Finite Group Databasebeta

← #34102842

Subgroups — #34102842

Order 20096.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097757C211yesmaximal, characteristic
2C2×#19097736C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D157C211yesmaximal, characteristic
4#23757177C211yesmaximal, characteristic, complemented
5#23757173C211yesmaximal, characteristic
6#23757167C211yesmaximal, characteristic, complemented
7#23757166C211yesmaximal, characteristic, complemented
8#23757163C211yesmaximal, characteristic, complemented
9#23757162C211yesmaximal, characteristic, complemented
10#23757147C211yesmaximal, characteristic, complemented
11#23757146C211yesmaximal, characteristic, complemented
12#23757144C211yesmaximal, characteristic, complemented
13#23757142C211yesmaximal, characteristic, complemented
14#23757118C211yesmaximal, characteristic, complemented
15#385×C157C211yesmaximal, characteristic, complemented
16C22×Dic314C2211yescharacteristic
17C22×C22⋊D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19#19097757C418yescomplemented
20#19097736—24no—
21C4×C22⋊D157—24no—
22C2×C4⋊Dic157C2211yescharacteristic
23C2×#14578933C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×#14578933C2211yescharacteristic
27C2×#14578933C2211yescharacteristic
28C2×C22⋊Dic157C2211yescharacteristic
29C2×C22⋊Dic157C2211yescharacteristic
30C2×C22⋊Dic157C2211yescharacteristic
31C2×#14578931C2211yescharacteristic
32C2×#14578931C2211yescharacteristic
33C2×#14578931C2211yescharacteristic
34C2×#14578931C2211yescharacteristic
35C2×#14578931C2211yescharacteristic
36C2×C4×Dic157C2211yescharacteristic
37C2×C4×Dic157C2211yescharacteristic
38C2×C4×Dic157C2211yescharacteristic
39C2×C4×Dic157C2211yescharacteristic
40C2×C4×Dic157C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45#19097678C2211yescharacteristic
46#19097678C2211yescharacteristic
47C23×C4×C157C2211yescharacteristic
48C2×C4⋊C4×C157C2211yescharacteristic
49C2×C4⋊C4×C157C2211yescharacteristic
50C2×C22⋊C4×C157C2211yescharacteristic
51C2×C22⋊C4×C157C2211yescharacteristic
52#95×C157C2211yescharacteristic
53#95×C157C2211yescharacteristic
54C23×D157C2311yescharacteristic
55C2×Dic314—22no—
56C2×Dic314—22no—
57C2×Dic314C2×C414yescomplemented
58C2×C22⋊D157—22no—
59C2×C22⋊D157—22no—
60C2×C22⋊D157C2×C414yes—
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C22×Dic157C2311yescharacteristic
65C22×Dic157C2311yescharacteristic
66C22×Dic157C2311yescharacteristic
67C22×Dic157C2311yescharacteristic
68C2×C4×D157—22no—
69C2×C4×D157—41no—
70C2×C4×D157—41no—
71C4⋊Dic157—41no—
72#14578933—22no—
73#14578933—22no—
74#14578933—22no—
75#14578933—41no—
76#14578933—41no—
77C22⋊Dic157—22no—
78C22⋊Dic157—41no—
79C22⋊Dic157C2×C414yescomplemented
80#14578931—22no—
81#14578931—41no—
82#14578931—41no—
83#14578931C2×C414yescomplemented
84#14578931C2×C414yescomplemented
85C4×Dic157—22no—
86C4×Dic157—22no—
87C4×Dic157—22no—
88C4×Dic157C2×C414yescomplemented
89C4×Dic157D414yes—
90C24×C157C2311yescharacteristic
91C22×C4×C157—22no—
92C22×C4×C157—41no—
93C22×C4×C157—41no—
94C22×C4×C157C2311yescharacteristic
95C22×C4×C157C2311yescharacteristic
96C22×C4×C157C2311yescharacteristic
97C22×C4×C157C2311yescharacteristic
