Finite Group Databasebeta

← #36336223

Subgroups — #36336223

Order 21376.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923977C211yesmaximal, characteristic, complemented
2#24923913C211yesmaximal, characteristic
3#24923627C211yesmaximal, characteristic, complemented
4#24923605C211yesmaximal, characteristic, complemented
5#24923581C211yesmaximal, characteristic, complemented
6#24923547C211yesmaximal, characteristic, complemented
7#24923512C211yesmaximal, characteristic, complemented
8#24923508C211yesmaximal, characteristic, complemented
9#24923489C211yesmaximal, characteristic, complemented
10#24923428C211yesmaximal, characteristic, complemented
11#24923370C211yesmaximal, characteristic, complemented
12#24923337C211yesmaximal, characteristic, complemented
13#24923267C211yesmaximal, characteristic, complemented
14C2×#19128753C211yesmaximal, characteristic, complemented
15#450×C167C211yesmaximal, characteristic, complemented
16C2×#14581949C2211yescharacteristic
17C2×#14581940C2211yescharacteristic, complemented
18C2×#14581933C2211yescharacteristic, complemented
19C22×C167⋊C8C2211yescharacteristic
20#19128831C2211yescharacteristic, complemented
21#19128822—21no—
22#19128819C2211yescharacteristic
23#19128813—21no—
24#19128812—21no—
25#19128811C2211yescharacteristic
26#19128809—21no—
27#19128808C2211yescharacteristic, complemented
28#19128804—21no—
29#19128803C2211yescharacteristic
30#19128801—21no—
31Q8×Dic167C2211yescharacteristic, complemented
32D4×Dic167C2211yescharacteristic
33C4×C22⋊D167C2211yescharacteristic
34(C22⋊D167)⋊C4—21no—
35C2×C4⋊Dic167C2211yescharacteristic
36C4⋊C4×D167C2211yescharacteristic, complemented
37C2×C22⋊Dic167—21no—
38#19128766C2211yescharacteristic, complemented
39#19128766C2211yescharacteristic, complemented
40#19128765C2211yescharacteristic, complemented
41Dic167⋊C8C2211yescharacteristic, complemented
42#19128757C2211yescharacteristic, complemented
43#19128757C2211yescharacteristic, complemented
44#19128756C2211yescharacteristic, complemented
45#19128755C2211yescharacteristic, complemented
46#19128753C2211yescharacteristic
47#19128753C2211yescharacteristic
48#19128753C2211yescharacteristic
49#19128753C2211yescharacteristic
50(C2×D167)⋊C8C2211yescharacteristic, complemented
51#19128742C2211yescharacteristic, complemented
52C22×D4×C167C2211yescharacteristic
53C2×SD16×C167C2211yescharacteristic
54C2×SD16×C167C2211yescharacteristic
55C22⋊Q8×C167C2211yescharacteristic
56#102×C167C2211yescharacteristic
57#102×C167C2211yescharacteristic
58C22⋊C8×C167C2211yescharacteristic
59#14581949—22no—
60#14581949—41no—
61C2×Dic334—21no—
62C2×Dic334C2311yescharacteristic
63C2×C22⋊D167—21no—
64C2×C22⋊D167—21no—
65C2×C22⋊D167D411yescharacteristic
66C22×Dic167—21no—
67C22×Dic167—21no—
68C22×Dic167—21no—
69C2×C4×D167—21no—
70C2×C4×D167C2311yescharacteristic
71#14581940—41no—
72#14581933—41no—
73C2×C167⋊C8—41no—
74C2×C167⋊C8C2311yescharacteristic
75C2×C167⋊C8C2311yescharacteristic
