Finite Group Databasebeta

← #38743013

Subgroups — #38743013

Order 25216.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D197C211yesmaximal, characteristic, complemented
2#27161591C211yesmaximal, characteristic
3C2×#20263254C211yesmaximal, characteristic, complemented
4C2×#20263252C211yesmaximal, characteristic, complemented
5C4×#15691335C211yesmaximal, characteristic, complemented
6#27160838C211yesmaximal, characteristic, complemented
7#339×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic
9C22×C4×D197C2211yescharacteristic
10C2×#15691342C2211yescharacteristic
11C2×#15691341—21no—
12M16×D197—22no—
13M16×D197—22no—
14C2×C8×D197—21no—
15#20263310—21no—
16#20263306C411yescharacteristic, complemented
17#20263306C411yescharacteristic, complemented
18#20263301C411yescharacteristic, complemented
19#20263301C411yescharacteristic, complemented
20#20263296—21no—
21C4×D788—21no—
22C4×D788—21no—
23C2×#15691335—21no—
24C2×C4×Dic197C2211yescharacteristic
25#20263254—22no—
26#20263252—22no—
27#20263233—21no—
28C2×M16×C197C2211yescharacteristic
29#117×C197C2211yescharacteristic
30C2×C42×C197C2211yescharacteristic
31C23×D197D411yescharacteristic
32#15691358—21no—
33#15691358—21no—
34#15691358—21no—
35#15691358—21no—
36C2×Dic394C2×C411yescharacteristic
37C2×D788—21no—
38C2×D788—21no—
39C2×D788C2×C411yescharacteristic
40C2×C22⋊D197—21no—
41C22×Dic197—21no—
42C22×Dic197D411yescharacteristic
43C2×C4×D197—21no—
44C2×C4×D197—21no—
45C2×C4×D197—21no—
46C2×C4×D197—21no—
47C2×C4×D197—21no—
48C2×C4×D197C2×C411yescharacteristic
49C2×C4×D197C2×C411yescharacteristic
50C2×C4×D197D411yescharacteristic
51C2×C4×D197D411yescharacteristic
52#15691342—21no—
53#15691342D412yescomplemented
54#15691341—24no—
55C2×#13096802—21no—
56C8×D197—24no—
57C4⋊Dic197—21no—
58C4⋊Dic197—21no—
59#15691335—24no—
60#15691335—21no—
61#15691335—21no—
62#15691335—21no—
63#15691335—21no—
64C4×Dic197—21no—
65C4×Dic197—21no—
66C22×C4×C197—21no—
67C22×C4×C197C2311yescharacteristic
68M16×C197—21no—
69M16×C197D412yescomplemented
70C2×C8×C197—21no—
71C4⋊C4×C197C2×C411yescharacteristic
72C4⋊C4×C197C2×C411yescharacteristic
73C22⋊C4×C197—21no—
74C42×C197—21no—
75C42×C197—21no—
76C42×C197—21no—
77C22×D197—21no—
78C22×D197—21no—
79C22×D197—22no—
80C22×D197—21no—
81C22×D197—41no—
82C22×D197C22⋊C411yescharacteristic
83C22×D197C22⋊C411yescharacteristic
84Dic394—21no—
85Dic394—21no—
86D788—21no—
87D788—21no—
88D788—42no—
89D788—42no—
90C22⋊D197—41no—
91C22⋊D197—41no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—22no—
95C2×Dic197—21no—
96C2×Dic197—41no—
97C2×Dic197—41no—
98C2×Dic197C22⋊C411yescharacteristic
99C2×Dic197C22⋊C411yescharacteristic
100C4×D197—21no—
101C4×D197—21no—
102C4×D197—21no—
103C4×D197—21no—
104C4×D197—21no—
105C4×D197—21no—
106C4×D197—21no—
107C4×D197—21no—
108C4×D197—41no—
109C4×D197—41no—
110#13096802—22no—
111C23×C197C2×D411yescharacteristic
112C2×C4×C197—22no—
113C2×C4×C197—21no—
114C2×C4×C197—21no—
115C2×C4×C197—41no—
116C2×C4×C197—41no—
117C2×C4×C197C22×C411yescharacteristic
118C2×C4×C197C2×D411yescharacteristic
119C2×C4×C197C2×D411yescharacteristic
120C2×C4×C197C2×D411yescharacteristic
121C2×C4×C197#4111yescharacteristic
122C2×C4×C197#4111yescharacteristic
123C8×C197—22no—
124C22×C197—21no—
125C22×C197—21no—
126C22×C197C2×C22⋊C411yescharacteristic
127C22×C197#12111yescharacteristic
128C22×C197#12111yescharacteristic
129C2×D197—21no—
130C2×D197—21no—
131C2×D197—21no—
132C2×D197—21no—
133C2×D197—21no—
134C2×D197—42no—
135C2×D197—42no—
136C2×D197—41no—
137C2×D197—41no—
138C2×D197—41no—
139C2×D197#10412yescomplemented
140C4×C197—42no—
141C4×C197—41no—
142C4×C197—41no—
143C4×C197C4×D411yescharacteristic
144C4×C197C4×D411yescharacteristic
145C4×C197#12011yescharacteristic
146C4×C197#12311yescharacteristic
147Dic197—21no—
148Dic197—21no—
149Dic197—42no—
150Dic197—41no—
151C2×C197—21no—
152C2×C197—21no—
153C2×C197#38611yescharacteristic
154C2×C197C2×#10411yescharacteristic
