Finite Group Databasebeta

← #34056521

Subgroups — #34056521

Order 19072.

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#23750309C211yesmaximal, characteristic, complemented
2#23750272C211yesmaximal, characteristic
3#23749978C211yesmaximal, characteristic, complemented
4#23749957C211yesmaximal, characteristic, complemented
5#23749946C211yesmaximal, characteristic, complemented
6#23749924C211yesmaximal, characteristic, complemented
7#23749865C211yesmaximal, characteristic, complemented
8#23749845C211yesmaximal, characteristic, complemented
9#23749785C211yesmaximal, characteristic, complemented
10#23749774C211yesmaximal, characteristic, complemented
11#23749710C211yesmaximal, characteristic, complemented
12C8×C22⋊D149C211yesmaximal, characteristic, complemented
13#23749622C211yesmaximal, characteristic, complemented
14#102×D149C211yesmaximal, characteristic, complemented
15#465×C149C211yesmaximal, characteristic, complemented
16C2×#14558736C2211yescharacteristic
17C2×D4×D149C2211yescharacteristic
18C2×#14558726C2211yescharacteristic, complemented
19C2×#14558723C2211yescharacteristic
20C2×C8×D149C2211yescharacteristic
21#18011904—21no—
22#18011902C2211yescharacteristic, complemented
23#18011901—21no—
24#18011898C2211yescharacteristic
25#18011893—21no—
26#18011893—21no—
27#18011891C2211yescharacteristic
28#18011887C2211yescharacteristic
29#18011882—21no—
30#18011879—21no—
31Q8×Dic149C2211yescharacteristic, complemented
32#18011873C2211yescharacteristic
33C4×C22⋊D149C2211yescharacteristic
34#18011870C2211yescharacteristic, complemented
35#18011869—21no—
36C4⋊C4×D149C2211yescharacteristic
37C22⋊C4×D149—21no—
38#18011844C2211yescharacteristic, complemented
39#18011842C2211yescharacteristic, complemented
40#18011840C2211yescharacteristic, complemented
41#18011836C2211yescharacteristic, complemented
42#18011836C2211yescharacteristic, complemented
43#18011834C2211yescharacteristic, complemented
44#18011833C2211yescharacteristic
45#18011833C2211yescharacteristic
46#18011832C2211yescharacteristic
47#18011832C2211yescharacteristic
48#18011831C2211yescharacteristic
49C22⋊#11924746C2211yescharacteristic, complemented
50#18011823C2211yescharacteristic
51C8×Dic149C2211yescharacteristic, complemented
52C2×SD16×C149C2211yescharacteristic
53C22×C8×C149C2211yescharacteristic, complemented
54C22⋊Q8×C149C2211yescharacteristic, complemented
55#121×C149C2211yescharacteristic
56#106×C149C2211yescharacteristic, complemented
57#103×C149C2211yescharacteristic, complemented
58#102×C149C2211yescharacteristic
59C23×D149—21no—
60#14558736—22no—
61#14558736—41no—
62C2×Dic298C2311yescharacteristic
63C2×D596C2311yescharacteristic
64C2×C22⋊D149—21no—
65C2×C22⋊D149—21no—
66C2×C22⋊D149D411yescharacteristic
67D4×D149—22no—
68D4×D149—41no—
69C2×C4×D149—21no—
70C2×C4×D149—21no—
71C2×C4×D149—21no—
72C2×C4×D149C2311yescharacteristic
73#14558726—41no—
74#14558723—22no—
75C2×#11924746C2311yescharacteristic
76C2×#11924746C2311yescharacteristic
77C8×D149—41no—
78C4⋊Dic149—21no—
79C4⋊Dic149—21no—
80C4⋊Dic149—21no—
81C4⋊Dic149C2311yescharacteristic
82C4⋊Dic149C2311yescharacteristic
83#14558715—21no—
84#14558715—21no—
85#14558715—21no—
86#14558715D411yescharacteristic
87C22⋊Dic149—21no—
88C22⋊Dic149—21no—
89C22⋊Dic149—21no—
90C22⋊Dic149D411yescharacteristic
91#14558713—21no—
92#14558713—21no—
93#14558713—21no—
94#14558713D411yescharacteristic
95C4×Dic149—21no—
