Finite Group Databasebeta

← #34079021

Subgroups — #34079021

Order 19328.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23752719C211yesmaximal, characteristic
2#117×D151C211yesmaximal, characteristic, complemented
3#23752552C211yesmaximal, characteristic, complemented
4#23752551C211yesmaximal, characteristic, complemented
5#23752546C211yesmaximal, characteristic, complemented
6#23752541C211yesmaximal, characteristic, complemented
7#23752535C211yesmaximal, characteristic, complemented
8#23752534C211yesmaximal, characteristic, complemented
9#23752529C211yesmaximal, characteristic, complemented
10#23752524C211yesmaximal, characteristic, complemented
11#23752250C211yesmaximal, characteristic, complemented
12#23752248C211yesmaximal, characteristic, complemented
13#23752208C211yesmaximal, characteristic, complemented
14C2×(C2×D151)⋊C8C211yesmaximal, characteristic, complemented
15#416×C151C211yesmaximal, characteristic, complemented
16C22×C4×D151C2211yescharacteristic
17C22×C151⋊C8C2211yescharacteristic
18#18012720—21no—
19#18012715C2211yescharacteristic
20#18012715C2211yescharacteristic
21#18012710C2211yescharacteristic
22#18012710C2211yescharacteristic
23#18012707—21no—
24C4×C22⋊D151—21no—
25C4×C22⋊D151—21no—
26#18012698C2211yescharacteristic
27#18012698C2211yescharacteristic
28#18012697C2211yescharacteristic
29#18012697C2211yescharacteristic
30#18012696—21no—
31#18012695—21no—
32C4⋊C4×D151C2211yescharacteristic
33C4⋊C4×D151C2211yescharacteristic
34C22⋊C4×D151—21no—
35C42×D151—21no—
36#18012673C2211yescharacteristic, complemented
37#18012673C2211yescharacteristic, complemented
38#18012672C2211yescharacteristic, complemented
39#18012672C2211yescharacteristic, complemented
40#18012671C2211yescharacteristic, complemented
41#18012671C2211yescharacteristic, complemented
42#18012670C2211yescharacteristic, complemented
43#18012670C2211yescharacteristic, complemented
44#18012663C2211yescharacteristic, complemented
45#18012663C2211yescharacteristic, complemented
46#18012660C2211yescharacteristic, complemented
47#18012660C2211yescharacteristic, complemented
48(C2×D151)⋊C8C2211yescharacteristic
49(C2×D151)⋊C8C2211yescharacteristic
50(C2×D151)⋊C8C2211yescharacteristic
51(C2×D151)⋊C8C2211yescharacteristic
52C2×#41×C151C2211yescharacteristic
53C22×C8×C151C2211yescharacteristic
54#117×C151C2211yescharacteristic
55#103×C151C2211yescharacteristic, complemented
56#103×C151C2211yescharacteristic, complemented
57#102×C151C2211yescharacteristic, complemented
58#102×C151C2211yescharacteristic, complemented
59C23×D151D411yescharacteristic
60C2×C22⋊D151—21no—
61C2×C22⋊D151—21no—
62C22×Dic151D411yescharacteristic
63C2×C4×D151—21no—
64C2×C4×D151—21no—
65C2×C4×D151—22no—
66C2×C4×D151—22no—
67C2×C4×D151D411yescharacteristic
68C2×C4×D151D411yescharacteristic
69C2×C4×D151C2311yescharacteristic
70C2×C4×D151C2311yescharacteristic
71C2×C151⋊C8—22no—
72C2×C151⋊C8C2311yescharacteristic
73C2×C151⋊C8C2311yescharacteristic
74C4⋊Dic151C2311yescharacteristic
75C4⋊Dic151C2311yescharacteristic
76C4⋊Dic151C2311yescharacteristic
77C4⋊Dic151C2311yescharacteristic
78Dic151⋊C4—21no—
79Dic151⋊C4—21no—
80Dic151⋊C4—21no—
81Dic151⋊C4—21no—
82C22⋊Dic151—21no—
83C22⋊Dic151—21no—
84C22⋊Dic151—21no—
85C22⋊Dic151—21no—
86(C2×D151)⋊C4—21no—
87(C2×D151)⋊C4—21no—
88(C2×D151)⋊C4—21no—
89(C2×D151)⋊C4—21no—
90C4×Dic151—21no—
91C4×Dic151—21no—
92C4×Dic151—21no—
93C4×Dic151—21no—
94#41×C151—22no—
95#41×C151—22no—
