Finite Group Databasebeta

← #36336174

Subgroups — #36336174

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#24923989C211yesmaximal, characteristic, complemented
2#24923628C211yesmaximal, characteristic, complemented
3#24923607C211yesmaximal, characteristic, complemented
4#24923590C211yesmaximal, characteristic, complemented
5#24923559C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923496C211yesmaximal, characteristic, complemented
8#24923488C211yesmaximal, characteristic, complemented
9#24923431C211yesmaximal, characteristic, complemented
10#24923420C211yesmaximal, characteristic, complemented
11#24923369C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923274C211yesmaximal, characteristic, complemented
14#24923265C211yesmaximal, characteristic, complemented
15#546×C167C211yesmaximal, characteristic
16C2×#14581940C2211yescharacteristic, complemented
17C2×#14581938C2211yescharacteristic
18C2×#14581938C2211yescharacteristic
19C2×#14581934C2211yescharacteristic
20C22×C167⋊C8C2211yescharacteristic, complemented
21#19128832C2211yescharacteristic, complemented
22#19128825—21no—
23#19128825—21no—
24#19128819—21no—
25#19128811C2211yescharacteristic
26#19128811C2211yescharacteristic, complemented
27C4×Dic334C2211yescharacteristic, complemented
28#19128797—21no—
29#19128791C2211yescharacteristic
30#19128791C2211yescharacteristic, complemented
31#19128765C2211yescharacteristic, complemented
32#19128763C2211yescharacteristic, complemented
33#19128763C2211yescharacteristic, complemented
34C4⋊(C167⋊C8)C2211yescharacteristic, complemented
35#19128756C2211yescharacteristic, complemented
36#19128756C2211yescharacteristic, complemented
37#19128756C2211yescharacteristic, complemented
38#19128755C2211yescharacteristic, complemented
39#19128753C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128753C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic, complemented
47#19128743C2211yescharacteristic, complemented
48C2×#41×C167C2211yescharacteristic
49C22×D4×C167C2211yescharacteristic
50#124×C167C2211yescharacteristic
51#123×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53#121×C167—21no—
54#121×C167C2211yescharacteristic
55#120×C167—21no—
56C4×D4×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58C2×C22⋊C4×C167—21no—
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61#14581940—41no—
62#14581938—22no—
63#14581938—41no—
64#14581934—41no—
65C2×C167⋊C8—41no—
66C2×C167⋊C8C2311yescharacteristic
67C2×C167⋊C8C2311yescharacteristic
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C4⋊Dic167—21no—
71C4⋊Dic167C2311yescharacteristic
72C4⋊Dic167C2311yescharacteristic
73C4⋊Dic167C2311yescharacteristic
74Dic167⋊C4—21no—
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77C22⋊Dic167—21no—
78C22⋊Dic167—21no—
79C22⋊Dic167—21no—
80C22⋊Dic167—21no—
81C4×Dic167—21no—
82C4×Dic167—21no—
83C4×Dic167—21no—
84C24×C167—21no—
85#41×C167—22no—
86#41×C167—41no—
87C2×Q8×C167C2311yescharacteristic
88C2×D4×C167—22no—
89C2×D4×C167—21no—
90C2×D4×C167—21no—
91C2×D4×C167—21no—
92C2×D4×C167—41no—
93C2×D4×C167D411yescharacteristic
94C2×D4×C167C2311yescharacteristic
95C2×D4×C167C2311yescharacteristic
96C22×C4×C167—21no—
97C22×C4×C167—21no—
98C22×C4×C167C2311yescharacteristic
99C22×C4×C167C2311yescharacteristic
100C4⋊C4×C167—21no—
101C4⋊C4×C167D411yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167—21no—
108C22⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C42×C167C2311yescharacteristic
111Dic334—21no—
112Dic334—21no—
113Dic334—42no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C167⋊C8—21no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C23×C167—21no—
123C23×C167—21no—
124C23×C167—42no—
125C23×C167—42no—
126C23×C167—81no—
127C23×C167C2×D411yescharacteristic
128C23×C167C2×D411yescharacteristic
129Q8×C167—21no—
130Q8×C167—21no—
131D4×C167—21no—
132D4×C167—21no—
133D4×C167—21no—
134D4×C167—41no—
135D4×C167—42no—
136D4×C167—42no—
137D4×C167—81no—
138D4×C167#4112yes—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167—42no—
146C2×C4×C167—81no—
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2411yescharacteristic
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—22no—
155C22×C167—21no—
156C22×C167—21no—
157C22×C167—21no—
158C22×C167—21no—
159C22×C167—41no—
160C22×C167—41no—
161C22×C167—41no—
162C22×C167—41no—
163C22×C167—41no—
164C22×C167—84no—
165C22×C167C22×D411yescharacteristic
166C4×C167—21no—
167C4×C167—21no—
168C4×C167—21no—
