Finite Group Databasebeta

← #34112113

Subgroups — #34112113

Order 20096.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757971C211yesmaximal, characteristic, complemented
2#23757922C211yesmaximal, characteristic, complemented
3#23757640C211yesmaximal, characteristic, complemented
4#23757613C211yesmaximal, characteristic, complemented
5#23757603C211yesmaximal, characteristic, complemented
6#23757571C211yesmaximal, characteristic, complemented
7#23757519C211yesmaximal, characteristic, complemented
8#23757505C211yesmaximal, characteristic, complemented
9#23757477C211yesmaximal, characteristic, complemented
10#23757419C211yesmaximal, characteristic, complemented
11#23757418C211yesmaximal, characteristic, complemented
12#23757382C211yesmaximal, characteristic, complemented
13#23757342C211yesmaximal, characteristic, complemented
14#23757266C211yesmaximal, characteristic, complemented
15#417×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×#14578943C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19C2×#14578939C2211yescharacteristic
20C2×C8×D157C2211yescharacteristic
21#19097769C2211yescharacteristic
22#19097766—21no—
23#19097762C2211yescharacteristic
24#19097756—21no—
25#19097754—21no—
26#19097752—21no—
27#19097750C2211yescharacteristic
28#19097747C2211yescharacteristic
29#19097746C2211yescharacteristic
30#19097745—21no—
31#19097745C2211yescharacteristic
32#19097743—21no—
33D4×Dic157C2211yescharacteristic
34#19097733C2211yescharacteristic
35C4×C22⋊D157—21no—
36C4×C22⋊D157C2211yescharacteristic
37#19097729C2211yescharacteristic
38C2×#14578933—21no—
39#19097705C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097702C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097696C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097694C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097681C2211yescharacteristic
52C2×#41×C157C2211yescharacteristic
53C2×M16×C157C2211yescharacteristic
54C2×C4⋊C4×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#102×C157C2211yescharacteristic
59#14578953—22no—
60#14578953—41no—
61C2×Dic314C2311yescharacteristic
62C2×D628—21no—
63C2×D628C2311yescharacteristic
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157D411yescharacteristic
67C22×Dic157—21no—
68C22×Dic157—21no—
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578943—41no—
72#14578941—41no—
73#14578939—22no—
74C2×#11929434C2311yescharacteristic
75C2×#11929434C2311yescharacteristic
76C8×D157—41no—
77C4⋊Dic157—21no—
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933—21no—
84#14578933—21no—
85#14578933D411yescharacteristic
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931D411yescharacteristic
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97#41×C157—22no—
98#41×C157—41no—
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105M16×C157—41no—
106C2×C8×C157C2311yescharacteristic
107C2×C8×C157C2311yescharacteristic
108C4⋊C4×C157—21no—
109C4⋊C4×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157C2×D411yescharacteristic
113Dic314—21no—
114Dic314—21no—
115D628—21no—
116D628—21no—
117D628—81no—
118C22⋊D157—41no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C2×Dic157—21no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—42no—
125C2×Dic157—42no—
126C2×Dic157C2×D411yescharacteristic
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C4×D157—21no—
130C4×D157—21no—
131C4×D157—81no—
132#11929434—21no—
133#11929434—21no—
134C23×C157—21no—
135C23×C157C2×D411yescharacteristic
136Q8×C157—21no—
137Q8×C157—21no—
138D4×C157—21no—
139D4×C157—42no—
140D4×C157#4112yes—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—42no—
147C2×C4×C157—81no—
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2411yescharacteristic
150C8×C157—21no—
151C8×C157—21no—
152C22×C157—21no—
153C22×C157—22no—
154C22×C157—21no—
155C22×C157—21no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C4×C157—21no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157—41no—
169C4×C157C2×#4111yescharacteristic
170C4×C157H(5,𝔽2)11yescharacteristic
171Dic157—21no—
172Dic157—21no—
173Dic157—21no—
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—41no—
178C2×C157—42no—
179C2×C157#54611yescharacteristic
