Finite Group Databasebeta

← #36334235

Subgroups — #36334235

Order 21376.

444 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923823C211yesmaximal, characteristic, complemented
2C4×D4×D167C211yesmaximal, characteristic
3#24923606C211yesmaximal, characteristic, complemented
4#24923603C211yesmaximal, characteristic, complemented
5#24923548C211yesmaximal, characteristic, complemented
6#24923547C211yesmaximal, characteristic, complemented
7#24923473C211yesmaximal, characteristic, complemented
8#24923472C211yesmaximal, characteristic, complemented
9#24923421C211yesmaximal, characteristic
10#24923418C211yesmaximal, characteristic
11#24923371C211yesmaximal, characteristic, complemented
12#24923369C211yesmaximal, characteristic, complemented
13#24923367C211yesmaximal, characteristic, complemented
14#24923324C211yesmaximal, characteristic, complemented
15#440×C167C211yesmaximal, characteristic, complemented
16C2×D4×D167C2211yescharacteristic
17C22×C4×D167—21no—
18C2×#14581939C2211yescharacteristic
19C2×#14581938C2211yescharacteristic
20C2×#14581937C2211yescharacteristic
21#19128831C2211yescharacteristic, complemented
22#19128830C2211yescharacteristic, complemented
23#19128822—21no—
24#19128812C2211yescharacteristic
25#19128810—21no—
26#19128808C2211yescharacteristic
27C4×Dic334—21no—
28C4×Dic334C2211yescharacteristic, complemented
29C4×D668C2211yescharacteristic
30D4×Dic167C2211yescharacteristic
31#19128794C2211yescharacteristic
32C4×C22⋊D167—21no—
33(C22⋊D167)⋊C4—21no—
34#19128789—21no—
35C4⋊C4×D167C2211yescharacteristic
36C22⋊C4×D167—21no—
37C42×D167C2211yescharacteristic
38#19128766C2211yescharacteristic, complemented
39#19128765C2211yescharacteristic, complemented
40#19128765C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42Dic167⋊C8C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128755C2211yescharacteristic, complemented
46#19128754C2211yescharacteristic
47#19128753C2211yescharacteristic
48#19128752C2211yescharacteristic
49(C2×D167)⋊C8C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic
51#19128743C2211yescharacteristic, complemented
52C2×SD16×C167C2211yescharacteristic
53C4×Q8×C167C2211yescharacteristic, complemented
54C4×D4×C167C2211yescharacteristic
55#107×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic
58#97×C167C2211yescharacteristic, complemented
59C23×D167—21no—
60C2×Dic334—21no—
61C2×Dic334C2311yescharacteristic
62C2×D668C2311yescharacteristic
63C2×C22⋊D167—21no—
64D4×D167—22no—
65D4×D167—22no—
66C22×Dic167—21no—
67C2×C4×D167—21no—
68C2×C4×D167—21no—
69C2×C4×D167—21no—
70C2×C4×D167—44no—
71C2×C4×D167D411yescharacteristic
72C2×C4×D167D411yescharacteristic
73C2×C4×D167C2311yescharacteristic
74#14581939—22no—
75#14581938—22no—
76#14581937—22no—
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C4⋊Dic167—21no—
80C4⋊Dic167C2311yescharacteristic
81C4⋊Dic167C2311yescharacteristic
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4—21no—
86Dic167⋊C4—21no—
87Dic167⋊C4—21no—
88C22⋊Dic167—21no—
89(C2×D167)⋊C4—21no—
90(C2×D167)⋊C4—21no—
91(C2×D167)⋊C4—21no—
92(C2×D167)⋊C4—21no—
93C4×Dic167—21no—
94C4×Dic167D411yescharacteristic
95C4×Dic167D411yescharacteristic
96C4×Dic167C2311yescharacteristic
