Finite Group Databasebeta

← #34113726

Subgroups — #34113726

Order 20096.

1112 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758005C213yesmaximal, complemented
2#23757999C211yesmaximal, characteristic, complemented
3#23757994C213yesmaximal, complemented
4#23757975C213yesmaximal, complemented
5#23757969C213yesmaximal, complemented
6#23757942C211yesmaximal, characteristic, complemented
7#23757934C211yesmaximal, characteristic, complemented
8#23757818C213yesmaximal, complemented
9#23757809C213yesmaximal, complemented
10#23757802C213yesmaximal, complemented
11#41×Dic157C211yesmaximal, characteristic, complemented
12C4×#14578955C211yesmaximal, characteristic, complemented
13C4×#14578953C213yesmaximal, complemented
14C2×Q8×Dic157C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578953C2213yes—
18#19097770C2213yes—
19#19097767C2213yes—
20#19097766C2216yes—
21#19097764C2216yes—
22#19097763C2213yes—
23#19097762C2213yes—
24#19097760C2213yes—
25#19097757C2216yes—
26#19097755C2216yes—
27#19097753C2216yes—
28#19097747C2213yes—
29#19097745C2213yes—
30#19097745C2213yes—
31#19097743C2216yes—
32#19097742C2213yes—
33Q8×Dic157C2211yescharacteristic
34Q8×Dic157C2211yescharacteristic
35Q8×Dic157C2213yes—
36Q8×Dic157C2211yescharacteristic
37Q8×Dic157C2213yes—
38C4×Dic314C2213yes—
39#19097736C2213yes—
40C4×D628C2213yes—
41D4×Dic157C2213yes—
42D4×Dic157C2213yes—
43D4×Dic157C2213yes—
44#19097733C2213yes—
45C4×C22⋊D157C2213yes—
46C4×C22⋊D157C2213yes—
47#19097731C2213yes—
48#19097731C2213yes—
49#19097731C2216yes—
50#19097730C2213yes—
51#19097729C2213yes—
52#19097729C2213yes—
53#19097727C2211yescharacteristic
54#19097727C2213yes—
55C2×C4⋊Dic157C2213yes—
56C2×C4×Dic157C2213yes—
57C2×C4×Dic157C2213yes—
58C42×D157C2213yes—
59C2×#41×C157C2211yescharacteristic
60C22×Q8×C157C2211yescharacteristic
61#124×C157C2213yes—
62C22⋊Q8×C157C2213yes—
63#121×C157C2213yes—
64C4×Q8×C157C2211yescharacteristic
65C4×D4×C157C2213yes—
66#14578955—24no—
67#14578953—212no—
68C2×Dic314C2313yes—
69C2×D628C2313yes—
70C2×C22⋊D157C2313yes—
71C2×C22⋊D157C2313yes—
72C2×C22⋊D157C2316yes—
73C22×Dic157C2313yes—
74C22×Dic157C2313yes—
75C22×Dic157C2313yes—
76C22×Dic157C2311yescharacteristic
77C2×C4×D157C2313yes—
78C2×C4×D157C2313yes—
79C2×C4×D157C2313yes—
80C4⋊Dic157C2313yes—
81C4⋊Dic157C2313yes—
82C4⋊Dic157C2316yes—
83C4⋊Dic157C2313yes—
84C4⋊Dic157C2313yes—
85#14578933C2311yescharacteristic
86#14578933C2313yes—
87#14578933C2311yescharacteristic
88#14578933C2313yes—
89#14578933C2316yes—
90C22⋊Dic157C2311yescharacteristic
91C22⋊Dic157C2313yes—
92C22⋊Dic157C2316yes—
93#14578931C2313yes—
94#14578931C2313yes—
95#14578931C2316yes—
96#14578931C2316yes—
97C4×Dic157C2311yescharacteristic
98C4×Dic157C2313yes—
99C4×Dic157C2313yes—
100C4×Dic157C2311yescharacteristic
101C4×Dic157C2313yes—
102C4×Dic157C2313yes—
103C4×Dic157C2313yes—
104C4×Dic157C2313yes—
105C4×Dic157C2313yes—
106#41×C157—24no—
107C2×Q8×C157—24no—
108C2×Q8×C157C2313yes—
