Finite Group Databasebeta

← #34113753

Subgroups — #34113753

Order 20096.

1094 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757999C211yesmaximal, characteristic, complemented
2#23757995C213yesmaximal, complemented
3#23757964C213yesmaximal, complemented
4#23757958C213yesmaximal, complemented
5Q8×C22⋊D157C211yesmaximal, characteristic, complemented
6#23757935C211yesmaximal, characteristic, complemented
7#23757934C211yesmaximal, characteristic, complemented
8#23757924C213yesmaximal, complemented
9#23757827C213yesmaximal, complemented
10#23757810C213yesmaximal, complemented
11#23757807C213yesmaximal, complemented
12C4×#14578955C211yesmaximal, characteristic, complemented
13C4×#14578954C213yesmaximal, complemented
14C4×Q8×D157C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954C2213yes—
18C2×Q8×D157C2211yescharacteristic
19#19097770C2213yes—
20#19097769C2213yes—
21#19097766C2216yes—
22#19097765C2216yes—
23#19097763C2213yes—
24#19097762C2213yes—
25#19097761C2216yes—
26#19097760C2213yes—
27#19097759C2213yes—
28#19097756C2216yes—
29#19097754C2216yes—
30#19097751C2213yes—
31#19097750C2213yes—
32#19097749C2216yes—
33#19097747C2213yes—
34#19097747C2213yes—
35#19097746C2213yes—
36#19097745C2213yes—
37Q8×Dic157C2211yescharacteristic
38Q8×Dic157C2211yescharacteristic
39C4×Dic314C2213yes—
40C4×Dic314C2213yes—
41C4×Dic314C2213yes—
42#19097736C2213yes—
43C4×D628C2213yes—
44C4×D628C2213yes—
45C4×D628C2213yes—
46#19097733C2213yes—
47C4×C22⋊D157C2213yes—
48C4×C22⋊D157C2213yes—
49C4×C22⋊D157C2211yescharacteristic
50C4×C22⋊D157C2213yes—
51#19097730C2213yes—
52#19097729C2213yes—
53#19097728C2216yes—
54#19097728C2213yes—
55#19097728C2213yes—
56C4⋊C4×D157C2213yes—
57C42×D157C2213yes—
58C42×D157C2213yes—
59C22×Q8×C157C2211yescharacteristic
60C4×Q8×C157C2213yes—
61C4×Q8×C157C2211yescharacteristic
62C4×Q8×C157C2213yes—
63C4×Q8×C157C2211yescharacteristic
64C2×C4⋊C4×C157C2213yes—
65C2×C42×C157C2213yes—
66#14578955—24no—
67#14578954—212no—
68C2×Dic314C2313yes—
69C2×Dic314C2313yes—
70Q8×D157—24no—
71C2×D628C2313yes—
72C2×D628C2313yes—
73C2×C22⋊D157C2313yes—
74C2×C22⋊D157C2311yescharacteristic
75C2×C4×D157C2311yescharacteristic
76C2×C4×D157C2313yes—
77C2×C4×D157C2313yes—
78C2×C4×D157C2313yes—
79C2×C4×D157C2313yes—
80C2×C4×D157C2313yes—
81C4⋊Dic157C2313yes—
82C4⋊Dic157C2313yes—
83C4⋊Dic157C2313yes—
84C4⋊Dic157C2313yes—
85#14578933C2316yes—
86#14578933C2311yescharacteristic
87#14578933C2313yes—
88#14578933C2313yes—
89#14578933C2313yes—
90#14578933C2316yes—
91#14578933C2316yes—
92C22⋊Dic157C2311yescharacteristic
93C22⋊Dic157C2313yes—
94#14578931C2311yescharacteristic
95#14578931C2313yes—
96#14578931C2313yes—
97#14578931C2313yes—
98#14578931C2316yes—
99#14578931C2316yes—
100#14578931C2316yes—
101C4×Dic157C2311yescharacteristic
102C4×Dic157C2313yes—
103C4×Dic157C2313yes—
104C4×Dic157C2313yes—
105C4×Dic157C2313yes—
106C4×Dic157C2313yes—
107C2×Q8×C157—24no—
108C2×Q8×C157C2313yes—
109C2×Q8×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111C22×C4×C157C2313yes—
