Finite Group Databasebeta

← #34114223

Subgroups — #34114223

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#23757999C211yesmaximal, characteristic, complemented
2#23757998C213yesmaximal, complemented
3#23757994C213yesmaximal, complemented
4#23757965C213yesmaximal, complemented
5#23757963C213yesmaximal, complemented
6Q8×C22⋊D157C211yesmaximal, characteristic, complemented
7Q8×C22⋊D157C211yesmaximal, characteristic, complemented
8#23757884C213yesmaximal, complemented
9#23757883C213yesmaximal, complemented
10#23757825C213yesmaximal, complemented
11#23757803C213yesmaximal
12#23757751C211yesmaximal, characteristic, complemented
13#23757749C211yesmaximal, characteristic, complemented
14C2×Q8×Dic157C211yesmaximal, characteristic, complemented
15D4×Q8×C157C211yesmaximal, characteristic, complemented
16C2×Q8×D157C2211yescharacteristic
17#19097770C2213yes—
18#19097769C2213yes—
19#19097769C2213yes—
20#19097766C2213yes—
21#19097766C2216yes—
22#19097764C2213yes—
23#19097764C2216yes—
24#19097763C2213yes—
25#19097762C2216yes—
26#19097762C2213yes—
27#19097761C2216yes—
28#19097761C2216yes—
29#19097761C2213yes—
30#19097757C2213yes—
31#19097755C2213yes—
32#19097752C2213yes—
33#19097750C2213yes—
34#19097750C2213yes—
35#19097748C2213yes—
36#19097747C2213yes—
37#19097747C2213yes—
38#19097743C2211yescharacteristic
39Q8×Dic157C2213yes—
40Q8×Dic157C2213yes—
41Q8×Dic157C2211yescharacteristic
42Q8×Dic157C2211yescharacteristic
43Q8×Dic157C2213yes—
44Q8×Dic157C2213yes—
45Q8×Dic157C2211yescharacteristic
46C4×Dic314C2213yes—
47C4×Dic314C2213yes—
48#19097736C2213yes—
49#19097736C2216yes—
50C4×C22⋊D157C2213yes—
51C4×C22⋊D157C2213yes—
52#19097730C2213yes—
53#19097730C2213yes—
54#19097729C2213yes—
55#19097728C2213yes—
56#19097727C2216yes—
57C2×C4⋊Dic157C2213yes—
58C2×C4×Dic157C2213yes—
59C22×Q8×C157C2211yescharacteristic
60C22×Q8×C157C2211yescharacteristic
61C4⋊Q8×C157C2213yes—
62C22⋊Q8×C157C2213yes—
63C22⋊Q8×C157C2213yes—
64C4×Q8×C157C2211yescharacteristic
65C4×D4×C157C2213yes—
66C2×Dic314C2313yes—
67C2×Dic314C2313yes—
68Q8×D157—24no—
69C2×C22⋊D157C2311yescharacteristic
70C2×C22⋊D157C2311yescharacteristic
71C22×Dic157C2311yescharacteristic
72C22×Dic157C2313yes—
73C2×C4×D157C2313yes—
74C4⋊Dic157C2313yes—
75C4⋊Dic157C2313yes—
76C4⋊Dic157C2316yes—
77C4⋊Dic157C2313yes—
78C4⋊Dic157C2313yes—
79C4⋊Dic157C2313yes—
80C4⋊Dic157C2313yes—
81#14578933C2316yes—
82#14578933C2316yes—
83#14578933C2313yes—
84#14578933C2313yes—
85#14578933C2313yes—
86#14578933C2313yes—
87#14578933C2316yes—
88#14578933C2311yescharacteristic
89#14578933C2313yes—
90C22⋊Dic157C2313yes—
91C22⋊Dic157C2311yescharacteristic
92C22⋊Dic157C2313yes—
93C22⋊Dic157C2313yes—
94#14578931C2311yescharacteristic
95#14578931C2313yes—
96#14578931C2313yes—
97#14578931C2313yes—
98C4×Dic157C2313yes—
99C4×Dic157C2313yes—
