Finite Group Databasebeta

← #34114226

Subgroups — #34114226

Order 20096.

992 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758001C213yesmaximal, complemented
2#23757999C211yesmaximal, characteristic, complemented
3#23757991C213yesmaximal, complemented
4#23757964C213yesmaximal, complemented
5#23757962C213yesmaximal, complemented
6Q8×D628C211yesmaximal, characteristic, complemented
7Q8×C22⋊D157C211yesmaximal, characteristic, complemented
8Q8×C22⋊D157C211yesmaximal, characteristic, complemented
9#23757885C213yesmaximal, complemented
10#23757881C213yesmaximal, complemented
11#23757827C213yesmaximal, complemented
12#23757801C213yesmaximal
13#23757749C211yesmaximal, characteristic, complemented
14C4×Q8×D157C211yesmaximal, characteristic, complemented
15#519×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18#19097771C2213yes—
19#19097770C2213yes—
20#19097769C2213yes—
21#19097769C2213yes—
22#19097766C2213yes—
23#19097765C2216yes—
24#19097764C2213yes—
25#19097764C2216yes—
26#19097763C2216yes—
27#19097763C2213yes—
28#19097762C2213yes—
29#19097761C2216yes—
30#19097761C2216yes—
31#19097761C2213yes—
32#19097757C2213yes—
33#19097756C2213yes—
34#19097753C2213yes—
35#19097751C2213yes—
36#19097751C2213yes—
37#19097750C2213yes—
38#19097750C2213yes—
39#19097749C2213yes—
40#19097747C2213yes—
41#19097747C2213yes—
42#19097744C2211yescharacteristic
43Q8×Dic157C2211yescharacteristic
44Q8×Dic157C2211yescharacteristic
45C4×Dic314C2213yes—
46C4×Dic314C2216yes—
47C4×Dic314C2213yes—
48C4×Dic314C2213yes—
49#19097736C2213yes—
50#19097736C2213yes—
51C4×D628C2213yes—
52C4×C22⋊D157C2213yes—
53C4×C22⋊D157C2213yes—
54#19097729C2213yes—
55#19097728C2216yes—
56#19097728C2213yes—
57C4⋊C4×D157C2213yes—
58C42×D157C2213yes—
59C22×Q8×C157C2211yescharacteristic
60C4×Q8×C157C2213yes—
61C4×Q8×C157C2211yescharacteristic
62C4×Q8×C157C2213yes—
63C4×Q8×C157C2211yescharacteristic
64#117×C157C2213yes—
65#117×C157C2213yes—
66C2×Dic314C2313yes—
67C2×Dic314C2313yes—
68C2×Dic314C2313yes—
69Q8×D157—24no—
70Q8×D157—24no—
71C2×D628C2311yescharacteristic
72C2×C22⋊D157C2311yescharacteristic
73C2×C22⋊D157C2311yescharacteristic
74C2×C4×D157C2311yescharacteristic
75C2×C4×D157C2313yes—
76C2×C4×D157C2313yes—
77C2×C4×D157C2313yes—
78C4⋊Dic157C2313yes—
79C4⋊Dic157C2313yes—
80C4⋊Dic157C2313yes—
81C4⋊Dic157C2316yes—
82C4⋊Dic157C2313yes—
83#14578933C2316yes—
84#14578933C2313yes—
85#14578933C2313yes—
86#14578933C2316yes—
87#14578933C2316yes—
88#14578933C2313yes—
89#14578933C2313yes—
90#14578933C2311yescharacteristic
91#14578933C2316yes—
92#14578933C2313yes—
93C22⋊Dic157C2313yes—
94C22⋊Dic157C2313yes—
95#14578931C2311yescharacteristic
96#14578931C2313yes—
97#14578931C2313yes—
98#14578931C2313yes—
99#14578931C2313yes—
100#14578931C2313yes—
101C4×Dic157C2313yes—
102C4×Dic157C2313yes—
103C4×Dic157C2313yes—
104C4×Dic157C2316yes—
105C4×Dic157C2313yes—
106C2×Q8×C157—24no—
