Finite Group Databasebeta

← #34112126

Subgroups — #34112126

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#23757931C211yesmaximal, characteristic, complemented
2#23757926C211yesmaximal, characteristic, complemented
3#23757607C211yesmaximal, characteristic, complemented
4#23757585C211yesmaximal, characteristic, complemented
5#23757565C211yesmaximal, characteristic, complemented
6#23757522C211yesmaximal, characteristic, complemented
7#23757508C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757419C211yesmaximal, characteristic, complemented
10#23757413C211yesmaximal, characteristic, complemented
11#23757375C211yesmaximal, characteristic, complemented
12#23757346C211yesmaximal, characteristic, complemented
13#23757271C211yesmaximal, characteristic, complemented
14#103×D157C211yesmaximal, characteristic, complemented
15#489×C157C211yesmaximal, characteristic, complemented
16C2×#14578955—21no—
17C2×#14578953C2211yescharacteristic
18C2×Q8×D157C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×#14578941C2211yescharacteristic
21C2×#14578939C2211yescharacteristic
22C2×#14578937C2211yescharacteristic
23C2×C8×D157C2211yescharacteristic
24#19097757—21no—
25#19097756—21no—
26#19097756—21no—
27#19097748—21no—
28#19097747—21no—
29#19097747C2211yescharacteristic
30#19097746C2211yescharacteristic
31#19097746C2211yescharacteristic
32#19097744—21no—
33#19097742C2211yescharacteristic
34C4×Dic314C2211yescharacteristic
35C4×D628C2211yescharacteristic
36C4×D628C2211yescharacteristic
37#19097731—21no—
38C4⋊C4×D157C2211yescharacteristic
39C4⋊C4×D157C2211yescharacteristic
40#19097705C2211yescharacteristic
41#19097704C2211yescharacteristic
42#19097704C2211yescharacteristic
43C4⋊#11929434C2211yescharacteristic
44#19097696C2211yescharacteristic
45#19097695C2211yescharacteristic
46#19097694C2211yescharacteristic
47#19097694C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097692C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097683C2211yescharacteristic
52C4⋊Q8×C157C2211yescharacteristic
53#124×C157C2211yescharacteristic
54#103×C157C2211yescharacteristic
55#103×C157C2211yescharacteristic
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58#97×C157C2211yescharacteristic
59#14578955—44no—
60#14578953—22no—
61#14578953—41no—
62C2×Dic314—21no—
63C2×Dic314C2311yescharacteristic
64Q8×D157—22no—
65Q8×D157—41no—
66C2×D628—21no—
67C2×D628—21no—
68C2×D628D411yescharacteristic
69C2×D628C2311yescharacteristic
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C22×Dic157—21no—
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77C2×C4×D157C2311yescharacteristic
78C2×C4×D157C2311yescharacteristic
79#14578943—22no—
80#14578941—41no—
81#14578939—41no—
82#14578937—41no—
83C2×#11929434C2311yescharacteristic
84C2×#11929434C2311yescharacteristic
85C8×D157—41no—
86C4⋊Dic157D411yescharacteristic
87C4⋊Dic157C2311yescharacteristic
88C4⋊Dic157C2311yescharacteristic
89#14578933—21no—
90#14578933—21no—
91#14578933—21no—
92#14578933—21no—
93C22⋊Dic157—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#14578931—21no—
98#14578931D411yescharacteristic
99#14578931D411yescharacteristic
100C4×Dic157—21no—
101C2×Q8×C157—21no—
102C2×Q8×C157C2311yescharacteristic
103C2×D4×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157—21no—
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C22⋊C4×C157—21no—
110C22⋊C4×C157—21no—
111C42×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157C2×D411yescharacteristic
114C22×D157C2×D411yescharacteristic
115Dic314—21no—
116Dic314—42no—
117Dic314#4112yescomplemented
118D628—22no—
119D628—21no—
120D628—21no—
121D628—42no—
122D628—42no—
123C22⋊D157—42no—
124C22⋊D157—42no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—21no—
128C2×Dic157—42no—
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C4×D157—21no—
132C4×D157—21no—
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×C157—21no—
142Q8×C157—21no—
143Q8×C157—21no—
144Q8×C157—42no—
145D4×C157—21no—
146D4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—21no—
