Finite Group Databasebeta

← #38754117

Subgroups — #38754117

Order 25216.

358 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#27161700C211yesmaximal, characteristic, complemented
2#27161383C211yesmaximal, characteristic, complemented
3#27161376C211yesmaximal, characteristic, complemented
4#27161369C211yesmaximal, characteristic, complemented
5#27161359C211yesmaximal, characteristic, complemented
6#27161330C211yesmaximal, characteristic, complemented
7#27161323C211yesmaximal, characteristic, complemented
8#27161313C211yesmaximal, characteristic, complemented
9#27161306C211yesmaximal, characteristic, complemented
10#27161294C211yesmaximal, characteristic, complemented
11#27161294C211yesmaximal, characteristic, complemented
12#27161276C211yesmaximal, characteristic, complemented
13#27161271C211yesmaximal, characteristic, complemented
14#27161106C211yesmaximal, characteristic, complemented
15#540×C197C211yesmaximal, characteristic, complemented
16C2×#15691346C2211yescharacteristic
17C2×#15691346C2211yescharacteristic
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691342C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22#20263326C2211yescharacteristic
23#20263321—21no—
24#20263314C2211yescharacteristic
25#20263312—21no—
26#20263308—21no—
27#20263305C2211yescharacteristic
28#20263304—21no—
29#20263300C2211yescharacteristic
30C4×Dic394C2211yescharacteristic
31C4×D788C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263259C2211yescharacteristic
34#20263257C2211yescharacteristic
35#20263257C2211yescharacteristic
36C4⋊#13096802C2211yescharacteristic
37C4⋊#13096802C2211yescharacteristic
38C4⋊#13096802C2211yescharacteristic
39C4⋊#13096802C2211yescharacteristic
40#20263249C2211yescharacteristic
41#20263249C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263247C2211yescharacteristic
45#20263247C2211yescharacteristic
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic
48#127×C197C2211yescharacteristic
49#125×C197C2211yescharacteristic
50#123×C197—21no—
51#123×C197—21no—
52#121×C197—21no—
53#121×C197—21no—
54#121×C197C2211yescharacteristic
55#121×C197C2211yescharacteristic
56C4×D4×C197C2211yescharacteristic
57C4×D4×C197C2211yescharacteristic
58#117×C197C2211yescharacteristic
59C2×Dic394—21no—
60C2×Dic394C2311yescharacteristic
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C4×D197—21no—
64#15691346—41no—
65#15691346—41no—
66#15691343—41no—
67#15691343—41no—
68#15691342—41no—
69C2×#13096802—41no—
70C2×#13096802C2311yescharacteristic
71C2×#13096802C2311yescharacteristic
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C4⋊Dic197—21no—
75C4⋊Dic197C2311yescharacteristic
76C4⋊Dic197C2311yescharacteristic
77#15691337—21no—
78#15691337—21no—
79C22⋊Dic197—21no—
80#15691335—21no—
81#15691335—21no—
82#15691335—21no—
83#15691335—21no—
84C4×Dic197—21no—
85C2×D4×C197—21no—
86C2×D4×C197—21no—
87C2×D4×C197—21no—
88C2×D4×C197—21no—
89C2×D4×C197C2311yescharacteristic
90C2×D4×C197C2311yescharacteristic
91C22×C4×C197—21no—
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
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×C197—21no—
103C22⋊C4×C197—21no—
104C22⋊C4×C197D411yescharacteristic
105C22⋊C4×C197D411yescharacteristic
106C42×C197C2311yescharacteristic
107C42×C197C2311yescharacteristic
108C22×D197—21no—
109Dic394—21no—
110Dic394—21no—
111Dic394—81no—
112D788—21no—
113D788—21no—
114C22⋊D197—81no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C4×D197—81no—
119#13096802—21no—
120#13096802—21no—
121#13096802—21no—
122#13096802—21no—
123C23×C197—21no—
124C23×C197—21no—
125C23×C197C2×D411yescharacteristic
126D4×C197—21no—
127D4×C197—21no—
128D4×C197—21no—
129D4×C197—21no—
130D4×C197—81no—
131D4×C197—81no—
132D4×C197—81no—
133D4×C197—81no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—81no—
139C2×C4×C197—81no—
140C2×C4×C197C2×D411yescharacteristic
141C2×C4×C197C2×D411yescharacteristic
142C2×C4×C197C2×D411yescharacteristic
143C2×C4×C197C2×D411yescharacteristic
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C22×C197—21no—
147C22×C197—21no—
148C22×C197—21no—
149C22×C197—41no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197—41no—
153C22×C197—41no—
154C22×C197—41no—
155C22×C197C22×D411yescharacteristic
156C2×D197—41no—
157C2×D197—41no—
158C2×D197—41no—
159C4×C197—21no—
160C4×C197—21no—
161C4×C197—21no—
162C4×C197—21no—
