Finite Group Databasebeta

← #34111997

Subgroups — #34111997

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D157C211yesmaximal, characteristic, complemented
2#23757924C211yesmaximal, characteristic
3#23757622C211yesmaximal, characteristic, complemented
4#23757603C211yesmaximal, characteristic, complemented
5#23757602C211yesmaximal, characteristic, complemented
6#23757556C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757477C211yesmaximal, characteristic, complemented
10#23757413C211yesmaximal, characteristic, complemented
11#23757412C211yesmaximal, characteristic, complemented
12#23757358C211yesmaximal, characteristic, complemented
13C8×C22⋊D157C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15C2×#103×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×Q8×D157C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22#19097770C2211yescharacteristic
23#19097769—21no—
24#19097769C2211yescharacteristic
25#19097756—21no—
26#19097754—21no—
27#19097750—21no—
28#19097750C2211yescharacteristic
29#19097749—21no—
30#19097747—21no—
31#19097747C2211yescharacteristic
32#19097746C2211yescharacteristic
33#19097745C2211yescharacteristic
34Q8×Dic157C2211yescharacteristic
35#19097733C2211yescharacteristic
36C4×C22⋊D157—21no—
37C4×C22⋊D157C2211yescharacteristic
38C4⋊C4×D157C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097702C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097693C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C22×Q8×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54C2×C4⋊C4×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#14578954—44no—
60C2×Dic314—21no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63Q8×D157—22no—
64Q8×D157—41no—
65C2×D628—21no—
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157D411yescharacteristic
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157C2311yescharacteristic
73#14578943—41no—
74#14578941—22no—
75#14578941—41no—
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C8×D157—41no—
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88#14578933D411yescharacteristic
89C22⋊Dic157—21no—
90C22⋊Dic157D411yescharacteristic
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931D411yescharacteristic
96C4×Dic157—21no—
97C4×Dic157C2311yescharacteristic
98C2×Q8×C157—21no—
99C2×Q8×C157—22no—
100C2×Q8×C157—41no—
101C2×Q8×C157C2311yescharacteristic
102C2×Q8×C157C2311yescharacteristic
103C22×C4×C157—21no—
104C22×C4×C157—21no—
105C22×C4×C157C2311yescharacteristic
106C2×C8×C157—41no—
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157—21no—
110C4⋊C4×C157C2311yescharacteristic
111C4⋊C4×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157C2×D411yescharacteristic
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117Dic314—42no—
118D628—21no—
119D628—21no—
120D628—42no—
121C22⋊D157—42no—
122C22⋊D157SD1614yescomplemented
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—42no—
132C4×D157—42no—
133C4×D157—42no—
134#11929434—21no—
135#11929434—21no—
136C23×C157C2×D411yescharacteristic
137Q8×C157—21no—
138Q8×C157—21no—
139Q8×C157—21no—
140Q8×C157—42no—
141Q8×C157#4112yescomplemented
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—42no—
149C2×C4×C157—42no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2411yescharacteristic
152C8×C157—21no—
153C8×C157—21no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157C2×SD1612yes—
157C22×C157C22×D411yescharacteristic
158C2×D157—21no—
159C2×D157—21no—
160C2×D157—41no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157C2×SD1612yes—
164C4×C157—21no—
165C4×C157—22no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—21no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157C2×SD1612yes—
177C2×C157—22no—
178C2×C157#54611yescharacteristic
179C2×C157C22×SD1611yescharacteristic
180C2×C157#57811yescharacteristic
181D157—22no—
182D157—81no—
183C157#329811yescharacteristic, complemented, Sylow
184#3298—1571nomaximal, Sylow
