Finite Group Databasebeta

← #34080939

Subgroups — #34080939

Order 19328.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752858C211yesmaximal, characteristic
2#23752569C211yesmaximal, characteristic, complemented
3#23752542C211yesmaximal, characteristic, complemented
4#23752536C211yesmaximal, characteristic, complemented
5#23752509C211yesmaximal, characteristic, complemented
6#23752451C211yesmaximal, characteristic, complemented
7#23752443C211yesmaximal, characteristic, complemented
8#23752434C211yesmaximal, characteristic, complemented
9#23752372C211yesmaximal, characteristic, complemented
10#23752362C211yesmaximal, characteristic, complemented
11C4×#14558991C211yesmaximal, characteristic, complemented
12D4×C151⋊C8C211yesmaximal, characteristic, complemented
13#23752206C211yesmaximal, characteristic, complemented
14C2×#18012659C211yesmaximal, characteristic, complemented
15#552×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic
17C22×D604C2211yescharacteristic
18C2×#14558994C2211yescharacteristic, complemented
19C2×#14558991C2211yescharacteristic
20C2×#14558991C2211yescharacteristic
21C22×C151⋊C8C2211yescharacteristic
22C22×C151⋊C8C2211yescharacteristic, complemented
23#18012727C2211yescharacteristic
24#18012724—21no—
25#18012718C2211yescharacteristic
26#18012713C2211yescharacteristic
27#18012712—21no—
28#18012707—21no—
29C4×D604C2211yescharacteristic
30C22⋊C4×D151—21no—
31#18012672C2211yescharacteristic, complemented
32#18012670C2211yescharacteristic, complemented
33#18012670C2211yescharacteristic, complemented
34C4⋊(C151⋊C8)C2211yescharacteristic, complemented
35#18012663C2211yescharacteristic, complemented
36#18012662C2211yescharacteristic, complemented
37#18012662C2211yescharacteristic, complemented
38#18012662C2211yescharacteristic, complemented
39#18012659C2211yescharacteristic
40#18012659C2211yescharacteristic
41#18012659C2211yescharacteristic
42#18012659C2211yescharacteristic
43#18012659C2211yescharacteristic
44#18012659C2211yescharacteristic
45C22⋊(C151⋊C8)C2211yescharacteristic
46C22⋊(C151⋊C8)C2211yescharacteristic, complemented
47C4×C151⋊C8C2211yescharacteristic, complemented
48#126×C151—21no—
49#125×C151—21no—
50#125×C151C2211yescharacteristic, complemented
51#123×C151—21no—
52C22⋊Q8×C151C2211yescharacteristic
53C22⋊Q8×C151C2211yescharacteristic, complemented
54C4×Q8×C151C2211yescharacteristic, complemented
55C4×D4×C151C2211yescharacteristic
56#117×C151—21no—
57#117×C151C2211yescharacteristic, complemented
58C2×C4⋊C4×C151C2211yescharacteristic
59C23×D151—21no—
60#14559004—22no—
61#14559004—41no—
62C2×Dic302C2311yescharacteristic
63C2×D604—22no—
64C2×D604—21no—
65C2×D604—41no—
66C2×D604C2311yescharacteristic
67C2×D604C2311yescharacteristic
68C2×C22⋊D151—21no—
69C2×C22⋊D151—21no—
70C2×C4×D151—21no—
71C2×C4×D151—21no—
72#14558994—41no—
73#14558991—22no—
74#14558991—41no—
75C2×C151⋊C8—41no—
76C2×C151⋊C8—41no—
77C2×C151⋊C8C2311yescharacteristic
78C2×C151⋊C8C2311yescharacteristic
79C2×C151⋊C8C2311yescharacteristic
80C2×C151⋊C8C2311yescharacteristic
81C4⋊Dic151C2311yescharacteristic
82Dic151⋊C4—21no—
83C22⋊Dic151—21no—
84(C2×D151)⋊C4—21no—
85(C2×D151)⋊C4—21no—
86(C2×D151)⋊C4—21no—
87(C2×D151)⋊C4—21no—
88C2×Q8×C151C2311yescharacteristic
89C2×D4×C151D411yescharacteristic
90C22×C4×C151—21no—
91C22×C4×C151C2311yescharacteristic
92C22×C4×C151C2311yescharacteristic
93C4⋊C4×C151—21no—
94C4⋊C4×C151—21no—