107C2×Q8×C157C2313yes—
108C2×Q8×C157C2311yescharacteristic
109C22×C4×C157C2313yes—
110C4⋊C4×C157C2313yes—
111C4⋊C4×C157C2313yes—
112C4⋊C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C22⋊C4×C157C2311yescharacteristic
115C22⋊C4×C157C2313yes—
116C42×C157C2313yes—
117C42×C157C2316yes—
118C42×C157C2313yes—
119C22×D157C2411yescharacteristic
120C22×D157C2411yescharacteristic
121Dic314—26no—
122Dic314—26no—
123Dic314—26no—
124D628—22no—
125C22⋊D157—22no—
126C22⋊D157—22no—
127C2×Dic157C2413yes—
128C2×Dic157C2413yes—
129C2×Dic157C2413yes—
130C2×Dic157C2411yescharacteristic
131C2×Dic157C2413yes—
132C2×Dic157C2411yescharacteristic
133C4×D157—26no—
134C4×D157—26no—
135C4×D157—26no—
136C4×D157—22no—
137C23×C157C2411yescharacteristic
138Q8×C157—26no—
139Q8×C157#4114yes—
140C2×C4×C157—26no—
141C2×C4×C157C2411yescharacteristic
142C2×C4×C157C2413yes—
143C2×C4×C157C2411yescharacteristic
144C2×C4×C157C2413yes—
145C2×C4×C157C2413yes—
146C2×C4×C157C2413yes—
147C22×C157—21no—
148C22×C157—21no—
149C22×C157—21no—
150C22×C157C2511yescharacteristic
151C2×D157—21no—
152C2×D157—21no—
153C2×D157—21no—
154C2×D157—21no—
155C2×D157—21no—
156C2×D1572−1+412yes—
157C4×C157—23no—
158C4×C157—23no—
159C4×C157—21no—
160C4×C157—21no—
161C4×C157—23no—
162C4×C157C2×#4116yes—
163Dic157—23no—
164Dic157—23no—
165Dic157—23no—
166Dic157—21no—
167Dic157—23no—
168Dic1572−1+412yes—
169C2×C157—41no—
170C2×C157C22×#4111yescharacteristic
171C2×C157C2×2−1+411yescharacteristic
172C2×C157C2×2−1+411yescharacteristic
173D157—22no—
174D157—41no—
175C157#352311yescharacteristic, complemented, Sylow
176#3523—1571nomaximal, Sylow
177#557—1571no—
178#557—1573no—
179#556—1573no—
180#552—1573no—
181#552—1573no—
182D4×Q8—1571no—
183D4×Q8—1571no—
184D4×Q8—1571no—
185#541—1573no—
186#541—1573no—
187#533—1573no—
188#529—1573no—
189#519—1571no—
190#519D15711yescharacteristic, complemented
191C2×C4×Q8—1571no—
192C22×Q8—1571no—
193C22×Q8—1571no—
194C22×Q8C2×D15711yescharacteristic
195C4⋊Q8—1573no—
196C4⋊Q8—1573no—
197C4⋊Q8—1573no—
198C4⋊Q8—1573no—
199#126—1573no—
200#126—1573no—
201#126—1576no—
202#126—1576no—
203#125—1573no—
204#125—1576no—
205#125—1576no—
206#125—1573no—
207#125—1576no—
208#125—1573no—
209#124—1573no—
210#123—1573no—
211#123—1573no—
212C22⋊Q8—1573no—
213C22⋊Q8—1573no—
214C22⋊Q8—1573no—
215C22⋊Q8—1573no—
216C22⋊Q8—1573no—
217C22⋊Q8—1573no—
218C22⋊Q8—1573no—
219#121—1571no—
220C4×Q8—1571no—
221C4×Q8—1571no—
222C4×Q8—1573no—
223C4×Q8—1573no—
224C4×Q8—1573no—
225C4×Q8—1573no—
226C4×Q8—1573no—
227C4×Q8—1576no—
228C4×Q8C2×D15711yescharacteristic
229C4×Q8C2×D15713yes—
230C4×Q8C2×D15711yescharacteristic
231C4×Q8C2×D15713yes—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1573no—
235#117—1573no—
236#117—1573no—
237#117—1576no—
238#117C2×D15713yes—
239#117C2×D15713yes—
240C2×C4⋊C4—1573no—
241C2×C42—1573no—
242C2×Q8—24no—
243C2×Q8—3144no—
244C2×Q8—3144no—
245C2×Q8—1573no—