180C2×C157#57511yescharacteristic
181C2×C157#57711yescharacteristic
182D157—41no—
183D157—81no—
184C157#330711yescharacteristic, complemented, Sylow
185#3307—1571nomaximal, Sylow
186#553—1571no—
187#547—1571no—
188#491—1571no—
189#485—1571no—
190#483—1571no—
191#478—1571no—
192#468—1571no—
193#465—1571no—
194#461—1571no—
195#449—1571no—
196#449—1571no—
197#442—1571no—
198#435—1571no—
199#417D15711yescharacteristic, complemented
200#416—1571no—
201C2×#41—1571no—
202C2×#41C2×D15711yescharacteristic
203C2×SD16—1571no—
204C2×SD16—1571no—
205C2×D8—1571no—
206C2×M16C2×D15711yescharacteristic
207C22×C8—1571no—
208C4⋊Q8—1571no—
209#126—3141no—
210#125—1571no—
211#123—3141no—
212#123—3141no—
213#123—3141no—
214C22⋊Q8—1571no—
215C22⋊Q8—1571no—
216#121—3141no—
217#121—3141no—
218#121—1571no—
219#121—1571no—
220C4×D4—3141no—
221C4×D4—1571no—
222C4×D4—1571no—
223C4×D4—1571no—
224#117—1571no—
225C2×C4⋊C4—3141no—
226C2×C4⋊C4C2×D15711yescharacteristic
227#107—1571no—
228#106—1571no—
229#106—1571no—
230C4⋊C8—1571no—
231#103—1571no—
232#103—1571no—
233#103—1571no—
234#103—1571no—
235#103C2×D15711yescharacteristic
236#103C2×D15711yescharacteristic
237#102—1571no—
238#102—1571no—
239#102C2×D15711yescharacteristic
240#102C2×D15711yescharacteristic
241C22⋊C8—1571no—
242C22⋊C8—1571no—
243#97—1571no—
244#41—22no—
245#41—41no—
246#41—3142no—
247#41—6281no—
248C2×Q8—1571no—
249C2×Q8C22×D15711yescharacteristic
250C2×D4—21no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—1571no—
255C2×D4—1571no—
256C2×D4C22×D15711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3141no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4—1571no—
263C22×C4C22×D15711yescharacteristic
264SD16—6281no—
265SD16—6281no—
266D8—3142no—
267M16—41no—
268C2×C8—6281no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8C22×D15711yescharacteristic
272C2×C8C22×D15711yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—3141no—
275C4⋊C4—3141no—
276C4⋊C4—3141no—
277C4⋊C4—3141no—
278C4⋊C4—3141no—
279C4⋊C4—3141no—
280C4⋊C4—1571no—
281C4⋊C4—1571no—
282C4⋊C4—1571no—
283C4⋊C4C22×D15711yescharacteristic
284C4⋊C4C22×D15711yescharacteristic
285C22⋊C4—3141no—
286C22⋊C4—3141no—
287C22⋊C4—3141no—
288C22⋊C4—3141no—
289C22⋊C4—3141no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—1571no—
293C22⋊C4—1571no—
294C42—3141no—
295C42—1571no—
296C23—21no—
297C23—3141no—
298C23—1571no—
299C23D4×D15711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3141no—
303Q8—3141no—
304D4—21no—
305D4—42no—
306D4—3141no—
307D4—3141no—
308D4—6281no—
309D4—6282no—
310D4—6282no—
311D4—12561no—
312D4#1457895412yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—6282no—
326C2×C4—6282no—
327C2×C4—1571no—
328C2×C4—1571no—
329C2×C4—1571no—
330C2×C4—12561no—
331C2×C4D4×D15711yescharacteristic
332C2×C4C23×D15711yescharacteristic
333C8—21no—
334C8—21no—
335C8—3141no—
336C8—3141no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—3141no—
344C22—3141no—
345C22—3141no—
346C22—6281no—
347C22—6281no—
348C22—6281no—
349C22C2×D4×D15711yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—3141no—
355C4—3141no—
356C4—3141no—
357C4—6281no—
358C4—6281no—
359C4—6281no—
360C4C2×#1457895411yescharacteristic
361C4#1909781611yescharacteristic
362C2—41no—
363C2—42no—
364C2—6281no—
365C2—12561no—
366C2#2375791311yescharacteristic
367C2#2375807211yescharacteristic
368C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
hypercenter#417D157
hypocenterC157#3307
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #23757971C2semidirect
2. #23757922C2semidirect
3. #23757640C2semidirect
4. #23757613C2semidirect
5. #23757603C2semidirect
6. #23757571C2semidirect
7. #23757519C2semidirect
8. #23757505C2semidirect
9. #23757477C2semidirect
10. #23757419C2semidirect
11. #23757418C2semidirect
12. #23757382C2semidirect
13. #23757342C2semidirect
14. #23757266C2semidirect
15. #417×C157C2semidirect
184. C157#3307semidirect
199. #417D157semidirect

Maximal subgroups

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

subgroupindex
#237579712
#237579222
#237576402
#237576132
#237576032
#237575712
#237575192
#237575052
#237574772
#237574192
#237574182
#237573822
#237573422
#237572662
#417×C1572
#3307157