95C4⋊C4×C151—21no—
96C4⋊C4×C151—21no—
97C4⋊C4×C151—21no—
98C4⋊C4×C151—21no—
99C4⋊C4×C151D411yescharacteristic
100C4⋊C4×C151C2311yescharacteristic
101C4⋊C4×C151C2311yescharacteristic
102C4⋊C4×C151C2311yescharacteristic
103C22⋊C4×C151—21no—
104C22⋊C4×C151—21no—
105C22⋊C4×C151—21no—
106C22⋊C4×C151D411yescharacteristic
107C22⋊C4×C151D411yescharacteristic
108C42×C151—21no—
109C42×C151—21no—
110C42×C151C2311yescharacteristic
111C22×D151—21no—
112C22×D151—21no—
113C22×D151—42no—
114C22×D151—42no—
115C22×D151—81no—
116Dic302—21no—
117Dic302—21no—
118D604—21no—
119D604—21no—
120D604—21no—
121D604—42no—
122D604#4112yes—
123C22⋊D151—42no—
124C22⋊D151—81no—
125C2×Dic151—21no—
126C2×Dic151—21no—
127C4×D151—42no—
128C4×D151—81no—
129C151⋊C8—21no—
130C151⋊C8—21no—
131C151⋊C8—21no—
132C151⋊C8—21no—
133C23×C151C2×D411yescharacteristic
134C23×C151C2×D411yescharacteristic
135Q8×C151—21no—
136Q8×C151—21no—
137D4×C151—22no—
138C2×C4×C151—21no—
139C2×C4×C151—21no—
140C2×C4×C151—21no—
141C2×C4×C151—21no—
142C2×C4×C151—21no—
143C2×C4×C151—21no—
144C2×C4×C151—21no—
145C2×C4×C151—21no—
146C2×C4×C151—42no—
147C2×C4×C151C2×D411yescharacteristic
148C2×C4×C151C2×D411yescharacteristic
149C2×C4×C151C2×D411yescharacteristic
150C2×C4×C151C2×D411yescharacteristic
151C2×C4×C151C2411yescharacteristic
152C22×C151—21no—
153C22×C151—21no—
154C22×C151—21no—
155C22×C151—21no—
156C22×C151—21no—
157C22×C151#13512yes—
158C22×C151C22×D411yescharacteristic
159C2×D151—22no—
160C2×D151—41no—
161C2×D151—41no—
162C2×D151—41no—
163C2×D151—41no—
164C2×D151—41no—
165C2×D151—84no—
166C4×C151—21no—
167C4×C151—21no—
168C4×C151—21no—
169C4×C151—21no—
170C4×C151—41no—
171C4×C151—41no—
172C4×C151—41no—
173C4×C151—41no—
174C4×C151C2×#4111yescharacteristic
175C4×C151H(5,𝔽2)11yescharacteristic
176Dic151—41no—
177Dic151—41no—
178C2×C151—22no—
179C2×C151—41no—
180C2×C151#54611yescharacteristic
181C2×C151C2×#13511yescharacteristic
182C2×C151#57711yescharacteristic
183D151—42no—
184D151—81no—
185C151#329611yescharacteristic, complemented, Sylow
186#3296—1511nomaximal, Sylow
187#552D15111yescharacteristic, complemented
188#546—1511no—
189#488—1511no—
190#483—1511no—
191#482—1511no—
192#478—1511no—
193#467—1511no—
194#465—1511no—
195#464—1511no—
196#452—1511no—
197#450—1511no—
198C4×SD16—1511no—
199C8×D4—1511no—
200#416—1511no—
201C2×#102—1511no—
202C2×#41—1511no—
203C22×D4—1511no—
204C2×Q16—1511no—
205C2×SD16—1511no—
206C2×SD16—1511no—
207C22×C8—1511no—
208C22×C8—1511no—
209#126—21no—
210#125—21no—
211#125C2×D15111yescharacteristic, complemented
212#124—1511no—
213#123—21no—
214#123—3021no—
215C22⋊Q8—1511no—
216C22⋊Q8C2×D15111yescharacteristic
217C22⋊Q8C2×D15111yescharacteristic, complemented
218#121—3021no—
219#121—1511no—
220#120—3021no—
221C4×Q8C2×D15111yescharacteristic, complemented
222C4×D4—1511no—
223C4×D4C2×D15111yescharacteristic
224#117—21no—
225#117C2×D15111yescharacteristic, complemented
226C2×C4⋊C4C2×D15111yescharacteristic
227C2×C22⋊C4—3021no—
228#107—1511no—
229#106—1511no—
230#106—1511no—
231C4⋊C8—1511no—
232#103—1511no—
233#103—1511no—
234#103—1511no—
235#103—1511no—
236#102—1511no—
237#102—1511no—
238#102—1511no—
239#102—1511no—
240#102—1511no—
241#102—1511no—
242C22⋊C8—1511no—
243C22⋊C8—1511no—
244C4×C8—1511no—
245C24—3021no—
246#41—3022no—
247#41—6041no—
248C2×Q8—1511no—
249C2×Q8C22×D15111yescharacteristic
250C2×D4—3022no—
251C2×D4—3021no—
252C2×D4—3021no—
253C2×D4—3021no—
