Finite Group Databasebeta

← #16830999

Subgroups — #16830999

Order 3712.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745280C211yesmaximal, characteristic, complemented
2C22×#90773C211yesmaximal, characteristic, complemented
3#14271826C211yesmaximal, characteristic, complemented
4#14271804C211yesmaximal, characteristic
5#14271556C211yesmaximal, characteristic, complemented
6#14271410C211yesmaximal, characteristic, complemented
7#14271388C211yesmaximal, characteristic, complemented
8#14271367C211yesmaximal, characteristic, complemented
9#14271328C211yesmaximal, characteristic, complemented
10#14271318C211yesmaximal, characteristic, complemented
11#14271313C211yesmaximal, characteristic, complemented
12#14271303C211yesmaximal, characteristic, complemented
13#14271262C211yesmaximal, characteristic, complemented
14#14271234C211yesmaximal, characteristic, complemented
15C29×#467C211yesmaximal, characteristic, complemented
16C2×#90809C2211yescharacteristic
17C2×Q8×D29C2211yescharacteristic, complemented
18C22×D116C2211yescharacteristic
19C22×Dic58C2211yescharacteristic
20C2×Q16×C29C2211yescharacteristic, complemented
21C22×C8×C29C2211yescharacteristic
22C29×C22⋊Q8C2211yescharacteristic
23C29×C22⋊Q8C2211yescharacteristic
24#11745309C2211yescharacteristic
25#11745306C2211yescharacteristic, complemented
26C2×#90785C2211yescharacteristic, complemented
27C2×#90785C2211yescharacteristic, complemented
28C2×#90784C2211yescharacteristic
29C2×#90784C2211yescharacteristic
30C4×C22⋊D29C2211yescharacteristic
31#11745280—24no—
32C2×#90773—24no—
33C2×#90773C2211yescharacteristic
34C2×#90773C2211yescharacteristic
35C2×#90773C2211yescharacteristic
36C2×#90773C2211yescharacteristic
37C2×#90772C2211yescharacteristic, complemented
38#11745259C2211yescharacteristic, complemented
39#11745258—21no—
40#11745257—21no—
41#11745257C2211yescharacteristic
42#11745256—21no—
43#11745255C2211yescharacteristic
44#11745250—21no—
45#11745249C2211yescharacteristic, complemented
46#11745244—21no—
47#11745243—21no—
48C22⋊C4×D29—21no—
49#11745240—21no—
50C29×#107C2211yescharacteristic, complemented
51C29×#103C2211yescharacteristic, complemented
52C29×#103C2211yescharacteristic, complemented
53C22⋊#6597C2211yescharacteristic
54#11745168C2211yescharacteristic, complemented
55#11745163C2211yescharacteristic, complemented
56#11745162C2211yescharacteristic, complemented
57#11745158C2211yescharacteristic, complemented
58#11745158C2211yescharacteristic, complemented
59#11745157C2211yescharacteristic
60#11745157C2211yescharacteristic
61C23×D29—21no—
62C2×Q8×C29C2311yescharacteristic
63C2×Q8×C29C2311yescharacteristic
64C22×C4×C29C2311yescharacteristic
65C2×C22⋊D29D411yescharacteristic
66C22×Dic29—21no—
67#90809—22no—
68#90809—41no—
69Q8×D29—22no—
70Q8×D29—41no—
71C2×D116—22no—
72C2×D116—21no—
73C2×D116—21no—
74C2×D116—21no—
75C2×D116—41no—
76C2×D116C2311yescharacteristic
77C2×D116C2311yescharacteristic
78C2×C4×D29—21no—
79C2×C4×D29—21no—
80C2×C4×D29—21no—
81C2×C4×D29C2311yescharacteristic
82C2×Dic58—22no—
83C2×Dic58—21no—
84C2×Dic58—21no—
85C2×Dic58—41no—
86C2×Dic58C2311yescharacteristic
87C2×Dic58C2311yescharacteristic
88Q16×C29—22no—
89C2×C8×C29—22no—
90C2×C8×C29C2311yescharacteristic
91C2×C8×C29C2311yescharacteristic
92C4⋊C4×C29—21no—
93C4⋊C4×C29—21no—
94C4⋊C4×C29C2311yescharacteristic
95C4⋊C4×C29C2311yescharacteristic
96C22⋊C4×C29—21no—
97C22⋊C4×C29—21no—
98C22⋊Dic29D411yescharacteristic
99#90785—22no—
100#90785—41no—
101#90784—22no—
102#90784—41no—
103#90781—21no—
104#90781—21no—
105#90781—21no—