246C2×Q8—1573no—
247C2×Q8—1573no—
248C2×Q8C22×D15713yes—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—1571no—
251C2×D4—1571no—
252C2×D4—1571no—
253C22×C4—1571no—
254C22×C4—1573no—
255C22×C4—1573no—
256C22×C4—1573no—
257C22×C4C22×D15713yes—
258C4⋊C4—1573no—
259C4⋊C4—1576no—
260C4⋊C4—1573no—
261C4⋊C4—1573no—
262C4⋊C4—1573no—
263C4⋊C4—1576no—
264C4⋊C4—1576no—
265C4⋊C4—1573no—
266C4⋊C4—1573no—
267C4⋊C4—1573no—
268C4⋊C4—1571no—
269C4⋊C4—1576no—
270C4⋊C4—1576no—
271C4⋊C4—1573no—
272C4⋊C4—1573no—
273C4⋊C4C22×D15713yes—
274C4⋊C4C22×D15713yes—
275C4⋊C4C22×D15713yes—
276C4⋊C4C22×D15713yes—
277C22⋊C4—1573no—
278C22⋊C4—1573no—
279C22⋊C4—1571no—
280C22⋊C4—1573no—
281C22⋊C4—1573no—
282C22⋊C4—1573no—
283C22⋊C4—1573no—
284C22⋊C4—1573no—
285C22⋊C4C22×D15711yescharacteristic
286C22⋊C4C22×D15713yes—
287C42—1573no—
288C42—1573no—
289C42—1573no—
290C42—1576no—
291C42—1573no—
292C42C22×D15713yes—
293C42C22×D15716yes—
294C42C22×D15713yes—
295C23—1571no—
296C23—1571no—
297C23C23×D15711yescharacteristic
298Q8—26no—
299Q8—3146no—
300Q8—3146no—
301Q8—3146no—
302Q8#1457895414yes—
303D4—3142no—
304D4—3142no—
305D4—3142no—
306C2×C4—26no—
307C2×C4—3146no—
308C2×C4—3142no—
309C2×C4—3146no—
310C2×C4—3146no—
311C2×C4—1573no—
312C2×C4—1571no—
313C2×C4—1573no—
314C2×C4—1573no—
315C2×C4—1573no—
316C2×C4—1571no—
317C2×C4C23×D15713yes—
318C2×C4C23×D15711yescharacteristic
319C2×C4C23×D15713yes—
320C2×C4C23×D15711yescharacteristic
321C2×C4C23×D15713yes—
322C2×C4C23×D15713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—3141no—
327C22—3141no—
328C22—3141no—
329C22—3141no—
330C22—3141no—
331C22—1572no—
332C22C24×D15711yescharacteristic
333C4—23no—
334C4—23no—
335C4—21no—
336C4—21no—
337C4—23no—
338C4—3143no—
339C4—3143no—
340C4—3143no—
341C4—3141no—
342C4—3143no—
343C4—1572no—
344C4C2×#1457895416yes—
345C2—41no—
346C2—3142no—
347C2—6281no—
348C2C22×#1457895411yescharacteristic
349C22−1+4×D15711yescharacteristic
350C22−1+4×D15711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#519×C157C2
prime radical#19097744C22
hypercenter#519D157
hypocenterC157#3523
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #519

Complements

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

NH ≅ G/Ntype
1. #23758001C2semidirect
2. #23757999C2semidirect
3. #23757991C2semidirect
4. #23757964C2semidirect
5. #23757962C2semidirect
6. Q8×D628C2semidirect
7. Q8×C22⋊D157C2semidirect
8. Q8×C22⋊D157C2semidirect
9. #23757885C2semidirect
10. #23757881C2semidirect
11. #23757827C2semidirect
13. #23757749C2semidirect
14. C4×Q8×D157C2semidirect
15. #519×C157C2semidirect
175. C157#3523semidirect
190. #519D157semidirect

Maximal subgroups

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

subgroupindex
#237580012
#237579992
#237579912
#237579642
#237579622
Q8×D6282
Q8×C22⋊D1572
#237578852
#237578812
#237578272
#237578012
#237577492
C4×Q8×D1572
#519×C1572
#3523157