100C4×Dic157C2313yes—
101C4×Dic157C2313yes—
102C4×Dic157C2316yes—
103C4×Dic157C2316yes—
104C4×Dic157C2313yes—
105C2×Q8×C157—24no—
106C2×Q8×C157—24no—
107C2×Q8×C157C2313yes—
108C2×Q8×C157C2313yes—
109C2×Q8×C157C2311yescharacteristic
110C2×D4×C157C2311yescharacteristic
111C22×C4×C157C2313yes—
112C22×C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C4⋊C4×C157C2316yes—
116C22⋊C4×C157C2313yes—
117C22⋊C4×C157C2313yes—
118C42×C157C2313yes—
119C22×D157C2411yescharacteristic
120Dic314—26no—
121Dic314—26no—
122C22⋊D157—22no—
123C22⋊D157—22no—
124C2×Dic157—26no—
125C2×Dic157—22no—
126C2×Dic157C2413yes—
127C2×Dic157C2411yescharacteristic
128C2×Dic157C2413yes—
129C2×Dic157C2413yes—
130C2×Dic157C2411yescharacteristic
131C2×Dic157C2413yes—
132C2×Dic157C2411yescharacteristic
133C4×D157—26no—
134C23×C157C2411yescharacteristic
135C23×C157C2411yescharacteristic
136Q8×C157—26no—
137Q8×C157—26no—
138Q8×C157#4114yes—
139D4×C157—22no—
140C2×C4×C157—26no—
141C2×C4×C157—26no—
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×C157—21no—
151C22×C157—21no—
152C22×C1572−1+412yes—
153C22×C157C2511yescharacteristic
154C2×D157—21no—
155C2×D157—21no—
156C2×D157—21no—
157C4×C157—23no—
158C4×C157—23no—
159C4×C157—21no—
160C4×C157—23no—
161C4×C157C2×#4116yes—
162Dic157—23no—
163Dic157—21no—
164Dic157—23no—
165Dic157—23no—
166Dic157—21no—
167Dic157—23no—
168Dic1572−1+412yes—
169C2×C157—22no—
170C2×C157—41no—
171C2×C157C22×#4111yescharacteristic
172C2×C157C2×2−1+411yescharacteristic
173C2×C157C2×2−1+411yescharacteristic
174D157—41no—
175C157#352311yescharacteristic, complemented, Sylow
176#3523—1571nomaximal, Sylow
177#557—1573no—
178#557—1571no—
179#556—1573no—
180#552—1573no—
181#552—1573no—
182D4×Q8—1571no—
183D4×Q8—1571no—
184D4×Q8D15711yescharacteristic, complemented
185#541—1573no—
186#541—1573no—
187#533—1573no—
188#529—1573no—
189#519—1571no—
190#519—1571no—
191C2×C4×Q8—1571no—
192C22×Q8—1571no—
193C22×Q8C2×D15711yescharacteristic
194C22×Q8C2×D15711yescharacteristic
195C4⋊Q8—1573no—
196C4⋊Q8—1573no—
197C4⋊Q8—1573no—
198C4⋊Q8C2×D15713yes—
199#126—1573no—
200#126—1576no—
201#126—1573no—
202#126—1576no—
203#125—1573no—
204#125—1576no—
205#125—1576no—
206#125—1576no—
207#125—1573no—
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⋊Q8C2×D15713yes—
218C22⋊Q8C2×D15713yes—
219#121—1571no—
220C4×Q8—1571no—
221C4×Q8—1573no—
222C4×Q8—1571no—
223C4×Q8—1573no—
224C4×Q8—1573no—
225C4×Q8—1571no—
226C4×Q8—1573no—
227C4×Q8—1573no—
228C4×Q8—1576no—
229C4×Q8—1573no—
230C4×Q8—1573no—
231C4×Q8C2×D15711yescharacteristic
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4C2×D15713yes—
235#117—1573no—
236#117—1573no—
237#117—1573no—
238#117—1576no—
239#117—1573no—
240C2×C4⋊C4—1573no—
241C2×C42—1573no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—3144no—
245C2×Q8—1573no—
