Finite Group Databasebeta

← #38745790

Subgroups — #38745790

Order 25216.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263303C211yesmaximal, characteristic
2#27160985C211yesmaximal, characteristic, complemented
3#27160983C211yesmaximal, characteristic, complemented
4#27160966C211yesmaximal, characteristic, complemented
5#27160966C211yesmaximal, characteristic, complemented
6#27160963C211yesmaximal, characteristic, complemented
7#27160962C211yesmaximal, characteristic, complemented
8#27160954C211yesmaximal, characteristic
9#27160952C211yesmaximal, characteristic
10C4×#15691337C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic197C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic197C211yesmaximal, characteristic
13C22⋊C4×Dic197C211yesmaximal, characteristic
14C2×#20263233C211yesmaximal, characteristic
15C4×C22⋊C4×C197C211yesmaximal, characteristic
16C22×Dic394C2211yescharacteristic
17C23×Dic197C2211yescharacteristic
18C23×Dic197C2211yescharacteristic
19#20263303C418yescomplemented
20C2×C4⋊Dic197C2211yescharacteristic
21C2×C4⋊Dic197C2211yescharacteristic
22C2×C4⋊Dic197C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×C22⋊Dic197C2211yescharacteristic
28C2×C22⋊Dic197C2211yescharacteristic
29C2×C4×Dic197C2211yescharacteristic
30C2×C4×Dic197C2211yescharacteristic
31C2×C4×Dic197C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33#20263233C2211yescharacteristic
34#20263233C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C23×C4×C197C2211yescharacteristic
46C2×C22⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48C2×C42×C197C2211yescharacteristic
49C2×C42×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C2×Dic394—41no—
53C2×Dic394—41no—
54C2×Dic394C2×C414yescomplemented
55C22×Dic197—22no—
56C22×Dic197—22no—
57C22×Dic197—41no—
58C22×Dic197—41no—
59C22×Dic197—41no—
60C22×Dic197C2×C414yes—
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C22×Dic197C2311yescharacteristic
67C22×Dic197C2311yescharacteristic
68C22×Dic197C2311yescharacteristic
69C4⋊Dic197—22no—
70C4⋊Dic197—41no—
71C4⋊Dic197C2×C414yescomplemented
72#15691337—22no—
73#15691337—41no—
74#15691337C2×C414yescomplemented
75#15691337C2×C414yescomplemented
76C22⋊Dic197—22no—
77C22⋊Dic197C2×C414yes—
78C4×Dic197—22no—
79C4×Dic197—41no—
80C4×Dic197—41no—
81C4×Dic197—41no—
82C24×C197C2311yescharacteristic
83C22×C4×C197—22no—
84C22×C4×C197—22no—
85C22×C4×C197—41no—
86C22×C4×C197C2311yescharacteristic
87C22×C4×C197C2311yescharacteristic
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22⋊C4×C197C2×C414yes—
93C22⋊C4×C197Q814yescomplemented
94C42×C197—22no—
95C42×C197—41no—
96Dic394—44no—
97C2×Dic197—21no—
98C2×Dic197—21no—
99C2×Dic197—21no—
100C2×Dic197—21no—
101C2×Dic197—21no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197—21no—
105C2×Dic197—21no—
106C2×Dic197—21no—
107C2×Dic197—21no—
108C2×Dic197—21no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—44no—
