Finite Group Databasebeta

← #34103291

Subgroups — #34103291

Order 20096.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097746C211yesmaximal, characteristic
2#23757175C211yesmaximal, characteristic, complemented
3#23757174C211yesmaximal, characteristic, complemented
4#23757164C211yesmaximal, characteristic, complemented
5#23757162C211yesmaximal, characteristic, complemented
6#23757159C211yesmaximal, characteristic, complemented
7#23757151C211yesmaximal, characteristic, complemented
8#23757151C211yesmaximal, characteristic, complemented
9#23757151C211yesmaximal, characteristic, complemented
10#23757151C211yesmaximal, characteristic, complemented
11#23757147C211yesmaximal, characteristic, complemented
12#23757144C211yesmaximal, characteristic, complemented
13#23757135C211yesmaximal, characteristic, complemented
14#23757132C211yesmaximal, characteristic, complemented
15#387×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20C22×C4×D157C2211yescharacteristic
21C22×C4×D157C2211yescharacteristic
22#19097746C418yescomplemented
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×#14578931C2211yescharacteristic
27C2×#14578931C2211yescharacteristic
28C2×#14578931C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×#14578931C2211yescharacteristic
37C2×#14578931C2211yescharacteristic
38C2×C4×Dic157C2211yescharacteristic
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
54C23×D157C2311yescharacteristic
55C2×D628—41no—
56C2×D628—41no—
57C2×D628—41no—
58C2×D628—41no—
59C2×D628—41no—
60C2×D628—41no—
61C2×D628C2×C414yes—
62C2×D628C2×C414yescomplemented
63C2×D628C2×C414yescomplemented
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C22×Dic157C2311yescharacteristic
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×D157—41no—
76C2×C4×D157—41no—
77C2×C4×D157C2×C414yescomplemented
78C4⋊Dic157—41no—
79#14578933—41no—
80#14578933—41no—
81#14578931—41no—
82#14578931—41no—
83#14578931—41no—
84#14578931—41no—
85#14578931—41no—
86#14578931—41no—
87#14578931—41no—
88#14578931—41no—
89#14578931—41no—
90#14578931—41no—
91#14578931C2×C414yescomplemented
92#14578931C2×C414yescomplemented
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—41no—
102C4⋊C4×C157C2×C414yescomplemented
103C42×C157—41no—
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×D157—21no—
115C22×D157—21no—
116C22×D157—21no—
117C22×D157—21no—
118C22×D157—21no—
119C22×D157—21no—
120C22×D157—21no—
121C22×D157—21no—
122C22×D157C22×C412yes—
123C22×D157C22×C412yes—
124C22×D157C22×C412yes—
125D628—44no—
126D628—44no—
127D628—44no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—21no—
140C2×Dic157—21no—
141C2×Dic157—21no—
142C2×Dic157C22×C412yes—
143C4×D157—44no—
144C4×D157—44no—
145C4×D157—44no—
146C23×C157C2411yescharacteristic
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×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157C22×C412yes—
166C2×C4×C157C22×C412yes—
167C2×C4×C157C22×C412yes—
168C22×C157C23×C411yescharacteristic
169C22×C157H(5,𝔽2)11yescharacteristic
170C22×C157H(5,𝔽2)11yescharacteristic
171C22×C157H(5,𝔽2)11yescharacteristic
172C22×C157H(5,𝔽2)11yescharacteristic
173C22×C157H(5,𝔽2)11yescharacteristic
174C22×C1572−1+411yescharacteristic
175C2×D157—22no—
176C2×D157—22no—
177C2×D157—22no—
178C2×D157—22no—
179C2×D157—22no—
180C2×D157—22no—
181C2×D157—22no—
182C2×D157—22no—
183C2×D157—22no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—41no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—41no—
196C4×C157—22no—
197C4×C157—22no—
198C4×C157—22no—
199C4×C157—41no—
200C4×C157—41no—
201C4×C157—41no—
202C4×C157—41no—
203Dic157—22no—
204Dic157—41no—
205Dic157—41no—
206Dic157—41no—
207Dic157—41no—
208C2×C157#51811yescharacteristic
209C2×C157#51811yescharacteristic
210C2×C157#52011yescharacteristic
211C2×C157#56011yescharacteristic
212C2×C157#56011yescharacteristic
213C2×C157#56211yescharacteristic
214C2×C157#56211yescharacteristic
215D157—42no—
216D157—42no—
217D157—42no—
218C157#237711yescharacteristic, complemented, Sylow
219#2377—1571nomaximal, Sylow
220C2×#121—1571no—
221#390—1571no—
222#390—1571no—
223#388—1571no—
224#388—1571no—
225#387—1571no—
226#387D15711yescharacteristic, complemented
227#386—1571no—
228#386—1571no—
229#386—1571no—
230#386—1571no—
231#385—1571no—
232#385—1571no—
233#381—1571no—
234#380—1571no—
235C22×D4—1571no—
236C22×D4—1571no—
237C22×D4—1571no—
238C23×C4—1571no—
239C23×C4—1571no—
240C23×C4—1571no—
241#121—1578no—
