Finite Group Databasebeta

← #34109606

Subgroups — #34109606

Order 20096.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×#19097800C211yesmaximal, characteristic, complemented
3#23757822C211yesmaximal, characteristic
4C4×Q8×D157C211yesmaximal, characteristic, complemented
5#23757496C211yesmaximal, characteristic, complemented
6#23757495C211yesmaximal, characteristic, complemented
7#23757492C211yesmaximal, characteristic, complemented
8#23757482C211yesmaximal, characteristic, complemented
9#23757481C211yesmaximal, characteristic, complemented
10#23757472C211yesmaximal, characteristic, complemented
11#23757471C211yesmaximal, characteristic, complemented
12#23757295C211yesmaximal, characteristic, complemented
13#23757273C211yesmaximal, characteristic, complemented
14#23757271C211yesmaximal, characteristic, complemented
15#464×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19#19097803—24no—
20#19097800—24no—
21C2×Dic628C2211yescharacteristic
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578943C2211yescharacteristic
25C2×#14578942C2211yescharacteristic
26C2×#14578941C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×#14578937C2211yescharacteristic
29#19097769C2211yescharacteristic
30#19097767C2211yescharacteristic
31#19097759C2211yescharacteristic
32#19097744—21no—
33#19097744—21no—
34Q8×Dic157C2211yescharacteristic
35C4×Dic314—21no—
36C4×Dic314C2211yescharacteristic
37#19097736—21no—
38#19097736C2211yescharacteristic
39#19097731—21no—
40#19097731—21no—
41C4⋊C4×D157—21no—
42C4⋊C4×D157C2211yescharacteristic
43C42×D157—21no—
44C42×D157C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic
46#19097700C2211yescharacteristic
47#19097700C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097695C2211yescharacteristic
50#19097694C2211yescharacteristic
51#19097693C2211yescharacteristic
52#19097692C2211yescharacteristic
53#19097691C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic
59#103×C157C2211yescharacteristic
60#102×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—22no—
63#14578953—22no—
64#14578953—22no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68Q8×D157—22no—
69Q8×D157D414yescomplemented
70C2×D628—21no—
71C2×D628C2311yescharacteristic
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C22×Dic157—21no—
76C2×C4×D157—21no—
77C2×C4×D157—21no—
78C2×C4×D157—21no—
79C2×C4×D157D411yescharacteristic
80C2×C4×D157D411yescharacteristic
81C2×C4×D157C2311yescharacteristic
82Dic628—22no—
83#14578944—22no—
84#14578944—22no—
85#14578943—22no—
86#14578942—22no—
87#14578941—22no—
88#14578937—22no—
89#14578937—22no—
90C2×#11929434C2311yescharacteristic
91C2×#11929434C2311yescharacteristic
92C4⋊Dic157—21no—
93C4⋊Dic157C2311yescharacteristic
94#14578933—21no—
95#14578933—21no—
96#14578933—21no—
97#14578933—21no—
98#14578933—21no—
99#14578931—21no—
100#14578931—21no—
101C4×Dic157—21no—
102C4×Dic157—21no—
103C4×Dic157—21no—
104C4×Dic157D411yescharacteristic
105C4×Dic157D411yescharacteristic
106C4×Dic157C2311yescharacteristic
107C2×Q8×C157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110Q16×C157—22no—
111SD16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C42×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157—21no—
121C22×D157C2×D411yescharacteristic
122Dic314—21no—
123Dic314—24no—
124Dic314—21no—
125Dic314—21no—
126Dic314C2×D412yes—
127D628—21no—
128D628—21no—
129D628—42no—
130C22⋊D157—42no—
131C22⋊D157—42no—
132C22⋊D157—42no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—42no—
137C2×Dic157C2×D411yescharacteristic
138C2×Dic157C2×D411yescharacteristic
139C2×Dic157C2×D411yescharacteristic
140C4×D157—24no—
141C4×D157—22no—
142C4×D157—41no—
143C4×D157—42no—
144C4×D157—42no—
145C4×D157C2×D412yes—
146C4×D157#4112yes—
147#11929434—21no—
148#11929434—21no—
149C23×C157—21no—
150Q8×C157—21no—
151Q8×C157—21no—
152Q8×C157—21no—
153Q8×C157C2×D412yes—
154D4×C157—21no—
155D4×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157C2×D411yescharacteristic
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157C2411yescharacteristic
162C8×C157—21no—
163C8×C157—21no—
164C22×C157—41no—
165C22×C157—41no—
166C22×C157—41no—
167C22×C157C22×D411yescharacteristic
168C2×D157—21no—
169C2×D157—21no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157#12112yes—
174C4×C157—22no—
175C4×C157—21no—
176C4×C157—21no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157C22×D411yescharacteristic
180C4×C157C2×#4111yescharacteristic
181Dic157—22no—
182Dic157—21no—
183Dic157—41no—
184Dic157—41no—
185Dic157#12112yes—
186Dic157#13712yescomplemented
187C2×C157—81no—
