Finite Group Databasebeta

← #38752911

Subgroups — #38752911

Order 25216.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161684C211yesmaximal, characteristic, complemented
2#27161397C211yesmaximal, characteristic, complemented
3#27161392C211yesmaximal, characteristic, complemented
4#27161386C211yesmaximal, characteristic, complemented
5#27161379C211yesmaximal, characteristic, complemented
6#27161279C211yesmaximal, characteristic, complemented
7#27161274C211yesmaximal, characteristic, complemented
8#27161224C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic, complemented
10#27161189C211yesmaximal, characteristic, complemented
11#27161177C211yesmaximal, characteristic, complemented
12C4×#15691345C211yesmaximal, characteristic, complemented
13C4×#15691343C211yesmaximal, characteristic, complemented
14#27161140C211yesmaximal, characteristic, complemented
15#520×C197C211yesmaximal, characteristic, complemented
16C2×#15691345C2211yescharacteristic
17C2×#15691345C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20#20263323C2211yescharacteristic
21#20263318C2211yescharacteristic
22#20263311—21no—
23#20263311—21no—
24#20263300C2211yescharacteristic
25#20263300C2211yescharacteristic
26#20263299—21no—
27#20263299—21no—
28C4×D788C2211yescharacteristic
29C4×D788C2211yescharacteristic
30#20263259C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263257C2211yescharacteristic
33#20263257C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35C4⋊#13096802C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263247C2211yescharacteristic
39#20263247C2211yescharacteristic
40#20263246C2211yescharacteristic
41#20263246C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44C22⋊#13096802C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46#20263237C2211yescharacteristic
47C4×#13096802C2211yescharacteristic
48C2×#41×C197C2211yescharacteristic
49C4×Q8×C197C2211yescharacteristic
50C4×Q8×C197C2211yescharacteristic
51C4×D4×C197—21no—
52C4×D4×C197—21no—
53C4×D4×C197C2211yescharacteristic
54C4×D4×C197C2211yescharacteristic
55#117×C197—21no—
56#117×C197C2211yescharacteristic
57C2×C4⋊C4×C197—21no—
58C2×C4⋊C4×C197C2211yescharacteristic
59C2×D788—21no—
60C2×D788—21no—
61C2×D788C2311yescharacteristic
62C2×D788C2311yescharacteristic
63C2×C22⋊D197—21no—
64C2×C22⋊D197—21no—
65C2×C4×D197—21no—
66C2×C4×D197—21no—
67#15691345—22no—
68#15691345—22no—
69#15691343—22no—
70#15691343—22no—
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77#15691337—21no—
78#15691337—21no—
79#15691335—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83#41×C197—22no—
84#41×C197—22no—
85C2×Q8×C197C2311yescharacteristic
86C2×D4×C197—21no—
87C2×D4×C197C2311yescharacteristic
88C22×C4×C197—21no—
89C22×C4×C197—21no—
90C22×C4×C197—21no—
91C22×C4×C197—21no—
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—21no—
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
96C4⋊C4×C197D411yescharacteristic
97C4⋊C4×C197D411yescharacteristic
98C4⋊C4×C197C2311yescharacteristic
99C4⋊C4×C197C2311yescharacteristic
100C22⋊C4×C197—21no—
101C22⋊C4×C197—21no—
102C22⋊C4×C197D411yescharacteristic
103C22⋊C4×C197D411yescharacteristic
104C42×C197—21no—
105C42×C197—21no—
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109C22×D197—21no—
110D788—21no—
111D788—21no—
112D788—21no—
113D788—21no—
114D788—81no—
115D788—81no—
116C22⋊D197—81no—
117C22⋊D197—81no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C4×D197—81no—
121C4×D197—81no—
122#13096802—21no—
123#13096802—21no—
124#13096802—21no—
125#13096802—21no—
126C23×C197—21no—
127C23×C197C2×D411yescharacteristic
128Q8×C197—21no—
129Q8×C197#4112yes—
130D4×C197—24no—
131D4×C197—21no—
132D4×C197#4112yes—
133C2×C4×C197—24no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—81no—
140C2×C4×C197—81no—
141C2×C4×C197—81no—
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2×D411yescharacteristic
146C2×C4×C197C2×D411yescharacteristic
147C2×C4×C197C2411yescharacteristic
148C22×C197—21no—
149C22×C197—22no—
150C22×C197—21no—
151C22×C197—21no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197C22×D411yescharacteristic
155C2×D197—41no—
156C2×D197—41no—
157C2×D197—41no—
158C2×D197—41no—
159C2×D197—41no—
160C2×D197—41no—
161C4×C197—21no—
162C4×C197—22no—
163C4×C197—21no—
164C4×C197—21no—
165C4×C197—21no—
166C4×C197—21no—
167C4×C197—41no—
168C4×C197—41no—
169C4×C197C2×#4111yescharacteristic
