Finite Group Databasebeta

← #34113204

Subgroups — #34113204

Order 20096.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758008C211yesmaximal, characteristic, complemented
2#23757996C211yesmaximal, characteristic, complemented
3#23757667C211yesmaximal, characteristic, complemented
4#23757662C211yesmaximal, characteristic
5#23757657C211yesmaximal, characteristic, complemented
6#23757496C211yesmaximal, characteristic, complemented
7#23757496C211yesmaximal, characteristic, complemented
8#23757489C211yesmaximal, characteristic, complemented
9#23757489C211yesmaximal, characteristic, complemented
10#23757479C211yesmaximal, characteristic, complemented
11#23757474C211yesmaximal, characteristic, complemented
12#23757386C211yesmaximal, characteristic, complemented
13#23757381C211yesmaximal, characteristic, complemented
14C4×#14578943C211yesmaximal, characteristic, complemented
15#446×C157C211yesmaximal, characteristic, complemented
16C2×Dic628C2211yescharacteristic
17C2×Dic628C2211yescharacteristic
18C2×#14578943C2211yescharacteristic
19C2×#14578943C2211yescharacteristic
20C2×#14578943C2211yescharacteristic
21C2×#14578943C2211yescharacteristic
22C2×#14578943C2211yescharacteristic
23C2×D1256C2211yescharacteristic
24C2×D1256C2211yescharacteristic
25#19097771C2211yescharacteristic
26#19097768C2211yescharacteristic
27#19097766—21no—
28#19097766—21no—
29#19097760—21no—
30#19097759C2211yescharacteristic
31#19097759C2211yescharacteristic
32#19097759C2211yescharacteristic
33#19097759C2211yescharacteristic
34#19097756—21no—
35#19097756—21no—
36#19097747—21no—
37C4×Dic314C2211yescharacteristic
38C4×Dic314C2211yescharacteristic
39C4×Dic314C2211yescharacteristic
40C4×D628C2211yescharacteristic
41C4×D628C2211yescharacteristic
42C4×D628C2211yescharacteristic
43#19097733—21no—
44#19097729—21no—
45#19097703C2211yescharacteristic
46#19097696C2211yescharacteristic
47#19097696C2211yescharacteristic
48#19097696C2211yescharacteristic
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097693C2211yescharacteristic
52C4×Q8×C157C2211yescharacteristic
53C4⋊C8×C157C2211yescharacteristic
54C4⋊C8×C157C2211yescharacteristic
55C4⋊C8×C157C2211yescharacteristic
56#97×C157C2211yescharacteristic
57#97×C157C2211yescharacteristic
58C4×C8×C157C2211yescharacteristic
59C2×Dic314C2311yescharacteristic
60C2×Dic314C2311yescharacteristic
61C2×Dic314C2311yescharacteristic
62C2×D628—21no—
63C2×D628—21no—
64C2×D628C2311yescharacteristic
65C2×D628C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×C4×D157—21no—
68C2×C4×D157—21no—
69C2×C4×D157—21no—
70Dic628—22no—
71Dic628—41no—
72#14578943—22no—
73#14578943—41no—
74#14578943—41no—
75#14578943—41no—
76#14578943—41no—
77D1256—22no—
78D1256—41no—
79C4⋊Dic157—21no—
80C4⋊Dic157—21no—
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97C4×Dic157—21no—
98C4×Dic157—21no—
99C4×Dic157—21no—
100C2×Q8×C157D411yescharacteristic
101C2×C8×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C4⋊C4×C157D411yescharacteristic
106C4⋊C4×C157D411yescharacteristic
107C4⋊C4×C157D411yescharacteristic
108C42×C157C2311yescharacteristic
109C42×C157C2311yescharacteristic
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—21no—
114Dic314—21no—
115Dic314—21no—
116Dic314—21no—
117Dic314—21no—
118Dic314—21no—
119Dic314—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—21no—
124D628—21no—
125D628—21no—
126D628—42no—
127D628—81no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C4×D157—42no—
134C4×D157—81no—
135C4×D157—81no—
136Q8×C157—41no—
137C2×C4×C157C2×D411yescharacteristic
138C2×C4×C157C2×D411yescharacteristic
139C2×C4×C157C2×D411yescharacteristic
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2×D411yescharacteristic
142C2×C4×C157C2×D411yescharacteristic
143C2×C4×C157C2411yescharacteristic
144C8×C157—21no—
145C8×C157—21no—
146C8×C157—21no—
147C8×C157#4112yes—
148C22×C157C22×D411yescharacteristic
149C2×D157—41no—
150C2×D157—41no—
151C2×D157—41no—
152C2×D157—41no—
153C2×D157—41no—
154C2×D157—41no—
155C2×D157—41no—
156C2×D157—41no—
157C2×D157—41no—
158C4×C157—21no—
159C4×C157—21no—
160C4×C157—21no—
161C4×C157—21no—
162C4×C157—21no—
163C4×C157—21no—
164C4×C157C2×#4111yescharacteristic
165C4×C157H(5,𝔽2)11yescharacteristic
166Dic157—41no—
167Dic157—41no—
168Dic157—41no—
169Dic157—41no—
170Dic157—41no—
171C2×C157#55011yescharacteristic
172C2×C157#57511yescharacteristic
