Finite Group Databasebeta

← #34102809

Subgroups — #34102809

Order 20096.

632 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097756C211yesmaximal, characteristic
2C2×C4×D628C211yesmaximal, characteristic
3C2×C4⋊C4×D157C211yesmaximal, characteristic, complemented
4#23757172C211yesmaximal, characteristic, complemented
5#23757170C211yesmaximal, characteristic, complemented
6#23757165C211yesmaximal, characteristic, complemented
7#23757162C211yesmaximal, characteristic, complemented
8#23757157C211yesmaximal, characteristic, complemented
9#23757151C211yesmaximal, characteristic
10#23757147C211yesmaximal, characteristic, complemented
11#23757147C211yesmaximal, characteristic, complemented
12#23757144C211yesmaximal, characteristic, complemented
13#23757135C211yesmaximal, characteristic, complemented
14#23757120C211yesmaximal, characteristic, complemented
15#387×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21#19097756C418yescomplemented
22C4×D628—24no—
23C2×C4⋊Dic157C2211yescharacteristic
24C2×C4⋊Dic157C2211yescharacteristic
25C2×C4⋊Dic157C2211yescharacteristic
26C2×#14578933C2211yescharacteristic
27C2×#14578933C2211yescharacteristic
28C2×#14578933C2211yescharacteristic
29C2×#14578933C2211yescharacteristic
30C4⋊C4×D157—24no—
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×#14578931C2211yescharacteristic
37C2×#14578931C2211yescharacteristic
38C2×#14578931C2211yescharacteristic
39C2×C4×Dic157C2211yescharacteristic
40C2×C4×Dic157C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45#19097678C2211yescharacteristic
46#19097678C2211yescharacteristic
47C2×C4⋊C4×C157C2211yescharacteristic
48C2×C4⋊C4×C157C2211yescharacteristic
49C2×C42×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52#95×C157C2211yescharacteristic
53#95×C157C2211yescharacteristic
54C23×D157C2311yescharacteristic
55C23×D157C2311yescharacteristic
56C2×D628—22no—
57C2×D628—22no—
58C2×D628C2×C414yes—
59C22×Dic157C2311yescharacteristic
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C2×C4×D157—22no—
66C2×C4×D157—22no—
67C2×C4×D157—22no—
68C2×C4×D157—22no—
69C2×C4×D157—41no—
70C2×C4×D157—41no—
71C2×C4×D157—41no—
72C2×C4×D157—41no—
73C2×C4×D157—41no—
74C2×C4×D157—41no—
75C2×C4×D157C2×C414yescomplemented
76C2×C4×D157D414yes—
77C4⋊Dic157—22no—
78C4⋊Dic157—22no—
79C4⋊Dic157—41no—
80#14578933—22no—
81#14578933—22no—
82#14578933—41no—
83#14578933C2×C414yescomplemented
84#14578931—22no—
85#14578931—22no—
86#14578931—22no—
87#14578931—41no—
88#14578931—41no—
89#14578931C2×C414yescomplemented
90#14578931C2×C414yescomplemented
91#14578931C2×C414yescomplemented
92C4×Dic157—41no—
93C4×Dic157—41no—
94C22×C4×C157C2311yescharacteristic
95C22×C4×C157C2311yescharacteristic
96C22×C4×C157C2311yescharacteristic
97C22×C4×C157C2311yescharacteristic
98C22×C4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101C4⋊C4×C157—22no—
102C4⋊C4×C157C2×C414yescomplemented
103C42×C157—22no—
104C22×D157—21no—
105C22×D157—21no—
106C22×D157—21no—
107C22×D157—21no—
108C22×D157—21no—
109C22×D157—21no—
110C22×D157—21no—
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157C22×C412yes—
115C22×D157C22×C412yes—