169C4×C167—21no—
170C4×C167—41no—
171C4×C167—41no—
172C4×C167C2×#4111yescharacteristic
173C4×C167H(5,𝔽2)11yescharacteristic
174Dic167—41no—
175Dic167—41no—
176Dic167—41no—
177Dic167—41no—
178C2×C167—41no—
179C2×C167—41no—
180C2×C167—42no—
181C2×C167—81no—
182C2×C167#54611yescharacteristic
183C2×C167#57511yescharacteristic
184C2×C167#57711yescharacteristic
185C167#331111yescharacteristic, complemented, Sylow
186#3311—1671nomaximal, Sylow
187#556—1671no—
188#546D16711yescharacteristic
189#489—1671no—
190#485—1671no—
191#482—1671no—
192#476—1671no—
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×#41C2×D16711yescharacteristic
203C22×D4C2×D16711yescharacteristic
204C2×Q16—1671no—
205C2×SD16—1671no—
206C2×SD16—1671no—
207C2×M16—1671no—
208C22×C8—1671no—
209C4⋊Q8—1671no—
210#126—3341no—
211#126—3341no—
212#124—3341no—
213#124C2×D16711yescharacteristic
214#123—21no—
215C22⋊Q8—1671no—
216C22⋊Q8—1671no—
217C22⋊Q8C2×D16711yescharacteristic
218#121—21no—
219#121C2×D16711yescharacteristic
220#120—21no—
221C4×Q8—3341no—
222C4×Q8—1671no—
223C4×D4C2×D16711yescharacteristic
224C4×D4C2×D16711yescharacteristic
225#117—1671no—
226#117—1671no—
227C2×C22⋊C4—21no—
228#107—1671no—
229#106—1671no—
230#106—1671no—
231C4⋊C8—1671no—
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#102—1671no—
241#102—1671no—
242C22⋊C8—1671no—
243C22⋊C8—1671no—
244#97—1671no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—3341no—
249C2×Q8—1671no—
250C2×Q8C22×D16711yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4C22⋊D16711yescharacteristic
257C2×D4C22×D16711yescharacteristic
258C2×D4C22×D16711yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C22×D16711yescharacteristic
262C22×C4C22×D16711yescharacteristic
263Q16—6681no—
264SD16—3342no—
265SD16—6681no—
266M16—6681no—
267C2×C8—6681no—
268C2×C8—1671no—
269C2×C8—1671no—
270C2×C8—1671no—
271C2×C8—1671no—
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⋊C4C22⋊D16711yescharacteristic
281C4⋊C4C22×D16711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3341no—
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4C22⋊D16711yescharacteristic
292C22⋊C4C22⋊D16711yescharacteristic
293C42—3341no—
294C42—3341no—
295C42—3341no—
296C42C22×D16711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D16711yescharacteristic
303C23C2×C22⋊D16711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3341no—
307Q8—3341no—
308Q8—6682no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—81no—
316D4#1458194912yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—3341no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—3341no—
329C2×C4C2×C22⋊D16711yescharacteristic
330C2×C4C2×C22⋊D16711yescharacteristic
331C2×C4C2×C22⋊D16711yescharacteristic
332C2×C4C2×C22⋊D16711yescharacteristic
333C2×C4C23×D16711yescharacteristic
334C8—3341no—
335C8—3341no—
336C8—3341no—
337C8—3341no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×C22⋊D16711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—6681no—
359C4—6681no—
360C4—6681no—
361C4—6681no—
362C4C2×#1458194911yescharacteristic
363C4#1912887711yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#2492391311yescharacteristic
369C2#2492406711yescharacteristic
370C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radical#546×C167C2
socleC22×C167C22×D4
hypercenter#546D167
hypocenterC167#3311
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #24923989C2semidirect
2. #24923628C2semidirect
3. #24923607C2semidirect
4. #24923590C2semidirect
5. #24923559C2semidirect
6. #24923525C2semidirect
7. #24923496C2semidirect
8. #24923488C2semidirect
9. #24923431C2semidirect
10. #24923420C2semidirect
11. #24923369C2semidirect
12. #24923341C2semidirect
13. #24923274C2semidirect
14. #24923265C2semidirect
16. C2×#14581940C22semidirect
20. C22×C167⋊C8C22semidirect
21. #19128832C22semidirect
26. #19128811C22semidirect
27. C4×Dic334C22semidirect
30. #19128791C22semidirect
31. #19128765C22semidirect
32. #19128763C22semidirect
33. #19128763C22semidirect
34. C4⋊(C167⋊C8)C22semidirect
35. #19128756C22semidirect
36. #19128756C22semidirect
37. #19128756C22semidirect
38. #19128755C22semidirect
46. C22⋊(C167⋊C8)C22semidirect
47. #19128743C22semidirect
185. C167#3311semidirect

Maximal subgroups

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

subgroupindex
#249239892
#249236282
#249236072
#249235902
#249235592
#249235252
#249234962
#249234882
#249234312
#249234202
#249233692
#249233412
#249232742
#249232652
#546×C1672
#3311167