Finite Group Databasebeta

← #36336178

Subgroups — #36336178

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#24923990C211yesmaximal, characteristic, complemented
2#24923907C211yesmaximal, characteristic
3#24923630C211yesmaximal, characteristic, complemented
4#24923609C211yesmaximal, characteristic, complemented
5#24923594C211yesmaximal, characteristic, complemented
6#24923557C211yesmaximal, characteristic, complemented
7#24923524C211yesmaximal, characteristic, complemented
8#24923500C211yesmaximal, characteristic, complemented
9#24923493C211yesmaximal, characteristic, complemented
10#24923418C211yesmaximal, characteristic, complemented
11#24923370C211yesmaximal, characteristic, complemented
12#24923337C211yesmaximal, characteristic, complemented
13#24923272C211yesmaximal, characteristic, complemented
14#24923265C211yesmaximal, characteristic, complemented
15#452×C167C211yesmaximal, characteristic, complemented
16C2×D4×D167C2211yescharacteristic
17C2×#14581937C2211yescharacteristic
18C2×#14581933C2211yescharacteristic
19C22×C167⋊C8C2211yescharacteristic, complemented
20#19128831C2211yescharacteristic, complemented
21#19128827—21no—
22#19128825—21no—
23#19128821C2211yescharacteristic
24#19128818—21no—
25#19128815—21no—
26#19128808C2211yescharacteristic
27#19128808C2211yescharacteristic
28#19128806C2211yescharacteristic
29#19128804—21no—
30#19128800—21no—
31Q8×Dic167C2211yescharacteristic, complemented
32#19128797—21no—
33D4×Dic167C2211yescharacteristic
34C4×C22⋊D167C2211yescharacteristic
35#19128791C2211yescharacteristic, complemented
36#19128790C2211yescharacteristic
37C2×(C2×D167)⋊C4—21no—
38#19128766C2211yescharacteristic, complemented
39#19128764C2211yescharacteristic, complemented
40#19128763C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42#19128756C2211yescharacteristic, complemented
43#19128756C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128754C2211yescharacteristic
46#19128754C2211yescharacteristic
47#19128753C2211yescharacteristic
48#19128753C2211yescharacteristic
49#19128752C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic
51#19128742C2211yescharacteristic, complemented
52C2×#41×C167C2211yescharacteristic
53C2×Q16×C167C2211yescharacteristic, complemented
54C2×SD16×C167C2211yescharacteristic
55C22⋊Q8×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic
58C22⋊C8×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C2×Dic334—21no—
61C2×D668C2311yescharacteristic
62C2×C22⋊D167—21no—
63C2×C22⋊D167—21no—
64C2×C22⋊D167D411yescharacteristic
65D4×D167—22no—
66D4×D167—41no—
67C22×Dic167—21no—
68C2×C4×D167C2311yescharacteristic
69#14581937—41no—
70#14581933—41no—
71C2×C167⋊C8—41no—
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C4⋊Dic167—21no—
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77C4⋊Dic167C2311yescharacteristic
78Dic167⋊C4—21no—
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81Dic167⋊C4D411yescharacteristic
82C22⋊Dic167—21no—
83C22⋊Dic167—21no—
84C22⋊Dic167D411yescharacteristic
85(C2×D167)⋊C4—21no—
86(C2×D167)⋊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)⋊C4D411yescharacteristic
92C4×Dic167—21no—
93C4×Dic167—21no—
94C4×Dic167—21no—
95C4×Dic167C2311yescharacteristic
96#41×C167—22no—
97#41×C167—41no—
98C2×Q8×C167C2311yescharacteristic
99C2×Q8×C167C2311yescharacteristic
100C2×D4×C167—21no—
101C2×D4×C167C2311yescharacteristic
102C22×C4×C167—21no—
103C22×C4×C167C2311yescharacteristic
104Q16×C167—41no—
105SD16×C167—22no—
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167—21no—
109C4⋊C4×C167C2311yescharacteristic
110C22⋊C4×C167—21no—
111C22×D167—21no—
112C22×D167—42no—
113C22×D167—42no—
114C22×D167—81no—
115C22×D167C2×D411yescharacteristic
116Dic334—42no—
117D668—21no—
118D668—21no—
119C22⋊D167—41no—
120C22⋊D167—42no—
121C22⋊D167—81no—
122C2×Dic167—21no—
123C2×Dic167—21no—
124C2×Dic167—21no—
125C2×Dic167—81no—
126C2×Dic167C2×D411yescharacteristic
127C2×Dic167C2×D411yescharacteristic
128C2×Dic167C2×D411yescharacteristic
129C4×D167—21no—
130C4×D167—21no—
131C167⋊C8—21no—
132C167⋊C8—21no—
133C23×C167—21no—
134C23×C167C2×D411yescharacteristic
135Q8×C167—21no—