116C22×D157C2×D412yes—
117C22×D157#4112yes—
118D628—28no—
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×Dic157—21no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157C22×C412yes—
135C2×Dic157C22×C412yes—
136C2×Dic157C2×D412yes—
137C4×D157—28no—
138C4×D157—44no—
139C4×D157—44no—
140C4×D157—44no—
141C23×C157C2411yescharacteristic
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×C157—21no—
155C2×C4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157C22×C412yes—
160C2×C4×C157C22×C412yes—
161C2×C4×C157C22×C412yes—
162C2×C4×C157C2×D412yes—
163C22×C157C23×C411yescharacteristic
164C22×C157C22×D411yescharacteristic
165C22×C157C2×#4111yescharacteristic
166C22×C157H(5,𝔽2)11yescharacteristic
167C22×C157H(5,𝔽2)11yescharacteristic
168C22×C1572−1+411yescharacteristic
169C22×C1572−1+411yescharacteristic
170C2×D157—22no—
171C2×D157—22no—
172C2×D157—22no—
173C2×D157—22no—
174C2×D157—22no—
175C2×D157—22no—
176C2×D157—22no—
177C2×D157—22no—
178C2×D157—22no—
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157C4×D414yes—
184C4×C157—22no—
185C4×C157—22no—
186C4×C157—22no—
187C4×C157—22no—
188C4×C157—41no—
189C4×C157—41no—
190C4×C157—41no—
191Dic157—22no—
192Dic157—22no—
193Dic157—22no—
194Dic157—41no—
195Dic157—41no—
196Dic157—41no—
197C2×C157C2×C4×D411yescharacteristic
198C2×C157#52011yescharacteristic
199C2×C157#52011yescharacteristic
200C2×C157#53411yescharacteristic
201C2×C157#53611yescharacteristic
202C2×C157#53911yescharacteristic
203C2×C157#53911yescharacteristic
204D157—24no—
205D157—42no—
206C157#232211yescharacteristic, complemented, Sylow
207#2322—1571nomaximal, Sylow
208C2×#123—1571no—
209C2×C4×D4—1571no—
210C22×C4⋊C4—1571no—
211#389—1571no—
212#389—1571no—
213#388—1571no—
214#388—1571no—
215#387—1571no—
216#387D15711yescharacteristic, complemented
217#386—1571no—
218#385—1571no—
219#385—1571no—
220#385—1571no—
221#381—1571no—
222C4×C22⋊C4—1571no—
223C22×D4—1571no—
224C23×C4—1571no—
225C23×C4—1571no—
226C23×C4—1571no—
227C23×C4—1571no—
228#123—1578no—
229C4×D4—3144no—
230C2×C4⋊C4—3144no—
231C2×C4⋊C4—1571no—
232C2×C4⋊C4—1571no—
233C2×C4⋊C4—1571no—
234C2×C4⋊C4—1571no—
235C2×C4⋊C4—1571no—
236C2×C4⋊C4—1571no—
237C2×C4⋊C4—1571no—
238C2×C4⋊C4C2×D15711yescharacteristic
239C2×C4⋊C4C2×D15711yescharacteristic
240C2×C22⋊C4—1571no—
241C2×C22⋊C4—1571no—
242C2×C22⋊C4—1571no—
243C2×C22⋊C4—1571no—
244C2×C22⋊C4—1571no—
245C2×C22⋊C4—1571no—
246C2×C22⋊C4—1571no—
247C2×C22⋊C4—1571no—
248C2×C42—1571no—
249C2×C42—1571no—
250C2×C42C2×D15711yescharacteristic
251#95—1571no—
252#95—1571no—
253#95—1571no—
254#95—1571no—
255#95—1571no—
256#95—1571no—
257#95C2×D15711yescharacteristic
258#95C2×D15711yescharacteristic
259#95C2×D15711yescharacteristic
260#95C2×D15711yescharacteristic
261C24—1571no—
262C24—1571no—
263C2×D4—3142no—
264C2×D4—3142no—
265C2×D4—1574no—
266C22×C4—3142no—
267C22×C4—3142no—
268C22×C4—3142no—
269C22×C4—3142no—
270C22×C4—6281no—
271C22×C4—6281no—
272C22×C4—6281no—
273C22×C4—6281no—
274C22×C4—6281no—
275C22×C4—6281no—
276C22×C4—1571no—
277C22×C4—1571no—
278C22×C4—1571no—
279C22×C4—1571no—
280C22×C4—1571no—
281C22×C4—1571no—
282C22×C4—1574no—
283C22×C4—1574no—
