Finite Group Databasebeta

← #34108978

Subgroups — #34108978

Order 20096.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic
2#23757755C211yesmaximal, characteristic
3#23757749C211yesmaximal, characteristic, complemented
4#23757377C211yesmaximal, characteristic, complemented
5#23757376C211yesmaximal, characteristic, complemented
6#23757375C211yesmaximal, characteristic, complemented
7Q16×Dic157C211yesmaximal, characteristic, complemented
8C4×#14578944C211yesmaximal, characteristic, complemented
9C4×#14578941C211yesmaximal, characteristic, complemented
10#23757360C211yesmaximal, characteristic, complemented
11#23757312C211yesmaximal, characteristic, complemented
12#23757268C211yesmaximal, characteristic
13#103×D157C211yesmaximal, characteristic, complemented
14#23757230C211yesmaximal, characteristic, complemented
15#441×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18#19097803C418yescomplemented
19C2×#14578944C2211yescharacteristic
20C2×#14578943C2211yescharacteristic
21C2×#14578941C2211yescharacteristic
22C2×#14578937C2211yescharacteristic
23C2×C8×D157C2211yescharacteristic
24Q8×Dic157C2211yescharacteristic
25Q8×Dic157C2211yescharacteristic
26C4×Dic314—21no—
27C4×Dic314C2211yescharacteristic
28#19097736—21no—
29#19097736C2211yescharacteristic
30C4×D628—21no—
31C4×D628C2211yescharacteristic
32#19097733—21no—
33#19097733C2211yescharacteristic
34#19097729—21no—
35#19097729C2211yescharacteristic
36#19097728—21no—
37#19097728—21no—
38#19097728C2211yescharacteristic
39C4⋊C4×D157—21no—
40C4⋊C4×D157C2211yescharacteristic
41#19097705C2211yescharacteristic
42#19097704C2211yescharacteristic
43#19097702C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097695C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097681C2211yescharacteristic
51C4×#11929434C2211yescharacteristic
52C8×Dic157C2211yescharacteristic
53C2×Q16×C157C2211yescharacteristic
54C4×Q8×C157C2211yescharacteristic
55C4×Q8×C157C2211yescharacteristic
56#106×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic
59#97×C157C2211yescharacteristic
60#14578955—22no—
61#14578955C2×C414yes—
62C2×Dic314—21no—
63C2×Dic314C2311yescharacteristic
64Q8×D157—22no—
65Q8×D157C2×C414yescomplemented
66C2×D628—21no—
67C2×D628C2311yescharacteristic
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70C2×C4×D157—21no—
71C2×C4×D157—21no—
72C2×C4×D157—21no—
73C2×C4×D157C2311yescharacteristic
74#14578944C2×C414yescomplemented
75#14578943C2×C414yescomplemented
76#14578941C2×C414yescomplemented
77#14578937C2×C414yescomplemented
78C2×#11929434C2311yescharacteristic
79C2×#11929434C2311yescharacteristic
80C8×D157—22no—
81C4⋊Dic157—21no—
82C4⋊Dic157—21no—
83C4⋊Dic157C2311yescharacteristic
84C4⋊Dic157C2311yescharacteristic
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88#14578933D411yescharacteristic
89#14578933D411yescharacteristic
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931D411yescharacteristic
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157—21no—
97C4×Dic157—21no—
98C4×Dic157—21no—
99C4×Dic157—21no—
100C4×Dic157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C2×Q8×C157C2311yescharacteristic
103Q16×C157C2×C414yescomplemented
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157—21no—
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C42×C157—21no—
111C42×C157—21no—
112C42×C157C2311yescharacteristic
113C22×D157—21no—
114C22×D157C2×D411yescharacteristic
115Dic314—21no—
116Dic314—24no—
117Dic314C22×C412yes—
118D628—24no—
119D628—21no—
120D628C22×C412yes—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157C2×D411yescharacteristic
125C2×Dic157C2×D411yescharacteristic
126C2×Dic157C2×D411yescharacteristic
127C4×D157—24no—
128C4×D157—24no—
129C4×D157—81no—
130C4×D157—81no—
131C4×D157—81no—
132C4×D157C22×C412yes—
133C4×D157#4112yes—
134#11929434—21no—
135#11929434C22×C412yes—
136Q8×C157—21no—
137Q8×C157—21no—
138Q8×C157C22×C412yes—
139Q8×C157C22×C412yes—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2411yescharacteristic
147C8×C157—21no—
148C8×C157C22×C412yes—
149C22×C157C22×D411yescharacteristic
150C2×D157—21no—
151C2×D157—21no—
152C2×D157—22no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157C4×D412yes—
156C4×C157—22no—
157C4×C157—22no—
158C4×C157—21no—
159C4×C157—21no—
160C4×C157—41no—
161C4×C157—41no—
162C4×C157C23×C411yescharacteristic
163C4×C157C2×#4111yescharacteristic
164Dic157—22no—
165Dic157—21no—
166Dic157—21no—
167Dic157—41no—
168Dic157—41no—
169Dic157C4×D412yes—