76C4⋊Dic167—21no—
77C4⋊Dic167—21no—
78C4⋊Dic167C2311yescharacteristic
79C4⋊Dic167C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81Dic167⋊C4—21no—
82Dic167⋊C4—21no—
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85Dic167⋊C4D411yescharacteristic
86C22⋊Dic167—21no—
87C22⋊Dic167—21no—
88C22⋊Dic167—21no—
89C22⋊Dic167—21no—
90C22⋊Dic167—21no—
91C22⋊Dic167D411yescharacteristic
92(C2×D167)⋊C4—21no—
93(C2×D167)⋊C4D411yescharacteristic
94C4×Dic167—21no—
95C4×Dic167C2311yescharacteristic
96C24×C167—21no—
97C2×Q8×C167C2311yescharacteristic
98C2×D4×C167—22no—
99C2×D4×C167—21no—
100C2×D4×C167—41no—
101C2×D4×C167C2311yescharacteristic
102C2×D4×C167C2311yescharacteristic
103C22×C4×C167C2311yescharacteristic
104SD16×C167—22no—
105SD16×C167—41no—
106C2×C8×C167C2311yescharacteristic
107C2×C8×C167C2311yescharacteristic
108C4⋊C4×C167—21no—
109C4⋊C4×C167C2311yescharacteristic
110C22⋊C4×C167—21no—
111C22×D167C2×D411yescharacteristic
112Dic334—21no—
113Dic334—21no—
114Dic334—81no—
115C22⋊D167—22no—
116C22⋊D167—42no—
117C22⋊D167—81no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167—42no—
123C2×Dic167—81no—
124C2×Dic167—81no—
125C2×Dic167C2×D411yescharacteristic
126C2×Dic167C2×D411yescharacteristic
127C2×Dic167C2×D411yescharacteristic
128C4×D167—21no—
129C4×D167—21no—
130C4×D167—81no—
131C167⋊C8—21no—
132C167⋊C8—21no—
133C23×C167—21no—
134C23×C167—21no—
135C23×C167—42no—
136C23×C167—42no—
137C23×C167—81no—
138C23×C167C2×D411yescharacteristic
139Q8×C167—21no—
140Q8×C167—21no—
141D4×C167—21no—
142D4×C167—21no—
143D4×C167—21no—
144D4×C167—42no—
145D4×C167#4112yes—
146C2×C4×C167—21no—
147C2×C4×C167—21no—
148C2×C4×C167—21no—
149C2×C4×C167—21no—
150C2×C4×C167C2×D411yescharacteristic
151C2×C4×C167C2411yescharacteristic
152C8×C167—21no—
153C8×C167—21no—
154C22×C167—21no—
155C22×C167—21no—
156C22×C167—22no—
157C22×C167—41no—
158C22×C167—41no—
159C22×C167—41no—
160C22×C167—41no—
161C22×C167—41no—
162C22×C167—84no—
163C22×C167#13712yes—
164C22×C167C22×D411yescharacteristic
165C2×D167—21no—
166C2×D167—21no—
167C2×D167—21no—
168C4×C167—21no—
169C4×C167—41no—
170C4×C167—41no—
171C4×C167C2×#4111yescharacteristic
172C4×C167H(5,𝔽2)11yescharacteristic
173Dic167—21no—
174Dic167—21no—
175Dic167—21no—
176Dic167—41no—
177Dic167—41no—
178Dic167—41no—
179Dic167—41no—
180C2×C167—22no—
181C2×C167—42no—
182C2×C167—81no—
183C2×C167#54611yescharacteristic
184C2×C167C2×#13711yescharacteristic
185C2×C167#57611yescharacteristic
186D167—41no—
187C167#331411yescharacteristic, complemented, Sylow
188#3314—1671nomaximal, Sylow
189#554—1671no—
190#546—1671no—
191#489—1671no—
192#484—1671no—
193#480—1671no—
194#474—1671no—
195#468—1671no—
196#467—1671no—
197#464—1671no—
198#452—1671no—
199#450D16711yescharacteristic, complemented
200#440—1671no—
201#435—1671no—
202#417—1671no—
203C2×#102—1671no—
204C2×#41—1671no—
205C22×D4C2×D16711yescharacteristic
206C2×Q16—1671no—
207C2×SD16C2×D16711yescharacteristic
208C2×SD16C2×D16711yescharacteristic
