Finite Group Databasebeta

← #36334049

Subgroups — #36334049

Order 21376.

420 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923776C211yesmaximal, characteristic
2#117×D167C211yesmaximal, characteristic, complemented
3#24923609C211yesmaximal, characteristic, complemented
4#24923608C211yesmaximal, characteristic, complemented
5#24923603C211yesmaximal, characteristic, complemented
6#24923598C211yesmaximal, characteristic, complemented
7#24923592C211yesmaximal, characteristic, complemented
8#24923591C211yesmaximal, characteristic, complemented
9#24923586C211yesmaximal, characteristic, complemented
10#24923581C211yesmaximal, characteristic, complemented
11#24923307C211yesmaximal, characteristic, complemented
12#24923305C211yesmaximal, characteristic, complemented
13#24923265C211yesmaximal, characteristic, complemented
14C2×(C2×D167)⋊C8C211yesmaximal, characteristic, complemented
15#416×C167C211yesmaximal, characteristic, complemented
16C22×C4×D167C2211yescharacteristic
17C22×C167⋊C8C2211yescharacteristic
18#19128813—21no—
19#19128808C2211yescharacteristic
20#19128808C2211yescharacteristic
21#19128803C2211yescharacteristic
22#19128803C2211yescharacteristic
23#19128800—21no—
24C4×C22⋊D167—21no—
25C4×C22⋊D167—21no—
26#19128791C2211yescharacteristic
27#19128791C2211yescharacteristic
28#19128790C2211yescharacteristic
29#19128790C2211yescharacteristic
30#19128789—21no—
31#19128788—21no—
32C4⋊C4×D167C2211yescharacteristic
33C4⋊C4×D167C2211yescharacteristic
34C22⋊C4×D167—21no—
35C42×D167—21no—
36#19128766C2211yescharacteristic, complemented
37#19128766C2211yescharacteristic, complemented
38#19128765C2211yescharacteristic, complemented
39#19128765C2211yescharacteristic, complemented
40#19128764C2211yescharacteristic, complemented
41#19128764C2211yescharacteristic, complemented
42#19128763C2211yescharacteristic, complemented
43#19128763C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128756C2211yescharacteristic, complemented
46#19128753C2211yescharacteristic, complemented
47#19128753C2211yescharacteristic, complemented
48(C2×D167)⋊C8C2211yescharacteristic
49(C2×D167)⋊C8C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic
51(C2×D167)⋊C8C2211yescharacteristic
52C2×#41×C167C2211yescharacteristic
53C22×C8×C167C2211yescharacteristic
54#117×C167C2211yescharacteristic
55#103×C167C2211yescharacteristic, complemented
56#103×C167C2211yescharacteristic, complemented
57#102×C167C2211yescharacteristic, complemented
58#102×C167C2211yescharacteristic, complemented
59C23×D167D411yescharacteristic
60C2×C22⋊D167—21no—
61C2×C22⋊D167—21no—
62C22×Dic167D411yescharacteristic
63C2×C4×D167—21no—
64C2×C4×D167—21no—
65C2×C4×D167—22no—
66C2×C4×D167—22no—
67C2×C4×D167D411yescharacteristic
68C2×C4×D167D411yescharacteristic
69C2×C4×D167C2311yescharacteristic
70C2×C4×D167C2311yescharacteristic
71C2×C167⋊C8—22no—
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C4⋊Dic167C2311yescharacteristic
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77C4⋊Dic167C2311yescharacteristic
78Dic167⋊C4—21no—
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81Dic167⋊C4—21no—
82C22⋊Dic167—21no—
83C22⋊Dic167—21no—
84C22⋊Dic167—21no—
85C22⋊Dic167—21no—
86(C2×D167)⋊C4—21no—
87(C2×D167)⋊C4—21no—
88(C2×D167)⋊C4—21no—
89(C2×D167)⋊C4—21no—
90C4×Dic167—21no—
91C4×Dic167—21no—
92C4×Dic167—21no—
93C4×Dic167—21no—
94#41×C167—22no—
