Finite Group Databasebeta

← #36335974

Subgroups — #36335974

Order 21376.

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#24923958C211yesmaximal, characteristic, complemented
2#24923907C211yesmaximal, characteristic
3#24923625C211yesmaximal, characteristic, complemented
4#24923598C211yesmaximal, characteristic, complemented
5#24923593C211yesmaximal, characteristic, complemented
6#24923571C211yesmaximal, characteristic, complemented
7#24923509C211yesmaximal, characteristic, complemented
8#24923498C211yesmaximal, characteristic, complemented
9#24923494C211yesmaximal, characteristic, complemented
10#24923431C211yesmaximal, characteristic, complemented
11#24923421C211yesmaximal, characteristic, complemented
12#24923360C211yesmaximal, characteristic, complemented
13C8×C22⋊D167C211yesmaximal, characteristic, complemented
14#102×D167C211yesmaximal, characteristic, complemented
15#416×C167C211yesmaximal, characteristic, complemented
16C2×D4×D167C2211yescharacteristic
17C2×#14581940C2211yescharacteristic, complemented
18C2×#14581939C2211yescharacteristic
19C2×#14581938C2211yescharacteristic
20C2×C8×D167C2211yescharacteristic
21#19128825—21no—
22#19128824—21no—
23#19128823C2211yescharacteristic, complemented
24#19128821C2211yescharacteristic
25#19128815—21no—
26#19128814—21no—
27#19128812C2211yescharacteristic
28#19128811C2211yescharacteristic, complemented
29#19128808C2211yescharacteristic
30#19128806C2211yescharacteristic
31#19128804—21no—
32#19128800—21no—
33#19128797C2211yescharacteristic, complemented
34D4×Dic167C2211yescharacteristic
35C4×C22⋊D167C2211yescharacteristic
36#19128789—21no—
37C4⋊C4×D167C2211yescharacteristic
38C2×(C2×D167)⋊C4—21no—
39#19128765C2211yescharacteristic, complemented
40#19128764C2211yescharacteristic, complemented
41#19128763C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic
46#19128753C2211yescharacteristic
47#19128752C2211yescharacteristic
48#19128752C2211yescharacteristic
49C22⋊(C167⋊C8)C2211yescharacteristic, complemented
50(C2×D167)⋊C8C2211yescharacteristic
51C8×Dic167C2211yescharacteristic, complemented
52C2×#41×C167C2211yescharacteristic
53C22×C8×C167C2211yescharacteristic, complemented
54#117×C167C2211yescharacteristic, complemented
55#103×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic
58#102×C167C2211yescharacteristic
59C23×D167—21no—
60C2×Dic334C2311yescharacteristic
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64C2×C22⋊D167D411yescharacteristic
65D4×D167—22no—
66D4×D167—41no—
67C22×Dic167—21no—
68C2×C4×D167—21no—
69C2×C4×D167C2311yescharacteristic
70#14581940—41no—
71#14581939—41no—
72#14581938—22no—
73C2×C167⋊C8C2311yescharacteristic
74C2×C167⋊C8C2311yescharacteristic
75C8×D167—41no—
76C4⋊Dic167—21no—
77C4⋊Dic167—21no—
78C4⋊Dic167C2311yescharacteristic
79C4⋊Dic167C2311yescharacteristic
80Dic167⋊C4—21no—
81Dic167⋊C4—21no—
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4D411yescharacteristic
86C22⋊Dic167—21no—
87C22⋊Dic167—21no—
88C22⋊Dic167D411yescharacteristic
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4D411yescharacteristic
95C4×Dic167—21no—
96C4×Dic167C2311yescharacteristic
97#41×C167—22no—
98#41×C167—41no—
99C2×Q8×C167C2311yescharacteristic
100C2×D4×C167—21no—
101C2×D4×C167C2311yescharacteristic