98C22×C4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C4⋊C4×C157—22no—
101C4⋊C4×C157—41no—
102C22⋊C4×C157—41no—
103C22⋊C4×C157C2×C414yescomplemented
104C22×D157—21no—
105C22×D157—21no—
106C22×D157—21no—
107C22×D157—21no—
108C22×D157—21no—
109C22×D157—21no—
110C22×D157C22×C412yes—
111Dic314—28no—
112C22⋊D157—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×Dic157C22×C412yes—
129C2×Dic157C22×C412yes—
130C2×Dic157C22×C412yes—
131C2×Dic157C2×D412yes—
132C2×Dic157C2×D412yes—
133C2×Dic157#4112yes—
134C4×D157—44no—
135C23×C157—21no—
136C23×C157—21no—
137C23×C157—21no—
138C23×C157—21no—
139C23×C157—21no—
140C23×C157—21no—
141C23×C157C22×C412yes—
142C23×C157C2411yescharacteristic
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—44no—
159C2×C4×C157C22×C412yes—
160C2×C4×C157C22×C412yes—
161C2×C4×C157C2×D412yes—
162C22×C157—22no—
163C22×C157—22no—
164C22×C157—22no—
165C22×C157—41no—
166C22×C157—41no—
167C22×C157—41no—
168C22×C157—41no—
169C22×C157C23×C411yescharacteristic
170C22×C157C22×D411yescharacteristic
171C22×C157C2×#4111yescharacteristic
172C22×C157H(5,𝔽2)11yescharacteristic
173C22×C157H(5,𝔽2)11yescharacteristic
174C22×C1572−1+411yescharacteristic
175C22×C1572−1+411yescharacteristic
176C2×D157—22no—
177C2×D157—22no—
178C2×D157—22no—
179C2×D157—41no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C4×C157—22no—
184C4×C157—22no—
185C4×C157—22no—
186C4×C157—41no—
187C4×C157—41no—
188C4×C157—41no—
189Dic157—22no—
190Dic157—22no—
191Dic157—22no—
192Dic157—41no—
193Dic157—41no—
194Dic157—41no—
195Dic157C4×D414yes—
196C2×C157—42no—
197C2×C157C2×C4×D411yescharacteristic
198C2×C157#52011yescharacteristic
199C2×C157#52011yescharacteristic
200C2×C157#53511yescharacteristic
201C2×C157#53611yescharacteristic
202C2×C157#54211yescharacteristic
203C2×C157#54211yescharacteristic
204D157—42no—
205C157#232611yescharacteristic, complemented, Sylow
206#2326—1571nomaximal, Sylow
207C2×#124—1571no—
208C2×C4×Q8—1571no—
209C2×C4×D4—1571no—
210#391—1571no—
211#390—1571no—
212#388—1571no—
213#388—1571no—
214#388—1571no—
215#388—1571no—
216#385—1571no—
217#385—1571no—
218#385—1571no—
219#385D15711yescharacteristic, complemented
220#384—1571no—
221#376—1571no—
222C22×Q8—1571no—
223C22×D4—1571no—
224C23×C4—1571no—
225C23×C4C2×D15711yescharacteristic
226#124—1578no—
227C4×Q8—3144no—
228C4×D4—3144no—
229C2×C4⋊C4—1571no—
230C2×C4⋊C4—1571no—
231C2×C4⋊C4—1571no—
232C2×C4⋊C4—1571no—
233C2×C4⋊C4—1571no—
234C2×C4⋊C4—1571no—
235C2×C4⋊C4C2×D15711yescharacteristic
236C2×C4⋊C4C2×D15711yescharacteristic
237C2×C22⋊C4—1571no—
238C2×C22⋊C4—1571no—
239C2×C22⋊C4—1571no—
240C2×C22⋊C4—1571no—
241C2×C22⋊C4—1571no—
242C2×C22⋊C4—1571no—
243C2×C22⋊C4—1571no—
244C2×C22⋊C4—1571no—
245C2×C22⋊C4C2×D15711yescharacteristic
246C2×C22⋊C4C2×D15711yescharacteristic
247C2×C42—1571no—
248C2×C42—1571no—
249C2×C42—1571no—
250C2×C42—1571no—
251C2×C42—1571no—
252#95—1571no—
253#95—1571no—
254#95—1571no—
255#95—1571no—
256#95—1571no—
257#95—1571no—
258#95C2×D15711yescharacteristic
259#95C2×D15711yescharacteristic
260C24—1571no—
261C24C22×D15711yescharacteristic
262C2×Q8—3142no—
263C2×Q8—3142no—
264C2×Q8—1574no—
265C2×D4—3142no—
266C2×D4—3142no—