170C2×C157C2×C4×D411yescharacteristic
171C2×C157#57711yescharacteristic
172C2×C157#57811yescharacteristic
173D157—22no—
174D157—42no—
175C157#294711yescharacteristic, complemented, Sylow
176#2947—1571nomaximal, Sylow
177C2×#137—1571no—
178#520—1571no—
179#519—1571no—
180#441—1571no—
181#441D15711yescharacteristic, complemented
182#440—1571no—
183#440—1571no—
184C4×Q16—1571no—
185C4×Q16—1571no—
186C4×SD16—1571no—
187C4×SD16—1571no—
188#428—1571no—
189#416—1571no—
190C2×#103—1571no—
191#405—1571no—
192C2×#41—1571no—
193C22×Q8—1571no—
194#137—1578no—
195C2×Q16—1571no—
196C2×Q16C2×D15711yescharacteristic
197C2×SD16—1571no—
198C2×SD16—1571no—
199C2×M16—1571no—
200C22×C8—1571no—
201C4×Q8—3141no—
202C4×Q8—3141no—
203C4×Q8—1571no—
204C4×Q8—1571no—
205C4×Q8—1571no—
206C4×Q8—1571no—
207C4×Q8C2×D15711yescharacteristic
208C4×Q8C2×D15711yescharacteristic
209C4×D4—3141no—
210C4×D4—3141no—
211C4×D4—1571no—
212C4×D4—1571no—
213#117—3141no—
214#117—3141no—
215#117—3141no—
216#117—1571no—
217#117—1571no—
218C2×C4⋊C4—3141no—
219C2×C4⋊C4—1571no—
220#107—1571no—
221#107—1571no—
222#106—1571no—
223#106C2×D15711yescharacteristic
224#103—1571no—
225#103—1571no—
226#103—1571no—
227#103—1571no—
228#103C2×D15711yescharacteristic
229#103C2×D15711yescharacteristic
230#102—1571no—
231#102—1571no—
232#97—1571no—
233#97C2×D15711yescharacteristic
234C4×C8—1571no—
235C4×C8—1571no—
236#41—3142no—
237#41—1574no—
238C2×Q8—3141no—
239C2×Q8—3142no—
240C2×Q8—1574no—
241C2×Q8—1571no—
242C2×Q8C22×D15711yescharacteristic
243C2×Q8C22×D15711yescharacteristic
244C2×D4—3141no—
245C2×D4—1571no—
246C22×C4—3141no—
247C22×C4—3141no—
248C22×C4—3141no—
249C22×C4—3141no—
250C22×C4—3141no—
251C22×C4—1571no—
252Q16—1574no—
253Q16C4×D15714yescomplemented
254SD16—1574no—
255SD16—1574no—
256M16—1574no—
257C2×C8—3142no—
258C2×C8—1571no—
259C2×C8—1571no—
260C2×C8C22×D15711yescharacteristic
261C2×C8C22×D15711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—3141no—
268C4⋊C4—3141no—
269C4⋊C4—1571no—
270C4⋊C4—1571no—
271C4⋊C4—1571no—
272C4⋊C4—1571no—
273C4⋊C4C22×D15711yescharacteristic
274C4⋊C4C22×D15711yescharacteristic
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—1571no—
279C22⋊C4—1571no—
280C42—21no—
281C42—21no—
282C42—3141no—
283C42—3141no—
284C42—3141no—
285C42—3141no—
286C42—3141no—
287C42—1571no—
288C42C22×D15711yescharacteristic
289C23—3141no—
290C23—1571no—
291Q8—21no—
292Q8—21no—
293Q8—3144no—
294Q8—3141no—
295Q8—1572no—
296Q8C2×C4×D15712yes—
297Q8C2×C4×D15712yes—
298D4—3144no—
299D4—3141no—
300D4—1572no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—3144no—
306C2×C4—3144no—
307C2×C4—3141no—
308C2×C4—3141no—
309C2×C4—3141no—
310C2×C4—1571no—
311C2×C4—1571no—
312C2×C4—1571no—
313C2×C4—1572no—
314C2×C4—1572no—
315C2×C4—12561no—
316C2×C4—12561no—
317C2×C4—12561no—
318C2×C4D4×D15711yescharacteristic
319C2×C4D4×D15711yescharacteristic
320C2×C4C23×D15711yescharacteristic
321C8—21no—
322C8—3141no—
323C8—1572no—
324C8C2×C4×D15712yes—
325C22—3141no—
326C22—3141no—
327C22—3142no—
328C22—6281no—
329C22—6281no—
330C22—1572no—
331C22C2×D4×D15711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—3141no—
339C4—3141no—
340C4—3142no—
341C4—6281no—
342C4—6281no—
343C4—1572no—
344C4C22×C4×D15711yescharacteristic
345C4#41×D15711yescharacteristic
346C2—3142no—
347C2—6282no—
348C2C4×D4×D15711yescharacteristic
349C2#2375808211yescharacteristic
350C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutatorC4×C157C23×C4
FrattiniC2×C4C23×D157
Fitting#441×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578955C22
cosocleC2×C4×C157C24
hypercenter#441D157
hypocenterC157#2947
LCSG ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
3. #23757749C2semidirect
4. #23757377C2semidirect
5. #23757376C2semidirect
6. #23757375C2semidirect
7. Q16×Dic157C2semidirect
8. C4×#14578944C2semidirect
9. C4×#14578941C2semidirect
10. #23757360C2semidirect
11. #23757312C2semidirect
13. #103×D157C2semidirect
14. #23757230C2semidirect
15. #441×C157C2semidirect
18. #19097803C4semidirect
65. Q8×D157C2×C4semidirect
74. #14578944C2×C4semidirect
75. #14578943C2×C4semidirect
76. #14578941C2×C4semidirect
77. #14578937C2×C4semidirect
103. Q16×C157C2×C4semidirect
175. C157#2947semidirect
181. #441D157semidirect
253. Q16C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
#237577552
#237577492
#237573772
#237573762
#237573752
Q16×Dic1572
C4×#145789442
C4×#145789412
#237573602
#237573122
#237572682
#103×D1572
#237572302
#441×C1572
#2947157