188C2×C157C2×#12111yescharacteristic
189C2×C157C2×#13711yescharacteristic
190C2×C157#57511yescharacteristic
191D157—22no—
192D157—81no—
193C157#303711yescharacteristic, complemented, Sylow
194#3037—1571nomaximal, Sylow
195C2×#137—1571no—
196C2×#137—1571no—
197#532—1571no—
198C2×C4×Q8—1571no—
199#464—1571no—
200#464—1571no—
201#464—1571no—
202#464D15711yescharacteristic, complemented
203#462—1571no—
204#462—1571no—
205#460—1571no—
206#460—1571no—
207#423—1571no—
208#417—1571no—
209#417—1571no—
210C2×#41—1571no—
211C2×#41—1571no—
212C22×Q8—1571no—
213#137—3144no—
214#137—3144no—
215C2×Q16—1571no—
216C2×Q16—1571no—
217C2×Q16—1571no—
218C2×Q16C2×D15711yescharacteristic
219C2×SD16—1571no—
220C2×SD16—1571no—
221C2×SD16—1571no—
222C2×SD16C2×D15711yescharacteristic
223C2×M16—1571no—
224C2×M16—1571no—
225C4⋊Q8—1571no—
226#127—1571no—
227#124—1571no—
228#124C2×D15711yescharacteristic
229#121—3141no—
230#121—3141no—
231C4×Q8—3141no—
232C4×Q8—3141no—
233C4×Q8—1571no—
234C4×Q8—1571no—
235C4×Q8—1571no—
236C4×Q8C2×D15711yescharacteristic
237C4×D4—3141no—
238C4×D4—3141no—
239C2×C4⋊C4—3141no—
240C2×C4⋊C4—1571no—
241C2×C42—3141no—
242C2×C42—1571no—
243C4⋊C8—1571no—
244C4⋊C8—1571no—
245C4⋊C8—1571no—
246C4⋊C8C2×D15711yescharacteristic
247#103—1571no—
248#103—1571no—
249#103—1571no—
250#103C2×D15711yescharacteristic
251#102—1571no—
252#102—1571no—
253#102—1571no—
254#102C2×D15711yescharacteristic
255#41—3142no—
256#41—3142no—
257#41—3142no—
258#41—3142no—
259C2×Q8—3141no—
260C2×Q8—3142no—
261C2×Q8—1574no—
262C2×Q8—1571no—
263C2×Q8—1571no—
264C2×Q8C22×D15711yescharacteristic
265C2×Q8C22×D15711yescharacteristic
266C2×D4—3141no—
267C2×D4—3141no—
268C2×D4—3141no—
269C2×D4—3141no—
270C2×D4—1571no—
271C2×D4C22×D15711yescharacteristic
272C22×C4—3141no—
273C22×C4—3141no—
274C22×C4—3141no—
275C22×C4—3141no—
276C22×C4—1571no—
277C22×C4—1571no—
278C22×C4—1571no—
279Q16—22no—
280Q16—3142no—
281Q16—3142no—
282Q16—3142no—
283SD16—22no—
284SD16—3142no—
285SD16—3142no—
286SD16—3142no—
287M16—3142no—
288M16—3142no—
289C2×C8—1571no—
290C2×C8—1571no—
291C2×C8C22×D15711yescharacteristic
292C2×C8C22×D15711yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—3141no—
295C4⋊C4—3141no—
296C4⋊C4—3141no—
297C4⋊C4—3141no—
298C4⋊C4—3141no—
299C4⋊C4—3141no—
300C4⋊C4—1571no—
301C4⋊C4C22×D15711yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—3141no—
305C22⋊C4—3141no—
306C42—21no—
307C42—3141no—
308C42—3141no—
309C42—3141no—
310C42—1571no—
311C42—1571no—
312C42—1571no—
313C42C22×D15711yescharacteristic
314C23—21no—
315C23—3141no—
316C23—1571no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—3144no—
321Q8—3141no—
322Q8—3141no—
323Q8—3141no—
324Q8—1572no—
325Q8D4×D15712yes—
326D4—21no—
327D4—21no—
328D4—3141no—
329D4—3141no—
330D4—6282no—
331D4—6282no—
332D4—6282no—
333D4—6282no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—3144no—
338C2×C4—3142no—
339C2×C4—3141no—
340C2×C4—3141no—
341C2×C4—3141no—
342C2×C4—6281no—
343C2×C4—6282no—
344C2×C4—6282no—
345C2×C4—6282no—
346C2×C4—1571no—
347C2×C4—1571no—
348C2×C4—1571no—
349C2×C4—1572no—
350C2×C4—1572no—
351C2×C4D4×D15711yescharacteristic
352C2×C4D4×D15711yescharacteristic
353C2×C4C23×D15711yescharacteristic
354C8—21no—
355C8—21no—
356C8—3141no—
357C8—3141no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3141no—
362C22—3141no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—1572no—
367C22C2×D4×D15711yescharacteristic
368C4—22no—
369C4—21no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—3142no—
374C4—3141no—
375C4—6281no—
376C4—6281no—
377C4—1572no—
378C4—1572no—
379C4C2×D4×D15711yescharacteristic
380C4#41×D15711yescharacteristic
381C2—81no—
382C2—3142no—
383C2—12561no—
384C2#121×D15711yescharacteristic
385C2#137×D15711yescharacteristic
386C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#464×C157C2
socleC22×C157C22×D4
prime radical#23757822C2
hypercenter#464D157
hypocenterC157#3037
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×#19097800C2semidirect
4. C4×Q8×D157C2semidirect
5. #23757496C2semidirect
6. #23757495C2semidirect
7. #23757492C2semidirect
8. #23757482C2semidirect
9. #23757481C2semidirect
10. #23757472C2semidirect
11. #23757471C2semidirect
12. #23757295C2semidirect
13. #23757273C2semidirect
14. #23757271C2semidirect
15. #464×C157C2semidirect
69. Q8×D157D4semidirect
186. Dic157#137semidirect
193. C157#3037semidirect
202. #464D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×#190978002
#237578222
C4×Q8×D1572
#237574962
#237574952
#237574922
#237574822
#237574812
#237574722
#237574712
#237572952
#237572732
#237572712
#464×C1572
#3037157