—
80#14578944—22no—
81#14578944—22no—
82#14578942—22no—
83#14578937—22no—
84C2×#11929434C2311yescharacteristic
85C2×#11929434C2311yescharacteristic
86C8×D157—22no—
87C4⋊Dic157—21no—
88C4⋊Dic157—21no—
89C4⋊Dic157—21no—
90C4⋊Dic157C2311yescharacteristic
91#14578933—21no—
92#14578933—21no—
93#14578933—21no—
94#14578933—21no—
95#14578933D411yescharacteristic
96#14578933D411yescharacteristic
97C22⋊Dic157—21no—
98C22⋊Dic157—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×D4×C157C2311yescharacteristic
110Q16×C157—22no—
111SD16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C42×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157C2×D411yescharacteristic
121Dic314—21no—
122Dic314—24no—
123Dic314—21no—
124Dic314—21no—
125Dic314—21no—
126Dic314—21no—
127Dic314—42no—
128Dic314C2×D412yes—
129C22⋊D157—42no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—42no—
135C2×Dic157C2×D411yescharacteristic
136C2×Dic157C2×D411yescharacteristic
137C2×Dic157C2×D411yescharacteristic
138C4×D157—24no—
139C4×D157—42no—
140C4×D157C2×D412yes—
141C4×D157#4112yes—
142#11929434—21no—
143#11929434—21no—
144C23×C157—21no—
145Q8×C157—21no—
146Q8×C157—21no—
147Q8×C157—21no—
148Q8×C157C2×D412yes—
149D4×C157—21no—
150D4×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×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157C22×D411yescharacteristic
163C2×D157—21no—
164C2×D157—21no—
165C2×D157#12112yes—
166C4×C157—22no—
167C4×C157—21no—
168C4×C157—21no—
169C4×C157—41no—
170C4×C157—41no—
171C4×C157C22×D411yescharacteristic
172C4×C157C2×#4111yescharacteristic
173Dic157—22no—
174Dic157—21no—
175Dic157—21no—
176Dic157—41no—
177Dic157—41no—
178Dic157—41no—
179Dic157#12112yes—
180C2×C157—81no—
181C2×C157C2×#12111yescharacteristic
182C2×C157#57811yescharacteristic
183C2×C157#57811yescharacteristic
184D157—22no—
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#464D15711yescharacteristic, complemented
195#462—1571no—
196#462—1571no—
197#462—1571no—
198#462—1571no—
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×Q16—1571no—
212C2×Q16C2×D15711yescharacteristic
213C2×SD16—1571no—
214C2×SD16C2×D15711yescharacteristic
215C2×M16—1571no—
216C22×C8—1571no—
217C4⋊Q8—1571no—
218C4⋊Q8—1571no—
219#124—1571no—
220#124C2×D15711yescharacteristic
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#103—1571no—
244#103C2×D15711yescharacteristic
245#102—1571no—
246#102C2×D15711yescharacteristic
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×Q8—1571no—
258C2×Q8C22×D15711yescharacteristic
259C2×Q8C22×D15711yescharacteristic
260C2×D4—3141no—
261C2×D4C22×D15711yescharacteristic
262C22×C4—3141no—
263C22×C4—3141no—
264C22×C4—3141no—
265C22×C4—1571no—
266Q16—22no—
267Q16—3142no—
268Q16—3142no—
269Q16—3142no—
270Q16—3142no—
271Q16—3142no—
272SD16—22no—
273SD16—3142no—
274M16—3142no—
275C2×C8—3142no—
276C2×C8—1571no—
277C2×C8—1571no—
278C2×C8C22×D15711yescharacteristic
279C2×C8C22×D15711yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—3141no—
282C4⋊C4—3141no—
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—21no—
293C22⋊C4—21no—
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—21no—
307C23—1571no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—3144no—
312Q8—3141no—
313Q8—3141no—
314Q8—3141no—
315Q8—3141no—
316Q8—3141no—
317Q8—6282no—
318Q8—1572no—
319Q8D4×D15712yes—
320D4—21no—
321D4—21no—
322D4—6282no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—3144no—
327C2×C4—3141no—
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—41no—
346C22—41no—
347C22—41no—
348C22—3141no—
349C22—3141no—
350C22—1572no—
351C22C2×D4×D15711yescharacteristic
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—3141no—
358C4—3141no—
359C4—3142no—
360C4—6281no—
361C4—6281no—
362C4—6281no—
363C4—1572no—
364C4C2×D4×D15711yescharacteristic
365C4#41×D15711yescharacteristic
366C2—81no—
367C2—3142no—
368C2#121×D15711yescharacteristic
369C2#2375808711yescharacteristic
370C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#464×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578953C22
hypercenter#464D157
hypocenterC157#3044
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. C2×Q16×D157C2semidirect
4. #23757749C2semidirect
5. #23757494C2semidirect
6. #23757493C2semidirect
7. #23757492C2semidirect
8. #23757484C2semidirect
9. #23757483C2semidirect
10. #23757482C2semidirect
11. #23757481C2semidirect
12. #23757298C2semidirect
13. #23757280C2semidirect
15. #464×C157C2semidirect
185. C157#3044semidirect
194. #464D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C2×Q16×D1572
#237578432
#237577492
#237574942
#237574932
#237574922
#237574842
#237574832
#237574822
#237574812
#237572982
#237572802
#237572662
#464×C1572
#3044157