Finite Group Databasebeta

← #36336171

Subgroups — #36336171

Order 21376.

402 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#24923993C211yesmaximal, characteristic, complemented
2#24923914C211yesmaximal, characteristic
3#24923631C211yesmaximal, characteristic, complemented
4#24923613C211yesmaximal, characteristic, complemented
5#24923595C211yesmaximal, characteristic, complemented
6#24923528C211yesmaximal, characteristic, complemented
7#24923499C211yesmaximal, characteristic, complemented
8#24923492C211yesmaximal, characteristic, complemented
9#24923432C211yesmaximal, characteristic, complemented
10#24923421C211yesmaximal, characteristic, complemented
11#24923369C211yesmaximal, characteristic, complemented
12#24923339C211yesmaximal, characteristic, complemented
13#24923272C211yesmaximal, characteristic, complemented
14#24923262C211yesmaximal, characteristic, complemented
15#476×C167C211yesmaximal, characteristic, complemented
16C2×#14581949C2211yescharacteristic
17C2×D4×D167C2211yescharacteristic
18C2×Dic668C2211yescharacteristic, complemented
19C2×#14581939C2211yescharacteristic
20C2×#14581938C2211yescharacteristic
21C2×#14581933C2211yescharacteristic
22C2×C8×D167C2211yescharacteristic, complemented
23#19128827—21no—
24#19128827—21no—
25#19128821—21no—
26#19128820C2211yescharacteristic
27#19128816—21no—
28#19128812C2211yescharacteristic
29#19128812C2211yescharacteristic
30#19128812C2211yescharacteristic, complemented
31#19128807C2211yescharacteristic
32#19128806—21no—
33#19128802—21no—
34Q8×Dic167—21no—
35C4×Dic334C2211yescharacteristic, complemented
36C4×D668C2211yescharacteristic
37#19128790C2211yescharacteristic
38#19128790C2211yescharacteristic, complemented
39C2×(C2×D167)⋊C4—21no—
40#19128765C2211yescharacteristic, complemented
41#19128757C2211yescharacteristic, complemented
42#19128755C2211yescharacteristic, complemented
43#19128755C2211yescharacteristic, complemented
44#19128755C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic
46#19128753C2211yescharacteristic
47#19128752C2211yescharacteristic
48#19128752C2211yescharacteristic
49(C2×D167)⋊C8C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51#19128743C2211yescharacteristic, complemented
52C4⋊Q8×C167C2211yescharacteristic, complemented
53C4×D4×C167C2211yescharacteristic
54#106×C167C2211yescharacteristic, complemented
55#106×C167C2211yescharacteristic, complemented
56C4⋊C8×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic
58#102×C167C2211yescharacteristic
59C23×D167—21no—
60#14581949—22no—
61#14581949—41no—
62C2×Dic334C2311yescharacteristic
63C2×Dic334C2311yescharacteristic
64C2×D668—21no—
65C2×D668D411yescharacteristic
66C2×D668C2311yescharacteristic
67C2×C22⋊D167—21no—
68C2×C22⋊D167—21no—
69D4×D167—22no—
70D4×D167—41no—
71C22×Dic167—21no—
72C2×C4×D167C2311yescharacteristic
73C2×C4×D167C2311yescharacteristic
74Dic668—41no—
75#14581939—41no—
76#14581938—22no—
77#14581933—41no—
78C2×C167⋊C8C2311yescharacteristic
79C2×C167⋊C8C2311yescharacteristic
80C8×D167—41no—
81C4⋊Dic167—21no—
82C4⋊Dic167—21no—
83C4⋊Dic167—21no—
84C4⋊Dic167D411yescharacteristic
85C4⋊Dic167C2311yescharacteristic
86Dic167⋊C4—21no—
87(C2×D167)⋊C4—21no—
88(C2×D167)⋊C4—21no—
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)⋊C4—21no—
95(C2×D167)⋊C4D411yescharacteristic
