Finite Group Databasebeta

← #34112017

Subgroups — #34112017

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#23757932C211yesmaximal, characteristic
2#23757630C211yesmaximal, characteristic, complemented
3#23757614C211yesmaximal, characteristic, complemented
4#23757586C211yesmaximal, characteristic, complemented
5#23757577C211yesmaximal, characteristic, complemented
6#23757512C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757495C211yesmaximal, characteristic, complemented
9#23757433C211yesmaximal, characteristic, complemented
10#23757428C211yesmaximal, characteristic, complemented
11C4×#14578944C211yesmaximal, characteristic, complemented
12D4×#11929434C211yesmaximal, characteristic, complemented
13#23757267C211yesmaximal, characteristic, complemented
14C2×#19097694C211yesmaximal, characteristic, complemented
15#552×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×Dic314C2211yescharacteristic
18C2×#14578944C2211yescharacteristic
19C2×#14578944C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21C22×#11929434C2211yescharacteristic
22C22×#11929434C2211yescharacteristic
23#19097759C2211yescharacteristic
24#19097757—21no—
25#19097750C2211yescharacteristic
26#19097749—21no—
27#19097745C2211yescharacteristic
28#19097743—21no—
29C4×Dic314C2211yescharacteristic
30#19097731—21no—
31#19097704C2211yescharacteristic
32#19097704C2211yescharacteristic
33#19097702C2211yescharacteristic
34C4⋊#11929434C2211yescharacteristic
35#19097695C2211yescharacteristic
36#19097694C2211yescharacteristic
37#19097694C2211yescharacteristic
38#19097694C2211yescharacteristic
39#19097694C2211yescharacteristic
40#19097694C2211yescharacteristic
41#19097694C2211yescharacteristic
42#19097691C2211yescharacteristic
43#19097691C2211yescharacteristic
44#19097691C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic
47C4×#11929434C2211yescharacteristic
48#126×C157—21no—
49#125×C157—21no—
50#125×C157C2211yescharacteristic
51#123×C157—21no—
52C22⋊Q8×C157C2211yescharacteristic
53C22⋊Q8×C157C2211yescharacteristic
54C4×Q8×C157C2211yescharacteristic
55C4×D4×C157C2211yescharacteristic
56#117×C157—21no—
57#117×C157C2211yescharacteristic
58C2×C4⋊C4×C157C2211yescharacteristic
59#14578954—22no—
60#14578954—41no—
61C2×Dic314—22no—
62C2×Dic314—21no—
63C2×Dic314—41no—
64C2×Dic314C2311yescharacteristic
65C2×Dic314C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C22×Dic157—21no—
70C2×C4×D157—21no—
71#14578944—22no—
72#14578944—41no—
73#14578941—41no—
74C2×#11929434—41no—
75C2×#11929434—41no—
76C2×#11929434C2311yescharacteristic
77C2×#11929434C2311yescharacteristic
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81#14578933—21no—
82#14578933—21no—
83C22⋊Dic157—21no—
84#14578931—21no—
85#14578931—21no—
86#14578931—21no—
87C4×Dic157—21no—
88C2×Q8×C157C2311yescharacteristic
89C2×D4×C157D411yescharacteristic
90C22×C4×C157—21no—
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C4⋊C4×C157—21no—
94C4⋊C4×C157—21no—
95C4⋊C4×C157—21no—
96C4⋊C4×C157—21no—
97C4⋊C4×C157—21no—
98C4⋊C4×C157—21no—
99C4⋊C4×C157D411yescharacteristic
100C4⋊C4×C157C2311yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C22⋊C4×C157—21no—
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157D411yescharacteristic
107C22⋊C4×C157D411yescharacteristic
108C42×C157—21no—
109C42×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112Dic314—21no—
113Dic314—21no—
114Dic314—21no—
115Dic314—42no—
116Dic314#4112yes—
117D628—21no—
118D628—21no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—42no—
125C4×D157—42no—
126#11929434—21no—
127#11929434—21no—
128#11929434—21no—
129#11929434—21no—
130C23×C157C2×D411yescharacteristic
131C23×C157C2×D411yescharacteristic
132Q8×C157—21no—
133Q8×C157—21no—
134D4×C157—22no—
135C2×C4×C157—21no—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—42no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2411yescharacteristic
149C22×C157—21no—
150C22×C157—21no—
151C22×C157—21no—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157#13512yes—
155C22×C157C22×D411yescharacteristic
156C2×D157—41no—
157C2×D157—41no—
158C2×D157—41no—
159C4×C157—21no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—21no—
163C4×C157—41no—
164C4×C157—41no—
165C4×C157—41no—
