Finite Group Databasebeta

← #34102768

Subgroups — #34102768

Order 20096.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097733C211yesmaximal, characteristic
2#23757164C211yesmaximal, characteristic, complemented
3#23757162C211yesmaximal, characteristic, complemented
4#23757159C211yesmaximal, characteristic, complemented
5#23757151C211yesmaximal, characteristic
6#23757151C211yesmaximal, characteristic
7#23757147C211yesmaximal, characteristic, complemented
8#23757144C211yesmaximal, characteristic, complemented
9#23757140C211yesmaximal, characteristic, complemented
10#23757137C211yesmaximal, characteristic, complemented
11#23757135C211yesmaximal, characteristic, complemented
12#23757130C211yesmaximal, characteristic, complemented
13C4×#14578931C211yesmaximal, characteristic, complemented
14#95×D157C211yesmaximal, characteristic, complemented
15#387×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21#19097733C418yescomplemented
22C2×C4⋊Dic157C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578931C2211yescharacteristic
26C2×#14578931C2211yescharacteristic
27C2×#14578931C2211yescharacteristic
28C2×#14578931C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×C4×Dic157C2211yescharacteristic
34C2×C4×Dic157C2211yescharacteristic
35#19097678C2211yescharacteristic
36#19097678C2211yescharacteristic
37#19097678C2211yescharacteristic
38#19097678C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45C2×C4⋊C4×C157C2211yescharacteristic
46C2×C4⋊C4×C157C2211yescharacteristic
47C2×C42×C157C2211yescharacteristic
48#95×C157C2211yescharacteristic
49#95×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52C23×D157C2311yescharacteristic
53C23×D157C2311yescharacteristic
54C2×D628—22no—
55C2×D628—22no—
56C2×D628C2×C414yes—
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2311yescharacteristic
59C22×Dic157C2311yescharacteristic
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C2×C4×D157—22no—
64C2×C4×D157—22no—
65C2×C4×D157—41no—
66C2×C4×D157—41no—
67C2×C4×D157—41no—
68C2×C4×D157—41no—
69C2×C4×D157—41no—
70C2×C4×D157—41no—
71C2×C4×D157—41no—
72C2×C4×D157—41no—
73C2×C4×D157C2×C414yes—
74C2×C4×D157C2×C414yescomplemented
75C4⋊Dic157—41no—
76#14578933—41no—
77#14578933—41no—
78#14578931—22no—
79#14578931—22no—
80#14578931—41no—
81#14578931—41no—
82#14578931—41no—
83#14578931—41no—
84#14578931C2×C414yes—
85#14578931C2×C414yescomplemented
86C4×Dic157—41no—
87C4×Dic157C2×C414yescomplemented
88C22×C4×C157C2311yescharacteristic
89C22×C4×C157C2311yescharacteristic
90C22×C4×C157C2311yescharacteristic
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C22×C4×C157C2311yescharacteristic
94C22×C4×C157C2311yescharacteristic
95C4⋊C4×C157—41no—
96C4⋊C4×C157C2×C414yescomplemented
97C42×C157—41no—
98C22×D157—21no—
99C22×D157—21no—
100C22×D157—21no—
101C22×D157—21no—
102C22×D157—21no—
103C22×D157—21no—
104C22×D157—21no—
105C22×D157—21no—
106C22×D157—21no—
107C22×D157—21no—
108C22×D157C22×C412yes—
109C22×D157C22×C412yes—
110C22×D157#4112yes—
111C22×D157#4112yes—