242C2×C4⋊C4—1571no—
243C2×C4⋊C4—1571no—
244C2×C4⋊C4—1571no—
245C2×C4⋊C4C2×D15711yescharacteristic
246C2×C4⋊C4C2×D15711yescharacteristic
247C2×C22⋊C4—1571no—
248C2×C22⋊C4—1571no—
249C2×C22⋊C4—1571no—
250C2×C22⋊C4—1571no—
251C2×C22⋊C4—1571no—
252C2×C22⋊C4—1571no—
253C2×C22⋊C4—1571no—
254C2×C22⋊C4—1571no—
255C2×C22⋊C4—1571no—
256C2×C22⋊C4—1571no—
257C2×C22⋊C4—1571no—
258C2×C22⋊C4—1571no—
259C2×C42—1571no—
260C2×C42C2×D15711yescharacteristic
261#95—1571no—
262#95—1571no—
263#95—1571no—
264#95—1571no—
265#95—1571no—
266#95—1571no—
267#95C2×D15711yescharacteristic
268#95C2×D15711yescharacteristic
269#95C2×D15711yescharacteristic
270#95C2×D15711yescharacteristic
271C24—1571no—
272C24—1571no—
273C24—1571no—
274C2×D4—6281no—
275C2×D4—6281no—
276C2×D4—6281no—
277C2×D4—6281no—
278C2×D4—6281no—
279C2×D4—6281no—
280C2×D4—1574no—
281C2×D4—1574no—
282C2×D4—1574no—
283C22×C4—6281no—
284C22×C4—6281no—
285C22×C4—6281no—
286C22×C4—6281no—
287C22×C4—6281no—
288C22×C4—6281no—
289C22×C4—6281no—
290C22×C4—6281no—
291C22×C4—1571no—
292C22×C4—1571no—
293C22×C4—1571no—
294C22×C4—1571no—
295C22×C4—1571no—
296C22×C4—1574no—
297C22×C4C22×D15711yescharacteristic
298C22×C4C22×D15711yescharacteristic
299C22×C4C22×D15711yescharacteristic
300C22×C4C22×D15711yescharacteristic
301C22×C4C22×D15711yescharacteristic
302C22×C4C22×D15711yescharacteristic
303C22×C4C22×D15711yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—6281no—
306C4⋊C4—6281no—
307C4⋊C4—6281no—
308C4⋊C4C4×D15714yescomplemented
309C22⋊C4—6281no—
310C22⋊C4—6281no—
311C22⋊C4—6281no—
312C22⋊C4—6281no—
313C22⋊C4—6281no—
314C22⋊C4—6281no—
315C22⋊C4—6281no—
316C22⋊C4—6281no—
317C22⋊C4—6281no—
318C22⋊C4—6281no—
319C22⋊C4—1574no—
320C22⋊C4—1574no—
321C42—41no—
322C42—6281no—
323C23—3141no—
324C23—3141no—
325C23—3141no—
326C23—3141no—
327C23—3141no—
328C23—3141no—
329C23—3141no—
330C23—3141no—
331C23—3141no—
332C23—3141no—
333C23—3141no—
334C23—3141no—
335C23—3141no—
336C23—3141no—
337C23—3141no—
338C23—3141no—
339C23—3141no—
340C23—3141no—
341C23—1572no—
342C23—1572no—
343C23—1572no—
344C23C23×D15711yescharacteristic
345D4—6284no—
346D4—6284no—
347D4—6284no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—3141no—
369C2×C4—3141no—
370C2×C4—3141no—
371C2×C4—3141no—
372C2×C4—3141no—
373C2×C4—3141no—
374C2×C4—3141no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—3141no—
380C2×C4—6284no—
381C2×C4—6284no—
382C2×C4—6284no—
383C2×C4—1572no—
384C2×C4C2×C4×D15712yes—
385C2×C4C2×C4×D15712yes—
386C2×C4C2×C4×D15712yes—
387C22—3142no—
388C22—3142no—
389C22—3142no—
390C22—3142no—
391C22—3142no—
392C22—3142no—
393C22—3142no—
394C22—3142no—
395C22—3142no—
396C22—6281no—
397C22—6281no—
398C22—6281no—
399C22—6281no—
400C22—6281no—
401C22—6281no—
402C22—6281no—
403C22—6281no—
404C22—6281no—
405C22—6281no—
406C22—6281no—
407C22—6281no—
408C22C22×C4×D15711yescharacteristic
409C22#1909781611yescharacteristic
410C22#1909781711yescharacteristic
411C22#1909781711yescharacteristic
412C22#1909781711yescharacteristic
413C22#1909781711yescharacteristic
414C22#1909781911yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—3142no—
423C4—6281no—
424C4—6281no—
425C4—6281no—
426C4—6281no—
427C2—6282no—
428C2—6282no—
429C2—6282no—
430C2#2375774511yescharacteristic
431C2#2375774711yescharacteristic
432C2#2375775511yescharacteristic
433C2#2375801311yescharacteristic
434C2#2375801411yescharacteristic
435C2#2375801911yescharacteristic
436C2#2375802111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#387×C157C2
socle, cosocleC23×C157C24
prime radicalC2×#19097746C2
hypercenter#387D157
hypocenterC157#2377
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #23757175C2semidirect
3. #23757174C2semidirect
4. #23757164C2semidirect
5. #23757162C2semidirect
6. #23757159C2semidirect
7. #23757151C2semidirect
8. #23757151C2semidirect
9. #23757151C2semidirect
10. #23757151C2semidirect
11. #23757147C2semidirect
12. #23757144C2semidirect
13. #23757135C2semidirect
14. #23757132C2semidirect
15. #387×C157C2semidirect
22. #19097746C4semidirect
62. C2×D628C2×C4semidirect
63. C2×D628C2×C4semidirect
77. C2×C4×D157C2×C4semidirect
91. #14578931C2×C4semidirect
92. #14578931C2×C4semidirect
102. C4⋊C4×C157C2×C4semidirect
218. C157#2377semidirect
226. #387D157semidirect
308. C4⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977462
#237571752
#237571742
#237571642
#237571622
#237571592
#237571512
#237571472
#237571442
#237571352
#237571322
#387×C1572
#2377157