Finite Group Databasebeta

← #34111690

Subgroups — #34111690

Order 20096.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×#19097792C211yesmaximal, characteristic, complemented
3#23757931C211yesmaximal, characteristic, complemented
4#23757926C211yesmaximal, characteristic, complemented
5#23757516C211yesmaximal, characteristic, complemented
6#23757511C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757480C211yesmaximal, characteristic, complemented
9#23757434C211yesmaximal, characteristic, complemented
10#23757432C211yesmaximal, characteristic, complemented
11#23757427C211yesmaximal, characteristic, complemented
12#23757417C211yesmaximal, characteristic, complemented
13#23757375C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#497×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578955C2211yescharacteristic
18C2×#14578953C2211yescharacteristic
19C2×Q8×D157C2211yescharacteristic
20#19097803—24no—
21#19097792—24no—
22C2×#14578944C2211yescharacteristic
23C2×#14578944C2211yescharacteristic
24C2×#14578943C2211yescharacteristic
25C2×#14578941C2211yescharacteristic
26C2×#14578939C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097757—21no—
30#19097756—21no—
31#19097756—21no—
32#19097748—21no—
33#19097747C2211yescharacteristic
34#19097747C2211yescharacteristic
35#19097747C2211yescharacteristic
36#19097746—21no—
37#19097744—21no—
38#19097742C2211yescharacteristic
39C4×Dic314C2211yescharacteristic
40C4×D628C2211yescharacteristic
41C4×D628C2211yescharacteristic
42#19097731—21no—
43C4⋊C4×D157—21no—
44#19097705C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic
46#19097696C2211yescharacteristic
47#19097695C2211yescharacteristic
48#19097695C2211yescharacteristic
49#19097695C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097692C2211yescharacteristic
52#19097683C2211yescharacteristic
53#19097683C2211yescharacteristic
54C2×Q16×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic
56C2×SD16×C157C2211yescharacteristic
57C2×SD16×C157C2211yescharacteristic
58C4⋊Q8×C157C2211yescharacteristic
59#124×C157C2211yescharacteristic
60#97×C157C2211yescharacteristic
61#14578955—22no—
62#14578955—22no—
63#14578955—41no—
64#14578955—41no—
65#14578953—22no—
66#14578953—41no—
67C2×Dic314—21no—
68C2×Dic314C2311yescharacteristic
69Q8×D157—22no—
70Q8×D157—41no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628D411yescharacteristic
74C2×D628C2311yescharacteristic
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C22×Dic157—21no—
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157—21no—
82C2×C4×D157C2311yescharacteristic
83C2×C4×D157C2311yescharacteristic
84#14578944—22no—
85#14578944—22no—
86#14578943D414yescomplemented
87#14578941—22no—
88#14578939—22no—
89#14578937—22no—
90C2×#11929434C2311yescharacteristic
91C2×#11929434C2311yescharacteristic
92C8×D157—22no—
93C4⋊Dic157D411yescharacteristic
94C4⋊Dic157C2311yescharacteristic
95C4⋊Dic157C2311yescharacteristic
96#14578933—21no—
97#14578933—21no—
98#14578933—21no—
99#14578933—21no—
100C22⋊Dic157—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931D411yescharacteristic
106#14578931D411yescharacteristic
107C4×Dic157—21no—