117C2×Dic197—44no—
118C2×Dic197C22×C412yes—
119C2×Dic197C22×C412yes—
120C2×Dic197C22×C412yes—
121C2×Dic197C22×C412yes—
122C2×Dic197#4112yes—
123C23×C197—21no—
124C23×C197—21no—
125C23×C197—21no—
126C23×C197—21no—
127C23×C197C22×C412yes—
128C23×C197C2×Q812yes—
129C23×C197#4112yes—
130C23×C197C2411yescharacteristic
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—44no—
145C2×C4×C197C22×C412yes—
146C2×C4×C197C22×C412yes—
147C2×C4×C197C2×Q812yes—
148C2×C4×C197C2×Q812yes—
149C2×C4×C197#4112yes—
150C22×C197—22no—
151C22×C197—22no—
152C22×C197—22no—
153C22×C197—22no—
154C22×C197—22no—
155C22×C197—22no—
156C22×C197C4×Q814yes—
157C22×C197C23×C411yescharacteristic
158C22×C197C22×Q811yescharacteristic
159C22×C197C2×#4111yescharacteristic
160C22×C197C2×#4111yescharacteristic
161C22×C197C2×#4111yescharacteristic
162C22×C197H(5,𝔽2)11yescharacteristic
163C22×C197H(5,𝔽2)11yescharacteristic
164C4×C197—22no—
165C4×C197—22no—
166C4×C197—22no—
167C4×C197—22no—
168C4×C197—22no—
169C4×C197—41no—
170Dic197—22no—
171Dic197—22no—
172Dic197—22no—
173Dic197—22no—
174Dic197—22no—
175Dic197—41no—
176Dic197—41no—
177Dic197—41no—
178C2×C197—24no—
179C2×C197C2×C4×Q811yescharacteristic
180C2×C197C4×#4111yescharacteristic
181C2×C197#51811yescharacteristic
182C2×C197#55111yescharacteristic
183C2×C197#55111yescharacteristic
184C2×C197#55411yescharacteristic
185C2×C197#55411yescharacteristic
186C197#236811yescharacteristic, complemented, Sylow
187#2368—1971nomaximal, Sylow
188C2×C22⋊Q8—1971no—
189#391—1971no—
190#391—1971no—
191#387—1971no—
192#387—1971no—
193#387—1971no—
194#387—1971no—
195#385—1971no—
196#385—1971no—
197C4×C4⋊C4—1971no—
198C4×C4⋊C4—1971no—
199C4×C22⋊C4—1971no—
200C4×C22⋊C4—1971no—
201C4×C22⋊C4D19711yescharacteristic
202C2×#95—1971no—
203C22×Q8—1971no—
204C23×C4—1971no—
205C23×C4—1971no—
206C23×C4C2×D19711yescharacteristic
207C22⋊Q8—1978no—
208C2×C4⋊C4—1971no—
209C2×C4⋊C4—1971no—
210C2×C4⋊C4—1971no—
211C2×C4⋊C4—1971no—
212C2×C4⋊C4—1971no—
213C2×C4⋊C4—1971no—
214C2×C4⋊C4—1971no—
215C2×C22⋊C4—1971no—
216C2×C22⋊C4—1971no—
217C2×C22⋊C4C2×D19711yescharacteristic
218C2×C22⋊C4C2×D19711yescharacteristic
219C2×C42—1971no—
220C2×C42—1971no—
221C2×C42—1971no—
222C2×C42—1971no—
223C2×C42C2×D19711yescharacteristic
224C2×C42C2×D19711yescharacteristic
225#95—1971no—
226#95—1971no—
227#95—1971no—
228#95—1971no—
229#95—1971no—
230#95—1971no—
231#95—1971no—
232#95—1971no—
233#95—1971no—
234#95—1971no—
235#95—1971no—
236#95—1971no—
237#95C2×D19711yescharacteristic
238#95C2×D19711yescharacteristic
239C24C22×D19711yescharacteristic
240C2×Q8—7881no—
241C2×Q8—7881no—
242C2×Q8—1974no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—7881no—
247C22×C4—7881no—
248C22×C4—7881no—
249C22×C4—3942no—
250C22×C4—3942no—
251C22×C4—1974no—
252C22×C4—1971no—
253C22×C4—1971no—
254C22×C4—1971no—
255C22×C4—1971no—
256C22×C4—1971no—
257C22×C4—1971no—
258C22×C4—1971no—
259C22×C4—1971no—