106#90781—21no—
107#90781—21no—
108#90781D411yescharacteristic
109C4⋊Dic29—21no—
110#90779—21no—
111#90779—21no—
112#90779—21no—
113#90779D411yescharacteristic
114C4×Dic29—21no—
115C4×Dic29C2311yescharacteristic
116C2×#6597C2311yescharacteristic
117C2×#6597C2311yescharacteristic
118#90773—22no—
119#90773—22no—
120#90773—22no—
121#90773D414yescomplemented
122#90772—22no—
123C23×C29C2×D411yescharacteristic
124C22×D29—21no—
125C22×D29—21no—
126C22×D29—42no—
127C22×D29—42no—
128C22×D29—81no—
129C22×D29C2×D411yescharacteristic
130Q8×C29—21no—
131Q8×C29—21no—
132Q8×C29—21no—
133Q8×C29—21no—
134C2×C4×C29—21no—
135C2×C4×C29—21no—
136C2×C4×C29—21no—
137C2×C4×C29—21no—
138C2×C4×C29—21no—
139C2×C4×C29—21no—
140C2×C4×C29C2×D411yescharacteristic
141C2×C4×C29C2411yescharacteristic
142C22⋊D29—22no—
143C2×Dic29—21no—
144C2×Dic29—21no—
145C2×Dic29—42no—
146C2×Dic29C2×D411yescharacteristic
147C2×Dic29C2×D411yescharacteristic
148C2×Dic29C2×D411yescharacteristic
149D116—21no—
150D116—21no—
151D116—21no—
152D116—42no—
153D116—42no—
154D116—81no—
155D116C2×D412yes—
156C4×D29—21no—
157C4×D29—21no—
158C4×D29—42no—
159C4×D29—42no—
160C4×D29—81no—
161Dic58—21no—
162Dic58—21no—
163Dic58—21no—
164Dic58—42no—
165Dic58—42no—
166Dic58C2×D412yes—
167C8×C29—21no—
168C8×C29C2×D412yes—
169#6597—21no—
170#6597—21no—
171#3266—291nomaximal, Sylow
172C22×C29—21no—
173C22×C29—21no—
174C22×C29#13712yescomplemented
175C22×C29C22×D411yescharacteristic
176C2×D29—21no—
177C2×D29—22no—
178C2×D29—21no—
179C2×D29—21no—
180C2×D29—41no—
181C2×D29—41no—
182C2×D29—41no—
183C2×D29—41no—
184C2×D29—41no—
185C2×D29—84no—
186C4×C29—21no—
187C4×C29—41no—
188C4×C29—41no—
189C4×C29—41no—
190C4×C29—41no—
191C4×C29C22×D411yescharacteristic
192C4×C29H(5,𝔽2)11yescharacteristic
193Dic29—22no—
194Dic29—21no—
195Dic29—21no—
196Dic29—21no—
197Dic29—41no—
198C2×#137—291no—
199C22×SD16—291no—
200D4×Q8—291no—
201#546—291no—
202#497—291no—
203#468—291no—
204#467D2911yescharacteristic, complemented
205#464—291no—
206#461—291no—
207#452—291no—
208#451—291no—
209#450—291no—
210#448—291no—
211#440—291no—
212#435—291no—
213C2×C29—22no—
214C2×C29D4211yescharacteristic
215C2×C29C2×#13711yescharacteristic
216C2×C29#57711yescharacteristic
217D29—41no—
218D29—42no—
219D29—81no—
220C2×#41—291no—
221C22×Q8—291no—
222C22×Q8—291no—
223C22×D4—291no—
224#137—584no—
225C2×Q16—291no—
226C2×Q16—291no—
227C2×Q16C2×D2911yescharacteristic, complemented
228C2×SD16—291no—
229C2×SD16—291no—
230C2×SD16—291no—
231C2×SD16—291no—
232C2×SD16—291no—
233C2×SD16—291no—
234C2×SD16—584no—
235C2×M16—291no—
236C22×C8C2×D2911yescharacteristic
237C4⋊Q8—291no—
238C4⋊Q8—581no—
239#124—291no—
240#123—581no—
241C22⋊Q8—291no—
242C22⋊Q8—581no—
243C22⋊Q8—581no—
244C22⋊Q8C2×D2911yescharacteristic
245C22⋊Q8C2×D2911yescharacteristic
246#121—291no—
247#121—581no—
248#120—581no—
249C4×Q8—291no—
250C4×D4—291no—
251C4×D4—291no—
252C4×D4—581no—
253C2×C22⋊C4—581no—
254#107C2×D2911yescharacteristic, complemented
255C4⋊C8—291no—
256#103—291no—
257#103—291no—
258#103C2×D2911yescharacteristic, complemented
259#103C2×D2911yescharacteristic, complemented
260#102—291no—
261#102—291no—
262C22⋊C8—291no—
263C22⋊C8—291no—
264#97—291no—
265C29#326611yescharacteristic, complemented, Sylow
266C24—581no—
267#41—582no—
268#41—1161no—
269C2×Q8—291no—
270C2×Q8—291no—
271C2×Q8—581no—
272C2×Q8—582no—
273C2×Q8—581no—
274C2×Q8—582no—
275C2×Q8—1161no—
276C2×Q8—1161no—
277C2×Q8C22×D2911yescharacteristic
278C2×Q8C22×D2911yescharacteristic
279C2×D4—291no—
280C2×D4—291no—
281C2×D4—291no—
