Finite Group Databasebeta

← #38769389

Subgroups — #38769389

Order 25472.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#27164577C211yesmaximal, characteristic
2#27164491C211yesmaximal, characteristic, complemented
3C2×#20263870C211yesmaximal, characteristic, complemented
4C2×#20263869C211yesmaximal, characteristic, complemented
5#27163785C211yesmaximal, characteristic, complemented
6#27163784C211yesmaximal, characteristic, complemented
7C2×#99×C199C211yesmaximal, characteristic, complemented
8C22×C22⋊D199C2211yescharacteristic
9#20263938C412yescomplemented
10#20263936—21no—
11#20263927—22no—
12#20263924C412yescomplemented
13#20263923—21no—
14C4×C22⋊D199—22no—
15(C22⋊D199)⋊C4—22no—
16#20263911—21no—
17C2×C22⋊Dic199C2211yescharacteristic
18C2×C22⋊Dic199C2211yescharacteristic
19C2×C22⋊Dic199C2211yescharacteristic
20C22⋊C4×D199—21no—
21#20263870—22no—
22#20263869—22no—
23C22×D4×C199C2211yescharacteristic
24C2×C22⋊C4×C199C2211yescharacteristic
25C2×C22⋊C4×C199C2211yescharacteristic
26#99×C199—22no—
27C23×D199D411yescharacteristic
28C2×C22⋊D199—22no—
29C2×C22⋊D199—21no—
30C2×C22⋊D199—21no—
31C2×C22⋊D199—21no—
32C2×C22⋊D199—21no—
33C2×C22⋊D199C2×C412yes—
34C2×C22⋊D199D412yes—
35C22×Dic199—21no—
36C22×Dic199—21no—
37C22×Dic199—21no—
38C22×Dic199D411yescharacteristic
39C2×C4×D199—22no—
40Dic199⋊C4—22no—
41Dic199⋊C4—22no—
42C22⋊Dic199—22no—
43C22⋊Dic199—21no—
44C22⋊Dic199—21no—
45C22⋊Dic199—21no—
46C22⋊Dic199—21no—
47C22⋊Dic199C2×C412yescomplemented
48C22⋊Dic199D412yescomplemented
49(C2×D199)⋊C4—22no—
50(C2×D199)⋊C4—22no—
51C4×Dic199—22no—
52C24×C199C2×C411yescharacteristic, complemented
53C24×C199C2311yescharacteristic
54C2×D4×C199—21no—
55C2×D4×C199—21no—
56C2×D4×C199—21no—
57C2×D4×C199—21no—
58C2×D4×C199—21no—
59C2×D4×C199—21no—
60C22×C4×C199—21no—
61C22×C4×C199—21no—
62C22×C4×C199C2×C411yescharacteristic, complemented
63C22⋊C4×C199—21no—
64C22⋊C4×C199—21no—
65C22⋊C4×C199—21no—
66C22⋊C4×C199D412yescomplemented
67C22×D199—21no—
68C22×D199—21no—
69C22×D199—81no—
70C22×D199C22⋊C412yes—
71C22⋊D199—22no—
72C22⋊D199—22no—
73C22⋊D199—22no—
74C22⋊D199—22no—
75C22⋊D199—82no—
76C2×Dic199—21no—
77C2×Dic199—21no—
78C2×Dic199—22no—
79C2×Dic199—22no—
80C2×Dic199—41no—
81C2×Dic199—81no—
82C2×Dic199—81no—
83C2×Dic199—81no—
84C2×Dic199C22⋊C412yes—
85C4×D199—82no—
86C23×C199—21no—
87C23×C199—21no—
88C23×C199—21no—
89C23×C199—81no—
90C23×C199—81no—
91C23×C199C22×C411yescharacteristic
92C23×C199C2×D411yescharacteristic
93C23×C199C2×D411yescharacteristic
94C23×C199C2×D411yescharacteristic
95C23×C199C2×D411yescharacteristic
96C23×C199#4111yescharacteristic
97C23×C199#4111yescharacteristic
98D4×C199—41no—
99D4×C199—41no—
100D4×C199—41no—
101D4×C199—41no—
102C2×C4×C199—21no—
103C2×C4×C199—21no—
104C2×C4×C199—21no—
105C2×C4×C199—22no—
106C2×C4×C199—41no—
107C2×C4×C199—81no—
108C2×C4×C199—81no—
109C22×C199—41no—
110C22×C199—41no—
111C22×C199—41no—
112C22×C199—41no—
113C22×C199—41no—
114C22×C199—41no—
115C22×C199—41no—
116C22×C199—42no—
117C22×C199—41no—
118C22×C199—81no—
119C22×C199—81no—
120C22×C199C2×C22⋊C411yescharacteristic
121C22×C199C4×D411yescharacteristic
122C22×C199C4×D411yescharacteristic
123C22×C199#12011yescharacteristic
124C22×C199#12111yescharacteristic
125C22×C199#12111yescharacteristic
126C22×C199#12311yescharacteristic
127C2×D199—22no—
128C2×D199—42no—
129C2×D199—42no—
130C2×D199—81no—
131C4×C199—42no—
132C4×C199—41no—
133C4×C199—81no—
134Dic199—22no—
135Dic199—42no—
136Dic199—42no—
137Dic199—81no—
138C2×C199—21no—
139C2×C199—21no—
140C2×C199—21no—
141C2×C199—21no—
142C2×C199—21no—
143C2×C199—21no—
144C2×C199—21no—
145C2×C199—81no—