108C2×Q8×C157C2311yescharacteristic
109C2×Q8×C157C2311yescharacteristic
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157C2311yescharacteristic
112Q16×C157—22no—
113Q16×C157—41no—
114SD16×C157—22no—
115SD16×C157—41no—
116C2×C8×C157C2311yescharacteristic
117C2×C8×C157C2311yescharacteristic
118C4⋊C4×C157—21no—
119C4⋊C4×C157—21no—
120C22⋊C4×C157—21no—
121C22⋊C4×C157—21no—
122C42×C157C2311yescharacteristic
123C22×D157—21no—
124C22×D157C2×D411yescharacteristic
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—42no—
128Dic314C2×D412yes—
129D628—21no—
130D628—41no—
131D628—42no—
132D628—42no—
133D628C2×D412yes—
134C22⋊D157—42no—
135C22⋊D157—42no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—21no—
139C2×Dic157—42no—
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C4×D157—21no—
143C4×D157—21no—
144C4×D157—21no—
145C4×D157—21no—
146C4×D157—42no—
147C4×D157—42no—
148C4×D157—42no—
149C4×D157—42no—
150#11929434—21no—
151#11929434—21no—
152C23×C157—21no—
153Q8×C157—21no—
154Q8×C157—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157—21no—
158Q8×C157—21no—
159D4×C157—21no—
160D4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157C2×D411yescharacteristic
165C2×C4×C157C2×D411yescharacteristic
166C2×C4×C157C2411yescharacteristic
167C8×C157—21no—
168C8×C157C2×D412yes—
169C22×C157—41no—
170C22×C157—41no—
171C22×C157—41no—
172C22×C157C22×D411yescharacteristic
173C2×D157—21no—
174C2×D157—21no—
175C2×D157—22no—
176C2×D157—21no—
177C2×D157—21no—
178C2×D157—21no—
179C2×D157—21no—
180C2×D157—41no—
181C2×D157—41no—
182C4×C157—21no—
183C4×C157—21no—
184C4×C157—41no—
185C4×C157—41no—
186C4×C157—41no—
187C4×C157C22×D411yescharacteristic
188C4×C157H(5,𝔽2)11yescharacteristic
189Dic157—21no—
190Dic157—21no—
191Dic157—22no—
192Dic157—41no—
193Dic157—41no—
194C2×C157—81no—
195C2×C157D4211yescharacteristic
196C2×C157#57511yescharacteristic
197C2×C157#57811yescharacteristic
198D157—41no—
199D157—41no—
200D157—42no—
201C157#326911yescharacteristic, complemented, Sylow
202#3269—1571nomaximal, Sylow
203C2×#137—1571no—
204C2×#135—1571no—
205#548—1571no—
206#547—1571no—
207#497D15711yescharacteristic, complemented
208#468—1571no—
209#467—1571no—
210#463—1571no—
211#462—1571no—
212#452—1571no—
213#452—1571no—
214#451—1571no—
215#449—1571no—
216#440—1571no—
217#435—1571no—
218C2×#41—1571no—
219C2×#41—1571no—
220C2×#41—1571no—
221C22×Q8—1571no—
222#137—3144no—
223#135—3144no—
224C2×Q16—1571no—
225C2×Q16—1571no—
226C2×Q16C2×D15711yescharacteristic
227C2×Q16C2×D15711yescharacteristic
228C2×SD16—1571no—
229C2×SD16—1571no—
230C2×SD16C2×D15711yescharacteristic
231C2×SD16C2×D15711yescharacteristic
232C2×D8—1571no—
233C2×M16—1571no—
234C22×C8—1571no—
235C4⋊Q8C2×D15711yescharacteristic
236#124—3141no—
237#124C2×D15711yescharacteristic
238#123—3141no—
239#123—3141no—
240C22⋊Q8—3141no—
241C22⋊Q8—1571no—
242C22⋊Q8—1571no—
243C22⋊Q8—1571no—
244#121—3141no—
245#121—3141no—
246#121—1571no—
247C4×Q8—1571no—
248C4×D4—3141no—
249C4×D4—1571no—
250C4×D4—1571no—
251C2×C4⋊C4—3141no—
252#107—1571no—
253C4⋊C8—1571no—
254#103—1571no—
255#103—1571no—
256#103—1571no—
257#103—1571no—
258#102—1571no—
259#102—1571no—
260C22⋊C8—1571no—
261C22⋊C8—1571no—