112C22×C4×C157C2313yes—
113C4⋊C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C4⋊C4×C157C2316yes—
116C42×C157C2313yes—
117C42×C157C2313yes—
118C42×C157C2313yes—
119C42×C157C2313yes—
120C22×D157C2413yes—
121C22×D157C2411yescharacteristic
122Dic314—26no—
123Dic314—26no—
124D628—26no—
125D628—26no—
126C22⋊D157—26no—
127C22⋊D157#4114yescomplemented
128C2×Dic157C2411yescharacteristic
129C2×Dic157C2413yes—
130C2×Dic157C2413yes—
131C2×Dic157C2411yescharacteristic
132C2×Dic157C2413yes—
133C2×Dic157C2411yescharacteristic
134C4×D157—26no—
135C4×D157—26no—
136C4×D157—26no—
137C4×D157—22no—
138C4×D157—26no—
139C4×D157—26no—
140C23×C157C2411yescharacteristic
141Q8×C157—26no—
142Q8×C157#4114yescomplemented
143C2×C4×C157—26no—
144C2×C4×C157—26no—
145C2×C4×C157—22no—
146C2×C4×C157C2413yes—
147C2×C4×C157C2413yes—
148C2×C4×C157C2411yescharacteristic
149C2×C4×C157C2413yes—
150C2×C4×C157C2413yes—
151C2×C4×C157C2411yescharacteristic
152C22×C157—21no—
153C22×C157—21no—
154C22×C157C2×#4112yes—
155C22×C157C2511yescharacteristic
156C2×D157—21no—
157C2×D157—21no—
158C2×D157—23no—
159C2×D157—23no—
160C2×D157—23no—
161C2×D157C2×#4112yes—
162C4×C157—23no—
163C4×C157—23no—
164C4×C157—23no—
165C4×C157—21no—
166C4×C157C2×#4116yes—
167C4×C1572−1+412yes—
168Dic157—21no—
169Dic157—23no—
170Dic157—23no—
171Dic157—21no—
172Dic157—23no—
173Dic157C2×#4112yes—
174C2×C157—22no—
175C2×C157C22×#4111yescharacteristic
176C2×C157C22×#4111yescharacteristic
177C2×C157C2×2−1+411yescharacteristic
178D157—22no—
179D157—43no—
180C157#348211yescharacteristic, complemented, Sylow
181#3482—1571nomaximal, Sylow
182#557—1571no—
183#556—1573no—
184#552—1573no—
185#552—1573no—
186D4×Q8—1571no—
187#548—1571no—
188#548—1571no—
189#547—1573no—
190#533—1573no—
191#529—1573no—
192#529—1573no—
193C4×#41—1571no—
194C4×#41—1573no—
195C2×C4×Q8—1571no—
196C2×C4×Q8D15711yescharacteristic, complemented
197C2×#41—1571no—
198C2×#41—1573no—
199C22×Q8—1571no—
200C22×Q8C2×D15711yescharacteristic
201C4⋊Q8—1573no—
202C4⋊Q8—1573no—
203#126—1576no—
204#126—1576no—
205#125—1573no—
206#125—1576no—
207#125—1573no—
208#124—1573no—
209#124—1573no—
210#123—1576no—
211#123—1576no—
212C22⋊Q8—1573no—
213C22⋊Q8—1573no—
214C22⋊Q8—1573no—
215C22⋊Q8—1576no—
216C22⋊Q8—1573no—
217#121—1573no—
218#121—1573no—
219C4×Q8—1571no—
220C4×Q8—1571no—
221C4×Q8—1573no—
222C4×Q8—1573no—
223C4×Q8—1573no—
224C4×Q8—1573no—
225C4×Q8C2×D15711yescharacteristic
226C4×Q8C2×D15713yes—
227C4×Q8C2×D15711yescharacteristic
228C4×Q8C2×D15713yes—
229C4×D4—1571no—
230C4×D4—1573no—
231C4×D4—1573no—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1573no—
235C4×D4—1573no—
236C4×D4—1573no—
237#117—1573no—
238#117—1573no—
239#117—1573no—
240#117—1573no—
241#117—1576no—
242C2×C4⋊C4—1573no—
243C2×C4⋊C4C2×D15713yes—
244C2×C42—1573no—
245C2×C42—1573no—
246C2×C42C2×D15713yes—
247#41—31412no—
248#41—3144no—
249C2×Q8—24no—
250C2×Q8—3144no—
251C2×Q8—1573no—
252C2×Q8—1573no—
253C2×Q8C22×D15713yes—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—1573no—