96C4×Dic149—21no—
97C4×Dic149C2311yescharacteristic
98C2×Q8×C149C2311yescharacteristic
99C2×D4×C149—21no—
100C2×D4×C149C2311yescharacteristic
101C22×C4×C149C2311yescharacteristic
102SD16×C149—41no—
103C2×C8×C149—41no—
104C2×C8×C149C2311yescharacteristic
105C2×C8×C149C2311yescharacteristic
106C4⋊C4×C149—21no—
107C4⋊C4×C149C2311yescharacteristic
108C4⋊C4×C149C2311yescharacteristic
109C22⋊C4×C149—21no—
110C22⋊C4×C149—21no—
111C22×D149—21no—
112C22×D149—42no—
113C22×D149—42no—
114C22×D149—81no—
115C22×D149C2×D411yescharacteristic
116Dic298—21no—
117Dic298—21no—
118D596—21no—
119D596#4112yes—
120C22⋊D149—22no—
121C22⋊D149—42no—
122C22⋊D149—42no—
123C2×Dic149—21no—
124C2×Dic149—21no—
125C2×Dic149—21no—
126C2×Dic149C2×D411yescharacteristic
127C2×Dic149C2×D411yescharacteristic
128C2×Dic149C2×D411yescharacteristic
129C4×D149—21no—
130C4×D149—21no—
131C4×D149—42no—
132C4×D149—42no—
133C4×D149—81no—
134#11924746—21no—
135#11924746—21no—
136C23×C149—21no—
137C23×C149C2×D411yescharacteristic
138Q8×C149—21no—
139Q8×C149—21no—
140D4×C149—21no—
141D4×C149—21no—
142D4×C149—81no—
143C2×C4×C149—21no—
144C2×C4×C149—21no—
145C2×C4×C149—21no—
146C2×C4×C149—21no—
147C2×C4×C149—21no—
148C2×C4×C149C2×D411yescharacteristic
149C2×C4×C149C2411yescharacteristic
150C8×C149—21no—
151C8×C149—21no—
152C22×C149—21no—
153C22×C149—21no—
154C22×C149—21no—
155C22×C149—41no—
156C22×C149—41no—
157C22×C149—41no—
158C22×C149C22×D411yescharacteristic
159C2×D149—21no—
160C2×D149—21no—
161C2×D149—22no—
162C2×D149—41no—
163C2×D149—41no—
164C2×D149—84no—
165C2×D149#13512yes—
166C4×C149—21no—
167C4×C149—41no—
168C4×C149—41no—
169C4×C149—41no—
170C4×C149C2×#4111yescharacteristic
171C4×C149H(5,𝔽2)11yescharacteristic
172Dic149—21no—
173Dic149—21no—
174Dic149—21no—
175Dic149—41no—
176Dic149—41no—
177Dic149—41no—
178C2×C149—41no—
179C2×C149—81no—
180C2×C149#54611yescharacteristic
181C2×C149C2×#13511yescharacteristic
182C2×C149#57711yescharacteristic
183D149—22no—
184D149—42no—
185C149#329611yescharacteristic, complemented, Sylow
186#3296—1491nomaximal, Sylow
187#552—1491no—
188#546—1491no—
189#488—1491no—
190#483—1491no—
191#482—1491no—
192#478—1491no—
193#467—1491no—
194#465D14911yescharacteristic, complemented
195#464—1491no—
196#452—1491no—
197#450—1491no—
198C4×SD16—1491no—
199C8×D4—1491no—
200#416—1491no—
201C2×#102—1491no—
202C2×#41—1491no—
203C22×D4—1491no—
204C2×Q16—1491no—
205C2×SD16—1491no—
206C2×SD16C2×D14911yescharacteristic
207C22×C8—1491no—
208C22×C8C2×D14911yescharacteristic, complemented
209#126—2981no—
210#125—2981no—
211#125—1491no—
212#124—1491no—
213#123—2981no—
214#123—2981no—
215C22⋊Q8—1491no—
216C22⋊Q8—1491no—
217C22⋊Q8C2×D14911yescharacteristic, complemented
218#121—2981no—
219#121C2×D14911yescharacteristic
220#120—2981no—
221C4×Q8—1491no—
222C4×D4—1491no—
223C4×D4—1491no—
224#117—2981no—
225#117—1491no—
226C2×C4⋊C4—1491no—
227C2×C22⋊C4—2981no—
228#107—1491no—
229#106—1491no—
230#106C2×D14911yescharacteristic, complemented
231C4⋊C8—1491no—
232#103—1491no—
233#103—1491no—
234#103—1491no—
235#103C2×D14911yescharacteristic, complemented
236#102—1491no—
237#102—1491no—
238#102—1491no—
239#102—1491no—
240#102—1491no—
241#102C2×D14911yescharacteristic
242C22⋊C8—1491no—
243C22⋊C8—1491no—
244C4×C8—1491no—
245C24—2981no—
246#41—2982no—
247#41—5961no—
248C2×Q8—1491no—
249C2×Q8C22×D14911yescharacteristic
250C2×D4—21no—
251C2×D4—2982no—
252C2×D4—2981no—