97C2×Q8×C167C2311yescharacteristic
98C2×D4×C167C2311yescharacteristic
99C22×C4×C167—21no—
100SD16×C167—22no—
101C2×C8×C167C2311yescharacteristic
102C2×C8×C167C2311yescharacteristic
103C4⋊C4×C167—21no—
104C4⋊C4×C167C2311yescharacteristic
105C4⋊C4×C167C2311yescharacteristic
106C22⋊C4×C167—21no—
107C42×C167—21no—
108C42×C167C2311yescharacteristic
109C22×D167—24no—
110C22×D167—21no—
111C22×D167—42no—
112C22×D167—42no—
113C22×D167C2×D411yescharacteristic
114Dic334—21no—
115Dic334—21no—
116Dic334—42no—
117D668—21no—
118D668#4112yes—
119C22⋊D167—24no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C2×Dic167—42no—
124C2×Dic167C2×D411yescharacteristic
125C2×Dic167C2×D411yescharacteristic
126C2×Dic167C2×D411yescharacteristic
127C4×D167—22no—
128C4×D167—21no—
129C4×D167—21no—
130C4×D167—22no—
131C4×D167—42no—
132C4×D167—42no—
133C4×D167—42no—
134C167⋊C8—21no—
135C167⋊C8—21no—
136C23×C167—21no—
137Q8×C167—21no—
138Q8×C167—21no—
139D4×C167—21no—
140D4×C167#4112yes—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—42no—
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2411yescharacteristic
148C8×C167—21no—
149C8×C167—21no—
150C22×C167—22no—
151C22×C167—41no—
152C22×C167—41no—
153C22×C167C22×D411yescharacteristic
154C2×D167—21no—
155C2×D167—21no—
156C2×D167—22no—
157C2×D167—48no—
158C2×D167—41no—
159C2×D167—41no—
160C2×D167#13712yes—
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167C2×#4111yescharacteristic
167C4×C167C2×#4111yescharacteristic
168Dic167—21no—
169Dic167—21no—
170Dic167—41no—
171Dic167—41no—
172Dic167—41no—
173Dic167AGL(1,ℤ/8)12yes—
174C2×C167—42no—
175C2×C167#52511yescharacteristic
176C2×C167C2×AGL(1,ℤ/8)11yescharacteristic
177C2×C167C2×#13711yescharacteristic
178D167—22no—
179D167—42no—
180C167#311011yescharacteristic, complemented, Sylow
181#3110—1671nomaximal, Sylow
182#533—1671no—
183C2×C4×D4—1671no—
184#484—1671no—
185#484—1671no—
186#474—1671no—
187#474—1671no—
188#461—1671no—
189#461—1671no—
190#450—1671no—
191#450—1671no—
192#440—1671no—
193#440—1671no—
194#440—1671no—
195#440D16711yescharacteristic, complemented
196#432—1671no—
197C22×D4—1671no—
198C23×C4—3341no—
199C2×SD16—1671no—
200C2×SD16—1671no—
201C2×SD16—1671no—
202C2×SD16C2×D16711yescharacteristic
203C4⋊Q8—1671no—
204C4⋊Q8—1671no—
205#125—3341no—
206C22⋊Q8—3341no—
207C22⋊Q8—1671no—
208C22⋊Q8—1671no—
209C4×Q8—3341no—
210C4×Q8—1671no—
211C4×Q8C2×D16711yescharacteristic, complemented
212C4×D4—3341no—
213C4×D4—3341no—
214C4×D4—1671no—
215C4×D4—1671no—
216C4×D4—1671no—
217C4×D4C2×D16711yescharacteristic
218#117—3341no—
219C2×C4⋊C4—1671no—
220C2×C22⋊C4—3341no—
221C2×C42—1671no—
222#107—1671no—
223#107—1671no—
224#107—1671no—
225#107C2×D16711yescharacteristic, complemented
226C4⋊C8—1671no—
227C4⋊C8—1671no—
228#103—1671no—
229#103—1671no—
230#103—1671no—
231#103C2×D16711yescharacteristic, complemented
232#102—1671no—
233#102—1671no—
234#102—1671no—
235#102C2×D16711yescharacteristic
236C22⋊C8—1671no—
237C22⋊C8—1671no—
238#97—1671no—
239#97C2×D16711yescharacteristic, complemented
240C24—3341no—
241C2×Q8—3341no—
242C2×Q8—1671no—
243C2×Q8C22×D16711yescharacteristic
244C2×D4—3342no—
245C2×D4—3342no—
246C2×D4—3341no—
247C2×D4—1671no—
248C2×D4C22×D16711yescharacteristic