109C2×Q8×C157C2311yescharacteristic
110C2×D4×C157C2313yes—
111C2×D4×C157C2313yes—
112C22×C4×C157C2313yes—
113C22×C4×C157C2313yes—
114C4⋊C4×C157C2313yes—
115C4⋊C4×C157C2313yes—
116C22⋊C4×C157C2311yescharacteristic
117C22⋊C4×C157C2313yes—
118C22⋊C4×C157C2316yes—
119C42×C157C2313yes—
120C22×D157C2413yes—
121Dic314—26no—
122D628—26no—
123C22⋊D157—26no—
124C22⋊D157—26no—
125C22⋊D157—46no—
126C2×Dic157—26no—
127C2×Dic157—43no—
128C2×Dic157—43no—
129C2×Dic157#4114yescomplemented
130C2×Dic157C2411yescharacteristic
131C2×Dic157C2413yes—
132C2×Dic157C2411yescharacteristic
133C2×Dic157C2413yes—
134C2×Dic157C2413yes—
135C2×Dic157C2411yescharacteristic
136C2×Dic157C2411yescharacteristic
137C4×D157—26no—
138C4×D157—26no—
139C4×D157—43no—
140C23×C157C2413yes—
141C23×C157C2411yescharacteristic
142Q8×C157—26no—
143Q8×C157#4114yescomplemented
144D4×C157—26no—
145D4×C157—26no—
146C2×C4×C157—26no—
147C2×C4×C157—26no—
148C2×C4×C157C2411yescharacteristic
149C2×C4×C157C2413yes—
150C2×C4×C157C2413yes—
151C2×C4×C157C2411yescharacteristic
152C2×C4×C157C2413yes—
153C22×C157—21no—
154C22×C157—21no—
155C22×C157—23no—
156C22×C157—23no—
157C22×C157—23no—
158C22×C157C2×#4112yes—
159C22×C157C2511yescharacteristic
160C2×D157—23no—
161C2×D157—23no—
162C2×D157—23no—
163C4×C157—21no—
164C4×C157—23no—
165C4×C157—23no—
166C4×C157—21no—
167C4×C157C2×#4116yes—
168Dic157—21no—
169Dic157—23no—
170Dic157—23no—
171Dic157—21no—
172Dic157—23no—
173Dic157C2×#4112yes—
174Dic157C2×#4112yes—
175C2×C157—22no—
176C2×C157—43no—
177C2×C157C22×#4111yescharacteristic
178C2×C157C22×#4111yescharacteristic
179C2×C157#58511yescharacteristic
180D157—43no—
181C157#348011yescharacteristic, complemented, Sylow
182#3480—1571nomaximal, Sylow
183#559—1573no—
184#557—1571no—
185#556—1573no—
186#553—1573no—
187#553—1573no—
188#550—1571no—
189#548—1571no—
190#548D15711yescharacteristic, complemented
191#532—1573no—
192#529—1573no—
193#529—1573no—
194C4×#41—1571no—
195C4×#41—1571no—
196C4×#41—1573no—
197C2×C4×Q8—1571no—
198C2×#41—1571no—
199C2×#41—1573no—
200C2×#41C2×D15711yescharacteristic
201C22×Q8C2×D15711yescharacteristic
202C4⋊Q8—1573no—
203#127—1573no—
204#126—1576no—
205#126—1576no—
206#125—1573no—
207#125—1573no—
208#124—1573no—
209#124—1576no—
210#124C2×D15713yes—
211#123—1576no—
212#123—1576no—
213C22⋊Q8—1573no—
214C22⋊Q8C2×D15713yes—
215#121—1573no—
216#121—1576no—
217#121—1573no—
218#121—1573no—
219#121C2×D15713yes—
220C4×Q8—1571no—
221C4×Q8—1573no—
222C4×Q8—1571no—
223C4×Q8—1573no—
224C4×Q8—1571no—
225C4×Q8—1573no—
226C4×Q8—1573no—
227C4×Q8C2×D15711yescharacteristic
228C4×D4—1573no—
229C4×D4—1573no—
230C4×D4—1573no—
231C4×D4—1573no—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1573no—
235C4×D4—1573no—
236C4×D4—1576no—
237C4×D4—1573no—
238C4×D4C2×D15713yes—
239#117—1573no—
240#117—1573no—
241#117—1573no—
242#117—1571no—
243#117—1573no—
244C2×C4⋊C4—1573no—
245C2×C42—1573no—
246C2×C42—1573no—
247C2×C42—1573no—
248#41—24no—
249#41—3144no—
250#41—31412no—