282C2×D4—582no—
283C2×D4—581no—
284C2×D4—581no—
285C2×D4—581no—
286C2×D4—1161no—
287C22×C4—291no—
288C22×C4—581no—
289C22×C4—581no—
290C22×C4—581no—
291C22×C4—581no—
292C22×C4C22×D2911yescharacteristic
293Q16—22no—
294Q16—582no—
295Q16—1161no—
296SD16—294no—
297SD16—582no—
298SD16—582no—
299SD16—582no—
300SD16—582no—
301SD16—1161no—
302M16—582no—
303C2×C8—22no—
304C2×C8—291no—
305C2×C8—291no—
306C2×C8C22×D2911yescharacteristic
307C2×C8C22×D2911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—291no—
311C4⋊C4—581no—
312C4⋊C4—581no—
313C4⋊C4—581no—
314C4⋊C4—581no—
315C4⋊C4C22×D2911yescharacteristic
316C4⋊C4C22×D2911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—291no—
320C22⋊C4—291no—
321C22⋊C4—581no—
322C22⋊C4—581no—
323C22⋊C4—581no—
324C22⋊C4—581no—
325C22⋊C4—581no—
326C42—291no—
327C42—581no—
328C23—291no—
329C23—581no—
330C23—581no—
331C23—1162no—
332C23—1162no—
333C23—2321no—
334C23D4×D2911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—292no—
340Q8—581no—
341Q8—581no—
342Q8—581no—
343Q8—1162no—
344Q8—1162no—
345D4—292no—
346D4—582no—
347D4—581no—
348D4—581no—
349D4—581no—
350D4—1162no—
351D4—1162no—
352D4—2321no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—291no—
360C2×C4—291no—
361C2×C4—291no—
362C2×C4—581no—
363C2×C4—581no—
364C2×C4—581no—
365C2×C4—581no—
366C2×C4—1162no—
367C2×C4—1162no—
368C2×C4—1162no—
369C2×C4—2321no—
370C2×C4D4×D2911yescharacteristic
371C2×C4C23×D2911yescharacteristic
372C8—21no—
373C8—581no—
374C8—581no—
375C8D4×D2912yes—
376C22—21no—
377C22—21no—
378C22—581no—
379C22—582no—
380C22—581no—
381C22—581no—
382C22—1161no—
383C22—1161no—
384C22—1161no—
385C22—1161no—
386C22—1161no—
387C22—2324no—
388C22#1174528012yescomplemented
389C22C2×D4×D2911yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—581no—
396C4—582no—
397C4—581no—
398C4—581no—
399C4—1161no—
400C4C2×D4×D2911yescharacteristic
401C4#1174536411yescharacteristic
402C2—22no—
403C2—1161no—
404C2—1162no—
405C2—2321no—
406C2#1427179711yescharacteristic
407C2C2×#1174528011yescharacteristic
408C2#1427197011yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#467C2
socleC22×C29C22×D4
prime radical#14271804C2
hypercenter#467D29
hypocenterC29#3266
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#11745280C2semidirect
2. C22×#90773C2semidirect
3. #14271826C2semidirect
5. #14271556C2semidirect
6. #14271410C2semidirect
7. #14271388C2semidirect
8. #14271367C2semidirect
9. #14271328C2semidirect
10. #14271318C2semidirect
11. #14271313C2semidirect
12. #14271303C2semidirect
13. #14271262C2semidirect
14. #14271234C2semidirect
15. C29×#467C2semidirect
17. C2×Q8×D29C22semidirect
20. C2×Q16×C29C22semidirect
25. #11745306C22semidirect
26. C2×#90785C22semidirect
27. C2×#90785C22semidirect
37. C2×#90772C22semidirect
38. #11745259C22semidirect
45. #11745249C22semidirect
50. C29×#107C22semidirect
51. C29×#103C22semidirect
52. C29×#103C22semidirect
54. #11745168C22semidirect
55. #11745163C22semidirect
56. #11745162C22semidirect
57. #11745158C22semidirect
58. #11745158C22semidirect
121. #90773D4semidirect
174. C22×C29#137semidirect
204. #467D29semidirect
227. C2×Q16C2×D29semidirect
254. #107C2×D29semidirect
258. #103C2×D29semidirect
259. #103C2×D29semidirect
265. C29#3266semidirect
388. C22#11745280semidirect

Maximal subgroups

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

subgroupindex
C2×#117452802
C22×#907732
#142718262
#142718042
#142715562
#142714102
#142713882
#142713672
#142713282
#142713182
#142713132
#142713032
#142712622
#142712342
C29×#4672
#326629