Finite Group Databasebeta

← #34112127

Subgroups — #34112127

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#23757927C211yesmaximal, characteristic, complemented
2#23757632C211yesmaximal, characteristic, complemented
3#23757606C211yesmaximal, characteristic, complemented
4#23757584C211yesmaximal, characteristic, complemented
5#23757568C211yesmaximal, characteristic, complemented
6#23757521C211yesmaximal, characteristic, complemented
7#23757507C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757418C211yesmaximal, characteristic, complemented
10#23757412C211yesmaximal, characteristic, complemented
11#23757371C211yesmaximal, characteristic, complemented
12#23757345C211yesmaximal, characteristic, complemented
13#23757274C211yesmaximal, characteristic, complemented
14C2×#19097695C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C2×#14578953—21no—
17C2×#14578941C2211yescharacteristic
18C2×#14578941C2211yescharacteristic
19C2×#14578939C2211yescharacteristic
20C2×#14578938C2211yescharacteristic
21C22×#11929434C2211yescharacteristic
22#19097771C2211yescharacteristic
23#19097755—21no—
24#19097755—21no—
25#19097751—21no—
26#19097745C2211yescharacteristic
27#19097745C2211yescharacteristic
28C4×D628C2211yescharacteristic
29C2×C4⋊Dic157C2211yescharacteristic
30C2×C4⋊Dic157C2211yescharacteristic
31#19097704C2211yescharacteristic
32#19097704C2211yescharacteristic
33#19097704C2211yescharacteristic
34C4⋊#11929434C2211yescharacteristic
35#19097695C2211yescharacteristic
36#19097695C2211yescharacteristic
37#19097695C2211yescharacteristic
38#19097695C2211yescharacteristic
39#19097695C2211yescharacteristic
40#19097695C2211yescharacteristic
41#19097692C2211yescharacteristic
42#19097692C2211yescharacteristic
43#19097692C2211yescharacteristic
44#19097691C2211yescharacteristic
45C22⋊#11929434C2211yescharacteristic
46C22⋊#11929434C2211yescharacteristic
47#19097682C2211yescharacteristic
48C2×#41×C157C2211yescharacteristic
49C22×Q8×C157C2211yescharacteristic
50#124×C157—21no—
51#124×C157C2211yescharacteristic
52C22⋊Q8×C157—21no—
53C22⋊Q8×C157C2211yescharacteristic
54#121×C157—21no—
55#121×C157C2211yescharacteristic
56C4×Q8×C157C2211yescharacteristic
57C4×D4×C157—21no—
58C4×D4×C157C2211yescharacteristic
59#14578953—44no—
60C2×Dic314—21no—
61C2×D628C2311yescharacteristic
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C22×Dic157—21no—
65C22×Dic157—21no—
66C2×C4×D157—21no—
67#14578941—22no—
68#14578941—41no—
69#14578939—41no—
70#14578938—41no—
71C2×#11929434—41no—
72C2×#11929434C2311yescharacteristic
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C2×#11929434C2311yescharacteristic
76C4⋊Dic157—21no—
77C4⋊Dic157—21no—
78C4⋊Dic157—21no—
79C4⋊Dic157C2311yescharacteristic
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82#14578931—21no—
83#14578931—21no—
84#14578931—21no—
85#41×C157—22no—
86#41×C157—41no—
87C2×Q8×C157—21no—
88C2×Q8×C157—22no—
89C2×Q8×C157—41no—
90C2×Q8×C157C2311yescharacteristic
91C2×Q8×C157C2311yescharacteristic
92C2×D4×C157—21no—
93C2×D4×C157—21no—
94C2×D4×C157D411yescharacteristic
95C2×D4×C157C2311yescharacteristic
96C22×C4×C157—21no—
97C22×C4×C157—21no—
98C22×C4×C157C2311yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C4⋊C4×C157—21no—
101C4⋊C4×C157—21no—
102C4⋊C4×C157D411yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C22⋊C4×C157—21no—
108C22⋊C4×C157D411yescharacteristic
109C22⋊C4×C157D411yescharacteristic
110C42×C157—21no—
111C42×C157C2311yescharacteristic
112C22×D157—21no—
113Dic314—42no—
114D628—21no—
115D628—21no—
116C22⋊D157—42no—
117C22⋊D157—42no—
118C2×Dic157—21no—
119C2×Dic157—21no—
120C2×Dic157—21no—
121C2×Dic157—42no—
122C2×Dic157—42no—
123C4×D157—42no—
124#11929434—21no—
125#11929434—21no—
126#11929434—21no—
127#11929434—21no—
128C23×C157—21no—
129C23×C157C2×D411yescharacteristic
130C23×C157C2×D411yescharacteristic
131Q8×C157—21no—
132Q8×C157—21no—
133Q8×C157—21no—
134Q8×C157—42no—
135Q8×C157#4112yescomplemented
136D4×C157—22no—
137D4×C157—21no—
138D4×C157—21no—
139D4×C157—42no—
140D4×C157—42no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—42no—
149C2×C4×C157—42no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2×D411yescharacteristic
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2411yescharacteristic
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157AGL(1,ℤ/8)12yes—
164C22×C157C22×D411yescharacteristic
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—41no—
168C4×C157—21no—