251C2×Q8—24no—
252C2×Q8—1573no—
253C2×Q8C22×D15713yes—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—1573no—
256C2×D4—1573no—
257C2×D4—1576no—
258C2×D4—1573no—
259C2×D4C22×D15713yes—
260C2×D4C22×D15713yes—
261C22×C4—1573no—
262C22×C4—1573no—
263C22×C4—1573no—
264C22×C4—1573no—
265C22×C4—1573no—
266C22×C4—1573no—
267C22×C4—1571no—
268C22×C4C22×D15713yes—
269C22×C4C22×D15713yes—
270C4⋊C4—1573no—
271C4⋊C4—1573no—
272C4⋊C4—1576no—
273C4⋊C4—1571no—
274C4⋊C4—1571no—
275C4⋊C4—1573no—
276C4⋊C4—1573no—
277C4⋊C4—1573no—
278C4⋊C4—1573no—
279C4⋊C4—1576no—
280C4⋊C4C22×D15713yes—
281C4⋊C4C22×D15713yes—
282C22⋊C4—1571no—
283C22⋊C4—1573no—
284C22⋊C4—1576no—
285C22⋊C4—1573no—
286C22⋊C4—1576no—
287C22⋊C4—1573no—
288C22⋊C4—1576no—
289C22⋊C4C22×D15711yescharacteristic
290C22⋊C4C22×D15713yes—
291C22⋊C4C22×D15716yes—
292C42—1571no—
293C42—1571no—
294C42—1573no—
295C42—1573no—
296C42—1573no—
297C42—1573no—
298C42—1573no—
299C42—1573no—
300C42—1573no—
301C42C22×D15713yes—
302C23—1573no—
303C23C23×D15713yes—
304C23C23×D15711yescharacteristic
305Q8—26no—
306Q8—3146no—
307Q8#1457895314yescomplemented
308D4—26no—
309D4—26no—
310D4—3146no—
311D4—3146no—
312D4—3146no—
313D4—6286no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3146no—
317C2×C4—3146no—
318C2×C4—3146no—
319C2×C4—6283no—
320C2×C4—6283no—
321C2×C4—6283no—
322C2×C4—1571no—
323C2×C4—1571no—
324C2×C4—1573no—
325C2×C4—1573no—
326C2×C4—1573no—
327C2×C4—1574no—
328C2×C4—1571no—
329C2×C4—1571no—
330C2×C4C23×D15711yescharacteristic
331C2×C4C23×D15713yes—
332C2×C4C23×D15711yescharacteristic
333C2×C4C23×D15713yes—
334C2×C4C23×D15713yes—
335C22—21no—
336C22—21no—
337C22—23no—
338C22—23no—
339C22—23no—
340C22—3143no—
341C22—3143no—
342C22—3143no—
343C22#41×D15712yes—
344C22C24×D15711yescharacteristic
345C4—23no—
346C4—21no—
347C4—23no—
348C4—21no—
349C4—3143no—
350C4—3141no—
351C4—3143no—
352C4—3141no—
353C4—3143no—
354C4—1572no—
355C4—1572no—
356C4C2×#1457895316yes—
357C2—22no—
358C2—43no—
359C2—6283no—
360C2C2×#41×D15711yescharacteristic
361C2C22×#1457895311yescharacteristic
362C2#2375811011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#548×C157C2
hypercenter#548D157
hypocenterC157#3480
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23758005C2semidirect
2. #23757999C2semidirect
3. #23757994C2semidirect
4. #23757975C2semidirect
5. #23757969C2semidirect
6. #23757942C2semidirect
7. #23757934C2semidirect
8. #23757818C2semidirect
9. #23757809C2semidirect
10. #23757802C2semidirect
11. #41×Dic157C2semidirect
12. C4×#14578955C2semidirect
13. C4×#14578953C2semidirect
14. C2×Q8×Dic157C2semidirect
15. #548×C157C2semidirect
129. C2×Dic157#41semidirect
143. Q8×C157#41semidirect
181. C157#3480semidirect
190. #548D157semidirect
307. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237580052
#237579992
#237579942
#237579752
#237579692
#237579422
#237579342
#237578182
#237578092
#237578022
#41×Dic1572
C4×#145789552
C4×#145789532
C2×Q8×Dic1572
#548×C1572
#3480157