Finite Group Databasebeta

← #34112130

Subgroups — #34112130

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757933C211yesmaximal, characteristic, complemented
2#23757924C211yesmaximal, characteristic, complemented
3#23757634C211yesmaximal, characteristic, complemented
4#23757588C211yesmaximal, characteristic, complemented
5#23757567C211yesmaximal, characteristic, complemented
6#23757521C211yesmaximal, characteristic, complemented
7#23757510C211yesmaximal, characteristic, complemented
8#23757491C211yesmaximal, characteristic, complemented
9#23757420C211yesmaximal, characteristic, complemented
10#23757413C211yesmaximal, characteristic, complemented
11#23757372C211yesmaximal, characteristic, complemented
12#23757347C211yesmaximal, characteristic, complemented
13#23757272C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#484×C157C211yesmaximal, characteristic, complemented
16C2×#14578954—21no—
17C2×#14578954C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21C2×D1256C2211yescharacteristic
22C2×#14578938C2211yescharacteristic
23C2×C8×D157C2211yescharacteristic
24#19097770C2211yescharacteristic
25#19097760C2211yescharacteristic
26#19097757—21no—
27#19097756—21no—
28#19097754—21no—
29#19097749—21no—
30#19097749—21no—
31#19097746C2211yescharacteristic
32#19097746C2211yescharacteristic
33#19097745C2211yescharacteristic
34#19097744—21no—
35#19097736C2211yescharacteristic
36#19097733—21no—
37#19097733C2211yescharacteristic
38C4×C22⋊D157C2211yescharacteristic
39C4⋊C4×D157C2211yescharacteristic
40#19097704C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097696C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097694C2211yescharacteristic
45#19097694C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097681C2211yescharacteristic
52C22⋊Q8×C157C2211yescharacteristic
53C2×C4⋊C4×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic
55#107×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59#14578954—22no—
60#14578954—41no—
61#14578954—44no—
62C2×Dic314—21no—
63C2×Dic314—21no—
64C2×Dic314C2311yescharacteristic
65Q8×D157—22no—
66Q8×D157—41no—
67C2×D628—21no—
68C2×D628—21no—
69C2×D628C2311yescharacteristic
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157D411yescharacteristic
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77C2×C4×D157—21no—
78C2×C4×D157C2311yescharacteristic
79#14578943—22no—
80#14578941—41no—
81D1256—41no—
82#14578938—41no—
83C2×#11929434C2311yescharacteristic
84C2×#11929434C2311yescharacteristic
85C8×D157—41no—
86C4⋊Dic157C2311yescharacteristic
87#14578933—21no—
88#14578933—21no—
89#14578933—21no—
90#14578933D411yescharacteristic
91C22⋊Dic157D411yescharacteristic
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#14578931—21no—
98#14578931D411yescharacteristic
99C4×Dic157—21no—
100C4×Dic157C2311yescharacteristic
101C2×Q8×C157C2311yescharacteristic
102C22×C4×C157—21no—
103C22×C4×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157—21no—
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157C2×D411yescharacteristic
115Dic314—21no—
116Dic314—42no—
117Dic314—42no—
118Dic314#4112yescomplemented
119D628—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—42no—
124D628—42no—
125C22⋊D157—22no—
126C22⋊D157—42no—
127C22⋊D157—42no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157C2×D411yescharacteristic
131C2×Dic157C2×D411yescharacteristic
132C2×Dic157C2×D411yescharacteristic
133C4×D157—21no—
134C4×D157—21no—
135C4×D157—42no—
136C4×D157—42no—
137C4×D157—42no—
138C4×D157—42no—
139#11929434—21no—
140#11929434—21no—
141C23×C157C2×D411yescharacteristic
142Q8×C157—21no—
143Q8×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—42no—
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—41no—
162C2×D157—41no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157AGL(1,ℤ/8)12yes—
168C4×C157—21no—
169C4×C157—41no—
170C4×C157—41no—
171C4×C157—41no—
172C4×C157—41no—
173C4×C157C2×#4111yescharacteristic
174C4×C157H(5,𝔽2)11yescharacteristic
175Dic157—21no—
176Dic157—22no—
177Dic157—21no—
178Dic157—21no—
179Dic157—41no—
180C2×C157—41no—
181C2×C157#54611yescharacteristic
182C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C157#57811yescharacteristic
184D157—22no—
185D157—81no—
186D157—81no—