96(C2×D167)⋊C4D411yescharacteristic
97C4×Dic167—21no—
98C4×Dic167—21no—
99C4×Dic167—21no—
100C2×Q8×C167—21no—
101C2×D4×C167C2311yescharacteristic
102C22×C4×C167—21no—
103C2×C8×C167C2311yescharacteristic
104C2×C8×C167C2311yescharacteristic
105C4⋊C4×C167—21no—
106C4⋊C4×C167C2311yescharacteristic
107C4⋊C4×C167C2311yescharacteristic
108C4⋊C4×C167C2311yescharacteristic
109C22⋊C4×C167—21no—
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112C22×D167—42no—
113C22×D167—42no—
114C22×D167—81no—
115C22×D167C2×D411yescharacteristic
116C22×D167C2×D411yescharacteristic
117Dic334—21no—
118Dic334—21no—
119Dic334—21no—
120Dic334—21no—
121D668—21no—
122D668—21no—
123D668—41no—
124D668—81no—
125C22⋊D167—42no—
126C22⋊D167—42no—
127C2×Dic167—21no—
128C2×Dic167—21no—
129C2×Dic167—21no—
130C2×Dic167—42no—
131C2×Dic167C2×D411yescharacteristic
132C2×Dic167C2×D411yescharacteristic
133C4×D167—21no—
134C4×D167—21no—
135C4×D167—21no—
136C4×D167—21no—
137C167⋊C8—21no—
138C167⋊C8—21no—
139C23×C167—21no—
140Q8×C167—42no—
141D4×C167—21no—
142D4×C167#4112yes—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—21no—
146C2×C4×C167—81no—
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2411yescharacteristic
150C8×C167—21no—
151C8×C167—21no—
152C22×C167—22no—
153C22×C167—41no—
154C22×C167—41no—
155C22×C167C22×D411yescharacteristic
156C2×D167—21no—
157C2×D167—21no—
158C2×D167—21no—
159C2×D167—21no—
160C2×D167—21no—
161C2×D167—21no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—84no—
166C4×C167—21no—
167C4×C167—21no—
168C4×C167—41no—
169C4×C167—41no—
170C4×C167—41no—
171C4×C167C2×#4111yescharacteristic
172C4×C167H(5,𝔽2)11yescharacteristic
173Dic167—21no—
174Dic167—21no—
175Dic167—41no—
176Dic167—41no—
177Dic167—41no—
178C2×C167—42no—
179C2×C167#54611yescharacteristic
180C2×C167#57511yescharacteristic
181C2×C167#57711yescharacteristic
182D167—41no—
183D167—41no—
184D167—81no—
185C167#331111yescharacteristic, complemented, Sylow
186#3311—1671nomaximal, Sylow
187#556—1671no—
188#546—1671no—
189#489—1671no—
190#485—1671no—
191#482—1671no—
192#476D16711yescharacteristic, complemented
193#470—1671no—
194#465—1671no—
195#464—1671no—
196#452—1671no—
197#450—1671no—
198#440—1671no—
199#435—1671no—
200#418—1671no—
201#416—1671no—
202C2×#41—1671no—
203C22×D4—1671no—
204C2×Q16—1671no—
205C2×SD16—1671no—
206C2×SD16—1671no—
207C2×M16—1671no—
208C22×C8—1671no—
209C4⋊Q8C2×D16711yescharacteristic, complemented
210#126—3341no—
211#126—3341no—
212#124—3341no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—1671no—
216C22⋊Q8—1671no—
217C22⋊Q8—1671no—
218#121—3341no—
219#121—1671no—
220#120—3341no—
221C4×Q8—3341no—
222C4×Q8—1671no—
223C4×D4—1671no—
224C4×D4C2×D16711yescharacteristic
225#117—1671no—
226#117—1671no—
227C2×C22⋊C4—3341no—
228#107—1671no—
229#106C2×D16711yescharacteristic, complemented
230#106C2×D16711yescharacteristic, complemented
231C4⋊C8C2×D16711yescharacteristic, complemented
232#103—1671no—
233#103—1671no—
234#103—1671no—
235#103—1671no—
236#102—1671no—
237#102—1671no—
238#102—1671no—
239#102—1671no—
240#102C2×D16711yescharacteristic
241#102C2×D16711yescharacteristic
242C22⋊C8—1671no—
243C22⋊C8—1671no—
244#97—1671no—
245C24—3341no—
246#41—3342no—
247#41—6681no—
248C2×Q8—21no—
249C2×Q8—1671no—
250C2×Q8—1671no—