185D4×Q8—1571no—
186#547—1571no—
187#487—1571no—
188#483—1571no—
189#483—1571no—
190#475—1571no—
191#465—1571no—
192#465—1571no—
193#461—1571no—
194#448—1571no—
195#448—1571no—
196C4×SD16—1571no—
197C8×D4—1571no—
198C2×#103—1571no—
199C2×#103D15711yescharacteristic, complemented
200C2×#41—3141no—
201C22×Q8—1571no—
202C22×Q8C2×D15711yescharacteristic
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×SD16—1571no—
206C22×C8—1571no—
207C22×C8C2×D15711yescharacteristic
208C4⋊Q8—3141no—
209C4⋊Q8—1571no—
210C4⋊Q8—1571no—
211#123—3141no—
212#123—3141no—
213C22⋊Q8—3141no—
214C22⋊Q8—3141no—
215C22⋊Q8—3141no—
216C22⋊Q8—1571no—
217C22⋊Q8—1571no—
218#121—1571no—
219#121—1571no—
220C4×Q8—1571no—
221C4×D4—3141no—
222C4×D4—1571no—
223C4×D4—1571no—
224C2×C4⋊C4—1571no—
225C2×C4⋊C4C2×D15711yescharacteristic
226#106—1571no—
227#106—1571no—
228#106—1571no—
229C4⋊C8—1571no—
230#103—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103C2×D15711yescharacteristic
236#103C2×D15711yescharacteristic
237#103C2×D15711yescharacteristic
238#103C2×D15711yescharacteristic
239#102—1571no—
240C22⋊C8—1571no—
241C22⋊C8—1571no—
242C4×C8—1571no—
243#41—6284no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—3141no—
248C2×Q8—3141no—
249C2×Q8—3142no—
250C2×Q8—6281no—
251C2×Q8—1571no—
252C2×Q8C22×D15711yescharacteristic
253C2×Q8C22×D15711yescharacteristic
254C2×D4—3141no—
255C2×D4—3141no—
256C2×D4—1571no—
257C2×D4—1571no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—3141no—
263C22×C4—1571no—
264C22×C4C22×D15711yescharacteristic
265SD16—3142no—
266SD16—6281no—
267SD16—6281no—
268C2×C8—41no—
269C2×C8—6281no—
270C2×C8—1571no—
271C2×C8—1571no—
272C2×C8C22×D15711yescharacteristic
273C2×C8C22×D15711yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—3141no—
280C4⋊C4—3141no—
281C4⋊C4—3141no—
282C4⋊C4—1571no—
283C4⋊C4—1571no—
284C4⋊C4—1571no—
285C4⋊C4C22×D15711yescharacteristic
286C4⋊C4C22×D15711yescharacteristic
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—1571no—
293C22⋊C4—1571no—
294C42—3141no—
295C42—1571no—
296C23—3141no—
297C23—1571no—
298C23D4×D15711yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—3141no—
304Q8—3141no—
305Q8—6282no—
306Q8—6282no—
307Q8#1457895412yescomplemented
308D4—3141no—
309D4—3141no—
310D4—6282no—
311D4—6282no—
312D4—1574no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—6282no—
327C2×C4—6282no—
328C2×C4—6282no—
329C2×C4—1571no—
330C2×C4—1571no—
331C2×C4—1571no—
332C2×C4D4×D15711yescharacteristic
333C2×C4C23×D15711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3141no—
337C8—3141no—
338C22—21no—
339C22—21no—
340C22—3141no—
341C22—3141no—
342C22—6281no—
343C22—6281no—
344C22—6281no—
345C22—1572no—
346C22SD16×D15712yes—
347C22C2×D4×D15711yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3141no—
354C4—3141no—
355C4—6281no—
356C4—6281no—
357C4—6281no—
358C4—1572no—
359C4C2×#1457895411yescharacteristic
360C4#1909781611yescharacteristic
361C2—22no—
362C2—3142no—
363C2—12561no—
364C2#2375791311yescharacteristic
365C2C2×SD16×D15711yescharacteristic
366C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#103×C157C2
socleC22×C157C22×D4
prime radical#23757924C2
hypercenterC2×#103D157
hypocenterC157#3298
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D157C2semidirect
3. #23757622C2semidirect
4. #23757603C2semidirect
5. #23757602C2semidirect
6. #23757556C2semidirect
7. #23757505C2semidirect
8. #23757502C2semidirect
9. #23757477C2semidirect
10. #23757413C2semidirect
11. #23757412C2semidirect
12. #23757358C2semidirect
13. C8×C22⋊D157C2semidirect
14. #103×D157C2semidirect
15. C2×#103×C157C2semidirect
122. C22⋊D157SD16semidirect
141. Q8×C157#41semidirect
183. C157#3298semidirect
199. C2×#103D157semidirect
307. Q8#14578954semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1572
#237579242
#237576222
#237576032
#237576022
#237575562
#237575052
#237575022
#237574772
#237574132
#237574122
#237573582
C8×C22⋊D1572
#103×D1572
C2×#103×C1572
#3298157