163C2×D157—21no—
164C2×D157—41no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157AGL(1,ℤ/8)12yes—
168C4×C157—21no—
169C4×C157—21no—
170C4×C157—41no—
171C4×C157—41no—
172C4×C157—41no—
173C4×C157C2×#4111yescharacteristic
174C4×C157H(5,𝔽2)11yescharacteristic
175Dic157—21no—
176Dic157—21no—
177Dic157—22no—
178Dic157—41no—
179Dic157—41no—
180C2×C157—81no—
181C2×C157#54611yescharacteristic
182C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
183C2×C157#57811yescharacteristic
184D157—22no—
185D157—41no—
186D157—81no—
187C157#330811yescharacteristic, complemented, Sylow
188#3308—1571nomaximal, Sylow
189#548—1571no—
190#547—1571no—
191#489D15711yescharacteristic, complemented
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×#41—1571no—
206C22×Q8—1571no—
207C2×SD16—1571no—
208C2×SD16—1571no—
209C2×D8—1571no—
210C2×M16—1571no—
211C22×C8—1571no—
212C4⋊Q8C2×D15711yescharacteristic
213#124—3141no—
214#124C2×D15711yescharacteristic
215#123—3141no—
216#123—3141no—
217C22⋊Q8—3141no—
218C22⋊Q8—3141no—
219C22⋊Q8—1571no—
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⋊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#103C2×D15711yescharacteristic
239#103C2×D15711yescharacteristic
240#102—1571no—
241#102—1571no—
242#102C2×D15711yescharacteristic
243#102C2×D15711yescharacteristic
244C22⋊C8—1571no—
245C22⋊C8—1571no—
246#97C2×D15711yescharacteristic
247#41—3142no—
248#41—6281no—
249#41—6284no—
250C2×Q8—21no—
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×D4C22×D15711yescharacteristic
263C22×C4—3141no—
264C22×C4—3141no—
265C22×C4—3141no—
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—1571no—
269C22×C4—1571no—
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—3141no—
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—3141no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—1571no—
297C22⋊C4—1571no—
298C42—3141no—
299C42C22×D15711yescharacteristic
300C23—21no—
301C23—3141no—
302C23—1571no—
303C23—1571no—
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—3141no—
308Q8—6282no—
309Q8—1572no—
310D4—21no—
311D4—21no—
312D4—3142no—
313D4—3141no—
314D4—3141no—
315D4—6282no—
316D4—6282no—
317D4—6282no—
318D4—6282no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—3141no—
326C2×C4—3141no—
327C2×C4—3141no—
328C2×C4—3141no—
329C2×C4—6282no—
330C2×C4—6282no—
331C2×C4—6282no—
332C2×C4—6282no—
333C2×C4—6282no—
334C2×C4—1571no—
335C2×C4—1571no—
336C2×C4D4×D15711yescharacteristic
337C2×C4D4×D15711yescharacteristic
338C2×C4C23×D15711yescharacteristic
339C8—21no—
340C8—21no—
341C8—3141no—
342C8—3141no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3141no—
347C22—3141no—
348C22—3141no—
349C22—3141no—
350C22—3141no—
351C22—6281no—
352C22—6281no—
353C22—6281no—
354C22—1572no—
355C22C2×D4×D15711yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3141no—
362C4—3141no—
363C4—3142no—
364C4—6281no—
365C4—6281no—
366C4#41×D15711yescharacteristic
367C4#1909781711yescharacteristic
368C2—81no—
369C2—3142no—
370C2—6281no—
371C2—12561no—
372C2#2375791611yescharacteristic
373C2AGL(1,ℤ/8)×D15711yescharacteristic
374C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#489×C157C2
socleC22×C157C22×D4
hypercenter#489D157
hypocenterC157#3308
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #23757931C2semidirect
2. #23757926C2semidirect
3. #23757607C2semidirect
4. #23757585C2semidirect
5. #23757565C2semidirect
6. #23757522C2semidirect
7. #23757508C2semidirect
8. #23757485C2semidirect
9. #23757419C2semidirect
10. #23757413C2semidirect
11. #23757375C2semidirect
12. #23757346C2semidirect
13. #23757271C2semidirect
14. #103×D157C2semidirect
15. #489×C157C2semidirect
117. Dic314#41semidirect
187. C157#3308semidirect
191. #489D157semidirect

Maximal subgroups

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

subgroupindex
#237579312
#237579262
#237576072
#237575852
#237575652
#237575222
#237575082
#237574852
#237574192
#237574132
#237573752
#237573462
#237572712
#103×D1572
#489×C1572
#3308157