Finite Group Databasebeta

← #34103305

Subgroups — #34103305

Order 20096.

520 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×#19097757C211yesmaximal, characteristic
2#23757179C211yesmaximal, characteristic, complemented
3#23757177C211yesmaximal, characteristic, complemented
4#23757164C211yesmaximal, characteristic, complemented
5#23757163C211yesmaximal, characteristic, complemented
6#23757161C211yesmaximal, characteristic, complemented
7#23757150C211yesmaximal, characteristic
8#23757149C211yesmaximal, characteristic
9#23757148C211yesmaximal, characteristic, complemented
10#23757147C211yesmaximal, characteristic, complemented
11#23757147C211yesmaximal, characteristic, complemented
12#23757145C211yesmaximal, characteristic, complemented
13#23757138C211yesmaximal, characteristic, complemented
14#23757137C211yesmaximal, characteristic, complemented
15#388×C157C211yesmaximal, characteristic, complemented
16C22×Dic314C2211yescharacteristic
17C22×C22⋊D157C2211yescharacteristic
18C23×Dic157C2211yescharacteristic
19C22×C4×D157C2211yescharacteristic
20#19097757C418yescomplemented
21C2×C4⋊Dic157C2211yescharacteristic
22C2×C4⋊Dic157C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×C22⋊Dic157C2211yescharacteristic
27C2×C22⋊Dic157C2211yescharacteristic
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×C22⋊C4×C157C2211yescharacteristic
47C2×C22⋊C4×C157C2211yescharacteristic
48C2×C22⋊C4×C157C2211yescharacteristic
49C2×C42×C157C2211yescharacteristic
50#95×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52C23×D157C2311yescharacteristic
53C2×Dic314—41no—
54C2×Dic314—41no—
55C2×Dic314C2×C414yescomplemented
56C2×C22⋊D157—41no—
57C2×C22⋊D157—41no—
58C2×C22⋊D157C2×C414yes—
59C22×Dic157—41no—
60C22×Dic157—41no—
61C22×Dic157—41no—
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C22×Dic157C2311yescharacteristic
69C2×C4×D157—41no—
70C2×C4×D157—41no—
71C2×C4×D157—41no—
72C4⋊Dic157—41no—
73C4⋊Dic157—41no—
74#14578933—41no—
75#14578933—41no—
76#14578933—41no—
77C22⋊Dic157—41no—
78C22⋊Dic157C2×C414yescomplemented
79#14578931—41no—
80#14578931—41no—
81#14578931—41no—
82#14578931C2×C414yescomplemented
83#14578931C2×C414yescomplemented
84C4×Dic157—41no—
85C4×Dic157C2×C414yescomplemented
86C24×C157C2311yescharacteristic
87C22×C4×C157C2311yescharacteristic
88C22×C4×C157C2311yescharacteristic
89C22×C4×C157C2311yescharacteristic
90C22×C4×C157C2311yescharacteristic
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C4⋊C4×C157—41no—
94C22⋊C4×C157—41no—
95C22⋊C4×C157—41no—
96C22⋊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×D157C22×C412yes—
105Dic314—44no—
106C22⋊D157—44no—
107C2×Dic157—21no—
108C2×Dic157—21no—
109C2×Dic157—21no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—21no—
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—44no—
126C2×Dic157C22×C412yes—
127C2×Dic157C22×C412yes—
128C2×Dic157C22×C412yes—
129C4×D157—44no—
130C23×C157—21no—
131C23×C157—21no—
132C23×C157—21no—
133C23×C157—21no—
134C23×C157—21no—
135C23×C157—21no—
136C23×C157C22×C412yes—
137C23×C157C2411yescharacteristic
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—
156C22×C157—22no—
157C22×C157—22no—
158C22×C157—22no—
159C22×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157C23×C411yescharacteristic
164C22×C157H(5,𝔽2)11yescharacteristic
165C22×C157H(5,𝔽2)11yescharacteristic
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—41no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C4×C157—22no—
178C4×C157—22no—
179C4×C157—41no—
180C4×C157—41no—
181C4×C157—41no—
182C4×C157—41no—
183Dic157—22no—
184Dic157—22no—
185Dic157—22no—
186Dic157—41no—
187Dic157—41no—
188Dic157—41no—
189Dic157—41no—
190C2×C157—42no—
191C2×C157#51811yescharacteristic
192C2×C157#52011yescharacteristic
193C2×C157#52011yescharacteristic
194C2×C157H(3,𝔽4)11yescharacteristic
195C2×C157#56211yescharacteristic
196C2×C157#56211yescharacteristic
197C2×C157#56311yescharacteristic
198D157—42no—
199C157#237911yescharacteristic, complemented, Sylow
200#2379—1571nomaximal, Sylow
201C2×#124—1571no—