96C2×Q8×C151C2311yescharacteristic
97C2×D4×C151—21no—
98C2×D4×C151C2311yescharacteristic
99C22×C4×C151—21no—
100C22×C4×C151C2311yescharacteristic
101C2×C8×C151—22no—
102C2×C8×C151C2311yescharacteristic
103C2×C8×C151C2311yescharacteristic
104C4⋊C4×C151C2311yescharacteristic
105C4⋊C4×C151C2311yescharacteristic
106C22⋊C4×C151—21no—
107C42×C151—21no—
108C22×D151—22no—
109C22×D151—22no—
110C22×D151—41no—
111C22×D151C2×D411yescharacteristic
112C22×D151C2×D411yescharacteristic
113C22⋊D151—42no—
114C22⋊D151—42no—
115C2×Dic151—22no—
116C2×Dic151—21no—
117C2×Dic151—21no—
118C2×Dic151—21no—
119C2×Dic151—21no—
120C2×Dic151C2×D411yescharacteristic
121C2×Dic151C2×D411yescharacteristic
122C4×D151—22no—
123C4×D151—22no—
124C4×D151—21no—
125C4×D151—21no—
126C4×D151—21no—
127C4×D151—21no—
128C4×D151—42no—
129C4×D151—42no—
130C151⋊C8—21no—
131C151⋊C8—21no—
132C23×C151—21no—
133C23×C151C2×D411yescharacteristic
134Q8×C151—21no—
135Q8×C151—21no—
136D4×C151—21no—
137D4×C151—21no—
138D4×C151—42no—
139C2×C4×C151—21no—
140C2×C4×C151—21no—
141C2×C4×C151—21no—
142C2×C4×C151—42no—
143C2×C4×C151C2×D411yescharacteristic
144C2×C4×C151#4112yes—
145C2×C4×C151#4112yes—
146C2×C4×C151C2411yescharacteristic
147C8×C151—21no—
148C8×C151—21no—
149C22×C151—21no—
150C22×C151—21no—
151C22×C151—41no—
152C22×C151—41no—
153C22×C151—41no—
154C22×C151#12312yes—
155C22×C151C22×D411yescharacteristic
156C2×D151—21no—
157C2×D151—21no—
158C2×D151—21no—
159C2×D151—21no—
160C2×D151—21no—
161C2×D151—44no—
162C2×D151#13512yes—
163C4×C151—41no—
164C4×C151—41no—
165C4×C151—41no—
166C4×C151#12312yes—
167C4×C151C2×#4111yescharacteristic
168C4×C151C2×#4111yescharacteristic
169Dic151—21no—
170Dic151—21no—
171Dic151—41no—
172Dic151—41no—
173Dic151—41no—
174Dic151—41no—
175C2×C151—22no—
176C2×C151—81no—
177C2×C151C2×#12311yescharacteristic
178C2×C151C2×#13511yescharacteristic
179C2×C151#57511yescharacteristic
180D151—22no—
181D151—41no—
182C151#309111yescharacteristic, complemented, Sylow
183#3091—1511nomaximal, Sylow
184#525—1511no—
185C2×#117—1511no—
186#485—1511no—
187#485—1511no—
188#484—1511no—
189#483—1511no—
190#482—1511no—
191#482—1511no—
192#481—1511no—
193#480—1511no—
194#427—1511no—
195#427—1511no—
196#416—1511no—
197#416D15111yescharacteristic, complemented
198C2×C22⋊C8—1511no—
199C2×#41C2×D15111yescharacteristic
200C23×C4—1511no—
201C22×C8—1511no—
202C22×C8C2×D15111yescharacteristic
203#123—3021no—
204C22⋊Q8—1511no—
205C22⋊Q8—1511no—
206#121—1511no—
207#121—1511no—
208#120—3021no—
209C4×D4—3021no—
210C4×D4—3021no—
211#117—3021no—
212#117—3021no—
213#117—1511no—
214#117—1511no—
215#117—1511no—
216#117—1511no—
217#117C2×D15111yescharacteristic
218C2×C4⋊C4—1511no—
219C2×C4⋊C4—1511no—
220C2×C22⋊C4—3021no—
221C2×C42—3021no—
222#107—1511no—
223#107—1511no—
224#107—1511no—
225#107—1511no—
226#106—1511no—
227#106—1511no—
228#106—1511no—
229#106—1511no—
230#103—1511no—
231#103—1511no—
232#103C2×D15111yescharacteristic, complemented
233#103C2×D15111yescharacteristic, complemented
234#102—1511no—
235#102—1511no—
236#102C2×D15111yescharacteristic, complemented
237#102C2×D15111yescharacteristic, complemented
238C22⋊C8—1511no—
239C22⋊C8—1511no—
240C22⋊C8—1511no—
241C22⋊C8—1511no—
242C24—1511no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D15111yescharacteristic
246C2×D4—21no—
247C2×D4—3021no—
248C2×D4—3021no—
249C2×D4C22×D15111yescharacteristic
250C22×C4—21no—
251C22×C4—3021no—
252C22×C4—3021no—