95#41×C167—22no—
96C2×Q8×C167C2311yescharacteristic
97C2×D4×C167—21no—
98C2×D4×C167C2311yescharacteristic
99C22×C4×C167—21no—
100C22×C4×C167C2311yescharacteristic
101C2×C8×C167—22no—
102C2×C8×C167C2311yescharacteristic
103C2×C8×C167C2311yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C4⋊C4×C167C2311yescharacteristic
106C22⋊C4×C167—21no—
107C42×C167—21no—
108C22×D167—22no—
109C22×D167—22no—
110C22×D167—41no—
111C22×D167C2×D411yescharacteristic
112C22×D167C2×D411yescharacteristic
113C22⋊D167—42no—
114C22⋊D167—42no—
115C2×Dic167—22no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167C2×D411yescharacteristic
121C2×Dic167C2×D411yescharacteristic
122C4×D167—22no—
123C4×D167—22no—
124C4×D167—21no—
125C4×D167—21no—
126C4×D167—21no—
127C4×D167—21no—
128C4×D167—42no—
129C4×D167—42no—
130C167⋊C8—21no—
131C167⋊C8—21no—
132C23×C167—21no—
133C23×C167C2×D411yescharacteristic
134Q8×C167—21no—
135Q8×C167—21no—
136D4×C167—21no—
137D4×C167—21no—
138D4×C167—42no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—42no—
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167#4112yes—
145C2×C4×C167#4112yes—
146C2×C4×C167C2411yescharacteristic
147C8×C167—21no—
148C8×C167—21no—
149C22×C167—21no—
150C22×C167—21no—
151C22×C167—41no—
152C22×C167—41no—
153C22×C167—41no—
154C22×C167#12312yes—
155C22×C167C22×D411yescharacteristic
156C2×D167—21no—
157C2×D167—21no—
158C2×D167—21no—
159C2×D167—21no—
160C2×D167—21no—
161C2×D167—44no—
162C2×D167#13512yes—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167#12312yes—
167C4×C167C2×#4111yescharacteristic
168C4×C167C2×#4111yescharacteristic
169Dic167—21no—
170Dic167—21no—
171Dic167—41no—
172Dic167—41no—
173Dic167—41no—
174Dic167—41no—
175C2×C167—22no—
176C2×C167—81no—
177C2×C167C2×#12311yescharacteristic
178C2×C167C2×#13511yescharacteristic
179C2×C167#57511yescharacteristic
180D167—22no—
181D167—41no—
182C167#309111yescharacteristic, complemented, Sylow
183#3091—1671nomaximal, Sylow
184#525—1671no—
185C2×#117—1671no—
186#485—1671no—
187#485—1671no—
188#484—1671no—
189#483—1671no—
190#482—1671no—
191#482—1671no—
192#481—1671no—
193#480—1671no—
194#427—1671no—
195#427—1671no—
196#416—1671no—
197#416D16711yescharacteristic, complemented
198C2×C22⋊C8—1671no—
199C2×#41C2×D16711yescharacteristic
200C23×C4—1671no—
201C22×C8—1671no—
202C22×C8C2×D16711yescharacteristic
203#123—3341no—
204C22⋊Q8—1671no—
205C22⋊Q8—1671no—
206#121—1671no—
207#121—1671no—
208#120—3341no—
209C4×D4—3341no—
210C4×D4—3341no—
211#117—3341no—
212#117—3341no—
213#117—1671no—
214#117—1671no—
215#117—1671no—
216#117—1671no—
217#117C2×D16711yescharacteristic
218C2×C4⋊C4—1671no—
219C2×C4⋊C4—1671no—
220C2×C22⋊C4—3341no—
221C2×C42—3341no—
222#107—1671no—
223#107—1671no—
224#107—1671no—
225#107—1671no—
226#106—1671no—
227#106—1671no—
228#106—1671no—
229#106—1671no—
230#103—1671no—
231#103—1671no—
232#103C2×D16711yescharacteristic, complemented
233#103C2×D16711yescharacteristic, complemented
234#102—1671no—
235#102—1671no—
236#102C2×D16711yescharacteristic, complemented
237#102C2×D16711yescharacteristic, complemented
238C22⋊C8—1671no—
239C22⋊C8—1671no—
240C22⋊C8—1671no—
241C22⋊C8—1671no—
242C24—1671no—
243#41—22no—
244#41—22no—
245C2×Q8C22×D16711yescharacteristic
246C2×D4—21no—
247C2×D4—3341no—
248C2×D4—3341no—
249C2×D4C22×D16711yescharacteristic
250C22×C4—21no—
251C22×C4—3341no—