251C2×D4—3341no—
252C2×D4—3342no—
253C2×D4—3341no—
254C2×D4—3341no—
255C2×D4—6681no—
256C2×D4—1671no—
257C2×D4—1671no—
258C2×D4C22×D16711yescharacteristic
259C22×C4—21no—
260C22×C4—3341no—
261C22×C4—1671no—
262C22×C4—1671no—
263Q16—6681no—
264SD16—3342no—
265SD16—6681no—
266M16—6681no—
267C2×C8—6681no—
268C2×C8—1671no—
269C2×C8—1671no—
270C2×C8C22×D16711yescharacteristic
271C2×C8C22×D16711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—3341no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—3341no—
277C4⋊C4—1671no—
278C4⋊C4—1671no—
279C4⋊C4C22×D16711yescharacteristic
280C4⋊C4C22×D16711yescharacteristic
281C4⋊C4C22×D16711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—1671no—
292C22⋊C4—1671no—
293C42—3341no—
294C42—3341no—
295C42—3341no—
296C42C22×D16711yescharacteristic
297C23—21no—
298C23—13361no—
299C23—3341no—
300C23—6682no—
301C23—6682no—
302C23—1671no—
303C23—1671no—
304Q8—42no—
305Q8—3341no—
306Q8—3341no—
307Q8—3341no—
308Q8—3341no—
309D4—21no—
310D4—13361no—
311D4—3341no—
312D4—3341no—
313D4—6681no—
314D4—6682no—
315D4—6682no—
316D4#1458195112yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—3341no—
322C2×C4—3341no—
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×C4D4×D16711yescharacteristic
332C2×C4D4×D16711yescharacteristic
333C2×C4C23×D16711yescharacteristic
334C8—21no—
335C8—21no—
336C8—3341no—
337C8—3341no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—13364no—
342C22—3341no—
343C22—3341no—
344C22—3341no—
345C22—3341no—
346C22—3341no—
347C22—3341no—
348C22—6681no—
349C22—6681no—
350C22—6681no—
351C22C2×D4×D16711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3341no—
358C4—3341no—
359C4—6681no—
360C4—6681no—
361C4—6681no—
362C4C2×#1458195111yescharacteristic
363C4#1912887811yescharacteristic
364C2—42no—
365C2—13361no—
366C2—6681no—
367C2—6681no—
368C2#2492391011yescharacteristic
369C2#2492406511yescharacteristic
370C2#2492407511yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#476×C167C2
socleC22×C167C22×D4
prime radical#24923914C2
hypercenter#476D167
hypocenterC167#3311
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #24923993C2semidirect
3. #24923631C2semidirect
4. #24923613C2semidirect
5. #24923595C2semidirect
6. #24923528C2semidirect
7. #24923499C2semidirect
8. #24923492C2semidirect
9. #24923432C2semidirect
10. #24923421C2semidirect
11. #24923369C2semidirect
12. #24923339C2semidirect
13. #24923272C2semidirect
14. #24923262C2semidirect
15. #476×C167C2semidirect
18. C2×Dic668C22semidirect
22. C2×C8×D167C22semidirect
30. #19128812C22semidirect
35. C4×Dic334C22semidirect
38. #19128790C22semidirect
40. #19128765C22semidirect
41. #19128757C22semidirect
42. #19128755C22semidirect
43. #19128755C22semidirect
44. #19128755C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. #19128743C22semidirect
52. C4⋊Q8×C167C22semidirect
54. #106×C167C22semidirect
55. #106×C167C22semidirect
56. C4⋊C8×C167C22semidirect
185. C167#3311semidirect
192. #476D167semidirect
209. C4⋊Q8C2×D167semidirect
229. #106C2×D167semidirect
230. #106C2×D167semidirect
231. C4⋊C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239932
#249239142
#249236312
#249236132
#249235952
#249235282
#249234992
#249234922
#249234322
#249234212
#249233692
#249233392
#249232722
#249232622
#476×C1672
#3311167