253C22×C4—3022no—
254C22×C4—3022no—
255C22×C4—1511no—
256C22×C4—1511no—
257C22×C4—1511no—
258C22×C4—1511no—
259C22×C4—1511no—
260C22×C4C22×D15111yescharacteristic
261C2×C8—22no—
262C2×C8—3022no—
263C2×C8—1511no—
264C2×C8—1511no—
265C2×C8C22×D15111yescharacteristic
266C2×C8C22×D15111yescharacteristic
267C4⋊C4—3021no—
268C4⋊C4—3021no—
269C4⋊C4—3021no—
270C4⋊C4—3021no—
271C4⋊C4—1511no—
272C4⋊C4—1511no—
273C4⋊C4—1511no—
274C4⋊C4—1511no—
275C4⋊C4C22×D15111yescharacteristic
276C4⋊C4C22×D15111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3021no—
279C22⋊C4—3021no—
280C22⋊C4—3021no—
281C22⋊C4—3021no—
282C22⋊C4—3021no—
283C22⋊C4—3021no—
284C22⋊C4—3021no—
285C22⋊C4—3021no—
286C42—21no—
287C42—3021no—
288C42—3021no—
289C42—3021no—
290C42—3021no—
291C23—21no—
292C23—3022no—
293C23—3022no—
294C23—6041no—
295C23—1511no—
296C23—1511no—
297C23D4×D15111yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—6042no—
304D4—6042no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3022no—
310C2×C4—3022no—
311C2×C4—3022no—
312C2×C4—3021no—
313C2×C4—3021no—
314C2×C4—3021no—
315C2×C4—3021no—
316C2×C4—3021no—
317C2×C4—3021no—
318C2×C4—3021no—
319C2×C4—3021no—
320C2×C4—6042no—
321C2×C4—6042no—
322C2×C4—1511no—
323C2×C4—1511no—
324C2×C4D4×D15111yescharacteristic
325C2×C4#1455900312yes—
326C2×C4#1455900412yes—
327C2×C4C23×D15111yescharacteristic
328C8—21no—
329C8—21no—
330C8—3021no—
331C8—3021no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—3021no—
338C22—3021no—
339C22—3021no—
340C22—3021no—
341C22—3021no—
342C22—6044no—
343C22—1512no—
344C22#1801272112yes—
345C22C2×D4×D15111yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3021no—
350C4—3021no—
351C4—6041no—
352C4—6041no—
353C4—6041no—
354C4—6041no—
355C4#1801272112yes—
356C4C2×#1455900311yescharacteristic
357C4C2×#1455900411yescharacteristic
358C2—22no—
359C2—81no—
360C2—3022no—
361C2—6041no—
362C2C2×#1801272111yescharacteristic
363C2#135×D15111yescharacteristic
364C2#2375300911yescharacteristic
subgroupG/H
centerC2×C4D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#416×C151C2
socleC22×C151C22×D4
prime radical#23752719C2
hypercenter#416D151
hypocenterC151#3091
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D151C2semidirect
3. #23752552C2semidirect
4. #23752551C2semidirect
5. #23752546C2semidirect
6. #23752541C2semidirect
7. #23752535C2semidirect
8. #23752534C2semidirect
9. #23752529C2semidirect
10. #23752524C2semidirect
11. #23752250C2semidirect
12. #23752248C2semidirect
13. #23752208C2semidirect
14. C2×(C2×D151)⋊C8C2semidirect
15. #416×C151C2semidirect
36. #18012673C22semidirect
37. #18012673C22semidirect
38. #18012672C22semidirect
39. #18012672C22semidirect
40. #18012671C22semidirect
41. #18012671C22semidirect
42. #18012670C22semidirect
43. #18012670C22semidirect
44. #18012663C22semidirect
45. #18012663C22semidirect
46. #18012660C22semidirect
47. #18012660C22semidirect
55. #103×C151C22semidirect
56. #103×C151C22semidirect
57. #102×C151C22semidirect
58. #102×C151C22semidirect
182. C151#3091semidirect
197. #416D151semidirect
232. #103C2×D151semidirect
233. #103C2×D151semidirect
236. #102C2×D151semidirect
237. #102C2×D151semidirect

Maximal subgroups

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

subgroupindex
#237527192
#117×D1512
#237525522
#237525512
#237525462
#237525412
#237525352
#237525342
#237525292
#237525242
#237522502
#237522482
#237522082
C2×(C2×D151)⋊C82
#416×C1512
#3091151