Finite Group Databasebeta

← #34110421

Subgroups — #34110421

Order 20096.

382 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757876C211yesmaximal, characteristic
2#23757751C211yesmaximal, characteristic, complemented
3#23757598C211yesmaximal, characteristic, complemented
4#23757594C211yesmaximal, characteristic, complemented
5#23757582C211yesmaximal, characteristic, complemented
6#23757578C211yesmaximal, characteristic, complemented
7#23757471C211yesmaximal, characteristic, complemented
8#23757471C211yesmaximal, characteristic, complemented
9#23757412C211yesmaximal, characteristic, complemented
10#23757376C211yesmaximal, characteristic, complemented
11#23757374C211yesmaximal, characteristic, complemented
12SD16×Dic157C211yesmaximal, characteristic, complemented
13#23757358C211yesmaximal, characteristic, complemented
14#23757331C211yesmaximal, characteristic, complemented
15#448×C157C211yesmaximal, characteristic, complemented
16C2×#14578941C2211yescharacteristic
17C2×#14578941C2211yescharacteristic
18#19097767C2211yescharacteristic
19#19097762C2211yescharacteristic
20#19097755—21no—
21#19097753—21no—
22#19097745C2211yescharacteristic
23#19097743—21no—
24#19097742—21no—
25Q8×Dic157—21no—
26Q8×Dic157—21no—
27Q8×Dic157C2211yescharacteristic
28Q8×Dic157C2211yescharacteristic
29Q8×Dic157C2211yescharacteristic
30Q8×Dic157C2211yescharacteristic
31D4×Dic157C2211yescharacteristic
32#19097733C2211yescharacteristic
33#19097730—21no—
34#19097730—21no—
35#19097730C2211yescharacteristic
36#19097730C2211yescharacteristic
37#19097705C2211yescharacteristic
38#19097704C2211yescharacteristic
39#19097703C2211yescharacteristic
40#19097702C2211yescharacteristic
41#19097700C2211yescharacteristic
42#19097700C2211yescharacteristic
43#19097695C2211yescharacteristic
44#19097695C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic
49C22⋊#11929434C2211yescharacteristic
50#19097681C2211yescharacteristic
51C8×Dic157C2211yescharacteristic
52C22×Q8×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic
56#103×C157C2211yescharacteristic
57#103×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic
59C2×D628C2311yescharacteristic
60C2×C22⋊D157—21no—
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C22×Dic157—21no—
64C2×C4×D157—21no—
65#14578941—22no—
66#14578941—22no—
67C2×#11929434C2311yescharacteristic
68C2×#11929434C2311yescharacteristic
69C4⋊Dic157—21no—
70C4⋊Dic157—21no—
71C4⋊Dic157—21no—
72C4⋊Dic157—21no—
73C4⋊Dic157—21no—
74C4⋊Dic157D411yescharacteristic
75C4⋊Dic157D411yescharacteristic
76C4⋊Dic157C2311yescharacteristic
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79#14578933—21no—
80C22⋊Dic157—21no—
81C22⋊Dic157—21no—
82C22⋊Dic157D411yescharacteristic
83C22⋊Dic157D411yescharacteristic
84#14578931—21no—
85#14578931—21no—
86C4×Dic157—21no—
87C4×Dic157—21no—
88C4×Dic157—21no—
89C4×Dic157—21no—
90C4×Dic157—21no—
91C4×Dic157C2311yescharacteristic
92C4×Dic157C2311yescharacteristic
93C2×Q8×C157—22no—
94C2×Q8×C157—22no—
95C2×Q8×C157—21no—
96C2×Q8×C157C2311yescharacteristic
97C2×Q8×C157C2311yescharacteristic
98C2×D4×C157—21no—
99C2×D4×C157C2311yescharacteristic
100C22×C4×C157—21no—
101C22×C4×C157C2311yescharacteristic
102SD16×C157—22no—
103SD16×C157—22no—
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C4⋊C4×C157C2311yescharacteristic
107C22⋊C4×C157—21no—
108C22×D157—21no—
109D628—21no—
110D628—21no—
111C22⋊D157—81no—
112C22⋊D157—81no—
113C22⋊D157—81no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—81no—
118C2×Dic157C2×D411yescharacteristic
119C2×Dic157C2×D411yescharacteristic
120C2×Dic157C2×D411yescharacteristic
121C2×Dic157C2×D411yescharacteristic
122C4×D157—81no—
123#11929434—21no—
124#11929434—21no—
125C23×C157—21no—
126C23×C157C2×D411yescharacteristic
127Q8×C157—24no—
128Q8×C157—21no—
129Q8×C157—21no—
130Q8×C157#4112yes—
131Q8×C157#4112yes—
132D4×C157—21no—
133D4×C157—21no—
134D4×C157—81no—
135C2×C4×C157—24no—
136C2×C4×C157—21no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2411yescharacteristic
143C8×C157—21no—
144C8×C157—21no—
145C22×C157—21no—
146C22×C157—21no—
147C22×C157—21no—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157C22×D411yescharacteristic
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C4×C157—22no—
156C4×C157—22no—
157C4×C157—21no—
158C4×C157—41no—
159C4×C157C2×#4111yescharacteristic
160C4×C157C2×#4111yescharacteristic
