Finite Group Databasebeta

← #38769151

Subgroups — #38769151

Order 25472.

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×D199C211yesmaximal, characteristic, complemented
2#27164530C211yesmaximal, characteristic
3C2×#20263883C211yesmaximal, characteristic, complemented
4C2×#20263881C211yesmaximal, characteristic, complemented
5C4×(C2×D199)⋊C4C211yesmaximal, characteristic, complemented
6#27163777C211yesmaximal, characteristic, complemented
7#339×C199C211yesmaximal, characteristic, complemented
8C2×#15691886C2211yescharacteristic
9C22×C4×D199C2211yescharacteristic
10C2×#15691870C2211yescharacteristic
11C2×#15691869—21no—
12M16×D199—22no—
13M16×D199—22no—
14C2×C8×D199—21no—
15#20263939—21no—
16#20263935C411yescharacteristic, complemented
17#20263935C411yescharacteristic, complemented
18#20263930C411yescharacteristic, complemented
19#20263930C411yescharacteristic, complemented
20#20263925—21no—
21C4×D796—21no—
22C4×D796—21no—
23C2×(C2×D199)⋊C4—21no—
24C2×C4×Dic199C2211yescharacteristic
25#20263883—22no—
26#20263881—22no—
27C199⋊#95—21no—
28C2×M16×C199C2211yescharacteristic
29#117×C199C2211yescharacteristic
30C2×C42×C199C2211yescharacteristic
31C23×D199D411yescharacteristic
32#15691886—21no—
33#15691886—21no—
34#15691886—21no—
35#15691886—21no—
36C2×Dic398C2×C411yescharacteristic
37C2×D796—21no—
38C2×D796—21no—
39C2×D796C2×C411yescharacteristic
40C2×C22⋊D199—21no—
41C22×Dic199—21no—
42C22×Dic199D411yescharacteristic
43C2×C4×D199—21no—
44C2×C4×D199—21no—
45C2×C4×D199—21no—
46C2×C4×D199—21no—
47C2×C4×D199—21no—
48C2×C4×D199C2×C411yescharacteristic
49C2×C4×D199C2×C411yescharacteristic
50C2×C4×D199D411yescharacteristic
51C2×C4×D199D411yescharacteristic
52#15691870—21no—
53#15691870D412yescomplemented
54#15691869—24no—
55C2×C199⋊C8—21no—
56C8×D199—24no—
57C4⋊Dic199—21no—
58C4⋊Dic199—21no—
59(C2×D199)⋊C4—24no—
60(C2×D199)⋊C4—21no—
61(C2×D199)⋊C4—21no—
62(C2×D199)⋊C4—21no—
63(C2×D199)⋊C4—21no—
64C4×Dic199—21no—
65C4×Dic199—21no—
66C22×C4×C199—21no—
67C22×C4×C199C2311yescharacteristic
68M16×C199—21no—
69M16×C199D412yescomplemented
70C2×C8×C199—21no—
71C4⋊C4×C199C2×C411yescharacteristic
72C4⋊C4×C199C2×C411yescharacteristic
73C22⋊C4×C199—21no—
74C42×C199—21no—
75C42×C199—21no—
76C42×C199—21no—
77C22×D199—21no—
78C22×D199—21no—
79C22×D199—22no—
80C22×D199—21no—
81C22×D199—41no—
82C22×D199C22⋊C411yescharacteristic
83C22×D199C22⋊C411yescharacteristic
84Dic398—21no—
85Dic398—21no—
86D796—21no—
87D796—21no—
88D796—42no—
89D796—42no—
90C22⋊D199—41no—
91C22⋊D199—41no—
92C2×Dic199—21no—
93C2×Dic199—21no—
94C2×Dic199—22no—
95C2×Dic199—21no—
96C2×Dic199—41no—
97C2×Dic199—41no—
98C2×Dic199C22⋊C411yescharacteristic
99C2×Dic199C22⋊C411yescharacteristic
100C4×D199—21no—
101C4×D199—21no—
102C4×D199—21no—
103C4×D199—21no—
104C4×D199—21no—
105C4×D199—21no—
106C4×D199—21no—
107C4×D199—21no—
108C4×D199—41no—
109C4×D199—41no—
110C199⋊C8—22no—
111C23×C199C2×D411yescharacteristic
112C2×C4×C199—22no—
113C2×C4×C199—21no—
114C2×C4×C199—21no—
115C2×C4×C199—41no—
116C2×C4×C199—41no—
117C2×C4×C199C22×C411yescharacteristic
118C2×C4×C199C2×D411yescharacteristic
119C2×C4×C199C2×D411yescharacteristic
120C2×C4×C199C2×D411yescharacteristic
121C2×C4×C199#4111yescharacteristic
122C2×C4×C199#4111yescharacteristic
123C8×C199—22no—
124C22×C199—21no—
125C22×C199—21no—
126C22×C199C2×C22⋊C411yescharacteristic
127C22×C199#12111yescharacteristic
128C22×C199#12111yescharacteristic
129C2×D199—21no—
130C2×D199—21no—
131C2×D199—21no—
132C2×D199—21no—
133C2×D199—21no—
134C2×D199—42no—
135C2×D199—42no—
136C2×D199—41no—
137C2×D199—41no—
138C2×D199—41no—
139C2×D199#10412yescomplemented
140C4×C199—42no—
141C4×C199—41no—
142C4×C199—41no—
143C4×C199C4×D411yescharacteristic
144C4×C199C4×D411yescharacteristic
145C4×C199#12011yescharacteristic
146C4×C199#12311yescharacteristic
147Dic199—21no—
148Dic199—21no—
149Dic199—42no—
150Dic199—41no—
151C2×C199—21no—
152C2×C199—21no—
153C2×C199#38611yescharacteristic
154C2×C199C2×#10411yescharacteristic