155C2×C197#42111yescharacteristic
156D197—22no—
157D197—41no—
158D197—81no—
159C197#187111yescharacteristic, complemented, Sylow
160#1871—1971nomaximal, Sylow
161C22×M16—1971no—
162#525—1971no—
163C2×#104—1971no—
164C2×#104—1971no—
165C4×C22⋊C4—1971no—
166#339—1971no—
167#339D19711yescharacteristic, complemented
168C2×#41—1971no—
169C23×C4—1971no—
170C2×M16—3941no—
171C2×M16—3942no—
172C2×M16—3942no—
173C2×M16—1971no—
174C2×M16C2×D19711yescharacteristic
175C22×C8—3941no—
176#123—3941no—
177C22⋊Q8—1971no—
178C22⋊Q8—1971no—
179#121—1971no—
180#121—1971no—
181#120—3941no—
182C4×D4—3941no—
183C4×D4—3941no—
184#117C2×D19711yescharacteristic
185C2×C22⋊C4—3941no—
186C2×C42—1971no—
187C2×C42C2×D19711yescharacteristic
188#104—3942no—
189#104—3942no—
190#95—3941no—
191C24—1971no—
192#41—3941no—
193#41—3941no—
194#41—3941no—
195#41—3941no—
196C2×Q8—1971no—
197C2×D4—3941no—
198C2×D4—3941no—
199C2×D4—3941no—
200C2×D4—1971no—
201C22×C4—21no—
202C22×C4—3941no—
203C22×C4—3941no—
204C22×C4—3941no—
205C22×C4—3941no—
206C22×C4—3941no—
207C22×C4—3941no—
208C22×C4—1971no—
209C22×C4—1971no—
210C22×C4—1971no—
211C22×C4—1971no—
212C22×C4—1971no—
213C22×C4C22×D19711yescharacteristic
214M16—21no—
215M16—3944no—
216M16—3941no—
217M16—1972no—
218M16D78812yescomplemented
219C2×C8—21no—
220C2×C8—3944no—
221C2×C8—3941no—
222C4⋊C4—3941no—
223C4⋊C4—3941no—
224C4⋊C4C4×D19711yescharacteristic
225C4⋊C4C4×D19711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3944no—
228C22⋊C4—3941no—
229C22⋊C4—3941no—
230C22⋊C4—3941no—
231C22⋊C4—3941no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—3941no—
236C42—3941no—
237C23—7881no—
238C23—3942no—
239C23—3941no—
240C23—3941no—
241C23—3941no—
242C23—1971no—
243C23—1971no—
244C23D4×D19711yescharacteristic
245Q8—3941no—
246Q8—3941no—
247D4—7882no—
248D4—7882no—
249D4—7881no—
250D4—7881no—
251D4—3941no—
252D4—3941no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—7881no—
259C2×C4—7881no—
260C2×C4—7881no—
261C2×C4—7881no—
262C2×C4—3942no—
263C2×C4—3941no—
264C2×C4—3941no—
265C2×C4—3941no—
266C2×C4—3941no—
267C2×C4—3941no—
268C2×C4—3941no—
269C2×C4—3941no—
270C2×C4—3941no—
271C2×C4—3941no—
272C2×C4—3941no—
273C2×C4—3941no—
274C2×C4—1971no—
275C2×C4—1971no—
276C2×C4C2×C4×D19711yescharacteristic
277C2×C4D4×D19711yescharacteristic
278C2×C4D4×D19711yescharacteristic
279C2×C4C2×D78811yescharacteristic
280C2×C4#1569135811yescharacteristic
281C2×C4#1569135911yescharacteristic
282C8—22no—
283C8—3942no—
284C22—21no—
285C22—21no—
286C22—7882no—
287C22—7882no—
288C22—7881no—
289C22—7881no—
290C22—7881no—
291C22—3941no—
292C22—3941no—
293C22—3941no—
294C22—3941no—
295C22—3941no—
296C22—1972no—
297C22C22⋊C4×D19711yescharacteristic
298C22#2026329911yescharacteristic
299C22#2026330111yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—7882no—
304C4—7881no—
305C4—3941no—
306C4—3941no—
307C4#2026328811yescharacteristic
308C4C4×D78811yescharacteristic
309C4#2026329611yescharacteristic
310C4#2026331111yescharacteristic
311C2—21no—
312C2—21no—
313C2—7881no—
314C2—15761no—
315C2—3942no—
316C2#2716095711yescharacteristic
317C2#104×D19711yescharacteristic
318C2#2716109411yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#339×C197C2
socleC22×C197C2×C22⋊C4
prime radical#27161591C2
cosocleC22×C4×C197C23
hypercenter#339D197
hypocenterC197#1871
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×M16×D197C2semidirect
3. C2×#20263254C2semidirect
4. C2×#20263252C2semidirect
5. C4×#15691335C2semidirect
6. #27160838C2semidirect
7. #339×C197C2semidirect
16. #20263306C4semidirect
17. #20263306C4semidirect
18. #20263301C4semidirect
19. #20263301C4semidirect
53. #15691342D4semidirect
69. M16×C197D4semidirect
139. C2×D197#104semidirect
159. C197#1871semidirect
167. #339D197semidirect
218. M16D788semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1972
#271615912
C2×#202632542
C2×#202632522
C4×#156913352
#271608382
#339×C1972
#1871197