170C4×C197C2×#4111yescharacteristic
171Dic197—41no—
172Dic197—41no—
173C2×C197—41no—
174C2×C197—42no—
175C2×C197#52511yescharacteristic
176C2×C197#57611yescharacteristic
177C2×C197#57711yescharacteristic
178D197—81no—
179D197—81no—
180C197#311911yescharacteristic, complemented, Sylow
181#3119—1971nomaximal, Sylow
182#540—1971no—
183#520D19711yescharacteristic, complemented
184#481—1971no—
185#480—1971no—
186#479—1971no—
187#478—1971no—
188#460—1971no—
189#459—1971no—
190#449—1971no—
191#449—1971no—
192#442—1971no—
193#440—1971no—
194C4×SD16—1971no—
195C4×D8—1971no—
196#433—1971no—
197C2×#41C2×D19711yescharacteristic
198C2×SD16—1971no—
199C2×SD16—1971no—
200C2×D8—1971no—
201C2×D8—1971no—
202#127—1971no—
203#125—1971no—
204#123—3941no—
205#123—3941no—
206#121—3941no—
207#121—3941no—
208#121—1971no—
209#121—1971no—
210C4×Q8C2×D19711yescharacteristic
211C4×Q8C2×D19711yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1971no—
215C4×D4—1971no—
216C4×D4C2×D19711yescharacteristic
217C4×D4C2×D19711yescharacteristic
218#117—21no—
219#117C2×D19711yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D19711yescharacteristic
222#107—1971no—
223#107—1971no—
224#106—1971no—
225#106—1971no—
226C4⋊C8—1971no—
227C4⋊C8—1971no—
228#103—1971no—
229#103—1971no—
230#102—1971no—
231#102—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102—1971no—
236C22⋊C8—1971no—
237C22⋊C8—1971no—
238#97—1971no—
239C4×C8—1971no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D19711yescharacteristic
243C2×D4—21no—
244C2×D4—3941no—
245C2×D4—3941no—
246C2×D4—3941no—
247C2×D4—3941no—
248C2×D4—1971no—
249C2×D4—1971no—
250C2×D4C22×D19711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3941no—
256C22×C4—3941no—
257C22×C4C22×D19711yescharacteristic
258SD16—3942no—
259SD16—3942no—
260D8—3942no—
261D8—3942no—
262C2×C8—1971no—
263C2×C8—1971no—
264C2×C8—1971no—
265C2×C8—1971no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—3941no—
270C4⋊C4—3941no—
271C4⋊C4—1971no—
272C4⋊C4—1971no—
273C4⋊C4C22⋊D19711yescharacteristic
274C4⋊C4C22⋊D19711yescharacteristic
275C4⋊C4C22×D19711yescharacteristic
276C4⋊C4C22×D19711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4C22⋊D19711yescharacteristic
284C22⋊C4C22⋊D19711yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D19711yescharacteristic
288C42C22×D19711yescharacteristic
289C23—21no—
290C23—3941no—
291C23—3941no—
292C23C2×C22⋊D19711yescharacteristic
293Q8—21no—
294Q8#1569135812yes—
295D4—24no—
296D4—21no—
297D4—15761no—
298D4—15761no—
299D4—15761no—
300D4—15761no—
301D4—3941no—
302D4—3941no—
303D4—3941no—
304D4—3941no—
305D4#1569135812yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—15761no—
316C2×C4—15761no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4C2×C22⋊D19711yescharacteristic
320C2×C4C2×C22⋊D19711yescharacteristic
321C2×C4C2×C22⋊D19711yescharacteristic
322C2×C4C2×C22⋊D19711yescharacteristic
323C2×C4C2×C22⋊D19711yescharacteristic
324C2×C4C23×D19711yescharacteristic
325C8—3941no—
326C8—3941no—
327C8—3941no—
328C8—3941no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—7881no—
336C22—7881no—
337C22—7881no—
338C22—7881no—
339C22—7881no—
340C22—7881no—
341C22C22×C22⋊D19711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—22no—
348C4—41no—
349C4—41no—
350C4—7881no—
351C4—7881no—
352C4C2×#1569135811yescharacteristic
353C4C2×#1569135811yescharacteristic
354C2—41no—
355C2—42no—
356C2—15761no—
357C2—15761no—
358C2#2716159011yescharacteristic
359C2#2716188211yescharacteristic
360C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#520×C197C2
socleC22×C197C22×D4
hypercenter#520D197
hypocenterC197#3119
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #27161684C2semidirect
2. #27161397C2semidirect
3. #27161392C2semidirect
4. #27161386C2semidirect
5. #27161379C2semidirect
6. #27161279C2semidirect
7. #27161274C2semidirect
8. #27161224C2semidirect
9. #27161224C2semidirect
10. #27161189C2semidirect
11. #27161177C2semidirect
12. C4×#15691345C2semidirect
13. C4×#15691343C2semidirect
14. #27161140C2semidirect
15. #520×C197C2semidirect
180. C197#3119semidirect
183. #520D197semidirect

Maximal subgroups

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

subgroupindex
#271616842
#271613972
#271613922
#271613862
#271613792
#271612792
#271612742
#271612242
#271611892
#271611772
C4×#156913452
C4×#156913432
#271611402
#520×C1972
#3119197