Finite Group Databasebeta

← #34113752

Subgroups — #34113752

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#23757992C213yesmaximal, complemented
3#23757963C213yesmaximal, complemented
4#23757957C213yesmaximal, complemented
5Q8×C22⋊D157C211yesmaximal, characteristic, complemented
6#23757930C211yesmaximal, characteristic, complemented
7#23757923C213yesmaximal, complemented
8#23757825C213yesmaximal, complemented
9#23757804C213yesmaximal, complemented
10#23757802C213yesmaximal, complemented
11#41×Dic157C211yesmaximal, characteristic, complemented
12C4×#14578953C213yesmaximal, complemented
13C2×Q8×Dic157C211yesmaximal, characteristic, complemented
14C4×Q8×D157C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2213yes—
17C2×Q8×D157C2211yescharacteristic
18#19097770C2213yes—
19#19097769C2213yes—
20#19097764C2216yes—
21#19097764C2216yes—
22#19097763C2213yes—
23#19097762C2213yes—
24#19097761C2216yes—
25#19097758C2213yes—
26#19097753C2216yes—
27#19097752C2216yes—
28#19097750C2213yes—
29#19097750C2213yes—
30#19097748C2216yes—
31#19097747C2213yes—
32#19097742C2213yes—
33Q8×Dic157C2213yes—
34Q8×Dic157C2211yescharacteristic
35Q8×Dic157C2211yescharacteristic
36Q8×Dic157C2213yes—
37Q8×Dic157C2211yescharacteristic
38C4×Dic314C2213yes—
39C4×Dic314C2213yes—
40#19097736C2213yes—
41#19097736C2213yes—
42D4×Dic157C2213yes—
43D4×Dic157C2213yes—
44D4×Dic157C2213yes—
45C4×C22⋊D157C2213yes—
46C4×C22⋊D157C2213yes—
47#19097731C2211yescharacteristic
48#19097731C2213yes—
49#19097730C2213yes—
50#19097729C2213yes—
51#19097727C2216yes—
52#19097727C2213yes—
53#19097727C2213yes—
54C2×C4⋊Dic157C2213yes—
55C4⋊C4×D157C2213yes—
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#14578953—212no—
67C2×Dic314C2313yes—
68C2×Dic314C2313yes—
69Q8×D157—24no—
70C2×C22⋊D157C2313yes—
71C2×C22⋊D157C2311yescharacteristic
72C22×Dic157C2311yescharacteristic
73C22×Dic157C2313yes—
74C22×Dic157C2313yes—
75C22×Dic157C2313yes—
76C2×C4×D157C2311yescharacteristic
77C2×C4×D157C2313yes—
78C2×C4×D157C2313yes—
79C4⋊Dic157C2313yes—
80C4⋊Dic157C2313yes—
81C4⋊Dic157C2316yes—
82C4⋊Dic157C2313yes—
83C4⋊Dic157C2313yes—
84#14578933C2316yes—
85#14578933C2311yescharacteristic
86#14578933C2313yes—
87#14578933C2313yes—
88#14578933C2313yes—
89#14578933C2316yes—
90#14578933C2316yes—
91C22⋊Dic157C2313yes—
92C22⋊Dic157C2313yes—
93C22⋊Dic157C2316yes—
94C22⋊Dic157C2316yes—
95#14578931C2311yescharacteristic
96#14578931C2313yes—
97C4×Dic157C2313yes—
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×D157C2411yescharacteristic
121Dic314—26no—
122Dic314—26no—
123C22⋊D157—26no—
124C22⋊D157#4114yescomplemented
125C2×Dic157—26no—
126C2×Dic157—26no—
127C2×Dic157—22no—
128C2×Dic157—26no—
129C2×Dic157C2413yes—
130C2×Dic157C2413yes—
131C2×Dic157C2413yes—
132C2×Dic157C2411yescharacteristic
133C2×Dic157C2413yes—
134C2×Dic157C2411yescharacteristic
135C2×Dic157C2411yescharacteristic
136C4×D157—26no—
137C4×D157—26no—
138C4×D157—22no—
139C23×C157C2413yes—
140C23×C157C2411yescharacteristic
141Q8×C157—26no—
142Q8×C157#4114yescomplemented
143D4×C157—26no—
144D4×C157—26no—
145C2×C4×C157—26no—
146C2×C4×C157—26no—
147C2×C4×C157C2413yes—
148C2×C4×C157C2411yescharacteristic
149C2×C4×C157C2413yes—
150C2×C4×C157C2411yescharacteristic
151C2×C4×C157C2413yes—
152C22×C157—21no—
153C22×C157—21no—
154C22×C157—23no—
155C22×C157—23no—
156C22×C157—23no—
157C22×C157C2×#4112yes—
158C22×C157C2511yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157C2×#4112yes—
162C4×C157—21no—
163C4×C157—23no—
164C4×C157—23no—
165C4×C157—21no—
166C4×C157C2×#4116yes—
167Dic157—23no—
168Dic157—23no—
169Dic157—23no—
170Dic157—21no—
171Dic157—23no—
172Dic157C2×#4112yes—
173Dic1572−1+412yes—
174C2×C157—22no—
175C2×C157—43no—
176C2×C157C22×#4111yescharacteristic
177C2×C157C22×#4111yescharacteristic
178C2×C157C2×2−1+411yescharacteristic
179D157—22no—
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#548D15711yescharacteristic, complemented
