Finite Group Databasebeta

← #34103281

Subgroups — #34103281

Order 20096.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097756C211yesmaximal, characteristic
2#23757174C211yesmaximal, characteristic
3#23757168C211yesmaximal, characteristic, complemented
4#23757165C211yesmaximal, characteristic, complemented
5#23757164C211yesmaximal, characteristic, complemented
6#23757164C211yesmaximal, characteristic, complemented
7#23757163C211yesmaximal, characteristic, complemented
8#23757159C211yesmaximal, characteristic, complemented
9#23757155C211yesmaximal, characteristic, complemented
10#23757151C211yesmaximal, characteristic
11#23757144C211yesmaximal, characteristic, complemented
12#23757144C211yesmaximal, characteristic, complemented
13#23757132C211yesmaximal, characteristic, complemented
14#23757130C211yesmaximal, characteristic, complemented
15#389×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19#19097756C418yescomplemented
20C2×C4⋊Dic157C2211yescharacteristic
21C2×#14578933C2211yescharacteristic
22C2×#14578933C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578931C2211yescharacteristic
26C2×#14578931C2211yescharacteristic
27C2×#14578931C2211yescharacteristic
28C2×#14578931C2211yescharacteristic
29C2×#14578931C2211yescharacteristic
30C2×#14578931C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×C4×Dic157C2211yescharacteristic
36C2×C4×Dic157C2211yescharacteristic
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×C4⋊C4×C157C2211yescharacteristic
48C2×C4⋊C4×C157C2211yescharacteristic
49C2×C42×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52C23×D157C2311yescharacteristic
53C23×D157C2311yescharacteristic
54C2×D628—41no—
55C2×D628—41no—
56C2×D628C2×C414yes—
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2311yescharacteristic
59C22×Dic157C2311yescharacteristic
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C2×C4×D157—41no—
64C2×C4×D157—41no—
65C2×C4×D157—41no—
66C2×C4×D157—41no—
67C2×C4×D157—41no—
68C2×C4×D157C2×C414yescomplemented
69C4⋊Dic157—41no—
70#14578933—41no—
71#14578933—41no—
72#14578933—41no—
73#14578933C2×C414yescomplemented
74#14578931—41no—
75#14578931—41no—
76#14578931—41no—
77#14578931—41no—
78#14578931—41no—
79#14578931—41no—
80#14578931—41no—
81#14578931C2×C414yes—
82#14578931C2×C414yes—
83#14578931C2×C414yescomplemented
84C4×Dic157—41no—
85C4×Dic157—41no—
86C22×C4×C157C2311yescharacteristic
87C22×C4×C157C2311yescharacteristic
88C22×C4×C157C2311yescharacteristic
89C22×C4×C157C2311yescharacteristic
90C22×C4×C157C2311yescharacteristic
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C4⋊C4×C157—41no—
94C4⋊C4×C157—41no—
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×D157—21no—
109C22×D157—21no—
110C22×D157C22×C412yes—
111C22×D157C22×C412yes—
112D628—44no—
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—
133C23×C157C2411yescharacteristic
134C2×C4×C157—21no—
135C2×C4×C157—21no—
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×C157C22×C412yes—
153C2×C4×C157C22×C412yes—
154C2×C4×C157C22×C412yes—
155C22×C157C23×C411yescharacteristic
156C22×C157H(5,𝔽2)11yescharacteristic
157C22×C157H(5,𝔽2)11yescharacteristic
158C22×C157H(5,𝔽2)11yescharacteristic
159C22×C157H(5,𝔽2)11yescharacteristic
160C22×C1572−1+411yescharacteristic
161C22×C1572−1+411yescharacteristic
162C2×D157—22no—
163C2×D157—22no—
164C2×D157—22no—
165C2×D157—22no—
166C2×D157—22no—
167C2×D157—22no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—41no—
174C2×D157—41no—
175C2×D157—41no—
176C4×C157—22no—
177C4×C157—22no—
178C4×C157—22no—
179C4×C157—41no—
180C4×C157—41no—
181C4×C157—41no—
182C4×C157—41no—
183Dic157—22no—
184Dic157—22no—
185Dic157—41no—
186Dic157—41no—
187Dic157—41no—
188Dic157—41no—
189C2×C157#51811yescharacteristic
190C2×C157#52011yescharacteristic
191C2×C157#52011yescharacteristic
192C2×C157#56011yescharacteristic
193C2×C157#56211yescharacteristic
194C2×C157#56211yescharacteristic
195C2×C157#56311yescharacteristic
196D157—42no—
197D157—42no—