161Dic157—21no—
162Dic157—21no—
163Dic157—21no—
164Dic157—21no—
165Dic157—41no—
166Dic157—41no—
167Dic157—41no—
168C2×C157—41no—
169C2×C157—81no—
170C2×C157#52511yescharacteristic
171C2×C157#57711yescharacteristic
172C2×C157#57711yescharacteristic
173D157—81no—
174C157#312611yescharacteristic, complemented, Sylow
175#3126—1571nomaximal, Sylow
176#540—1571no—
177#519—1571no—
178#482—1571no—
179#482—1571no—
180#479—1571no—
181#479—1571no—
182#460—1571no—
183#460—1571no—
184#448—1571no—
185#448D15711yescharacteristic, complemented
186#440—1571no—
187#440—1571no—
188C4×SD16—1571no—
189C4×SD16—1571no—
190#433—1571no—
191C22×Q8C2×D15711yescharacteristic
192C2×SD16—1571no—
193C2×SD16—1571no—
194C2×SD16C2×D15711yescharacteristic
195C2×SD16C2×D15711yescharacteristic
196#127—1571no—
197#125—1571no—
198#123—3141no—
199#123—3141no—
200#121—3141no—
201#121—3141no—
202#121—1571no—
203#121C2×D15711yescharacteristic
204C4×Q8—3141no—
205C4×Q8—3141no—
206C4×Q8—1571no—
207C4×Q8—1571no—
208C4×Q8—1571no—
209C4×Q8—1571no—
210C4×D4—1571no—
211C4×D4—1571no—
212#117—3141no—
213#117—3141no—
214#117—1571no—
215#117—1571no—
216#107—1571no—
217#107—1571no—
218#106—1571no—
219#106—1571no—
220C4⋊C8—1571no—
221C4⋊C8—1571no—
222#103—1571no—
223#103—1571no—
224#103C2×D15711yescharacteristic
225#103C2×D15711yescharacteristic
226#102—1571no—
227#102—1571no—
228#102—1571no—
229#102—1571no—
230C22⋊C8—1571no—
231C22⋊C8C2×D15711yescharacteristic
232#97—1571no—
233C4×C8—1571no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—21no—
237C2×Q8C22×D15711yescharacteristic
238C2×Q8C22×D15711yescharacteristic
239C2×D4—21no—
240C2×D4—3141no—
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4C22×D15711yescharacteristic
245C22×C4—21no—
246C22×C4—3141no—
247C22×C4—3141no—
248C22×C4C22×D15711yescharacteristic
249SD16—22no—
250SD16—22no—
251SD16—3142no—
252SD16—3142no—
253C2×C8—1571no—
254C2×C8—1571no—
255C2×C8C22×D15711yescharacteristic
256C2×C8C22×D15711yescharacteristic
257C4⋊C4—3141no—
258C4⋊C4—3141no—
259C4⋊C4—3141no—
260C4⋊C4—3141no—
261C4⋊C4—3141no—
262C4⋊C4—3141no—
263C4⋊C4—1571no—
264C4⋊C4—1571no—
265C4⋊C4—1571no—
266C4⋊C4—1571no—
267C4⋊C4—1571no—
268C4⋊C4C22×D15711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—3141no—
271C22⋊C4—3141no—
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—1571no—
275C22⋊C4—1571no—
276C42—3141no—
277C42—3141no—
278C42—3141no—
279C42—3141no—
280C42—3141no—
281C42—1571no—
282C42—1571no—
283C23—21no—
284C23—3141no—
285C23D4×D15711yescharacteristic
286Q8—24no—
287Q8—21no—
288Q8—21no—
289Q8#1457895312yes—
290Q8#1457895312yes—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—3141no—
295D4—3141no—
296D4—12561no—
297D4—12561no—
298D4—12561no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—3141no—
306C2×C4—3141no—
307C2×C4—3141no—
308C2×C4—1571no—
309C2×C4—1571no—
310C2×C4—1571no—
311C2×C4—1571no—
312C2×C4—12561no—
313C2×C4—12561no—
314C2×C4D4×D15711yescharacteristic
315C2×C4C23×D15711yescharacteristic
316C8—21no—
317C8—21no—
318C8—3141no—
319C8—3141no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—6281no—
327C22—6281no—
328C22—6281no—
329C22C2×D4×D15711yescharacteristic
330C4—22no—
331C4—22no—
332C4—21no—
333C4—41no—
334C4—3141no—
335C4—3141no—
336C4—3141no—
337C4—3141no—
338C4—6281no—
339C4—6281no—
340C4—6281no—
341C4C2×#1457895311yescharacteristic
342C4C2×#1457895311yescharacteristic
343C2—41no—
344C2—81no—
345C2—12561no—
346C2#2375777911yescharacteristic
347C2#2375807911yescharacteristic
348C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#448×C157C2
socleC22×C157C22×D4
prime radical#23757876C2
hypercenter#448D157
hypocenterC157#3126
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
2. #23757751C2semidirect
3. #23757598C2semidirect
4. #23757594C2semidirect
5. #23757582C2semidirect
6. #23757578C2semidirect
7. #23757471C2semidirect
8. #23757471C2semidirect
9. #23757412C2semidirect
10. #23757376C2semidirect
11. #23757374C2semidirect
12. SD16×Dic157C2semidirect
13. #23757358C2semidirect
14. #23757331C2semidirect
15. #448×C157C2semidirect
174. C157#3126semidirect
185. #448D157semidirect

Maximal subgroups

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

subgroupindex
#237578762
#237577512
#237575982
#237575942
#237575822
#237575782
#237574712
#237574122
#237573762
#237573742
SD16×Dic1572
#237573582
#237573312
#448×C1572
#3126157