Finite Group Databasebeta

← #34112049

Subgroups — #34112049

Order 20096.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757991C211yesmaximal, characteristic, complemented
2#23757922C211yesmaximal, characteristic
3#23757622C211yesmaximal, characteristic, complemented
4#23757614C211yesmaximal, characteristic, complemented
5#23757613C211yesmaximal, characteristic, complemented
6#23757551C211yesmaximal, characteristic, complemented
7#23757514C211yesmaximal, characteristic, complemented
8#23757513C211yesmaximal, characteristic, complemented
9#23757482C211yesmaximal, characteristic, complemented
10#23757436C211yesmaximal, characteristic, complemented
11#23757435C211yesmaximal, characteristic, complemented
12#23757364C211yesmaximal, characteristic, complemented
13C8×C22⋊D157C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic, complemented
15#416×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Dic628C2211yescharacteristic
18C2×#14578944C2211yescharacteristic
19C2×#14578942C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097770C2211yescharacteristic
22#19097769C2211yescharacteristic
23#19097765—21no—
24#19097764—21no—
25#19097757—21no—
26#19097754—21no—
27#19097752—21no—
28#19097751C2211yescharacteristic
29#19097750C2211yescharacteristic
30#19097750C2211yescharacteristic
31#19097747C2211yescharacteristic
32#19097743—21no—
33C4×Dic314—21no—
34#19097736C2211yescharacteristic
35D4×Dic157C2211yescharacteristic
36C4×C22⋊D157C2211yescharacteristic
37#19097729C2211yescharacteristic
38C2×#14578933—21no—
39#19097705C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097702C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic
54#117×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—41no—
61C2×Dic314—21no—
62C2×Dic314C2311yescharacteristic
63C2×Dic314C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157D411yescharacteristic
66C22×Dic157—21no—
67C22×Dic157—21no—
68C2×C4×D157C2311yescharacteristic
69Dic628—41no—
70#14578944—41no—
71#14578942—22no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C8×D157—41no—
75C4⋊Dic157—21no—
76C4⋊Dic157C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78#14578933—21no—
79#14578933—21no—
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933D411yescharacteristic
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157D411yescharacteristic
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931D411yescharacteristic
94C4×Dic157—21no—
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97#41×C157—22no—
98#41×C157—41no—
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157C2311yescharacteristic
104C2×C8×C157—41no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C42×C157—21no—
111C22×D157C2×D411yescharacteristic
112Dic314—21no—
113Dic314—21no—
114Dic314—21no—
115Dic314—21no—
116Dic314—42no—
117C22⋊D157—22no—
118C22⋊D157—42no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—42no—
124C2×Dic157—42no—
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C4×D157—21no—
129C4×D157—21no—
130#11929434—21no—
131#11929434—21no—
132C23×C157—21no—
133C23×C157C2×D411yescharacteristic
134Q8×C157—21no—
135Q8×C157—21no—
136D4×C157—21no—
137D4×C157—42no—
138D4×C157#4112yes—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—42no—
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2411yescharacteristic
147C8×C157—21no—
148C8×C157—21no—
149C22×C157—21no—
150C22×C157—22no—
151C22×C157—21no—
152C22×C157—21no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157C22×D411yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—21no—
159C4×C157—21no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157—41no—
163C4×C157C2×#4111yescharacteristic
164C4×C157H(5,𝔽2)11yescharacteristic
165Dic157—21no—
166Dic157—21no—
167Dic157—41no—
168Dic157—41no—
169Dic157—41no—
170Dic157—41no—
171Dic157#13512yes—
172C2×C157—41no—
173C2×C157—42no—
174C2×C157#54611yescharacteristic