246C2×Q8—1573no—
247C2×Q8C22×D15713yes—
248C2×Q8C22×D15713yes—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—1571no—
251C2×D4—1571no—
252C2×D4C22×D15711yescharacteristic
253C22×C4—1571no—
254C22×C4—1573no—
255C22×C4—1573no—
256C22×C4C22×D15713yes—
257C22×C4C22×D15713yes—
258C4⋊C4—1573no—
259C4⋊C4—1573no—
260C4⋊C4—1576no—
261C4⋊C4—1573no—
262C4⋊C4—1573no—
263C4⋊C4—1573no—
264C4⋊C4—1573no—
265C4⋊C4—1573no—
266C4⋊C4—1576no—
267C4⋊C4—1576no—
268C4⋊C4—1573no—
269C4⋊C4—1573no—
270C4⋊C4—1571no—
271C4⋊C4—1576no—
272C4⋊C4—1573no—
273C4⋊C4—1573no—
274C4⋊C4C22×D15713yes—
275C4⋊C4C22×D15713yes—
276C4⋊C4C22×D15716yes—
277C22⋊C4—1571no—
278C22⋊C4—1573no—
279C22⋊C4—1573no—
280C22⋊C4—1573no—
281C22⋊C4—1571no—
282C22⋊C4—1573no—
283C22⋊C4—1573no—
284C22⋊C4—1573no—
285C22⋊C4C22×D15713yes—
286C22⋊C4C22×D15713yes—
287C42—1573no—
288C42—1573no—
289C42—1573no—
290C42—1573no—
291C42—1576no—
292C42—1576no—
293C42—1573no—
294C42C22×D15713yes—
295C23—1571no—
296C23C23×D15711yescharacteristic
297C23C23×D15711yescharacteristic
298Q8—26no—
299Q8—26no—
300Q8—3146no—
301Q8—3146no—
302Q8#1457895314yes—
303D4—22no—
304D4—3142no—
305D4—3142no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—3146no—
309C2×C4—3142no—
310C2×C4—3146no—
311C2×C4—1571no—
312C2×C4—1573no—
313C2×C4—1573no—
314C2×C4—1571no—
315C2×C4—1573no—
316C2×C4—1573no—
317C2×C4—1571no—
318C2×C4C23×D15713yes—
319C2×C4C23×D15713yes—
320C2×C4C23×D15713yes—
321C2×C4C23×D15711yescharacteristic
322C2×C4C23×D15713yes—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—3141no—
329C22—3141no—
330C22—3141no—
331C22#1909781912yes—
332C22C24×D15711yescharacteristic
333C4—23no—
334C4—23no—
335C4—23no—
336C4—21no—
337C4—3143no—
338C4—3141no—
339C4—3143no—
340C4—3143no—
341C4—3143no—
342C4—3141no—
343C4—1572no—
344C4C2×#1457895316yes—
345C2—22no—
346C2—41no—
347C2—6281no—
348C2C22×#1457895311yescharacteristic
349C22−1+4×D15711yescharacteristic
350C2C2×#1909781911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
FittingD4×Q8×C157C2
prime radical#19097743C22
hypercenterD4×Q8D157
hypocenterC157#3523
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #23757999C2semidirect
2. #23757998C2semidirect
3. #23757994C2semidirect
4. #23757965C2semidirect
5. #23757963C2semidirect
6. Q8×C22⋊D157C2semidirect
7. Q8×C22⋊D157C2semidirect
8. #23757884C2semidirect
9. #23757883C2semidirect
10. #23757825C2semidirect
12. #23757751C2semidirect
13. #23757749C2semidirect
14. C2×Q8×Dic157C2semidirect
15. D4×Q8×C157C2semidirect
175. C157#3523semidirect
184. D4×Q8D157semidirect

Maximal subgroups

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

subgroupindex
#237579992
#237579982
#237579942
#237579652
#237579632
Q8×C22⋊D1572
#237578842
#237578832
#237578252
#237578032
#237577512
#237577492
C2×Q8×Dic1572
D4×Q8×C1572
#3523157