Finite Group Databasebeta

← #38745970

Subgroups — #38745970

Order 25216.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263286C211yesmaximal, characteristic
2C2×#20263282C211yesmaximal, characteristic, complemented
3#27161020C211yesmaximal, characteristic, complemented
4#27161014C211yesmaximal, characteristic, complemented
5#27161013C211yesmaximal, characteristic, complemented
6#27161008C211yesmaximal, characteristic, complemented
7#27161007C211yesmaximal, characteristic, complemented
8#27160995C211yesmaximal, characteristic
9#27160994C211yesmaximal, characteristic
10C4⋊C4×Dic197C211yesmaximal, characteristic, complemented
11#27160930C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
13#27160926C211yesmaximal, characteristic, complemented
14#27160926C211yesmaximal, characteristic, complemented
15#397×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197C2211yescharacteristic
17C23×Dic197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19#20263286—24no—
20#20263282—24no—
21C2×C4⋊Dic197C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×C22⋊Dic197C2211yescharacteristic
28C2×C22⋊Dic197C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35C2×C4×Dic197C2211yescharacteristic
36C2×C4×Dic197C2211yescharacteristic
37C2×C4×Dic197C2211yescharacteristic
38C2×C4×Dic197C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C22⋊C4×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53C23×D197C2311yescharacteristic
54C2×C22⋊D197—22no—
55C2×C22⋊D197—22no—
56C2×C22⋊D197—22no—
57C22×Dic197—22no—
58C22×Dic197—22no—
59C22×Dic197D414yes—
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C22×Dic197C2311yescharacteristic
67C2×C4×D197—22no—
68C2×C4×D197—22no—
69C2×C4×D197—22no—
70C4⋊Dic197—22no—
71#15691337—22no—
72#15691337—22no—
73#15691337—22no—
74#15691337—22no—
75#15691337—22no—
76C22⋊Dic197—22no—
77C22⋊Dic197—22no—
78#15691335—22no—
79#15691335—22no—
80#15691335—22no—
81#15691335—22no—
82#15691335—22no—
83C4×Dic197—22no—
84C4×Dic197—22no—
85C4×Dic197—22no—
86C4×Dic197—22no—
87C4×Dic197—22no—
88C24×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C22×C4×C197C2311yescharacteristic
95C4⋊C4×C197—22no—
96C22⋊C4×C197—22no—
97C22⋊C4×C197—22no—
98C22⋊C4×C197—22no—
99C22×D197—21no—
100C22×D197—21no—
101C22×D197—21no—
102C22×D197—21no—
103C22×D197—21no—
104C22×D197—21no—
105C22×D197#4112yes—
106C22⋊D197—28no—
107C2×Dic197—21no—
108C2×Dic197—21no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—28no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197C2×D412yes—
124C2×Dic197C2×D412yes—
125C2×Dic197#4112yes—
126C2×Dic197#4112yes—
127C2×Dic197#4112yes—
128C2×Dic197#4112yes—
129C4×D197—44no—
130C23×C197—21no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—21no—
134C23×C197—21no—
135C23×C197—21no—
136C23×C197C2×D412yes—
137C23×C197C2411yescharacteristic
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—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197—21no—
155C2×C4×C197#4112yes—
156C22×C197—22no—
157C22×C197—22no—
158C22×C197—22no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197C22×D411yescharacteristic
164C22×C197C2×#4111yescharacteristic
165C22×C197C2×#4111yescharacteristic
166C22×C197C2×#4111yescharacteristic
167C22×C197C2×#4111yescharacteristic
168C22×C197C2×#4111yescharacteristic
169C22×C197C2×#4111yescharacteristic
170C2×D197—22no—
171C2×D197—22no—
172C2×D197—22no—
173C2×D197—41no—
174C2×D197—41no—
175C2×D197—41no—
176C2×D197—41no—
177C4×C197—22no—
178C4×C197—41no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197—41no—
183Dic197—22no—
184Dic197—22no—
185Dic197—22no—
186Dic197—41no—
187Dic197—41no—
188Dic197—41no—
189Dic197#12314yes—
190C2×C197—42no—
191C2×C197C2×#12311yescharacteristic
192C2×C197#52511yescharacteristic
193C2×C197#52911yescharacteristic
194C2×C197#52911yescharacteristic
195C2×C197#52911yescharacteristic
196C2×C197#52911yescharacteristic
197C2×C197#53211yescharacteristic
198D197—42no—
199C197#239511yescharacteristic, complemented, Sylow
200#2395—1971nomaximal, Sylow
201C2×C4×D4—1971no—
202C2×#117—1971no—
203#400—1971no—