202#391—1571no—
203#391—1571no—
204#388—1571no—
205#388—1571no—
206#388—1571no—
207#388D15711yescharacteristic, complemented
208#386—1571no—
209#386—1571no—
210#385—1571no—
211#385—1571no—
212#385—1571no—
213#385—1571no—
214#382—1571no—
215#382—1571no—
216C22×Q8—1571no—
217C22×D4—1571no—
218C23×C4—1571no—
219C23×C4—1571no—
220#124—1578no—
221C2×C4⋊C4—1571no—
222C2×C4⋊C4—1571no—
223C2×C4⋊C4—1571no—
224C2×C4⋊C4—1571no—
225C2×C4⋊C4—1571no—
226C2×C4⋊C4C2×D15711yescharacteristic
227C2×C22⋊C4—1571no—
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⋊C4C2×D15711yescharacteristic
235C2×C22⋊C4C2×D15711yescharacteristic
236C2×C22⋊C4C2×D15711yescharacteristic
237C2×C42—1571no—
238C2×C42—1571no—
239C2×C42C2×D15711yescharacteristic
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#95—1571no—
250#95C2×D15711yescharacteristic
251#95C2×D15711yescharacteristic
252C24—1571no—
253C24C22×D15711yescharacteristic
254C2×Q8—6281no—
255C2×Q8—6281no—
256C2×Q8—1574no—
257C2×D4—6281no—
258C2×D4—6281no—
259C2×D4—1574no—
260C22×C4—6281no—
261C22×C4—6281no—
262C22×C4—6281no—
263C22×C4—6281no—
264C22×C4—6281no—
265C22×C4—6281no—
266C22×C4—1571no—
267C22×C4—1571no—
268C22×C4—1571no—
269C22×C4—1571no—
270C22×C4—1571no—
271C22×C4—1571no—
272C22×C4—1571no—
273C22×C4C22×D15711yescharacteristic
274C22×C4C22×D15711yescharacteristic
275C22×C4C22×D15711yescharacteristic
276C22×C4C22×D15711yescharacteristic
277C22×C4C22×D15711yescharacteristic
278C22×C4C22×D15711yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—6281no—
281C4⋊C4—6281no—
282C4⋊C4—6281no—
283C4⋊C4—6281no—
284C4⋊C4—6281no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—6281no—
288C22⋊C4—6281no—
289C22⋊C4—6281no—
290C22⋊C4—6281no—
291C22⋊C4—1574no—
292C22⋊C4—1574no—
293C22⋊C4—1574no—
294C22⋊C4C4×D15714yescomplemented
295C42—41no—
296C42—6281no—
297C42—1574no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—3141no—
305C23—3141no—
306C23—3141no—
307C23—3141no—
308C23—3141no—
309C23—3141no—
310C23—1572no—
311C23C2×C4×D15712yes—
312C23C23×D15711yescharacteristic
313Q8—6284no—
314D4—6284no—
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—3141no—
348C2×C4—3141no—
349C2×C4—6284no—
350C2×C4—6284no—
351C2×C4—1572no—
352C2×C4—1572no—
353C2×C4—1572no—
354C2×C4C2×C4×D15712yes—
355C2×C4C2×C4×D15712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3142no—
364C22—3142no—
365C22—3142no—
366C22—6281no—
367C22—6281no—
368C22—6281no—
369C22—6281no—
370C22C22×C4×D15711yescharacteristic
371C22#1909781611yescharacteristic
372C22#1909781611yescharacteristic
373C22#1909781711yescharacteristic
374C22#1909781711yescharacteristic
375C22#1909781811yescharacteristic
376C22#1909781811yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3142no—
384C4—3142no—
385C4—3142no—
386C4—6281no—
387C4—6281no—
388C4—6281no—
389C4—6281no—
390C2—42no—
391C2—6282no—
392C2#2375774511yescharacteristic
393C2#2375775211yescharacteristic
394C2#2375775311yescharacteristic
395C2#2375801511yescharacteristic
396C2#2375801811yescharacteristic
397C2#2375802011yescharacteristic
398C2#2375802511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#388×C157C2
socle, cosocleC23×C157C24
prime radicalC22×C22⋊D157C22
hypercenter#388D157
hypocenterC157#2379
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #23757179C2semidirect
3. #23757177C2semidirect
4. #23757164C2semidirect
5. #23757163C2semidirect
6. #23757161C2semidirect
9. #23757148C2semidirect
10. #23757147C2semidirect
11. #23757147C2semidirect
12. #23757145C2semidirect
13. #23757138C2semidirect
14. #23757137C2semidirect
15. #388×C157C2semidirect
20. #19097757C4semidirect
55. C2×Dic314C2×C4semidirect
78. C22⋊Dic157C2×C4semidirect
82. #14578931C2×C4semidirect
83. #14578931C2×C4semidirect
85. C4×Dic157C2×C4semidirect
96. C22⋊C4×C157C2×C4semidirect
199. C157#2379semidirect
207. #388D157semidirect
294. C22⋊C4C4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977572
#237571792
#237571772
#237571642
#237571632
#237571612
#237571502
#237571492
#237571482
#237571472
#237571452
#237571382
#237571372
#388×C1572
#2379157