102C22×C4×C167—21no—
103C22×C4×C167C2311yescharacteristic
104C2×C8×C167—41no—
105C2×C8×C167C2311yescharacteristic
106C2×C8×C167C2311yescharacteristic
107C4⋊C4×C167C2311yescharacteristic
108C4⋊C4×C167C2311yescharacteristic
109C22⋊C4×C167—21no—
110C42×C167—21no—
111C22×D167—21no—
112C22×D167—42no—
113C22×D167—42no—
114C22×D167—81no—
115C22×D167C2×D411yescharacteristic
116Dic334—21no—
117Dic334—21no—
118D668—21no—
119D668—21no—
120C22⋊D167—22no—
121C22⋊D167—42no—
122C22⋊D167—81no—
123C2×Dic167—21no—
124C2×Dic167—21no—
125C2×Dic167—21no—
126C2×Dic167—81no—
127C2×Dic167C2×D411yescharacteristic
128C2×Dic167C2×D411yescharacteristic
129C2×Dic167C2×D411yescharacteristic
130C4×D167—21no—
131C4×D167—21no—
132C4×D167—42no—
133C167⋊C8—21no—
134C167⋊C8—21no—
135C23×C167—21no—
136C23×C167C2×D411yescharacteristic
137Q8×C167—21no—
138Q8×C167—21no—
139D4×C167—21no—
140D4×C167—42no—
141D4×C167#4112yes—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—21no—
147C2×C4×C167—42no—
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2411yescharacteristic
150C8×C167—21no—
151C8×C167—21no—
152C22×C167—21no—
153C22×C167—22no—
154C22×C167—21no—
155C22×C167—21no—
156C22×C167—41no—
157C22×C167—41no—
158C22×C167C22×D411yescharacteristic
159C2×D167—21no—
160C2×D167—21no—
161C2×D167—41no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—84no—
165C2×D167#13512yes—
166C4×C167—21no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167—41no—
170C4×C167C2×#4111yescharacteristic
171C4×C167H(5,𝔽2)11yescharacteristic
172Dic167—21no—
173Dic167—21no—
174Dic167—21no—
175Dic167—41no—
176Dic167—41no—
177Dic167—41no—
178C2×C167—42no—
179C2×C167—41no—
180C2×C167#54611yescharacteristic
181C2×C167C2×#13511yescharacteristic
182C2×C167#57711yescharacteristic
183D167—22no—
184D167—81no—
185C167#329611yescharacteristic, complemented, Sylow
186#3296—1671nomaximal, Sylow
187#552—1671no—
188#546—1671no—
189#488—1671no—
190#483—1671no—
191#482—1671no—
192#478—1671no—
193#467—1671no—
194#465—1671no—
195#464—1671no—
196#452—1671no—
197#450—1671no—
198C4×SD16—1671no—
199C8×D4—1671no—
200#416D16711yescharacteristic, complemented
201C2×#102—1671no—
202C2×#41C2×D16711yescharacteristic
203C22×D4—1671no—
204C2×Q16—1671no—
205C2×SD16—1671no—
206C2×SD16—1671no—
207C22×C8—1671no—
208C22×C8C2×D16711yescharacteristic, complemented
209#126—3341no—
210#125—3341no—
211#125—1671no—
212#124—1671no—
213#123—3341no—
214#123—3341no—
215C22⋊Q8—1671no—
216C22⋊Q8—1671no—
217C22⋊Q8—1671no—
218#121—3341no—
219#121—1671no—
220#120—3341no—
221C4×Q8—1671no—
222C4×D4—1671no—
223C4×D4—1671no—
224#117—3341no—
225#117C2×D16711yescharacteristic, complemented
226C2×C4⋊C4—1671no—
227C2×C22⋊C4—3341no—
228#107—1671no—
229#106—1671no—
230#106—1671no—
231C4⋊C8—1671no—
232#103—1671no—
233#103—1671no—
234#103C2×D16711yescharacteristic, complemented
235#103C2×D16711yescharacteristic, complemented
236#102—1671no—
237#102—1671no—
238#102—1671no—
239#102—1671no—
240#102C2×D16711yescharacteristic
241#102C2×D16711yescharacteristic
242C22⋊C8—1671no—
243C22⋊C8—1671no—
244C4×C8—1671no—
245C24—3341no—
246#41—22no—
247#41—41no—
248C2×Q8—1671no—
249C2×Q8C22×D16711yescharacteristic
250C2×D4—21no—
251C2×D4—3341no—
252C2×D4—3342no—
253C2×D4—3341no—
254C2×D4—6681no—
255C2×D4—1671no—