112D628—28no—
113C2×Dic157—21no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—21no—
128C2×Dic157—21no—
129C2×Dic157C22×C412yes—
130C2×Dic157C22×C412yes—
131C4×D157—44no—
132C4×D157—44no—
133C4×D157—44no—
134C4×D157—44no—
135C23×C157C2411yescharacteristic
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157—21no—
154C2×C4×C157C22×C412yes—
155C2×C4×C157C22×C412yes—
156C2×C4×C157C22×C412yes—
157C22×C157C23×C411yescharacteristic
158C22×C157C2×#4111yescharacteristic
159C22×C157C2×#4111yescharacteristic
160C22×C157H(5,𝔽2)11yescharacteristic
161C22×C157H(5,𝔽2)11yescharacteristic
162C22×C157H(5,𝔽2)11yescharacteristic
163C22×C1572−1+411yescharacteristic
164C2×D157—22no—
165C2×D157—22no—
166C2×D157—22no—
167C2×D157—22no—
168C2×D157—22no—
169C2×D157—22no—
170C2×D157—22no—
171C2×D157—22no—
172C2×D157—22no—
173C2×D157—41no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157#11714yes—
178C4×C157—22no—
179C4×C157—22no—
180C4×C157—22no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157—41no—
185Dic157—22no—
186Dic157—22no—
187Dic157—41no—
188Dic157—41no—
189Dic157—41no—
190Dic157—41no—
191C2×C157C2×#11711yescharacteristic
192C2×C157#51811yescharacteristic
193C2×C157#52011yescharacteristic
194C2×C157#53811yescharacteristic
195C2×C157#53811yescharacteristic
196C2×C157#53911yescharacteristic
197C2×C157#53911yescharacteristic
198D157—24no—
199D157—42no—
200C157#231811yescharacteristic, complemented, Sylow
201#2318—1571nomaximal, Sylow
202C2×C4×D4—1571no—
203#388—1571no—
204#388—1571no—
205#387—1571no—
206#387D15711yescharacteristic, complemented
207#386—1571no—
208#386—1571no—
209#385—1571no—
210#385—1571no—
211#383—1571no—
212#382—1571no—
213#381—1571no—
214#380—1571no—
215C4×C22⋊C4—1571no—
216C2×#95—1571no—
217C22×D4—1571no—
218C23×C4—1571no—
219C23×C4—1571no—
220C23×C4—1571no—
221C23×C4—1571no—
222C4×D4—1578no—
223C2×C4⋊C4—1571no—
224C2×C4⋊C4—1571no—
225C2×C4⋊C4—1571no—
226C2×C4⋊C4C2×D15711yescharacteristic
227C2×C4⋊C4C2×D15711yescharacteristic
228C2×C22⋊C4—1571no—
229C2×C22⋊C4—1571no—
230C2×C22⋊C4—1571no—
231C2×C22⋊C4—1571no—
232C2×C22⋊C4—1571no—
233C2×C22⋊C4—1571no—
234C2×C22⋊C4—1571no—
235C2×C22⋊C4—1571no—
236C2×C42—1571no—
237C2×C42—1571no—
238C2×C42C2×D15711yescharacteristic
239#95—1571no—
240#95—1571no—
241#95—1571no—
242#95—1571no—
243#95—1571no—
244#95—1571no—
245#95—1571no—
246#95—1571no—
247#95—1571no—
248#95—1571no—
249#95C2×D15711yescharacteristic
250#95C2×D15711yescharacteristic
251#95C2×D15711yescharacteristic
252#95C2×D15711yescharacteristic
253C24—1571no—
254C24—1571no—
255C2×D4—3142no—
256C2×D4—3142no—
257C2×D4—1574no—
258C22×C4—3142no—
259C22×C4—3142no—
260C22×C4—6281no—
261C22×C4—6281no—
262C22×C4—6281no—
263C22×C4—6281no—
264C22×C4—6281no—
265C22×C4—6281no—
266C22×C4—6281no—
267C22×C4—6281no—
268C22×C4—1571no—
269C22×C4—1571no—
270C22×C4—1571no—
271C22×C4—1571no—
272C22×C4—1571no—
273C22×C4—1571no—
274C22×C4—1574no—
275C22×C4—1574no—
276C22×C4C22×D15711yescharacteristic
277C22×C4C22×D15711yescharacteristic