175C2×C157C2×#13511yescharacteristic
176C2×C157#57811yescharacteristic
177D157—41no—
178C157#330211yescharacteristic, complemented, Sylow
179#3302—1571nomaximal, Sylow
180#556—1571no—
181#547—1571no—
182#487—1571no—
183#485—1571no—
184#485—1571no—
185#474—1571no—
186#467—1571no—
187#467—1571no—
188#462—1571no—
189#452—1571no—
190#452—1571no—
191C4×SD16—1571no—
192C8×D4—1571no—
193#416—1571no—
194#416D15711yescharacteristic, complemented
195C2×#41—1571no—
196C2×#41C2×D15711yescharacteristic
197C2×Q16—1571no—
198C2×Q16—1571no—
199C2×SD16—1571no—
200C22×C8—1571no—
201C22×C8C2×D15711yescharacteristic
202C4⋊Q8—1571no—
203C4⋊Q8—1571no—
204#126—3141no—
205#126—3141no—
206#124—3141no—
207#123—3141no—
208#123—3141no—
209C22⋊Q8—1571no—
210C22⋊Q8—1571no—
211C22⋊Q8—1571no—
212C22⋊Q8—1571no—
213#121—3141no—
214C4×Q8—3141no—
215C4×Q8—1571no—
216C4×D4—1571no—
217C4×D4—1571no—
218#117—1571no—
219#117C2×D15711yescharacteristic
220C2×C4⋊C4—3141no—
221#107—1571no—
222#107—1571no—
223#106—1571no—
224C4⋊C8—1571no—
225#103—1571no—
226#103—1571no—
227#103—1571no—
228#103—1571no—
229#103—1571no—
230#103C2×D15711yescharacteristic
231#103C2×D15711yescharacteristic
232#102—1571no—
233#102C2×D15711yescharacteristic
234#102C2×D15711yescharacteristic
235C22⋊C8—1571no—
236C22⋊C8—1571no—
237C4×C8—1571no—
238#41—22no—
239#41—41no—
240#41—3142no—
241#41—6281no—
242C2×Q8—3141no—
243C2×Q8—1571no—
244C2×Q8—1571no—
245C2×Q8C22×D15711yescharacteristic
246C2×D4—21no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4C22×D15711yescharacteristic
250C22×C4—21no—
251C22×C4—3141no—
252C22×C4—3141no—
253C22×C4—1571no—
254C22×C4C22×D15711yescharacteristic
255Q16—6281no—
256Q16—6281no—
257SD16—3142no—
258C2×C8—41no—
259C2×C8—6281no—
260C2×C8—1571no—
261C2×C8—1571no—
262C2×C8C22×D15711yescharacteristic
263C2×C8C22×D15711yescharacteristic
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—3141no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—1571no—
273C4⋊C4—1571no—
274C4⋊C4—1571no—
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4—1571no—
285C22⋊C4—1571no—
286C42—21no—
287C42—3141no—
288C42—3141no—
289C42—1571no—
290C23—21no—
291C23—1571no—
292C23D4×D15711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3141no—
296Q8—3141no—
297Q8—3141no—
298Q8—3141no—
299Q8—6282no—
300D4—21no—
301D4—42no—
302D4—3142no—
303D4—6282no—
304D4#1457895412yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—6282no—
318C2×C4—6282no—
319C2×C4—1571no—
320C2×C4—1571no—
321C2×C4—1571no—
322C2×C4D4×D15711yescharacteristic
323C2×C4C23×D15711yescharacteristic
324C8—21no—
325C8—21no—
326C8—3141no—
327C8—3141no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—3141no—
335C22—3141no—
336C22—3141no—
337C22C2×D4×D15711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—3141no—
343C4—3141no—
344C4—6281no—
345C4—6281no—
346C4—6281no—
347C4—6281no—
348C4—1572no—
349C4C2×#1457895411yescharacteristic
350C4#1909781611yescharacteristic
351C2—41no—
352C2—42no—
353C2—6281no—
354C2#2375791311yescharacteristic
355C2#135×D15711yescharacteristic
356C2#2375808711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
prime radical#23757922C2
hypercenter#416D157
hypocenterC157#3302
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #23757991C2semidirect
3. #23757622C2semidirect
4. #23757614C2semidirect
5. #23757613C2semidirect
6. #23757551C2semidirect
7. #23757514C2semidirect
8. #23757513C2semidirect
9. #23757482C2semidirect
10. #23757436C2semidirect
11. #23757435C2semidirect
12. #23757364C2semidirect
13. C8×C22⋊D157C2semidirect
14. #23757266C2semidirect
15. #416×C157C2semidirect
178. C157#3302semidirect
194. #416D157semidirect

Maximal subgroups

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

subgroupindex
#237579912
#237579222
#237576222
#237576142
#237576132
#237575512
#237575142
#237575132
#237574822
#237574362
#237574352
#237573642
C8×C22⋊D1572
#237572662
#416×C1572
#3302157