260C22×C4C22×D19711yescharacteristic
261C22×C4C22×D19711yescharacteristic
262C22×C4C22×D19711yescharacteristic
263C22×C4C22×D19711yescharacteristic
264C22×C4C22×D19711yescharacteristic
265C22×C4C22×D19711yescharacteristic
266C4⋊C4—7881no—
267C4⋊C4—7881no—
268C4⋊C4—3942no—
269C4⋊C4—3942no—
270C4⋊C4—1974no—
271C4⋊C4—1974no—
272C4⋊C4—1974no—
273C22⋊C4—3942no—
274C22⋊C4—1974no—
275C22⋊C4C4×D19714yes—
276C22⋊C4Dic39414yescomplemented
277C42—22no—
278C42—41no—
279C42—7881no—
280C42—7881no—
281C42—7881no—
282C42—3942no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23C2×C4×D19712yes—
288C23C2×Dic39412yes—
289C23#1569135812yes—
290C23C23×D19711yescharacteristic
291Q8—7884no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—7884no—
307C2×C4—7884no—
308C2×C4—3941no—
309C2×C4—3941no—
310C2×C4—3941no—
311C2×C4—3941no—
312C2×C4—3941no—
313C2×C4—3941no—
314C2×C4—3941no—
315C2×C4—3941no—
316C2×C4—3941no—
317C2×C4—3941no—
318C2×C4—3941no—
319C2×C4—3941no—
320C2×C4—3941no—
321C2×C4—3941no—
322C2×C4—3941no—
323C2×C4—3941no—
324C2×C4—3941no—
325C2×C4—3941no—
326C2×C4—3941no—
327C2×C4—1972no—
328C2×C4—1972no—
329C2×C4—1972no—
330C2×C4—1972no—
331C2×C4—1972no—
332C2×C4C2×C4×D19712yes—
333C2×C4C2×C4×D19712yes—
334C2×C4C2×Dic39412yes—
335C2×C4C2×Dic39412yes—
336C2×C4#1569135712yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22C4×Dic39414yes—
344C22C22×C4×D19711yescharacteristic
345C22C22×Dic39411yescharacteristic
346C22#41×D19711yescharacteristic
347C22C2×#1569135711yescharacteristic
348C22C2×#1569135811yescharacteristic
349C22#2026337111yescharacteristic
350C22#2026337211yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—7881no—
358C4—7881no—
359C4—7881no—
360C4—3942no—
361C4—3942no—
362C4—3942no—
363C4—3942no—
364C4—3942no—
365C2—24no—
366C2C2×C4×Dic39411yescharacteristic
367C2C4×#1569135711yescharacteristic
368C2#2716155111yescharacteristic
369C2#2716175311yescharacteristic
370C2#2716175511yescharacteristic
371C2#2716178611yescharacteristic
372C2#2716178811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
FittingC4×C22⋊C4×C197C2
socle, cosocleC23×C197C24
prime radicalC24×C197C23
hypercenterC4×C22⋊C4D197
hypocenterC197#2368
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #27160985C2semidirect
3. #27160983C2semidirect
4. #27160966C2semidirect
5. #27160966C2semidirect
6. #27160963C2semidirect
7. #27160962C2semidirect
10. C4×#15691337C2semidirect
11. C4×C4⋊Dic197C2semidirect
19. #20263303C4semidirect
54. C2×Dic394C2×C4semidirect
71. C4⋊Dic197C2×C4semidirect
74. #15691337C2×C4semidirect
75. #15691337C2×C4semidirect
93. C22⋊C4×C197Q8semidirect
186. C197#2368semidirect
276. C22⋊C4Dic394semidirect

Maximal subgroups

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

subgroupindex
C2×#202633032
#271609852
#271609832
#271609662
#271609632
#271609622
#271609542
#271609522
C4×#156913372
C4×C4⋊Dic1972
C22⋊C4×Dic1972
C2×#202632332
C4×C22⋊C4×C1972
#2368197