163C4×C197—21no—
164C4×C197—41no—
165C4×C197—41no—
166C4×C197H(5,𝔽2)11yescharacteristic
167C4×C1972−1+411yescharacteristic
168Dic197—41no—
169Dic197—41no—
170Dic197—41no—
171C2×C197—41no—
172C2×C197—81no—
173C2×C197—81no—
174C2×C197#53711yescharacteristic
175C2×C197#57611yescharacteristic
176C2×C197#57711yescharacteristic
177D197—81no—
178C197#325711yescharacteristic, complemented, Sylow
179#3257—1971nomaximal, Sylow
180#542—1971no—
181#540D19711yescharacteristic, complemented
182#479—1971no—
183#478—1971no—
184#476—1971no—
185#474—1971no—
186#469—1971no—
187#468—1971no—
188#466—1971no—
189#465—1971no—
190#463—1971no—
191#463—1971no—
192#460—1971no—
193#459—1971no—
194#424—1971no—
195C2×SD16—1971no—
196C2×SD16—1971no—
197C2×D8—1971no—
198C2×D8—1971no—
199C2×M16—1971no—
200C22×C8—1971no—
201C4⋊Q8—1971no—
202#127C2×D19711yescharacteristic
203#126—3941no—
204#125C2×D19711yescharacteristic
205#124—3941no—
206#124—1971no—
207#123—21no—
208#123—21no—
209#123—3941no—
210C22⋊Q8—3941no—
211C22⋊Q8—1971no—
212#121—21no—
213#121—21no—
214#121—1971no—
215#121C2×D19711yescharacteristic
216#121C2×D19711yescharacteristic
217C4×Q8—1971no—
218C4×D4—1971no—
219C4×D4C2×D19711yescharacteristic
220C4×D4C2×D19711yescharacteristic
221#117C2×D19711yescharacteristic
222#107—1971no—
223#107—1971no—
224#106—1971no—
225#106—1971no—
226C4⋊C8—1971no—
227C4⋊C8—1971no—
228C4⋊C8—1971no—
229C4⋊C8—1971no—
230#103—1971no—
231#103—1971no—
232#102—1971no—
233#102—1971no—
234#102—1971no—
235#102—1971no—
236#102—1971no—
237#102—1971no—
238C2×Q8—3941no—
239C2×Q8—1971no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3941no—
245C2×D4—1971no—
246C2×D4C22×D19711yescharacteristic
247C2×D4C22×D19711yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3941no—
251C22×C4C22×D19711yescharacteristic
252SD16—7881no—
253SD16—7881no—
254D8—7881no—
255D8—7881no—
256M16—7881no—
257C2×C8—7881no—
258C2×C8—1971no—
259C2×C8—1971no—
260C2×C8—1971no—
261C2×C8—1971no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—3941no—
265C4⋊C4—3941no—
266C4⋊C4—3941no—
267C4⋊C4—1971no—
268C4⋊C4—1971no—
269C4⋊C4C22⋊D19711yescharacteristic
270C4⋊C4C22⋊D19711yescharacteristic
271C4⋊C4C22×D19711yescharacteristic
272C4⋊C4C22×D19711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—3941no—
278C22⋊C4—3941no—
279C22⋊C4—3941no—
280C22⋊C4—3941no—
281C22⋊C4—3941no—
282C22⋊C4C22⋊D19711yescharacteristic
283C22⋊C4C22⋊D19711yescharacteristic
284C42—3941no—
285C42C22×D19711yescharacteristic
286C42C22×D19711yescharacteristic
287C23—21no—
288C23—21no—
289C23—3941no—
290C23C2×C22⋊D19711yescharacteristic
291Q8—15761no—
292Q8—3941no—
293Q8—3941no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—15761no—
303D4—3941no—
304D4—3941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—15761no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4C2×C22⋊D19711yescharacteristic
316C2×C4C2×C22⋊D19711yescharacteristic
317C2×C4C2×C22⋊D19711yescharacteristic
318C2×C4C2×C22⋊D19711yescharacteristic
319C2×C4C2×C22⋊D19711yescharacteristic
320C2×C4C23×D19711yescharacteristic
321C8—3941no—
322C8—3941no—
323C8—3941no—
324C8—3941no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—7881no—
335C22—7881no—
336C22—7881no—
337C22C22×C22⋊D19711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—7881no—
346C4—7881no—
347C4—7881no—
348C4#2026337211yescharacteristic
349C4#2026337311yescharacteristic
350C2—41no—
351C2—81no—
352C2—81no—
353C2—15761no—
354C2#2716165711yescharacteristic
355C2#2716188211yescharacteristic
356C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#540×C197C2
socleC22×C197C22×D4
hypercenter#540D197
hypocenterC197#3257
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #27161700C2semidirect
2. #27161383C2semidirect
3. #27161376C2semidirect
4. #27161369C2semidirect
5. #27161359C2semidirect
6. #27161330C2semidirect
7. #27161323C2semidirect
8. #27161313C2semidirect
9. #27161306C2semidirect
10. #27161294C2semidirect
11. #27161294C2semidirect
12. #27161276C2semidirect
13. #27161271C2semidirect
14. #27161106C2semidirect
15. #540×C197C2semidirect
178. C197#3257semidirect
181. #540D197semidirect

Maximal subgroups

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

subgroupindex
#271617002
#271613832
#271613762
#271613692
#271613592
#271613302
#271613232
#271613132
#271613062
#271612942
#271612762
#271612712
#271611062
#540×C1972
#3257197