169C4×C157—21no—
170C4×C157—22no—
171C4×C157—21no—
172C4×C157—21no—
173C4×C157—41no—
174C4×C157—41no—
175C4×C157C2×#4111yescharacteristic
176C4×C157H(5,𝔽2)11yescharacteristic
177Dic157—41no—
178Dic157—41no—
179Dic157—41no—
180C2×C157—22no—
181C2×C157—41no—
182C2×C157—81no—
183C2×C157#54611yescharacteristic
184C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C157#57811yescharacteristic
186D157—81no—
187C157#330811yescharacteristic, complemented, Sylow
188#3308—1571nomaximal, Sylow
189#548D15711yescharacteristic, complemented
190#547—1571no—
191#489—1571no—
192#484—1571no—
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×#41C2×D15711yescharacteristic
206C22×Q8C2×D15711yescharacteristic
207C2×SD16—1571no—
208C2×SD16—1571no—
209C2×D8—1571no—
210C2×M16—1571no—
211C22×C8—1571no—
212C4⋊Q8—1571no—
213#124—21no—
214#124C2×D15711yescharacteristic
215#123—3141no—
216#123—3141no—
217C22⋊Q8—21no—
218C22⋊Q8—3141no—
219C22⋊Q8C2×D15711yescharacteristic
220#121—21no—
221#121—1571no—
222#121—1571no—
223#121C2×D15711yescharacteristic
224C4×Q8C2×D15711yescharacteristic
225C4×D4—21no—
226C4×D4—1571no—
227C4×D4C2×D15711yescharacteristic
228C2×C4⋊C4—1571no—
229C2×C4⋊C4—1571no—
230#107—1571no—
231#107—1571no—
232#107—1571no—
233C4⋊C8—1571no—
234#103—1571no—
235#103—1571no—
236#103—1571no—
237#103—1571no—
238#103—1571no—
239#103—1571no—
240#102—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244C22⋊C8—1571no—
245C22⋊C8—1571no—
246#97—1571no—
247#41—22no—
248#41—41no—
249#41—6284no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3141no—
254C2×Q8C22×D15711yescharacteristic
255C2×Q8C22×D15711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—1571no—
261C2×D4C22⋊D15711yescharacteristic
262C2×D4C22×D15711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3141no—
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4C22×D15711yescharacteristic
269C22×C4C22×D15711yescharacteristic
270SD16—3142no—
271SD16—6281no—
272D8—6281no—
273M16—6281no—
274C2×C8—6281no—
275C2×C8—1571no—
276C2×C8—1571no—
277C2×C8—1571no—
278C2×C8—1571no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3141no—
282C4⋊C4—3141no—
283C4⋊C4—3141no—
284C4⋊C4—1571no—
285C4⋊C4—1571no—
286C4⋊C4—1571no—
287C4⋊C4C22⋊D15711yescharacteristic
288C4⋊C4C22×D15711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4C22⋊D15711yescharacteristic
297C22⋊C4C22⋊D15711yescharacteristic
298C42—21no—
299C42C22×D15711yescharacteristic
300C23—21no—
301C23—3141no—
302C23C2×C22⋊D15711yescharacteristic
303C23C2×C22⋊D15711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—6282no—
309Q8#1457895312yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3141no—
316D4—3141no—
317D4—6282no—
318D4—6282no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3141no—
329C2×C4—3141no—
330C2×C4—3141no—
331C2×C4—6282no—
332C2×C4—6282no—
333C2×C4—6282no—
334C2×C4C2×C22⋊D15711yescharacteristic
335C2×C4C2×C22⋊D15711yescharacteristic
336C2×C4C2×C22⋊D15711yescharacteristic
337C2×C4C2×C22⋊D15711yescharacteristic
338C2×C4C23×D15711yescharacteristic
339C8—3141no—
340C8—3141no—
341C8—3141no—
342C8—3141no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—6281no—
352C22—6281no—
353C22—6281no—
354C22#1909779612yes—
355C22C22×C22⋊D15711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—6281no—
364C4—6281no—
365C4—6281no—
366C4C2×#1457895311yescharacteristic
367C4#1909781611yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—12561no—
372C2#2375791711yescharacteristic
373C2C2×#1909779611yescharacteristic
374C2#2375808811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#548×C157C2
socleC22×C157C22×D4
hypercenter#548D157
hypocenterC157#3308
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23757927C2semidirect
2. #23757632C2semidirect
3. #23757606C2semidirect
4. #23757584C2semidirect
5. #23757568C2semidirect
6. #23757521C2semidirect
7. #23757507C2semidirect
8. #23757485C2semidirect
9. #23757418C2semidirect
10. #23757412C2semidirect
11. #23757371C2semidirect
12. #23757345C2semidirect
13. #23757274C2semidirect
14. C2×#19097695C2semidirect
15. #548×C157C2semidirect
135. Q8×C157#41semidirect
187. C157#3308semidirect
189. #548D157semidirect
309. Q8#14578953semidirect

Maximal subgroups

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

subgroupindex
#237579272
#237576322
#237576062
#237575842
#237575682
#237575212
#237575072
#237574852
#237574182
#237574122
#237573712
#237573452
#237572742
C2×#190976952
#548×C1572
#3308157