254C2×D4—6041no—
255C2×D4—1511no—
256C2×D4—1511no—
257C2×D4C22⋊D15111yescharacteristic
258C22×C4—21no—
259C22×C4—3021no—
260C22×C4—3021no—
261C22×C4C22×D15111yescharacteristic
262C22×C4C22×D15111yescharacteristic
263Q16—6041no—
264SD16—3022no—
265SD16—6041no—
266C2×C8—6041no—
267C2×C8—6041no—
268C2×C8—1511no—
269C2×C8—1511no—
270C2×C8—1511no—
271C2×C8—1511no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3021no—
279C4⋊C4—1511no—
280C4⋊C4C22⋊D15111yescharacteristic
281C4⋊C4C22×D15111yescharacteristic
282C4⋊C4C22×D15111yescharacteristic
283C4⋊C4C22×D15111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3021no—
288C22⋊C4—3021no—
289C22⋊C4—3021no—
290C22⋊C4—3021no—
291C22⋊C4—3021no—
292C22⋊C4C22⋊D15111yescharacteristic
293C22⋊C4C22⋊D15111yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D15111yescharacteristic
297C23—3021no—
298C23—3021no—
299C23—6042no—
300C23—6042no—
301C23—12081no—
302C23C2×C22⋊D15111yescharacteristic
303C23C2×C22⋊D15111yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3021no—
307Q8—3021no—
308D4—22no—
309D4—3021no—
310D4—3021no—
311D4—3021no—
312D4—6042no—
313D4—6042no—
314D4—1512no—
315D4—12081no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—3021no—
326C2×C4—3021no—
327C2×C4—6042no—
328C2×C4—12081no—
329C2×C4C2×C22⋊D15111yescharacteristic
330C2×C4C2×C22⋊D15111yescharacteristic
331C2×C4C2×C22⋊D15111yescharacteristic
332C2×C4C2×C22⋊D15111yescharacteristic
333C2×C4C23×D15111yescharacteristic
334C8—3021no—
335C8—3021no—
336C8—3021no—
337C8—3021no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—3022no—
344C22—6041no—
345C22—6041no—
346C22—6041no—
347C22—6041no—
348C22—6041no—
349C22—12084no—
350C22#1801275912yes—
351C22C22×C22⋊D15111yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6041no—
361C4—6041no—
362C4#41×D15111yescharacteristic
363C4#1801278511yescharacteristic
364C2—22no—
365C2—41no—
366C2—6042no—
367C2—12081no—
368C2#2375285011yescharacteristic
369C2C2×#1801275911yescharacteristic
370C2#2375302011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#552×C151C2
socleC22×C151C22×D4
prime radical#23752858C2
hypercenter#552D151
hypocenterC151#3296
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23752569C2semidirect
3. #23752542C2semidirect
4. #23752536C2semidirect
5. #23752509C2semidirect
6. #23752451C2semidirect
7. #23752443C2semidirect
8. #23752434C2semidirect
9. #23752372C2semidirect
10. #23752362C2semidirect
11. C4×#14558991C2semidirect
12. D4×C151⋊C8C2semidirect
13. #23752206C2semidirect
14. C2×#18012659C2semidirect
15. #552×C151C2semidirect
18. C2×#14558994C22semidirect
22. C22×C151⋊C8C22semidirect
31. #18012672C22semidirect
32. #18012670C22semidirect
33. #18012670C22semidirect
34. C4⋊(C151⋊C8)C22semidirect
35. #18012663C22semidirect
36. #18012662C22semidirect
37. #18012662C22semidirect
38. #18012662C22semidirect
46. C22⋊(C151⋊C8)C22semidirect
47. C4×C151⋊C8C22semidirect
50. #125×C151C22semidirect
53. C22⋊Q8×C151C22semidirect
54. C4×Q8×C151C22semidirect
57. #117×C151C22semidirect
185. C151#3296semidirect
187. #552D151semidirect
211. #125C2×D151semidirect
217. C22⋊Q8C2×D151semidirect
221. C4×Q8C2×D151semidirect
225. #117C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237528582
#237525692
#237525422
#237525362
#237525092
#237524512
#237524432
#237524342
#237523722
#237523622
C4×#145589912
D4×C151⋊C82
#237522062
C2×#180126592
#552×C1512
#3296151