198C157#237611yescharacteristic, complemented, Sylow
199#2376—1571nomaximal, Sylow
200C2×#123—1571no—
201#390—1571no—
202#389—1571no—
203#389D15711yescharacteristic, complemented
204#388—1571no—
205#388—1571no—
206#388—1571no—
207#388—1571no—
208#387—1571no—
209#387—1571no—
210#386—1571no—
211#385—1571no—
212#385—1571no—
213#380—1571no—
214#380—1571no—
215C22×D4—1571no—
216C23×C4—1571no—
217C23×C4—1571no—
218#123—1578no—
219C2×C4⋊C4—1571no—
220C2×C4⋊C4—1571no—
221C2×C4⋊C4—1571no—
222C2×C4⋊C4—1571no—
223C2×C4⋊C4—1571no—
224C2×C4⋊C4C2×D15711yescharacteristic
225C2×C4⋊C4C2×D15711yescharacteristic
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×C22⋊C4—1571no—
237C2×C22⋊C4—1571no—
238C2×C42—1571no—
239C2×C42—1571no—
240C2×C42C2×D15711yescharacteristic
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
251C24—1571no—
252C24—1571no—
253C2×D4—6281no—
254C2×D4—6281no—
255C2×D4—1574no—
256C22×C4—6281no—
257C22×C4—6281no—
258C22×C4—6281no—
259C22×C4—6281no—
260C22×C4—6281no—
261C22×C4—1571no—
262C22×C4—1571no—
263C22×C4—1571no—
264C22×C4—1571no—
265C22×C4—1571no—
266C22×C4—1571no—
267C22×C4—1574no—
268C22×C4C22×D15711yescharacteristic
269C22×C4C22×D15711yescharacteristic
270C22×C4C22×D15711yescharacteristic
271C22×C4C22×D15711yescharacteristic
272C22×C4C22×D15711yescharacteristic
273C22×C4C22×D15711yescharacteristic
274C22×C4C22×D15711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—6281no—
279C4⋊C4—6281no—
280C4⋊C4—6281no—
281C4⋊C4—6281no—
282C4⋊C4—1574no—
283C4⋊C4C4×D15714yescomplemented
284C22⋊C4—6281no—
285C22⋊C4—6281no—
286C22⋊C4—6281no—
287C22⋊C4—6281no—
288C22⋊C4—6281no—
289C22⋊C4—6281no—
290C22⋊C4—6281no—
291C22⋊C4—1574no—
292C22⋊C4—1574no—
293C22⋊C4—1574no—
294C42—41no—
295C42—6281no—
296C42—6281no—
297C23—3141no—
298C23—3141no—
299C23—3141no—
300C23—3141no—
301C23—3141no—
302C23—3141no—
303C23—3141no—
304C23—3141no—
305C23—3141no—
306C23—3141no—
307C23—3141no—
308C23—3141no—
309C23—1572no—
310C23—1572no—
311C23C23×D15711yescharacteristic
312D4—6284no—
313C2×C4—21no—
314C2×C4—21no—
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—3141no—
332C2×C4—3141no—
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—6284no—
348C2×C4—6284no—
349C2×C4—1572no—
350C2×C4—1572no—
351C2×C4C2×C4×D15712yes—
352C2×C4C2×C4×D15712yes—
353C2×C4C2×C4×D15712yes—
354C22—3142no—
355C22—3142no—
356C22—3142no—
357C22—3142no—
358C22—3142no—
359C22—3142no—
360C22—6281no—
361C22—6281no—
362C22—6281no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—6281no—
368C22C22×C4×D15711yescharacteristic
369C22#1909781611yescharacteristic
370C22#1909781611yescharacteristic
371C22#1909781711yescharacteristic
372C22#1909781711yescharacteristic
373C22#1909781911yescharacteristic
374C22#1909781911yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—3142no—
383C4—3142no—
384C4—6281no—
385C4—6281no—
386C4—6281no—
387C4—6281no—
388C2—6282no—
389C2—6282no—
390C2#2375774511yescharacteristic
391C2#2375775411yescharacteristic
392C2#2375775511yescharacteristic
393C2#2375801111yescharacteristic
394C2#2375801911yescharacteristic
395C2#2375802111yescharacteristic
396C2#2375802211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#389×C157C2
socle, cosocleC23×C157C24
prime radicalC22×D628C22
hypercenter#389D157
hypocenterC157#2376
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #23757168C2semidirect
4. #23757165C2semidirect
5. #23757164C2semidirect
6. #23757164C2semidirect
7. #23757163C2semidirect
8. #23757159C2semidirect
9. #23757155C2semidirect
11. #23757144C2semidirect
12. #23757144C2semidirect
13. #23757132C2semidirect
14. #23757130C2semidirect
15. #389×C157C2semidirect
19. #19097756C4semidirect
68. C2×C4×D157C2×C4semidirect
73. #14578933C2×C4semidirect
83. #14578931C2×C4semidirect
96. C4⋊C4×C157C2×C4semidirect
198. C157#2376semidirect
203. #389D157semidirect
283. 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
#237571742
#237571682
#237571652
#237571642
#237571632
#237571592
#237571552
#237571512
#237571442
#237571322
#237571302
#389×C1572
#2376157