136Q8×C167—21no—
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—21no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—84no—
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×C167#57511yescharacteristic
182C2×C167#57711yescharacteristic
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#476—1671no—
193#470—1671no—
194#465—1671no—
195#464—1671no—
196#452D16711yescharacteristic, complemented
197#450—1671no—
198#440—1671no—
199#435—1671no—
200#418—1671no—
201#416—1671no—
202C2×#41C2×D16711yescharacteristic
203C22×D4—1671no—
204C2×Q16C2×D16711yescharacteristic, complemented
205C2×SD16—1671no—
206C2×SD16C2×D16711yescharacteristic
207C2×M16—1671no—
208C22×C8—1671no—
209C4⋊Q8—1671no—
210#126—3341no—
211#126—3341no—
212#124—3341no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—1671no—
216C22⋊Q8—1671no—
217C22⋊Q8C2×D16711yescharacteristic, complemented
218#121—3341no—
219#121—1671no—
220#120—3341no—
221C4×Q8—3341no—
222C4×Q8—1671no—
223C4×D4—1671no—
224C4×D4—1671no—
225#117—1671no—
226#117—1671no—
227C2×C22⋊C4—3341no—
228#107—1671no—
229#106—1671no—
230#106—1671no—
231C4⋊C8—1671no—
232#103—1671no—
233#103—1671no—
234#103—1671no—
235#103C2×D16711yescharacteristic, complemented
236#102—1671no—
237#102—1671no—
238#102—1671no—
239#102—1671no—
240#102—1671no—
241#102C2×D16711yescharacteristic
242C22⋊C8—1671no—
243C22⋊C8C2×D16711yescharacteristic, complemented
244#97—1671no—
245C24—3341no—
246#41—22no—
247#41—41no—
248C2×Q8—3341no—
249C2×Q8C22×D16711yescharacteristic
250C2×Q8C22×D16711yescharacteristic
251C2×D4—21no—
252C2×D4—3341no—
253C2×D4—3342no—
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×C4C22×D16711yescharacteristic
263Q16—41no—
264SD16—22no—
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⋊C4—1671no—
280C4⋊C4—1671no—
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—
296C42—1671no—
297C23—21no—
298C23—13361no—
299C23—3341no—
300C23—6682no—
301C23—6682no—
302C23—1671no—
303C23D4×D16711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—6682no—
309D4—21no—
310D4—42no—
311D4—13361no—
312D4—3341no—
313D4—3341no—
314D4—6681no—
315D4—6682no—
316D4#1458194912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—13361no—
324C2×C4—3341no—
325C2×C4—3341no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—3341no—
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—3341no—
348C22—6681no—
349C22—6681no—
350C22—6681no—
351C22C2×D4×D16711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3341no—
357C4—3341no—
358C4—3341no—
359C4—6681no—
360C4—6681no—
361C4—6681no—
362C4C2×#1458194911yescharacteristic
363C4#1912887711yescharacteristic
364C2—41no—
365C2—42no—
366C2—13361no—
367C2—6681no—
368C2#2492390611yescharacteristic
369C2#2492406611yescharacteristic
370C2#2492407811yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#452×C167C2
socleC22×C167C22×D4
prime radical#24923907C2
hypercenter#452D167
hypocenterC167#3311
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #24923990C2semidirect
3. #24923630C2semidirect
4. #24923609C2semidirect
5. #24923594C2semidirect
6. #24923557C2semidirect
7. #24923524C2semidirect
8. #24923500C2semidirect
9. #24923493C2semidirect
10. #24923418C2semidirect
11. #24923370C2semidirect
12. #24923337C2semidirect
13. #24923272C2semidirect
14. #24923265C2semidirect
15. #452×C167C2semidirect
19. C22×C167⋊C8C22semidirect
20. #19128831C22semidirect
31. Q8×Dic167C22semidirect
35. #19128791C22semidirect
38. #19128766C22semidirect
39. #19128764C22semidirect
40. #19128763C22semidirect
41. Dic167⋊C8C22semidirect
42. #19128756C22semidirect
43. #19128756C22semidirect
44. #19128756C22semidirect
51. #19128742C22semidirect
53. C2×Q16×C167C22semidirect
55. C22⋊Q8×C167C22semidirect
56. #103×C167C22semidirect
58. C22⋊C8×C167C22semidirect
185. C167#3311semidirect
196. #452D167semidirect
204. C2×Q16C2×D167semidirect
217. C22⋊Q8C2×D167semidirect
235. #103C2×D167semidirect
243. C22⋊C8C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239902
#249239072
#249236302
#249236092
#249235942
#249235572
#249235242
#249235002
#249234932
#249234182
#249233702
#249233372
#249232722
#249232652
#452×C1672
#3311167