155C2×C199#42111yescharacteristic
156D199—22no—
157D199—41no—
158D199—81no—
159C199#187111yescharacteristic, complemented, Sylow
160#1871—1991nomaximal, Sylow
161C22×M16—1991no—
162#525—1991no—
163C2×#104—1991no—
164C2×#104—1991no—
165C4×C22⋊C4—1991no—
166#339—1991no—
167#339D19911yescharacteristic, complemented
168C2×#41—1991no—
169C23×C4—1991no—
170C2×M16—3981no—
171C2×M16—3982no—
172C2×M16—3982no—
173C2×M16—1991no—
174C2×M16C2×D19911yescharacteristic
175C22×C8—3981no—
176#123—3981no—
177C22⋊Q8—1991no—
178C22⋊Q8—1991no—
179#121—1991no—
180#121—1991no—
181#120—3981no—
182C4×D4—3981no—
183C4×D4—3981no—
184#117C2×D19911yescharacteristic
185C2×C22⋊C4—3981no—
186C2×C42—1991no—
187C2×C42C2×D19911yescharacteristic
188#104—3982no—
189#104—3982no—
190#95—3981no—
191C24—1991no—
192#41—3981no—
193#41—3981no—
194#41—3981no—
195#41—3981no—
196C2×Q8—1991no—
197C2×D4—3981no—
198C2×D4—3981no—
199C2×D4—3981no—
200C2×D4—1991no—
201C22×C4—21no—
202C22×C4—3981no—
203C22×C4—3981no—
204C22×C4—3981no—
205C22×C4—3981no—
206C22×C4—3981no—
207C22×C4—3981no—
208C22×C4—1991no—
209C22×C4—1991no—
210C22×C4—1991no—
211C22×C4—1991no—
212C22×C4—1991no—
213C22×C4C22×D19911yescharacteristic
214M16—21no—
215M16—3984no—
216M16—3981no—
217M16—1992no—
218M16D79612yescomplemented
219C2×C8—21no—
220C2×C8—3984no—
221C2×C8—3981no—
222C4⋊C4—3981no—
223C4⋊C4—3981no—
224C4⋊C4C4×D19911yescharacteristic
225C4⋊C4C4×D19911yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—3984no—
228C22⋊C4—3981no—
229C22⋊C4—3981no—
230C22⋊C4—3981no—
231C22⋊C4—3981no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—3981no—
236C42—3981no—
237C23—7961no—
238C23—3982no—
239C23—3981no—
240C23—3981no—
241C23—3981no—
242C23—1991no—
243C23—1991no—
244C23D4×D19911yescharacteristic
245Q8—3981no—
246Q8—3981no—
247D4—7962no—
248D4—7962no—
249D4—7961no—
250D4—7961no—
251D4—3981no—
252D4—3981no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—7961no—
259C2×C4—7961no—
260C2×C4—7961no—
261C2×C4—7961no—
262C2×C4—3982no—
263C2×C4—3981no—
264C2×C4—3981no—
265C2×C4—3981no—
266C2×C4—3981no—
267C2×C4—3981no—
268C2×C4—3981no—
269C2×C4—3981no—
270C2×C4—3981no—
271C2×C4—3981no—
272C2×C4—3981no—
273C2×C4—3981no—
274C2×C4—1991no—
275C2×C4—1991no—
276C2×C4C2×C4×D19911yescharacteristic
277C2×C4D4×D19911yescharacteristic
278C2×C4D4×D19911yescharacteristic
279C2×C4C2×D79611yescharacteristic
280C2×C4#1569188611yescharacteristic
281C2×C4#1569188711yescharacteristic
282C8—22no—
283C8—3982no—
284C22—21no—
285C22—21no—
286C22—7962no—
287C22—7962no—
288C22—7961no—
289C22—7961no—
290C22—7961no—
291C22—3981no—
292C22—3981no—
293C22—3981no—
294C22—3981no—
295C22—3981no—
296C22—1992no—
297C22C22⋊C4×D19911yescharacteristic
298C22#2026392811yescharacteristic
299C22#2026393011yescharacteristic
300C4—42no—
301C4—41no—
302C4—41no—
303C4—7962no—
304C4—7961no—
305C4—3981no—
306C4—3981no—
307C4#2026391711yescharacteristic
308C4C4×D79611yescharacteristic
309C4#2026392511yescharacteristic
310C4#2026394011yescharacteristic
311C2—21no—
312C2—21no—
313C2—7961no—
314C2—15921no—
315C2—3982no—
316C2#2716389611yescharacteristic
317C2#104×D19911yescharacteristic
318C2#2716403311yescharacteristic
subgroupG/H
centerC2×C4D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#339×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164530C2
cosocleC22×C4×C199C23
hypercenter#339D199
hypocenterC199#1871
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×M16×D199C2semidirect
3. C2×#20263883C2semidirect
4. C2×#20263881C2semidirect
5. C4×(C2×D199)⋊C4C2semidirect
6. #27163777C2semidirect
7. #339×C199C2semidirect
16. #20263935C4semidirect
17. #20263935C4semidirect
18. #20263930C4semidirect
19. #20263930C4semidirect
53. #15691870D4semidirect
69. M16×C199D4semidirect
139. C2×D199#104semidirect
159. C199#1871semidirect
167. #339D199semidirect
218. M16D796semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1992
#271645302
C2×#202638832
C2×#202638812
C4×(C2×D199)⋊C42
#271637772
#339×C1992
#1871199