278C22×C4C22×D15711yescharacteristic
279C22×C4C22×D15711yescharacteristic
280C22×C4C22×D15711yescharacteristic
281C22×C4C22×D15711yescharacteristic
282C22×C4C22×D15711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—6281no—
285C4⋊C4—6281no—
286C4⋊C4—6281no—
287C4⋊C4C4×D15714yescomplemented
288C22⋊C4—3142no—
289C22⋊C4—3142no—
290C22⋊C4—6281no—
291C22⋊C4—6281no—
292C22⋊C4—6281no—
293C22⋊C4—6281no—
294C22⋊C4—1574no—
295C22⋊C4—1574no—
296C42—41no—
297C42—6281no—
298C42—1574no—
299C23—3141no—
300C23—3141no—
301C23—3141no—
302C23—3141no—
303C23—3141no—
304C23—3141no—
305C23—3141no—
306C23—3141no—
307C23—3141no—
308C23—3141no—
309C23—1572no—
310C23—1572no—
311C23—1572no—
312C23—1572no—
313C23C23×D15711yescharacteristic
314D4—3148no—
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—21no—
333C2×C4—3141no—
334C2×C4—3141no—
335C2×C4—3141no—
336C2×C4—3141no—
337C2×C4—3141no—
338C2×C4—3141no—
339C2×C4—3141no—
340C2×C4—3141no—
341C2×C4—3141no—
342C2×C4—3141no—
343C2×C4—3141no—
344C2×C4—3141no—
345C2×C4—3141no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—6284no—
350C2×C4—6284no—
351C2×C4—6284no—
352C2×C4—6284no—
353C2×C4—1572no—
354C2×C4—1572no—
355C2×C4C2×C4×D15712yes—
356C2×C4C2×C4×D15712yes—
357C2×C4C2×C4×D15712yes—
358C22—3142no—
359C22—3142no—
360C22—3142no—
361C22—3142no—
362C22—3142no—
363C22—3142no—
364C22—3142no—
365C22—3142no—
366C22—3142no—
367C22—6281no—
368C22—6281no—
369C22—6281no—
370C22—6281no—
371C22—1574no—
372C22C22×C4×D15711yescharacteristic
373C22#41×D15711yescharacteristic
374C22#41×D15711yescharacteristic
375C22#1909781611yescharacteristic
376C22#1909781711yescharacteristic
377C22#1909781711yescharacteristic
378C22#1909781911yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—3142no—
387C4—3142no—
388C4—6281no—
389C4—6281no—
390C4—6281no—
391C4—6281no—
392C2—3144no—
393C2—6282no—
394C2#117×D15711yescharacteristic
395C2#2375774511yescharacteristic
396C2#2375775511yescharacteristic
397C2#2375785911yescharacteristic
398C2#2375786211yescharacteristic
399C2#2375786911yescharacteristic
400C2#2375787111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#387×C157C2
socle, cosocleC23×C157C24
prime radicalC22×D628C22
hypercenter#387D157
hypocenterC157#2318
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #23757164C2semidirect
3. #23757162C2semidirect
4. #23757159C2semidirect
7. #23757147C2semidirect
8. #23757144C2semidirect
9. #23757140C2semidirect
10. #23757137C2semidirect
11. #23757135C2semidirect
12. #23757130C2semidirect
13. C4×#14578931C2semidirect
14. #95×D157C2semidirect
15. #387×C157C2semidirect
21. #19097733C4semidirect
74. C2×C4×D157C2×C4semidirect
85. #14578931C2×C4semidirect
87. C4×Dic157C2×C4semidirect
96. C4⋊C4×C157C2×C4semidirect
200. C157#2318semidirect
206. #387D157semidirect
287. C4⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977332
#237571642
#237571622
#237571592
#237571512
#237571472
#237571442
#237571402
#237571372
#237571352
#237571302
C4×#145789312
#95×D1572
#387×C1572
#2318157