267C2×D4—1574no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—3142no—
272C22×C4—6281no—
273C22×C4—6281no—
274C22×C4—1571no—
275C22×C4—1571no—
276C22×C4—1571no—
277C22×C4—1571no—
278C22×C4—1571no—
279C22×C4—1571no—
280C22×C4—1571no—
281C22×C4C22×D15711yescharacteristic
282C22×C4C22×D15711yescharacteristic
283C22×C4C22×D15711yescharacteristic
284C22×C4C22×D15711yescharacteristic
285C22×C4C22×D15711yescharacteristic
286C22×C4C22×D15711yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—3142no—
290C4⋊C4—3142no—
291C4⋊C4—3142no—
292C4⋊C4—6281no—
293C4⋊C4—6281no—
294C4⋊C4—6281no—
295C22⋊C4—41no—
296C22⋊C4—3142no—
297C22⋊C4—3142no—
298C22⋊C4—6281no—
299C22⋊C4—6281no—
300C22⋊C4—6281no—
301C22⋊C4—1574no—
302C22⋊C4—1574no—
303C22⋊C4—1574no—
304C22⋊C4C4×D15714yescomplemented
305C42—3142no—
306C42—3142no—
307C42—3142no—
308C42—1574no—
309C42—1574no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—3141no—
317C23—3141no—
318C23—3141no—
319C23—3141no—
320C23—3141no—
321C23—3141no—
322C23—1572no—
323C23C2×C4×D15712yes—
324C23C23×D15711yescharacteristic
325Q8—3148no—
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—44no—
343C2×C4—3141no—
344C2×C4—3141no—
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—6284no—
359C2×C4—1572no—
360C2×C4—1572no—
361C2×C4—1572no—
362C2×C4—1572no—
363C2×C4—1572no—
364C2×C4—1572no—
365C2×C4C2×C4×D15712yes—
366C2×C4C2×C4×D15712yes—
367C2×C4D4×D15712yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3142no—
376C22—3142no—
377C22—3142no—
378C22—6281no—
379C22—6281no—
380C22—6281no—
381C22—6281no—
382C22C22×C4×D15711yescharacteristic
383C22C2×D4×D15711yescharacteristic
384C22#41×D15711yescharacteristic
385C22#1909781611yescharacteristic
386C22#1909781611yescharacteristic
387C22#1909781811yescharacteristic
388C22#1909781911yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—3142no—
396C4—3142no—
397C4—3142no—
398C4—6281no—
399C4—6281no—
400C4—6281no—
401C4—1574no—
402C2—42no—
403C2—6282no—
404C2C4×D4×D15711yescharacteristic
405C2#2375775211yescharacteristic
406C2#2375775411yescharacteristic
407C2#2375783811yescharacteristic
408C2#2375784611yescharacteristic
409C2#2375789111yescharacteristic
410C2#2375789311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutatorC22×C157C23×C4
Fitting#385×C157C2
socle, cosocleC23×C157C24
prime radicalC22×C22⋊D157C22
hypercenter#385D157
hypocenterC157#2326
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#19097736C2semidirect
4. #23757177C2semidirect
6. #23757167C2semidirect
7. #23757166C2semidirect
8. #23757163C2semidirect
9. #23757162C2semidirect
10. #23757147C2semidirect
11. #23757146C2semidirect
12. #23757144C2semidirect
13. #23757142C2semidirect
14. #23757118C2semidirect
15. #385×C157C2semidirect
19. #19097757C4semidirect
57. C2×Dic314C2×C4semidirect
79. C22⋊Dic157C2×C4semidirect
83. #14578931C2×C4semidirect
84. #14578931C2×C4semidirect
88. C4×Dic157C2×C4semidirect
103. C22⋊C4×C157C2×C4semidirect
205. C157#2326semidirect
219. #385D157semidirect
304. 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
C2×#190977362
C2×C4×C22⋊D1572
#237571772
#237571732
#237571672
#237571662
#237571632
#237571622
#237571472
#237571462
#237571442
#237571422
#237571182
#385×C1572
#2326157