256C2×D4—1571no—
257C2×D4—1573no—
258C2×D4—1573no—
259C22×C4—1573no—
260C22×C4—1573no—
261C22×C4—1573no—
262C22×C4—1573no—
263C22×C4—1571no—
264C22×C4—1573no—
265C22×C4C22×D15713yes—
266C22×C4C22×D15713yes—
267C22×C4C22×D15711yescharacteristic
268C4⋊C4—1573no—
269C4⋊C4—1576no—
270C4⋊C4—1571no—
271C4⋊C4—1573no—
272C4⋊C4—1573no—
273C4⋊C4—1573no—
274C4⋊C4—1573no—
275C4⋊C4—1573no—
276C4⋊C4—1573no—
277C4⋊C4—1576no—
278C4⋊C4—1576no—
279C4⋊C4C22×D15713yes—
280C4⋊C4C22×D15713yes—
281C4⋊C4C22×D15716yes—
282C22⋊C4—1571no—
283C22⋊C4—1573no—
284C22⋊C4—1571no—
285C22⋊C4—1573no—
286C22⋊C4—1576no—
287C22⋊C4—1573no—
288C22⋊C4—1573no—
289C22⋊C4—1576no—
290C22⋊C4—1576no—
291C42—1573no—
292C42—1573no—
293C42—1573no—
294C42—1573no—
295C42—1571no—
296C42—1573no—
297C42C22×D15713yes—
298C42C22×D15713yes—
299C42C22×D15713yes—
300C42C22×D15713yes—
301C23—1573no—
302C23—1571no—
303C23C23×D15711yescharacteristic
304Q8—26no—
305Q8—3146no—
306Q8—3146no—
307Q8#1457895414yescomplemented
308D4—3146no—
309D4—3146no—
310D4—3146no—
311D4—1574no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—3146no—
316C2×C4—3146no—
317C2×C4—3146no—
318C2×C4—3146no—
319C2×C4—3142no—
320C2×C4—3146no—
321C2×C4—1571no—
322C2×C4—1571no—
323C2×C4—1573no—
324C2×C4—1573no—
325C2×C4—1573no—
326C2×C4—1571no—
327C2×C4C23×D15713yes—
328C2×C4C23×D15711yescharacteristic
329C2×C4C23×D15713yes—
330C2×C4C23×D15713yes—
331C2×C4C23×D15713yes—
332C2×C4C23×D15711yescharacteristic
333C22—21no—
334C22—21no—
335C22—3141no—
336C22—3141no—
337C22—3143no—
338C22—3143no—
339C22—3143no—
340C22—1572no—
341C22#41×D15712yes—
342C22C24×D15711yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—3143no—
348C4—3141no—
349C4—3143no—
350C4—3141no—
351C4—3143no—
352C4—1572no—
353C4C2×#1457895416yes—
354C4#1909781912yes—
355C2—22no—
356C2—3142no—
357C2—6283no—
358C2C2×#41×D15711yescharacteristic
359C2C22×#1457895411yescharacteristic
360C2C2×#1909781911yescharacteristic
subgroupG/H
centerC2×C4C23×D157
commutator, socle, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×C4×Q8×C157C2
hypercenterC2×C4×Q8D157
hypocenterC157#3482
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #23757999C2semidirect
2. #23757995C2semidirect
3. #23757964C2semidirect
4. #23757958C2semidirect
5. Q8×C22⋊D157C2semidirect
6. #23757935C2semidirect
7. #23757934C2semidirect
8. #23757924C2semidirect
9. #23757827C2semidirect
10. #23757810C2semidirect
11. #23757807C2semidirect
12. C4×#14578955C2semidirect
13. C4×#14578954C2semidirect
14. C4×Q8×D157C2semidirect
15. C2×C4×Q8×C157C2semidirect
127. C22⋊D157#41semidirect
142. Q8×C157#41semidirect
180. C157#3482semidirect
196. C2×C4×Q8D157semidirect
307. Q8#14578954semidirect

Maximal subgroups

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

subgroupindex
#237579992
#237579952
#237579642
#237579582
Q8×C22⋊D1572
#237579352
#237579342
#237579242
#237578272
#237578102
#237578072
C4×#145789552
C4×#145789542
C4×Q8×D1572
C2×C4×Q8×C1572
#3482157