204#399—1971no—
205#398—1971no—
206#397—1971no—
207#397—1971no—
208#397D19711yescharacteristic, complemented
209#394—1971no—
210#394—1971no—
211C4×C4⋊C4—1971no—
212C4×C4⋊C4—1971no—
213C4×C22⋊C4—1971no—
214C4×C22⋊C4—1971no—
215C4×C22⋊C4—1971no—
216C22×D4—1971no—
217C23×C4—1971no—
218C23×C4—1971no—
219C4×D4—3944no—
220#117—3944no—
221C2×C4⋊C4—1971no—
222C2×C4⋊C4—1971no—
223C2×C4⋊C4—1971no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4—1971no—
226C2×C4⋊C4—1971no—
227C2×C4⋊C4C2×D19711yescharacteristic
228C2×C22⋊C4—1971no—
229C2×C22⋊C4—1971no—
230C2×C22⋊C4—1971no—
231C2×C22⋊C4—1971no—
232C2×C22⋊C4—1971no—
233C2×C22⋊C4—1971no—
234C2×C22⋊C4—1971no—
235C2×C22⋊C4C2×D19711yescharacteristic
236C2×C22⋊C4C2×D19711yescharacteristic
237C2×C22⋊C4C2×D19711yescharacteristic
238C2×C42—1971no—
239C2×C42—1971no—
240C2×C42—1971no—
241C2×C42—1971no—
242C2×C42—1971no—
243#95—1971no—
244#95—1971no—
245#95—1971no—
246#95—1971no—
247#95—1971no—
248#95—1971no—
249#95—1971no—
250#95C2×D19711yescharacteristic
251#95C2×D19711yescharacteristic
252#95C2×D19711yescharacteristic
253C24—1971no—
254C24C22×D19711yescharacteristic
255C2×D4—3942no—
256C2×D4—3942no—
257C2×D4—3942no—
258C22×C4—3942no—
259C22×C4—3942no—
260C22×C4—3942no—
261C22×C4—3942no—
262C22×C4—3942no—
263C22×C4—1971no—
264C22×C4—1971no—
265C22×C4—1971no—
266C22×C4—1974no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4—1971no—
270C22×C4—1971no—
271C22×C4C22×D19711yescharacteristic
272C22×C4C22×D19711yescharacteristic
273C22×C4C22×D19711yescharacteristic
274C22×C4C22×D19711yescharacteristic
275C22×C4C22×D19711yescharacteristic
276C22×C4C22×D19711yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—3942no—
279C4⋊C4—3942no—
280C4⋊C4—3942no—
281C4⋊C4—3942no—
282C4⋊C4—3942no—
283C4⋊C4—3942no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—3942no—
288C22⋊C4—3942no—
289C22⋊C4—3942no—
290C22⋊C4—3942no—
291C22⋊C4—3942no—
292C22⋊C4—3942no—
293C22⋊C4—3942no—
294C42—3942no—
295C42—3942no—
296C42—3942no—
297C42—3942no—
298C42—3942no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—3941no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23—3941no—
310C23—3941no—
311C23—1972no—
312C23D4×D19712yes—
313C23C23×D19711yescharacteristic
314D4—3948no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—7884no—
333C2×C4—3941no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—3941no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3948no—
349C2×C4—1972no—
350C2×C4—1972no—
351C2×C4—1972no—
352C2×C4—1972no—
353C2×C4—1972no—
354C2×C4—1972no—
355C2×C4#1569135712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—7881no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—3942no—
368C22—3942no—
369C22—3942no—
370C22C2×D4×D19711yescharacteristic
371C22#41×D19711yescharacteristic
372C22#41×D19711yescharacteristic
373C22#41×D19711yescharacteristic
374C22#41×D19711yescharacteristic
375C22#41×D19711yescharacteristic
376C22C2×#1569135711yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—7881no—
384C4—7881no—
385C4—7881no—
386C4—3942no—
387C4—3942no—
388C4—3942no—
389C4—1974no—
390C2—42no—
391C2—7882no—
392C2#123×D19711yescharacteristic
393C2#2716158711yescharacteristic
394C2#2716160611yescharacteristic
395C2#2716160811yescharacteristic
396C2#2716161211yescharacteristic
397C2#2716161211yescharacteristic
398C2#2716162411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#397×C197C2
prime radicalC22×C22⋊D197C22
hypercenter#397D197
hypocenterC197#2395
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#20263282C2semidirect
3. #27161020C2semidirect
4. #27161014C2semidirect
5. #27161013C2semidirect
6. #27161008C2semidirect
7. #27161007C2semidirect
10. C4⋊C4×Dic197C2semidirect
11. #27160930C2semidirect
12. C22⋊C4×Dic197C2semidirect
13. #27160926C2semidirect
14. #27160926C2semidirect
15. #397×C197C2semidirect
199. C197#2395semidirect
208. #397D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632862
C2×#202632822
#271610202
#271610142
#271610132
#271610082
#271610072
#271609952
#271609942
C4⋊C4×Dic1972
#271609302
C22⋊C4×Dic1972
#271609262
#397×C1972
#2395197