284C22×C4C22×D15711yescharacteristic
285C22×C4C22×D15711yescharacteristic
286C22×C4C22×D15711yescharacteristic
287C22×C4C22×D15711yescharacteristic
288C22×C4C22×D15711yescharacteristic
289C22×C4C22×D15711yescharacteristic
290C22×C4C22×D15711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—3142no—
293C4⋊C4—3142no—
294C4⋊C4—3142no—
295C4⋊C4—3142no—
296C4⋊C4—6281no—
297C4⋊C4—6281no—
298C4⋊C4—1574no—
299C4⋊C4C4×D15714yescomplemented
300C22⋊C4—3142no—
301C22⋊C4—3142no—
302C22⋊C4—3142no—
303C22⋊C4—6281no—
304C22⋊C4—6281no—
305C22⋊C4—1574no—
306C22⋊C4—1574no—
307C22⋊C4—1574no—
308C42—22no—
309C42—6281no—
310C42—6281no—
311C23—3141no—
312C23—3141no—
313C23—3141no—
314C23—3141no—
315C23—3141no—
316C23—3141no—
317C23—3141no—
318C23—3141no—
319C23—3141no—
320C23—3141no—
321C23—1572no—
322C23—1572no—
323C23—1572no—
324C23—1572no—
325C23C23×D15711yescharacteristic
326D4—3148no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3148no—
345C2×C4—3141no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—3141no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—3141no—
358C2×C4—3141no—
359C2×C4—3141no—
360C2×C4—6284no—
361C2×C4—6284no—
362C2×C4—6284no—
363C2×C4—1572no—
364C2×C4—1572no—
365C2×C4—1572no—
366C2×C4C2×C4×D15712yes—
367C2×C4C2×C4×D15712yes—
368C2×C4C2×C4×D15712yes—
369C2×C4D4×D15712yes—
370C22—3142no—
371C22—3142no—
372C22—3142no—
373C22—3142no—
374C22—3142no—
375C22—3142no—
376C22—3142no—
377C22—3142no—
378C22—3142no—
379C22—6281no—
380C22—6281no—
381C22—6281no—
382C22—6281no—
383C22—1574no—
384C22C22×C4×D15711yescharacteristic
385C22C2×D4×D15711yescharacteristic
386C22#41×D15711yescharacteristic
387C22#1909781711yescharacteristic
388C22#1909781711yescharacteristic
389C22#1909781811yescharacteristic
390C22#1909781911yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—3142no—
399C4—3142no—
400C4—3142no—
401C4—6281no—
402C4—6281no—
403C4—6281no—
404C2—3144no—
405C2—6282no—
406C2C4×D4×D15711yescharacteristic
407C2#2375775311yescharacteristic
408C2#2375775511yescharacteristic
409C2#2375783411yescharacteristic
410C2#2375784611yescharacteristic
411C2#2375786411yescharacteristic
412C2#2375787111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#387×C157C2
socle, cosocleC23×C157C24
prime radicalC22×D628C22
hypercenter#387D157
hypocenterC157#2322
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊C4×D157C2semidirect
4. #23757172C2semidirect
5. #23757170C2semidirect
6. #23757165C2semidirect
7. #23757162C2semidirect
8. #23757157C2semidirect
10. #23757147C2semidirect
11. #23757147C2semidirect
12. #23757144C2semidirect
13. #23757135C2semidirect
14. #23757120C2semidirect
15. #387×C157C2semidirect
21. #19097756C4semidirect
75. C2×C4×D157C2×C4semidirect
83. #14578933C2×C4semidirect
89. #14578931C2×C4semidirect
90. #14578931C2×C4semidirect
91. #14578931C2×C4semidirect
102. C4⋊C4×C157C2×C4semidirect
206. C157#2322semidirect
216. #387D157semidirect
299. C4⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977562
C2×C4×D6282
C2×C4⋊C4×D1572
#237571722
#237571702
#237571652
#237571622
#237571572
#237571512
#237571472
#237571442
#237571352
#237571202
#387×C1572
#2322157