Finite Group Databasebeta

← #38751552

Subgroups — #38751552

Order 25216.

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×#20263355C211yesmaximal, characteristic
2#27161558C211yesmaximal, characteristic
3#27161555C211yesmaximal, characteristic, complemented
4#27161187C211yesmaximal, characteristic, complemented
5#27161183C211yesmaximal, characteristic, complemented
6#27161176C211yesmaximal, characteristic, complemented
7#27161173C211yesmaximal, characteristic, complemented
8C4×#15691348C211yesmaximal, characteristic, complemented
9SD16×Dic197C211yesmaximal, characteristic, complemented
10C4×#15691346C211yesmaximal, characteristic, complemented
11#27161120C211yesmaximal, characteristic, complemented
12#27161072C211yesmaximal, characteristic
13#103×D197C211yesmaximal, characteristic, complemented
14#27161036C211yesmaximal, characteristic, complemented
15#440×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×Q8×D197C2211yescharacteristic
18#20263355C418yescomplemented
19C2×Dic788C2211yescharacteristic
20C2×#15691348C2211yescharacteristic
21C2×#15691346C2211yescharacteristic
22C2×#15691341C2211yescharacteristic
23C2×C8×D197C2211yescharacteristic
24Q8×Dic197C2211yescharacteristic
25C4×Dic394—21no—
26C4×Dic394C2211yescharacteristic
27C4×Dic394C2211yescharacteristic
28#20263291—21no—
29#20263291C2211yescharacteristic
30#20263291C2211yescharacteristic
31D4×Dic197C2211yescharacteristic
32C4×C22⋊D197—21no—
33#20263286—21no—
34#20263284—21no—
35#20263284C2211yescharacteristic
36#20263283—21no—
37#20263283C2211yescharacteristic
38#20263282—21no—
39C2×#15691337—21no—
40C4⋊C4×D197C2211yescharacteristic
41#20263259C2211yescharacteristic
42#20263258C2211yescharacteristic
43#20263257C2211yescharacteristic
44#20263251C2211yescharacteristic
45#20263251C2211yescharacteristic
46#20263250C2211yescharacteristic
47#20263249C2211yescharacteristic
48#20263249C2211yescharacteristic
49#20263247C2211yescharacteristic
50#20263236C2211yescharacteristic
51C4×#13096802C2211yescharacteristic
52C8×Dic197C2211yescharacteristic
53C2×SD16×C197C2211yescharacteristic
54C4×Q8×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic
56#107×C197C2211yescharacteristic
57#103×C197C2211yescharacteristic
58#102×C197C2211yescharacteristic
59#97×C197C2211yescharacteristic
60#15691357—22no—
61#15691357C2×C414yes—
62C2×Dic394—21no—
63C2×Dic394C2311yescharacteristic
64C2×Dic394C2311yescharacteristic
65Q8×D197—22no—
66Q8×D197C2×C414yescomplemented
67C2×C22⋊D197—21no—
68C22×Dic197—21no—
69C22×Dic197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72C2×C4×D197C2311yescharacteristic
73Dic788C2×C414yescomplemented
74#15691348C2×C414yescomplemented
75#15691346C2×C414yescomplemented
76#15691341C2×C414yescomplemented
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C8×D197—22no—
80C4⋊Dic197—21no—
81C4⋊Dic197C2311yescharacteristic
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84#15691337—21no—
85#15691337—21no—
86#15691337—21no—
87#15691337—21no—
88#15691337D411yescharacteristic
89#15691337D411yescharacteristic
90C22⋊Dic197—21no—
91#15691335—21no—
92#15691335D411yescharacteristic
93#15691335D411yescharacteristic
94C4×Dic197—21no—
95C4×Dic197—21no—
96C4×Dic197—21no—
97C4×Dic197—21no—
98C4×Dic197—21no—
99C4×Dic197—21no—
100C4×Dic197C2311yescharacteristic
101C2×Q8×C197C2311yescharacteristic
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104SD16×C197C2×C414yescomplemented
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C22⋊C4×C197—21no—
111C42×C197—21no—
112C42×C197C2311yescharacteristic
113C22×D197C2×D411yescharacteristic
114Dic394—21no—
115Dic394—21no—
116Dic394—24no—
117Dic394C22×C412yes—
118Dic394C22×C412yes—
119C22⋊D197—24no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—24no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—81no—
126C2×Dic197C2×D411yescharacteristic
127C2×Dic197C2×D411yescharacteristic
128C2×Dic197C2×D411yescharacteristic
129C4×D197—24no—
130C4×D197—81no—
131C4×D197C22×C412yes—
132C4×D197#4112yes—
133#13096802—21no—
134#13096802C22×C412yes—
135C23×C197—21no—
136Q8×C197—21no—
137Q8×C197C22×C412yes—
138D4×C197—21no—
139D4×C197C22×C412yes—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—81no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2411yescharacteristic
147C8×C197—21no—
148C8×C197C22×C412yes—
149C22×C197—22no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197C22×D411yescharacteristic
153C2×D197—21no—
154C2×D197—21no—
155C2×D197C4×D412yes—
156C4×C197—22no—
157C4×C197—21no—
158C4×C197—21no—
159C4×C197—41no—
160C4×C197—41no—
161C4×C197C23×C411yescharacteristic
162C4×C197C2×#4111yescharacteristic
163Dic197—22no—