249C22×C4—21no—
250C22×C4—3341no—
251C22×C4—3341no—
252C22×C4—3341no—
253C22×C4—3341no—
254C22×C4—6684no—
255C22×C4—1671no—
256C22×C4—1671no—
257C22×C4—1671no—
258SD16—22no—
259SD16—3342no—
260SD16—3342no—
261SD16—3342no—
262C2×C8—1671no—
263C2×C8—1671no—
264C2×C8C22×D16711yescharacteristic
265C2×C8C22×D16711yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—3341no—
268C4⋊C4—3341no—
269C4⋊C4—3341no—
270C4⋊C4—3341no—
271C4⋊C4—3341no—
272C4⋊C4—3341no—
273C4⋊C4—3341no—
274C4⋊C4—1671no—
275C4⋊C4—1671no—
276C4⋊C4C22×D16711yescharacteristic
277C4⋊C4C22×D16711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C42—21no—
285C42—3341no—
286C42—1671no—
287C42—1671no—
288C42—1671no—
289C42C22×D16711yescharacteristic
290C23—21no—
291C23—3344no—
292C23—3341no—
293C23—6682no—
294C23—6682no—
295C23—1671no—
296Q8—21no—
297Q8—21no—
298Q8—3341no—
299Q8—3341no—
300Q8—6682no—
301D4—21no—
302D4—3344no—
303D4—3341no—
304D4—1672no—
305D4#1458195012yes—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—3342no—
311C2×C4—3342no—
312C2×C4—3341no—
313C2×C4—3341no—
314C2×C4—3341no—
315C2×C4—3341no—
316C2×C4—3341no—
317C2×C4—6682no—
318C2×C4—6682no—
319C2×C4—6682no—
320C2×C4—6682no—
321C2×C4—1671no—
322C2×C4—1671no—
323C2×C4—1671no—
324C2×C4D4×D16711yescharacteristic
325C2×C4D4×D16711yescharacteristic
326C2×C4C23×D16711yescharacteristic
327C8—21no—
328C8—21no—
329C8—3341no—
330C8—3341no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—3341no—
335C22—3341no—
336C22—3342no—
337C22—6688no—
338C22—6681no—
339C22—6681no—
340C22—1672no—
341C22C2×D4×D16711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3341no—
348C4—3341no—
349C4—6681no—
350C4—6681no—
351C4—6681no—
352C4—1672no—
353C4#41×D16711yescharacteristic
354C4C2×#1458195011yescharacteristic
355C2—42no—
356C2—3342no—
357C2—6682no—
358C2#2492377811yescharacteristic
359C2AGL(1,ℤ/8)×D16711yescharacteristic
360C2#137×D16711yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#440×C167C2
socleC22×C167C22×D4
prime radicalC2×D4×D167C22
hypercenter#440D167
hypocenterC167#3110
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #24923823C2semidirect
3. #24923606C2semidirect
4. #24923603C2semidirect
5. #24923548C2semidirect
6. #24923547C2semidirect
7. #24923473C2semidirect
8. #24923472C2semidirect
11. #24923371C2semidirect
12. #24923369C2semidirect
13. #24923367C2semidirect
14. #24923324C2semidirect
15. #440×C167C2semidirect
21. #19128831C22semidirect
22. #19128830C22semidirect
28. C4×Dic334C22semidirect
38. #19128766C22semidirect
39. #19128765C22semidirect
40. #19128765C22semidirect
41. Dic167⋊C8C22semidirect
42. Dic167⋊C8C22semidirect
43. #19128757C22semidirect
44. #19128756C22semidirect
45. #19128755C22semidirect
51. #19128743C22semidirect
53. C4×Q8×C167C22semidirect
55. #107×C167C22semidirect
56. #103×C167C22semidirect
58. #97×C167C22semidirect
180. C167#3110semidirect
195. #440D167semidirect
211. C4×Q8C2×D167semidirect
225. #107C2×D167semidirect
231. #103C2×D167semidirect
239. #97C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249238232
C4×D4×D1672
#249236062
#249236032
#249235482
#249235472
#249234732
#249234722
#249234212
#249234182
#249233712
#249233692
#249233672
#249233242
#440×C1672
#3110167