166C4×C157—41no—
167C4×C157C2×#4111yescharacteristic
168C4×C157H(5,𝔽2)11yescharacteristic
169Dic157—22no—
170Dic157—41no—
171Dic157—41no—
172C2×C157—22no—
173C2×C157—41no—
174C2×C157#54611yescharacteristic
175C2×C157C2×#13511yescharacteristic
176C2×C157#57811yescharacteristic
177D157—81no—
178C157#330011yescharacteristic, complemented, Sylow
179#3300—1571nomaximal, Sylow
180#552D15711yescharacteristic, complemented
181#548—1571no—
182#488—1571no—
183#485—1571no—
184#480—1571no—
185#479—1571no—
186#467—1571no—
187#465—1571no—
188#464—1571no—
189#452—1571no—
190#451—1571no—
191C4×Q16—1571no—
192C8×D4—1571no—
193#416—1571no—
194C2×#103—1571no—
195C2×#41—1571no—
196C22×Q8—1571no—
197C2×Q16—1571no—
198C2×Q16—1571no—
199C2×SD16—1571no—
200C22×C8—1571no—
201C22×C8—1571no—
202#126—21no—
203#125—21no—
204#125C2×D15711yescharacteristic
205#124—3141no—
206#124—1571no—
207#123—21no—
208C22⋊Q8—3141no—
209C22⋊Q8—1571no—
210C22⋊Q8C2×D15711yescharacteristic
211C22⋊Q8C2×D15711yescharacteristic
212#121—3141no—
213#121—1571no—
214C4×Q8—1571no—
215C4×Q8C2×D15711yescharacteristic
216C4×D4—3141no—
217C4×D4C2×D15711yescharacteristic
218#117—21no—
219#117C2×D15711yescharacteristic
220C2×C4⋊C4C2×D15711yescharacteristic
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#103—1571no—
231#103—1571no—
232#102—1571no—
233#102—1571no—
234#102—1571no—
235C22⋊C8—1571no—
236C22⋊C8—1571no—
237C4×C8—1571no—
238#41—3142no—
239#41—6281no—
240C2×Q8—3142no—
241C2×Q8—3141no—
242C2×Q8—6281no—
243C2×Q8—1571no—
244C2×Q8—1571no—
245C2×Q8C22×D15711yescharacteristic
246C2×D4—3141no—
247C2×D4—3141no—
248C2×D4—1571no—
249C2×D4C22⋊D15711yescharacteristic
250C22×C4—21no—
251C22×C4—3141no—
252C22×C4—3141no—
253C22×C4C22×D15711yescharacteristic
254C22×C4C22×D15711yescharacteristic
255Q16—3142no—
256Q16—6281no—
257SD16—6281no—
258C2×C8—6281no—
259C2×C8—6281no—
260C2×C8—1571no—
261C2×C8—1571no—
262C2×C8—1571no—
263C2×C8—1571no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—3141no—
271C4⋊C4—3141no—
272C4⋊C4—1571no—
273C4⋊C4C22⋊D15711yescharacteristic
274C4⋊C4C22×D15711yescharacteristic
275C4⋊C4C22×D15711yescharacteristic
276C4⋊C4C22×D15711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C22⋊C4C22⋊D15711yescharacteristic
285C22⋊C4C22⋊D15711yescharacteristic
286C42—21no—
287C42—21no—
288C42—3141no—
289C42C22×D15711yescharacteristic
290C23—3141no—
291C23C2×C22⋊D15711yescharacteristic
292C23C2×C22⋊D15711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3141no—
296Q8—3141no—
297Q8—3141no—
298Q8—6282no—
299Q8—1572no—
300D4—22no—
301D4—3141no—
302D4—3141no—
303D4—6282no—
304D4—6282no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—6282no—
318C2×C4—6282no—
319C2×C4C2×C22⋊D15711yescharacteristic
320C2×C4C2×C22⋊D15711yescharacteristic
321C2×C4C2×C22⋊D15711yescharacteristic
322C2×C4C2×C22⋊D15711yescharacteristic
323C2×C4C23×D15711yescharacteristic
324C8—3141no—
325C8—3141no—
326C8—3141no—
327C8—3141no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—6281no—
334C22—6281no—
335C22—6281no—
336C22#1909779112yes—
337C22C22×C22⋊D15711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3142no—
347C4—6281no—
348C4—6281no—
349C4#41×D15711yescharacteristic
350C4#1909781711yescharacteristic
351C2—22no—
352C2—41no—
353C2—12561no—
354C2#2375791111yescharacteristic
355C2C2×#1909779111yescharacteristic
356C2#2375808811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#552×C157C2
socleC22×C157C22×D4
prime radical#23757932C2
hypercenter#552D157
hypocenterC157#3300
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23757630C2semidirect
3. #23757614C2semidirect
4. #23757586C2semidirect
5. #23757577C2semidirect
6. #23757512C2semidirect
7. #23757504C2semidirect
8. #23757495C2semidirect
9. #23757433C2semidirect
10. #23757428C2semidirect
11. C4×#14578944C2semidirect
12. D4×#11929434C2semidirect
13. #23757267C2semidirect
14. C2×#19097694C2semidirect
15. #552×C157C2semidirect
178. C157#3300semidirect
180. #552D157semidirect

Maximal subgroups

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

subgroupindex
#237579322
#237576302
#237576142
#237575862
#237575772
#237575122
#237575042
#237574952
#237574332
#237574282
C4×#145789442
D4×#119294342
#237572672
C2×#190976942
#552×C1572
#3300157