187C157#330811yescharacteristic, complemented, Sylow
188#3308—1571nomaximal, Sylow
189#548—1571no—
190#547—1571no—
191#489—1571no—
192#484D15711yescharacteristic, complemented
193#480—1571no—
194#477—1571no—
195#468—1571no—
196#466—1571no—
197#463—1571no—
198#449—1571no—
199#448—1571no—
200#440—1571no—
201#435—1571no—
202#417—1571no—
203C2×#103—1571no—
204C2×#41—3141no—
205C2×#41—1571no—
206C22×Q8—1571no—
207C2×SD16—1571no—
208C2×SD16—1571no—
209C2×D8—1571no—
210C2×M16—1571no—
211C22×C8—1571no—
212C4⋊Q8—1571no—
213#124—3141no—
214#124—1571no—
215#123—3141no—
216#123—3141no—
217C22⋊Q8—3141no—
218C22⋊Q8—3141no—
219C22⋊Q8C2×D15711yescharacteristic
220#121—3141no—
221#121—1571no—
222#121—1571no—
223#121—1571no—
224C4×Q8—1571no—
225C4×D4—3141no—
226C4×D4—1571no—
227C4×D4—1571no—
228C2×C4⋊C4—1571no—
229C2×C4⋊C4C2×D15711yescharacteristic
230#107—1571no—
231#107C2×D15711yescharacteristic
232#107C2×D15711yescharacteristic
233C4⋊C8—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#103—1571no—
238#103C2×D15711yescharacteristic
239#103C2×D15711yescharacteristic
240#102—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244C22⋊C8—1571no—
245C22⋊C8C2×D15711yescharacteristic
246#97—1571no—
247#41—3142no—
248#41—6281no—
249#41—6284no—
250C2×Q8—3141no—
251C2×Q8—3141no—
252C2×Q8—3142no—
253C2×Q8—6281no—
254C2×Q8—1571no—
255C2×Q8C22×D15711yescharacteristic
256C2×D4—3141no—
257C2×D4—3141no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—1571no—
261C2×D4—1571no—
262C2×D4—1571no—
263C22×C4—21no—
264C22×C4—3141no—
265C22×C4—3141no—
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—1571no—
269C22×C4C22×D15711yescharacteristic
270SD16—3142no—
271SD16—6281no—
272D8—6281no—
273M16—6281no—
274C2×C8—6281no—
275C2×C8—1571no—
276C2×C8—1571no—
277C2×C8C22×D15711yescharacteristic
278C2×C8C22×D15711yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3141no—
282C4⋊C4—3141no—
283C4⋊C4—3141no—
284C4⋊C4—1571no—
285C4⋊C4—1571no—
286C4⋊C4C22×D15711yescharacteristic
287C4⋊C4C22×D15711yescharacteristic
288C4⋊C4C22×D15711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—1571no—
297C22⋊C4—1571no—
298C42—3141no—
299C42—1571no—
300C23—3141no—
301C23—3141no—
302C23—1571no—
303C23D4×D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3141no—
307Q8—6282no—
308Q8—6282no—
309Q8—1572no—
310D4—3142no—
311D4—3141no—
312D4—3141no—
313D4—3141no—
314D4—3141no—
315D4—6282no—
316D4—6282no—
317D4—6282no—
318D4—6282no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—3141no—
327C2×C4—3141no—
328C2×C4—3141no—
329C2×C4—3141no—
330C2×C4—6282no—
331C2×C4—6282no—
332C2×C4—6282no—
333C2×C4—6282no—
334C2×C4—1571no—
335C2×C4—1571no—
336C2×C4—1571no—
337C2×C4D4×D15711yescharacteristic
338C2×C4C23×D15711yescharacteristic
339C8—21no—
340C8—21no—
341C8—3141no—
342C8—3141no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—3141no—
347C22—3141no—
348C22—6281no—
349C22—6281no—
350C22—6281no—
351C22—6281no—
352C22—6281no—
353C22—6281no—
354C22—1572no—
355C22C2×D4×D15711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3141no—
362C4—3141no—
363C4—3142no—
364C4—3141no—
365C4—6281no—
366C4#41×D15711yescharacteristic
367C4#1909781711yescharacteristic
368C2—41no—
369C2—3142no—
370C2—12561no—
371C2—12561no—
372C2#2375791511yescharacteristic
373C2AGL(1,ℤ/8)×D15711yescharacteristic
374C2#2375808611yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#484×C157C2
socleC22×C157C22×D4
hypercenter#484D157
hypocenterC157#3308
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #23757933C2semidirect
2. #23757924C2semidirect
3. #23757634C2semidirect
4. #23757588C2semidirect
5. #23757567C2semidirect
6. #23757521C2semidirect
7. #23757510C2semidirect
8. #23757491C2semidirect
9. #23757420C2semidirect
10. #23757413C2semidirect
11. #23757372C2semidirect
12. #23757347C2semidirect
13. #23757272C2semidirect
14. #103×D157C2semidirect
15. #484×C157C2semidirect
118. Dic314#41semidirect
187. C157#3308semidirect
192. #484D157semidirect

Maximal subgroups

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

subgroupindex
#237579332
#237579242
#237576342
#237575882
#237575672
#237575212
#237575102
#237574912
#237574202
#237574132
#237573722
#237573472
#237572722
#103×D1572
#484×C1572
#3308157