252C22×C4—3341no—
253C22×C4—3342no—
254C22×C4—3342no—
255C22×C4—1671no—
256C22×C4—1671no—
257C22×C4—1671no—
258C22×C4—1671no—
259C22×C4—1671no—
260C22×C4C22×D16711yescharacteristic
261C2×C8—22no—
262C2×C8—3342no—
263C2×C8—1671no—
264C2×C8—1671no—
265C2×C8C22×D16711yescharacteristic
266C2×C8C22×D16711yescharacteristic
267C4⋊C4—3341no—
268C4⋊C4—3341no—
269C4⋊C4—3341no—
270C4⋊C4—3341no—
271C4⋊C4—1671no—
272C4⋊C4—1671no—
273C4⋊C4—1671no—
274C4⋊C4—1671no—
275C4⋊C4C22×D16711yescharacteristic
276C4⋊C4C22×D16711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—3341no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C42—21no—
287C42—3341no—
288C42—3341no—
289C42—3341no—
290C42—3341no—
291C23—21no—
292C23—3342no—
293C23—3342no—
294C23—6681no—
295C23—1671no—
296C23—1671no—
297C23D4×D16711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—21no—
301D4—21no—
302D4—42no—
303D4—6682no—
304D4—6682no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—3342no—
310C2×C4—3341no—
311C2×C4—3341no—
312C2×C4—3341no—
313C2×C4—3341no—
314C2×C4—3342no—
315C2×C4—3342no—
316C2×C4—3341no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4—6682no—
321C2×C4—6682no—
322C2×C4—1671no—
323C2×C4—1671no—
324C2×C4D4×D16711yescharacteristic
325C2×C4#1458194912yes—
326C2×C4#1458195012yes—
327C2×C4C23×D16711yescharacteristic
328C8—21no—
329C8—21no—
330C8—3341no—
331C8—3341no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—3341no—
338C22—3341no—
339C22—3341no—
340C22—3341no—
341C22—3341no—
342C22—6684no—
343C22—1672no—
344C22#1912881412yes—
345C22C2×D4×D16711yescharacteristic
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3341no—
350C4—3341no—
351C4—6681no—
352C4—6681no—
353C4—6681no—
354C4—6681no—
355C4#1912881412yes—
356C4C2×#1458194911yescharacteristic
357C4C2×#1458195011yescharacteristic
358C2—22no—
359C2—81no—
360C2—3342no—
361C2—6681no—
362C2C2×#1912881411yescharacteristic
363C2#135×D16711yescharacteristic
364C2#2492406611yescharacteristic
subgroupG/H
centerC2×C4D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#416×C167C2
socleC22×C167C22×D4
prime radical#24923776C2
hypercenter#416D167
hypocenterC167#3091
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #117×D167C2semidirect
3. #24923609C2semidirect
4. #24923608C2semidirect
5. #24923603C2semidirect
6. #24923598C2semidirect
7. #24923592C2semidirect
8. #24923591C2semidirect
9. #24923586C2semidirect
10. #24923581C2semidirect
11. #24923307C2semidirect
12. #24923305C2semidirect
13. #24923265C2semidirect
14. C2×(C2×D167)⋊C8C2semidirect
15. #416×C167C2semidirect
36. #19128766C22semidirect
37. #19128766C22semidirect
38. #19128765C22semidirect
39. #19128765C22semidirect
40. #19128764C22semidirect
41. #19128764C22semidirect
42. #19128763C22semidirect
43. #19128763C22semidirect
44. #19128756C22semidirect
45. #19128756C22semidirect
46. #19128753C22semidirect
47. #19128753C22semidirect
55. #103×C167C22semidirect
56. #103×C167C22semidirect
57. #102×C167C22semidirect
58. #102×C167C22semidirect
182. C167#3091semidirect
197. #416D167semidirect
232. #103C2×D167semidirect
233. #103C2×D167semidirect
236. #102C2×D167semidirect
237. #102C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249237762
#117×D1672
#249236092
#249236082
#249236032
#249235982
#249235922
#249235912
#249235862
#249235812
#249233072
#249233052
#249232652
C2×(C2×D167)⋊C82
#416×C1672
#3091167