256C2×D4—1671no—
257C2×D4C22×D16711yescharacteristic
258C22×C4—21no—
259C22×C4—3341no—
260C22×C4—3341no—
261C22×C4—1671no—
262C22×C4C22×D16711yescharacteristic
263Q16—6681no—
264SD16—3342no—
265SD16—6681no—
266C2×C8—41no—
267C2×C8—6681no—
268C2×C8—1671no—
269C2×C8—1671no—
270C2×C8C22×D16711yescharacteristic
271C2×C8C22×D16711yescharacteristic
272C4⋊C4—3341no—
273C4⋊C4—3341no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—3341no—
277C4⋊C4—3341no—
278C4⋊C4—3341no—
279C4⋊C4—1671no—
280C4⋊C4—1671no—
281C4⋊C4—1671no—
282C4⋊C4C22×D16711yescharacteristic
283C4⋊C4C22×D16711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—3341no—
292C22⋊C4—1671no—
293C22⋊C4—1671no—
294C42—21no—
295C42—3341no—
296C42—1671no—
297C23—21no—
298C23—13361no—
299C23—3341no—
300C23—6682no—
301C23—6682no—
302C23—1671no—
303C23D4×D16711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3341no—
307Q8—3341no—
308D4—21no—
309D4—42no—
310D4—13361no—
311D4—3342no—
312D4—3341no—
313D4—3341no—
314D4—6682no—
315D4#1458195012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—13361no—
323C2×C4—3341no—
324C2×C4—3341no—
325C2×C4—3341no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—6682no—
329C2×C4—1671no—
330C2×C4—1671no—
331C2×C4—1671no—
332C2×C4D4×D16711yescharacteristic
333C2×C4C23×D16711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3341no—
337C8—3341no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—13364no—
345C22—3341no—
346C22—3341no—
347C22—6681no—
348C22—6681no—
349C22—6681no—
350C22—1672no—
351C22C2×D4×D16711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3341no—
357C4—3341no—
358C4—3341no—
359C4—6681no—
360C4—6681no—
361C4—6681no—
362C4C2×#1458195011yescharacteristic
363C4#1912887711yescharacteristic
364C2—41no—
365C2—42no—
366C2—13361no—
367C2—3342no—
368C2#2492390911yescharacteristic
369C2#135×D16711yescharacteristic
370C2#2492407611yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#416×C167C2
socleC22×C167C22×D4
prime radical#24923907C2
hypercenter#416D167
hypocenterC167#3296
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #24923958C2semidirect
3. #24923625C2semidirect
4. #24923598C2semidirect
5. #24923593C2semidirect
6. #24923571C2semidirect
7. #24923509C2semidirect
8. #24923498C2semidirect
9. #24923494C2semidirect
10. #24923431C2semidirect
11. #24923421C2semidirect
12. #24923360C2semidirect
13. C8×C22⋊D167C2semidirect
14. #102×D167C2semidirect
15. #416×C167C2semidirect
17. C2×#14581940C22semidirect
23. #19128823C22semidirect
28. #19128811C22semidirect
33. #19128797C22semidirect
39. #19128765C22semidirect
40. #19128764C22semidirect
41. #19128763C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128757C22semidirect
44. #19128756C22semidirect
49. C22⋊(C167⋊C8)C22semidirect
51. C8×Dic167C22semidirect
53. C22×C8×C167C22semidirect
54. #117×C167C22semidirect
55. #103×C167C22semidirect
56. #103×C167C22semidirect
185. C167#3296semidirect
200. #416D167semidirect
208. C22×C8C2×D167semidirect
225. #117C2×D167semidirect
234. #103C2×D167semidirect
235. #103C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239582
#249239072
#249236252
#249235982
#249235932
#249235712
#249235092
#249234982
#249234942
#249234312
#249234212
#249233602
C8×C22⋊D1672
#102×D1672
#416×C1672
#3296167