253C2×D4—2981no—
254C2×D4—5961no—
255C2×D4—1491no—
256C2×D4—1491no—
257C2×D4C22×D14911yescharacteristic
258C22×C4—2981no—
259C22×C4—2981no—
260C22×C4—2981no—
261C22×C4—1491no—
262C22×C4C22×D14911yescharacteristic
263Q16—5961no—
264SD16—41no—
265SD16—2982no—
266C2×C8—41no—
267C2×C8—5961no—
268C2×C8—1491no—
269C2×C8—1491no—
270C2×C8C22×D14911yescharacteristic
271C2×C8C22×D14911yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2981no—
274C4⋊C4—2981no—
275C4⋊C4—2981no—
276C4⋊C4—2981no—
277C4⋊C4—2981no—
278C4⋊C4—2981no—
279C4⋊C4—1491no—
280C4⋊C4—1491no—
281C4⋊C4—1491no—
282C4⋊C4C22×D14911yescharacteristic
283C4⋊C4C22×D14911yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—2981no—
287C22⋊C4—2981no—
288C22⋊C4—2981no—
289C22⋊C4—2981no—
290C22⋊C4—2981no—
291C22⋊C4—2981no—
292C22⋊C4—1491no—
293C22⋊C4—1491no—
294C42—2981no—
295C42—2981no—
296C42—1491no—
297C23—21no—
298C23—2981no—
299C23—5962no—
300C23—5962no—
301C23—1491no—
302C23—11921no—
303C23D4×D14911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2981no—
307Q8—2981no—
308D4—21no—
309D4—21no—
310D4—81no—
311D4—2982no—
312D4—2981no—
313D4—5962no—
314D4—5962no—
315D4—1492no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2981no—
322C2×C4—2981no—
323C2×C4—2981no—
324C2×C4—2981no—
325C2×C4—2981no—
326C2×C4—5962no—
327C2×C4—5962no—
328C2×C4—1491no—
329C2×C4—1491no—
330C2×C4—1491no—
331C2×C4—11921no—
332C2×C4D4×D14911yescharacteristic
333C2×C4C23×D14911yescharacteristic
334C8—21no—
335C8—21no—
336C8—2981no—
337C8—2981no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—2981no—
345C22—2981no—
346C22—2982no—
347C22—5961no—
348C22—5961no—
349C22—1492no—
350C22—11924no—
351C22C2×D4×D14911yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—2981no—
357C4—2981no—
358C4—2981no—
359C4—5961no—
360C4—5961no—
361C4—5961no—
362C4#41×D14911yescharacteristic
363C4#1801195611yescharacteristic
364C2—41no—
365C2—81no—
366C2—2982no—
367C2—5962no—
368C2#2375026411yescharacteristic
369C2#135×D14911yescharacteristic
370C2#2375043111yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#465×C149C2
socleC22×C149C22×D4
prime radical#23750272C2
hypercenter#465D149
hypocenterC149#3296
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23750309C2semidirect
3. #23749978C2semidirect
4. #23749957C2semidirect
5. #23749946C2semidirect
6. #23749924C2semidirect
7. #23749865C2semidirect
8. #23749845C2semidirect
9. #23749785C2semidirect
10. #23749774C2semidirect
11. #23749710C2semidirect
12. C8×C22⋊D149C2semidirect
13. #23749622C2semidirect
14. #102×D149C2semidirect
15. #465×C149C2semidirect
18. C2×#14558726C22semidirect
22. #18011902C22semidirect
31. Q8×Dic149C22semidirect
34. #18011870C22semidirect
38. #18011844C22semidirect
39. #18011842C22semidirect
40. #18011840C22semidirect
41. #18011836C22semidirect
42. #18011836C22semidirect
43. #18011834C22semidirect
49. C22⋊#11924746C22semidirect
51. C8×Dic149C22semidirect
53. C22×C8×C149C22semidirect
54. C22⋊Q8×C149C22semidirect
56. #106×C149C22semidirect
57. #103×C149C22semidirect
185. C149#3296semidirect
194. #465D149semidirect
208. C22×C8C2×D149semidirect
217. C22⋊Q8C2×D149semidirect
230. #106C2×D149semidirect
235. #103C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503092
#237502722
#237499782
#237499572
#237499462
#237499242
#237498652
#237498452
#237497852
#237497742
#237497102
C8×C22⋊D1492
#237496222
#102×D1492
#465×C1492
#3296149