164Dic197—22no—
165Dic197—21no—
166Dic197—21no—
167Dic197—41no—
168Dic197—41no—
169Dic197C4×D412yes—
170C2×C197—42no—
171C2×C197C2×C4×D411yescharacteristic
172C2×C197#57711yescharacteristic
173C2×C197#57811yescharacteristic
174D197—22no—
175C197#294711yescharacteristic, complemented, Sylow
176#2947—1971nomaximal, Sylow
177C2×#137—1971no—
178#520—1971no—
179#519—1971no—
180#441—1971no—
181#441—1971no—
182#440—1971no—
183#440D19711yescharacteristic, complemented
184C4×Q16—1971no—
185C4×Q16—1971no—
186C4×SD16—1971no—
187C4×SD16—1971no—
188#428—1971no—
189#416—1971no—
190C2×#103—1971no—
191#405—1971no—
192C2×#41—1971no—
193C22×Q8—1971no—
194#137—1978no—
195C2×Q16—1971no—
196C2×Q16—1971no—
197C2×SD16—1971no—
198C2×SD16C2×D19711yescharacteristic
199C2×M16—1971no—
200C22×C8—1971no—
201C4×Q8—3941no—
202C4×Q8—3941no—
203C4×Q8—1971no—
204C4×Q8—1971no—
205C4×Q8—1971no—
206C4×Q8—1971no—
207C4×Q8—1971no—
208C4×Q8C2×D19711yescharacteristic
209C4×D4—3941no—
210C4×D4—3941no—
211C4×D4—1971no—
212C4×D4C2×D19711yescharacteristic
213#117—3941no—
214#117—3941no—
215#117—3941no—
216#117—1971no—
217#117—1971no—
218C2×C4⋊C4—3941no—
219C2×C4⋊C4—1971no—
220#107—1971no—
221#107C2×D19711yescharacteristic
222#106—1971no—
223#106—1971no—
224#103—1971no—
225#103—1971no—
226#103—1971no—
227#103—1971no—
228#103—1971no—
229#103C2×D19711yescharacteristic
230#102—1971no—
231#102C2×D19711yescharacteristic
232#97—1971no—
233#97C2×D19711yescharacteristic
234C4×C8—1971no—
235C4×C8—1971no—
236#41—3942no—
237#41—1974no—
238C2×Q8—3941no—
239C2×Q8—3942no—
240C2×Q8—1974no—
241C2×Q8—1971no—
242C2×Q8—1971no—
243C2×Q8C22×D19711yescharacteristic
244C2×D4—3941no—
245C2×D4C22×D19711yescharacteristic
246C22×C4—21no—
247C22×C4—3941no—
248C22×C4—3941no—
249C22×C4—3941no—
250C22×C4—3941no—
251C22×C4—1971no—
252Q16—1974no—
253Q16—1974no—
254SD16—1974no—
255SD16C4×D19714yescomplemented
256M16—1974no—
257C2×C8—3942no—
258C2×C8—1971no—
259C2×C8—1971no—
260C2×C8C22×D19711yescharacteristic
261C2×C8C22×D19711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—3941no—
264C4⋊C4—3941no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—3941no—
268C4⋊C4—3941no—
269C4⋊C4—1971no—
270C4⋊C4—1971no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4C22×D19711yescharacteristic
274C4⋊C4C22×D19711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—3941no—
277C22⋊C4—3941no—
278C22⋊C4—1971no—
279C22⋊C4—1971no—
280C42—21no—
281C42—3941no—
282C42—3941no—
283C42—3941no—
284C42—3941no—
285C42—3941no—
286C42—3941no—
287C42—1971no—
288C42C22×D19711yescharacteristic
289C23—21no—
290C23—1971no—
291Q8—21no—
292Q8—3944no—
293Q8—3941no—
294Q8—3941no—
295Q8—1972no—
296Q8—1972no—
297Q8C2×C4×D19712yes—
298D4—21no—
299D4—3944no—
300D4C2×C4×D19712yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—15761no—
306C2×C4—15761no—
307C2×C4—3944no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3944no—
313C2×C4—1971no—
314C2×C4—1971no—
315C2×C4—1971no—
316C2×C4—1972no—
317C2×C4—1972no—
318C2×C4D4×D19711yescharacteristic
319C2×C4D4×D19711yescharacteristic
320C2×C4C23×D19711yescharacteristic
321C8—21no—
322C8—3941no—
323C8—1972no—
324C8C2×C4×D19712yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—3941no—
329C22—3941no—
330C22—1972no—
331C22C2×D4×D19711yescharacteristic
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—7881no—
338C4—7881no—
339C4—3941no—
340C4—3941no—
341C4—3942no—
342C4—3942no—
343C4—1972no—
344C4C22×C4×D19711yescharacteristic
345C4#41×D19711yescharacteristic
346C2—42no—
347C2—3942no—
348C2C4×D4×D19711yescharacteristic
349C2#2716188911yescharacteristic
350C2#2716189511yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
Fitting#440×C197C2
socleC22×C197C22×D4
prime radicalC2×#15691357C22
cosocleC2×C4×C197C24
hypercenter#440D197
hypocenterC197#2947
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
3. #27161555C2semidirect
4. #27161187C2semidirect
5. #27161183C2semidirect
6. #27161176C2semidirect
7. #27161173C2semidirect
8. C4×#15691348C2semidirect
9. SD16×Dic197C2semidirect
10. C4×#15691346C2semidirect
11. #27161120C2semidirect
13. #103×D197C2semidirect
14. #27161036C2semidirect
15. #440×C197C2semidirect
18. #20263355C4semidirect
66. Q8×D197C2×C4semidirect
73. Dic788C2×C4semidirect
74. #15691348C2×C4semidirect
75. #15691346C2×C4semidirect
76. #15691341C2×C4semidirect
104. SD16×C197C2×C4semidirect
175. C197#2947semidirect
183. #440D197semidirect
255. SD16C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633552
#271615582
#271615552
#271611872
#271611832
#271611762
#271611732
C4×#156913482
SD16×Dic1972
C4×#156913462
#271611202
#271610722
#103×D1972
#271610362
#440×C1972
#2947197