262#97C2×D15711yescharacteristic
263#41—3142no—
264#41—3142no—
265#41—3142no—
266#41—6281no—
267#41—6281no—
268#41—6281no—
269C2×Q8—3141no—
270C2×Q8—3142no—
271C2×Q8—6281no—
272C2×Q8—1571no—
273C2×Q8C22×D15711yescharacteristic
274C2×Q8C22×D15711yescharacteristic
275C2×Q8C22×D15711yescharacteristic
276C2×D4—3141no—
277C2×D4—3141no—
278C2×D4—3141no—
279C2×D4—3141no—
280C2×D4—1571no—
281C2×D4—1571no—
282C2×D4C22×D15711yescharacteristic
283C22×C4—3141no—
284C22×C4—3141no—
285C22×C4—3141no—
286C22×C4—3141no—
287C22×C4—3141no—
288C22×C4—1571no—
289C22×C4—1571no—
290Q16—22no—
291Q16—41no—
292Q16—3142no—
293Q16—3142no—
294SD16—22no—
295SD16—41no—
296SD16—3142no—
297SD16—1574no—
298D8—3142no—
299M16—3142no—
300C2×C8—3142no—
301C2×C8—1571no—
302C2×C8—1571no—
303C2×C8C22×D15711yescharacteristic
304C2×C8C22×D15711yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—3141no—
308C4⋊C4—3141no—
309C4⋊C4—3141no—
310C4⋊C4—3141no—
311C4⋊C4—1571no—
312C4⋊C4—1571no—
313C4⋊C4—1571no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C22⋊C4—3141no—
320C22⋊C4—3141no—
321C22⋊C4—1571no—
322C22⋊C4—1571no—
323C42—3141no—
324C42C22×D15711yescharacteristic
325C23—21no—
326C23—3141no—
327C23—1571no—
328C23—1571no—
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—3141no—
336Q8—6282no—
337Q8—1572no—
338D4—21no—
339D4—21no—
340D4—3141no—
341D4—6281no—
342D4—6282no—
343D4—6282no—
344D4—6282no—
345D4—6282no—
346D4—1572no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—3141no—
354C2×C4—3141no—
355C2×C4—3141no—
356C2×C4—3141no—
357C2×C4—6282no—
358C2×C4—6282no—
359C2×C4—6282no—
360C2×C4—6282no—
361C2×C4—6282no—
362C2×C4—1571no—
363C2×C4—1571no—
364C2×C4D4×D15711yescharacteristic
365C2×C4D4×D15711yescharacteristic
366C2×C4C23×D15711yescharacteristic
367C8—21no—
368C8—3141no—
369C8—3141no—
370C8D4×D15712yes—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—3141no—
375C22—3141no—
376C22—3141no—
377C22—3141no—
378C22—3142no—
379C22—3141no—
380C22—3141no—
381C22—6281no—
382C22—6281no—
383C22C2×D4×D15711yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—3141no—
390C4—3141no—
391C4—3142no—
392C4—6281no—
393C4—6281no—
394C4C2×D4×D15711yescharacteristic
395C4#1909781611yescharacteristic
396C2—81no—
397C2—6281no—
398C2—6281no—
399C2—6282no—
400C2#2375790811yescharacteristic
401C2#2375807211yescharacteristic
402C2#2375808911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#497×C157C2
socleC22×C157C22×D4
hypercenter#497D157
hypocenterC157#3269
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×#19097792C2semidirect
3. #23757931C2semidirect
4. #23757926C2semidirect
5. #23757516C2semidirect
6. #23757511C2semidirect
7. #23757485C2semidirect
8. #23757480C2semidirect
9. #23757434C2semidirect
10. #23757432C2semidirect
11. #23757427C2semidirect
12. #23757417C2semidirect
13. #23757375C2semidirect
14. #23757346C2semidirect
15. #497×C157C2semidirect
86. #14578943D4semidirect
201. C157#3269semidirect
207. #497D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×#190977922
#237579312
#237579262
#237575162
#237575112
#237574852
#237574802
#237574342
#237574322
#237574272
#237574172
#237573752
#237573462
#497×C1572
#3269157