173C2×C157#57711yescharacteristic
174D157—81no—
175D157—81no—
176D157—81no—
177C157#340111yescharacteristic, complemented, Sylow
178#3401—1571nomaximal, Sylow
179#559—1571no—
180#556—1571no—
181#497—1571no—
182#496—1571no—
183#495—1571no—
184#464—1571no—
185#464—1571no—
186#463—1571no—
187#463—1571no—
188#461—1571no—
189#460—1571no—
190#446D15711yescharacteristic, complemented
191#442—1571no—
192#441—1571no—
193C4×SD16—1571no—
194C2×Q16—1571no—
195C2×Q16—1571no—
196C2×SD16—1571no—
197C2×SD16—1571no—
198C2×SD16—1571no—
199C2×SD16—1571no—
200C2×SD16—1571no—
201C2×D8—1571no—
202C2×D8—1571no—
203C4⋊Q8—1571no—
204#127—1571no—
205#126—3141no—
206#126—3141no—
207#124—3141no—
208#124—1571no—
209#124—1571no—
210#124—1571no—
211#124—1571no—
212#123—3141no—
213#123—3141no—
214C22⋊Q8—3141no—
215C4×Q8—1571no—
216C4×Q8—1571no—
217C4×Q8—1571no—
218C4×Q8C2×D15711yescharacteristic
219C4×D4—3141no—
220C4×D4—1571no—
221C4×D4—1571no—
222C4×D4—1571no—
223#117—3141no—
224#106—1571no—
225C4⋊C8C2×D15711yescharacteristic
226C4⋊C8C2×D15711yescharacteristic
227C4⋊C8C2×D15711yescharacteristic
228#103—1571no—
229#103—1571no—
230#103—1571no—
231#102—1571no—
232#102—1571no—
233#102—1571no—
234#97C2×D15711yescharacteristic
235#97C2×D15711yescharacteristic
236C4×C8C2×D15711yescharacteristic
237C2×Q8—1571no—
238C2×Q8—1571no—
239C2×Q8—1571no—
240C2×Q8D62811yescharacteristic
241C2×D4—3141no—
242C2×D4—3141no—
243C2×D4—1571no—
244C2×D4—1571no—
245C2×D4—1571no—
246C22×C4—3141no—
247C22×C4—3141no—
248C22×C4—3141no—
249Q16—3142no—
250Q16—6281no—
251SD16—3142no—
252SD16—6281no—
253SD16—6281no—
254SD16—6281no—
255SD16—6281no—
256D8—3142no—
257D8—6281no—
258C2×C8C22×D15711yescharacteristic
259C2×C8C22×D15711yescharacteristic
260C2×C8C22×D15711yescharacteristic
261C2×C8C22×D15711yescharacteristic
262C4⋊C4—3141no—
263C4⋊C4—3141no—
264C4⋊C4—3141no—
265C4⋊C4—3141no—
266C4⋊C4—3141no—
267C4⋊C4—1571no—
268C4⋊C4—1571no—
269C4⋊C4D62811yescharacteristic
270C4⋊C4D62811yescharacteristic
271C4⋊C4D62811yescharacteristic
272C22⋊C4—3141no—
273C22⋊C4—3141no—
274C22⋊C4—3141no—
275C22⋊C4—3141no—
276C22⋊C4—3141no—
277C22⋊C4—3141no—
278C22⋊C4—3141no—
279C22⋊C4—3141no—
280C22⋊C4—3141no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C42—3141no—
284C42—3141no—
285C42—3141no—
286C42C22×D15711yescharacteristic
287C42C22×D15711yescharacteristic
288C42C22×D15711yescharacteristic
289C23—3141no—
290C23—3141no—
291C23—3141no—
292Q8—41no—
293Q8—3141no—
294Q8—3141no—
295Q8—3141no—
296Q8—3141no—
297Q8—3141no—
298Q8—3141no—
299D4—3141no—
300D4—3141no—
301D4—3141no—
302D4—3141no—
303D4—3141no—
304D4—3141no—
305D4—6282no—
306D4—12561no—
307C2×C4—3141no—
308C2×C4—3141no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—6282no—
313C2×C4—12561no—
314C2×C4—12561no—
315C2×C4C2×D62811yescharacteristic
316C2×C4C2×D62811yescharacteristic
317C2×C4C2×D62811yescharacteristic
318C2×C4C2×D62811yescharacteristic
319C2×C4C2×D62811yescharacteristic
320C2×C4C2×D62811yescharacteristic
321C2×C4C23×D15711yescharacteristic
322C8—21no—
323C8—21no—
324C8—21no—
325C8#1457895512yes—
326C22—6281no—
327C22—6281no—
328C22—6281no—
329C22—6281no—
330C22—6281no—
331C22—6281no—
332C22—6281no—
333C22—6281no—
334C22—6281no—
335C22C22×D62811yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—6281no—
343C4—6281no—
344C4—6281no—
345C4—6281no—
346C4—6281no—
347C4C2×#1457895511yescharacteristic
348C4#1909781711yescharacteristic
349C2—12561no—
350C2—12561no—
351C2—12561no—
352C2#2375794511yescharacteristic
353C2#2375807311yescharacteristic
354C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#446×C157C2
socleC22×C157C22×D4
prime radical#23757662C2
hypercenter#446D157
hypocenterC157#3401
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #23758008C2semidirect
2. #23757996C2semidirect
3. #23757667C2semidirect
5. #23757657C2semidirect
6. #23757496C2semidirect
7. #23757496C2semidirect
8. #23757489C2semidirect
9. #23757489C2semidirect
10. #23757479C2semidirect
11. #23757474C2semidirect
12. #23757386C2semidirect
13. #23757381C2semidirect
14. C4×#14578943C2semidirect
15. #446×C157C2semidirect
177. C157#3401semidirect
190. #446D157semidirect

Maximal subgroups

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

subgroupindex
#237580082
#237579962
#237576672
#237576622
#237576572
#237574962
#237574892
#237574792
#237574742
#237573862
#237573812
C4×#145789432
#446×C1572
#3401157