189#547—1573no—
190#533—1573no—
191#529—1573no—
192#529—1573no—
193C4×#41—1571no—
194C4×#41—1573no—
195C2×C4×Q8—1571no—
196C2×C4×Q8—1571no—
197C2×#41—1573no—
198C2×#41C2×D15711yescharacteristic
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#124C2×D15713yes—
210#123—1576no—
211#123—1576no—
212C22⋊Q8—1573no—
213C22⋊Q8—1573no—
214C22⋊Q8—1573no—
215C22⋊Q8—1576no—
216C22⋊Q8C2×D15713yes—
217#121—1573no—
218#121C2×D15713yes—
219C4×Q8—1571no—
220C4×Q8—1573no—
221C4×Q8—1571no—
222C4×Q8—1573no—
223C4×Q8—1571no—
224C4×Q8—1573no—
225C4×Q8—1573no—
226C4×Q8—1573no—
227C4×Q8—1573no—
228C4×Q8C2×D15711yescharacteristic
229C4×D4—1573no—
230C4×D4—1573no—
231C4×D4—1573no—
232C4×D4—1573no—
233C4×D4—1573no—
234C4×D4—1571no—
235C4×D4—1573no—
236C4×D4C2×D15713yes—
237#117—1573no—
238#117—1573no—
239#117—1573no—
240#117—1573no—
241#117—1576no—
242C2×C4⋊C4—1573no—
243C2×C4⋊C4—1573no—
244C2×C42—1573no—
245C2×C42—1573no—
246C2×C42—1573no—
247#41—24no—
248#41—31412no—
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×D4C22×D15713yes—
258C2×D4C22×D15713yes—
259C22×C4—1571no—
260C22×C4—1573no—
261C22×C4—1573no—
262C22×C4—1573no—
263C22×C4—1571no—
264C22×C4—1573no—
265C22×C4—1573no—
266C22×C4C22×D15713yes—
267C22×C4C22×D15713yes—
268C4⋊C4—1573no—
269C4⋊C4—1573no—
270C4⋊C4—1576no—
271C4⋊C4—1576no—
272C4⋊C4—1571no—
273C4⋊C4—1573no—
274C4⋊C4—1573no—
275C4⋊C4—1573no—
276C4⋊C4—1573no—
277C4⋊C4—1573no—
278C4⋊C4—1576no—
279C4⋊C4—1576no—
280C4⋊C4C22×D15713yes—
281C4⋊C4C22×D15713yes—
282C22⋊C4—1573no—
283C22⋊C4—1576no—
284C22⋊C4—1573no—
285C22⋊C4—1576no—
286C22⋊C4—1571no—
287C22⋊C4—1573no—
288C22⋊C4C22×D15711yescharacteristic
289C22⋊C4C22×D15713yes—
290C22⋊C4C22×D15716yes—
291C42—1573no—
292C42—1571no—
293C42—1573no—
294C42—1573no—
295C42—1573no—
296C42—1573no—
297C42—1573no—
298C42—1573no—
299C42—1573no—
300C42C22×D15713yes—
301C23—1571no—
302C23C23×D15713yes—
303C23C23×D15711yescharacteristic
304Q8—26no—
305Q8—3146no—
306Q8—3146no—
307Q8#1457895314yescomplemented
308D4—26no—
309D4—26no—
310D4—3146no—
311D4—1574no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—3146no—
315C2×C4—3142no—
316C2×C4—3146no—
317C2×C4—3146no—
318C2×C4—3142no—
319C2×C4—3146no—
320C2×C4—3146no—
321C2×C4—1571no—
322C2×C4—1573no—
323C2×C4—1573no—
324C2×C4—1573no—
325C2×C4—1573no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4C23×D15711yescharacteristic
329C2×C4C23×D15711yescharacteristic
330C2×C4C23×D15713yes—
331C2×C4C23×D15713yes—
332C2×C4C23×D15713yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—3141no—
339C22—3141no—
340C22—1572no—
341C22#41×D15712yes—
342C22C24×D15711yescharacteristic
343C4—23no—
344C4—21no—
345C4—23no—
346C4—21no—
347C4—3143no—
348C4—3143no—
349C4—3143no—
350C4—3141no—
351C4—3143no—
352C4—1572no—
353C4—1572no—
354C4C2×#1457895316yes—
355C2—22no—
356C2—43no—
357C2—3142no—
358C2C2×#41×D15711yescharacteristic
359C2C22×#1457895311yescharacteristic
360C22−1+4×D15711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D157
commutator, socle, cosocleC22×C157C25
Fitting#548×C157C2
hypercenter#548D157
hypocenterC157#3482
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23757999C2semidirect
2. #23757992C2semidirect
3. #23757963C2semidirect
4. #23757957C2semidirect
5. Q8×C22⋊D157C2semidirect
6. #23757930C2semidirect
7. #23757923C2semidirect
8. #23757825C2semidirect
9. #23757804C2semidirect
10. #23757802C2semidirect
11. #41×Dic157C2semidirect
12. C4×#14578953C2semidirect
13. C2×Q8×Dic157C2semidirect
14. C4×Q8×D157C2semidirect
15. #548×C157C2semidirect
124. C22⋊D157#41semidirect
142. Q8×C157#41semidirect
180. C157#3482semidirect
188. #548D157semidirect
307. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237579992
#237579922
#237579632
#237579572
Q8×C22⋊D1572
#237579302
#237579232
#237578252
#237578042
#237578022
#41×Dic1572
C4×#145789532
C2×Q8×Dic1572
C4×Q8×D1572
#548×C1572
#3482157