146C2×C199#38611yescharacteristic
147D199—42no—
148C199#191311yescharacteristic, complemented, Sylow
149#1913—1991nomaximal, Sylow
150#534—1991no—
151#518—1991no—
152C2×#99—1991no—
153C2×#99—1991no—
154C2×#99D19911yescharacteristic, complemented
155#342—1991no—
156#342—1991no—
157C22×D4—1991no—
158C22×D4C2×D19911yescharacteristic
159#123—3981no—
160#123—1992no—
161#121—3982no—
162#120—3981no—
163#120—1992no—
164C4×D4—3982no—
165C4×D4—3982no—
166#117—3981no—
167C2×C22⋊C4—3981no—
168C2×C22⋊C4—1991no—
169C2×C22⋊C4—1991no—
170C2×C22⋊C4—1991no—
171C2×C22⋊C4C2×D19911yescharacteristic
172C2×C22⋊C4C2×D19911yescharacteristic
173#99—22no—
174#99—3982no—
175#99—3982no—
176C24—1991no—
177C24C4×D19911yescharacteristic, complemented
178C24C22×D19911yescharacteristic
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—3982no—
186C2×D4—3981no—
187C2×D4—3981no—
188C2×D4—3981no—
189C2×D4—3981no—
190C2×D4—1992no—
191C2×D4—1992no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—3982no—
195C22×C4—3981no—
196C22×C4—3981no—
197C22×C4—3981no—
198C22×C4—1991no—
199C22×C4C4×D19911yescharacteristic, complemented
200C4⋊C4—3982no—
201C4⋊C4—3982no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—3982no—
206C22⋊C4—3982no—
207C22⋊C4—3981no—
208C22⋊C4—3981no—
209C22⋊C4—3981no—
210C22⋊C4—3981no—
211C22⋊C4—3982no—
212C22⋊C4—1992no—
213C22⋊C4—1992no—
214C22⋊C4C22⋊D19912yescomplemented
215C42—3982no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—81no—
220C23—81no—
221C23—15921no—
222C23—3981no—
223C23—3981no—
224C23—1992no—
225C23C2×C4×D19911yescharacteristic
226C23D4×D19911yescharacteristic
227C23D4×D19911yescharacteristic
228C23D4×D19911yescharacteristic
229C23C2×C22⋊D19911yescharacteristic
230C23#1569188511yescharacteristic
231C23#1569188611yescharacteristic
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—15922no—
237D4—3982no—
238D4—3982no—
239D4—3982no—
240D4—3982no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—7961no—
249C2×C4—15922no—
250C2×C4—15921no—
251C2×C4—15921no—
252C2×C4—15921no—
253C2×C4—3982no—
254C2×C4—3982no—
255C2×C4—3981no—
256C2×C4—3981no—
257C2×C4—1992no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—81no—
268C22—81no—
269C22—7962no—
270C22—7962no—
271C22—15921no—
272C22—3982no—
273C22C22⋊C4×D19911yescharacteristic
274C22(C22⋊D199)⋊C411yescharacteristic
275C22C4×C22⋊D19911yescharacteristic
276C22#2026392311yescharacteristic
277C22#2026392611yescharacteristic
278C22#2026392811yescharacteristic
279C22#2026393711yescharacteristic
280C4—42no—
281C4—41no—
282C4—81no—
283C4—7962no—
284C4—7962no—
285C4—15921no—
286C4—3982no—
287C2—21no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—21no—
292C2—21no—
293C2—21no—
294C2—81no—
295C2—7962no—
296C2#2716389411yescharacteristic
subgroupG/H
centerC2#27163894
commutatorC23×C199C22×C4
FrattiniC24C22×D199
FittingC2×#99×C199C2
socleC2×C199#386
prime radical#27164577C2
cosocleC24×C199C23
hypercenterC2×#99D199
hypocenterC199#1913
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #27164491C2semidirect
3. C2×#20263870C2semidirect
4. C2×#20263869C2semidirect
5. #27163785C2semidirect
6. #27163784C2semidirect
7. C2×#99×C199C2semidirect
9. #20263938C4semidirect
12. #20263924C4semidirect
47. C22⋊Dic199C2×C4semidirect
48. C22⋊Dic199D4semidirect
52. C24×C199C2×C4semidirect
62. C22×C4×C199C2×C4semidirect
66. C22⋊C4×C199D4semidirect
148. C199#1913semidirect
154. C2×#99D199semidirect
177. C24C4×D199semidirect
199. C22×C4C4×D199semidirect
214. C22⋊C4C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271645772
#271644912
C2×#202638702
C2×#202638692
#271637852
#271637842
C2×#99×C1992
#1913199