Finite Group Databasebeta

← #36336176

Subgroups — #36336176

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#24923915C211yesmaximal, characteristic
3#24923629C211yesmaximal, characteristic, complemented
4#24923611C211yesmaximal, characteristic, complemented
5#24923594C211yesmaximal, characteristic, complemented
6#24923560C211yesmaximal, characteristic, complemented
7#24923527C211yesmaximal, characteristic, complemented
8#24923498C211yesmaximal, characteristic, complemented
9#24923492C211yesmaximal, characteristic, complemented
10#24923433C211yesmaximal, characteristic, complemented
11#24923422C211yesmaximal, characteristic, complemented
12#24923371C211yesmaximal, characteristic, complemented
13#24923275C211yesmaximal, characteristic, complemented
14#24923266C211yesmaximal, characteristic, complemented
15#435×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×Dic668C2211yescharacteristic, complemented
19C2×#14581939C2211yescharacteristic
20C2×#14581939C2211yescharacteristic
21#19128832C2211yescharacteristic, complemented
22#19128825—21no—
23#19128825—21no—
24#19128820C2211yescharacteristic
25#19128819—21no—
26#19128817—21no—
27#19128811C2211yescharacteristic
28#19128811C2211yescharacteristic
29#19128811C2211yescharacteristic, complemented
30#19128806C2211yescharacteristic
31#19128805—21no—
32#19128800—21no—
33C4×Dic334C2211yescharacteristic, complemented
34#19128797—21no—
35C4×D668C2211yescharacteristic
36#19128791C2211yescharacteristic
37#19128791C2211yescharacteristic, complemented
38C22⋊C4×D167—21no—
39#19128766C2211yescharacteristic, complemented
40#19128764C2211yescharacteristic, complemented
41#19128764C2211yescharacteristic, complemented
42#19128757C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128757C2211yescharacteristic, complemented
45#19128757C2211yescharacteristic, complemented
46#19128754C2211yescharacteristic
47#19128754C2211yescharacteristic
48#19128754C2211yescharacteristic
49#19128754C2211yescharacteristic
50#19128754C2211yescharacteristic
51#19128754C2211yescharacteristic
52C2×M16×C167C2211yescharacteristic
53C22×C8×C167C2211yescharacteristic, complemented
54C4×D4×C167C2211yescharacteristic
55C4⋊C8×C167C2211yescharacteristic, complemented
56C22⋊C8×C167C2211yescharacteristic
57C22⋊C8×C167C2211yescharacteristic, complemented
58#97×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60#14581950—22no—
61#14581950—41no—
62C2×Dic334—21no—
63C2×Dic334C2311yescharacteristic
64C2×Dic334C2311yescharacteristic
65C2×D668—22no—
66C2×D668—21no—
67C2×D668—41no—
68C2×D668C2311yescharacteristic
69C2×D668C2311yescharacteristic
70C2×C22⋊D167—21no—
71C2×C22⋊D167—21no—
72C2×C4×D167—21no—
73C2×C4×D167—21no—
74Dic668—41no—
75#14581939—22no—
76#14581939—41no—
77C4⋊Dic167—21no—
78C4⋊Dic167C2311yescharacteristic
79C4⋊Dic167C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86C22⋊Dic167—21no—
87C22⋊Dic167—21no—
88C22⋊Dic167—21no—
89C22⋊Dic167—21no—
90C22⋊Dic167—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95C4×Dic167—21no—
96C4×Dic167—21no—
97C4×Dic167—21no—
98C2×D4×C167D411yescharacteristic
99C22×C4×C167C2311yescharacteristic
100C22×C4×C167C2311yescharacteristic
101M16×C167—41no—
102C2×C8×C167—41no—
103C2×C8×C167C2311yescharacteristic
104C2×C8×C167C2311yescharacteristic
105C2×C8×C167C2311yescharacteristic
106C2×C8×C167C2311yescharacteristic
107C4⋊C4×C167D411yescharacteristic
108C22⋊C4×C167D411yescharacteristic
109C22⋊C4×C167D411yescharacteristic
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112C22×D167—21no—
113C22×D167—42no—
114C22×D167—42no—
115C22×D167—81no—
116Dic334—21no—
117Dic334—21no—
118Dic334—21no—
119Dic334—21no—
120Dic334—42no—
121D668—21no—
122D668—21no—
123D668—21no—
124D668—42no—
125D668#4112yes—
126C22⋊D167—42no—
127C22⋊D167—81no—
128C2×Dic167—21no—
129C2×Dic167—21no—
130C2×Dic167—21no—
131C2×Dic167—21no—
132C2×Dic167—21no—
133C2×Dic167—21no—
134C4×D167—42no—
135C4×D167—81no—
136C23×C167C2×D411yescharacteristic
137C23×C167C2×D411yescharacteristic
138D4×C167—41no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2411yescharacteristic
148C8×C167—21no—
149C8×C167—21no—
150C8×C167—21no—
151C8×C167—21no—
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—21no—
155C22×C167—21no—
156C22×C167—21no—
157C22×C167—21no—
158C22×C167C22×D411yescharacteristic
159C2×D167—22no—
160C2×D167—41no—
161C2×D167—41no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—84no—
166C4×C167—21no—
167C4×C167—21no—
168C4×C167—21no—
169C4×C167—21no—
170C4×C167C2×#4111yescharacteristic
171C4×C167H(5,𝔽2)11yescharacteristic
172Dic167—41no—
173Dic167—41no—
174Dic167—41no—
175Dic167—41no—
176Dic167—41no—
177Dic167—41no—
178C2×C167—41no—
179C2×C167—41no—
180C2×C167#54611yescharacteristic
181C2×C167#57511yescharacteristic
182C2×C167#57711yescharacteristic
183D167—42no—
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#452—1671no—
197#450—1671no—
198#440—1671no—
199#435D16711yescharacteristic, complemented
200#418—1671no—
201#416—1671no—
202C2×#41—1671no—
203C22×D4—1671no—
204C2×Q16—1671no—
205C2×SD16—1671no—
206C2×SD16—1671no—
207C2×M16C2×D16711yescharacteristic
208C22×C8C2×D16711yescharacteristic, complemented
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⋊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#106—1671no—
230#106—1671no—
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#102—1671no—
241#102—1671no—
242C22⋊C8C2×D16711yescharacteristic
243C22⋊C8C2×D16711yescharacteristic, complemented
244#97C2×D16711yescharacteristic, complemented
245C24—3341no—
246#41—3342no—
247#41—6681no—
248C2×Q8—3341no—
249C2×Q8—1671no—
250C2×Q8—1671no—
251C2×D4—3342no—
252C2×D4—3341no—
253C2×D4—3341no—
254C2×D4—3341no—
255C2×D4—6681no—
256C2×D4—1671no—
257C2×D4—1671no—
258C2×D4D66811yescharacteristic
259C22×C4—3341no—
260C22×C4—3341no—
261C22×C4C22×D16711yescharacteristic
262C22×C4C22×D16711yescharacteristic
263Q16—6681no—
264SD16—3342no—
265SD16—6681no—
266M16—41no—
267C2×C8—41no—
268C2×C8C22×D16711yescharacteristic
269C2×C8C22×D16711yescharacteristic
270C2×C8C22×D16711yescharacteristic
271C2×C8C22×D16711yescharacteristic
272C4⋊C4—3341no—
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⋊C4D66811yescharacteristic
282C22⋊C4—3341no—
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⋊C4D66811yescharacteristic
292C22⋊C4D66811yescharacteristic
293C42—3341no—
294C42—3341no—
295C42—3341no—
296C42C22×D16711yescharacteristic
297C23—13361no—
298C23—3341no—
299C23—3341no—
300C23—6682no—
301C23—6682no—
302C23C2×D66811yescharacteristic
303C23C2×D66811yescharacteristic
304Q8—3341no—
305Q8—3341no—
306Q8—3341no—
307Q8—3341no—
308Q8—6682no—
309D4—41no—
310D4—13361no—
311D4—3341no—
312D4—3341no—
313D4—3341no—
314D4—6682no—
315D4—6682no—
316D4—1672no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—13361no—
322C2×C4—3341no—
323C2×C4—3341no—
324C2×C4—3341no—
325C2×C4—3341no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—6682no—
329C2×C4C2×D66811yescharacteristic
330C2×C4C2×D66811yescharacteristic
331C2×C4C2×D66811yescharacteristic
332C2×C4C2×D66811yescharacteristic
333C2×C4C23×D16711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—13364no—
345C22—3342no—
346C22—6681no—
347C22—6681no—
348C22—6681no—
349C22—6681no—
350C22—6681no—
351C22C22×D66811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—6681no—
357C4—6681no—
358C4—6681no—
359C4—6681no—
360C4—6681no—
361C4—6681no—
362C4#41×D16711yescharacteristic
363C4#1912887711yescharacteristic
364C2—41no—
365C2—41no—
366C2—13361no—
367C2—6682no—
368C2#2492391411yescharacteristic
369C2#2492406911yescharacteristic
370C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#435×C167C2
socleC22×C167C22×D4
prime radical#24923915C2
hypercenter#435D167
hypocenterC167#3311
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #24923989C2semidirect
3. #24923629C2semidirect
4. #24923611C2semidirect
5. #24923594C2semidirect
6. #24923560C2semidirect
7. #24923527C2semidirect
8. #24923498C2semidirect
9. #24923492C2semidirect
10. #24923433C2semidirect
11. #24923422C2semidirect
12. #24923371C2semidirect
13. #24923275C2semidirect
14. #24923266C2semidirect
15. #435×C167C2semidirect
18. C2×Dic668C22semidirect
21. #19128832C22semidirect
29. #19128811C22semidirect
33. C4×Dic334C22semidirect
37. #19128791C22semidirect
39. #19128766C22semidirect
40. #19128764C22semidirect
41. #19128764C22semidirect
42. #19128757C22semidirect
43. #19128757C22semidirect
44. #19128757C22semidirect
45. #19128757C22semidirect
53. C22×C8×C167C22semidirect
55. C4⋊C8×C167C22semidirect
57. C22⋊C8×C167C22semidirect
58. #97×C167C22semidirect
185. C167#3311semidirect
199. #435D167semidirect
208. C22×C8C2×D167semidirect
231. C4⋊C8C2×D167semidirect
243. C22⋊C8C2×D167semidirect
244. #97C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239892
#249239152
#249236292
#249236112
#249235942
#249235602
#249235272
#249234982
#249234922
#249234332
#249234222
#249233712
#249232752
#249232662
#435×C1672
#3311167