209C2×M16—1671no—
210C22×C8—1671no—
211C4⋊Q8—1671no—
212#125—3341no—
213#124—1671no—
214#123—3341no—
215C22⋊Q8—3341no—
216C22⋊Q8—3341no—
217C22⋊Q8—1671no—
218C22⋊Q8—1671no—
219C22⋊Q8C2×D16711yescharacteristic
220#121—3341no—
221#121—1671no—
222#120—3341no—
223C4×Q8—1671no—
224C4×D4—3341no—
225C4×D4—1671no—
226C4×D4—1671no—
227C2×C4⋊C4—1671no—
228C2×C4⋊C4—1671no—
229C2×C22⋊C4—3341no—
230#107—1671no—
231#107—1671no—
232#107—1671no—
233C4⋊C8—1671no—
234#103—1671no—
235#103—1671no—
236#103—1671no—
237#103—1671no—
238#102—1671no—
239#102—1671no—
240#102—1671no—
241#102—1671no—
242#102C2×D16711yescharacteristic
243#102C2×D16711yescharacteristic
244C22⋊C8—1671no—
245C22⋊C8C2×D16711yescharacteristic
246#97—1671no—
247C24—21no—
248#41—3342no—
249#41—6681no—
250C2×Q8—3341no—
251C2×Q8—1671no—
252C2×Q8C22×D16711yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—3341no—
257C2×D4—3341no—
258C2×D4—1671no—
259C2×D4C22×D16711yescharacteristic
260C2×D4C22×D16711yescharacteristic
261C22×C4—3341no—
262C22×C4—3341no—
263C22×C4—3341no—
264C22×C4—3341no—
265C22×C4—1671no—
266C22×C4C22×D16711yescharacteristic
267Q16—6681no—
268SD16—22no—
269SD16—41no—
270M16—6681no—
271C2×C8—6681no—
272C2×C8—1671no—
273C2×C8—1671no—
274C2×C8C22×D16711yescharacteristic
275C2×C8C22×D16711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3341no—
278C4⋊C4—3341no—
279C4⋊C4—3341no—
280C4⋊C4—3341no—
281C4⋊C4—3341no—
282C4⋊C4—3341no—
283C4⋊C4—1671no—
284C4⋊C4—1671no—
285C4⋊C4—1671no—
286C4⋊C4—1671no—
287C4⋊C4C22×D16711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—3341no—
292C22⋊C4—3341no—
293C22⋊C4—3341no—
294C22⋊C4—3341no—
295C22⋊C4—1671no—
296C22⋊C4—1671no—
297C42—3341no—
298C42—1671no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—1671no—
305C23D4×D16711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—13361no—
309Q8—3341no—
310Q8—3341no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—13361no—
316D4—3342no—
317D4—6682no—
318D4#1458194912yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—13361no—
324C2×C4—13361no—
325C2×C4—13361no—
326C2×C4—3341no—
327C2×C4—3341no—
328C2×C4—3341no—
329C2×C4—3341no—
330C2×C4—3341no—
331C2×C4—3341no—
332C2×C4—6682no—
333C2×C4—1671no—
334C2×C4—1671no—
335C2×C4—1671no—
336C2×C4D4×D16711yescharacteristic
337C2×C4C23×D16711yescharacteristic
338C8—21no—
339C8—21no—
340C8—3341no—
341C8—3341no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—3341no—
352C22—3341no—
353C22—3341no—
354C22#1912886112yes—
355C22C2×D4×D16711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—3341no—
360C4—3341no—
361C4—3341no—
362C4—6681no—
363C4—6681no—
364C4—6681no—
365C4—6681no—
366C4C2×#1458194911yescharacteristic
367C4#1912887711yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—6681no—
372C2#2492390611yescharacteristic
373C2C2×#1912886111yescharacteristic
374C2#2492407311yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#450×C167C2
socleC22×C167C22×D4
prime radical#24923913C2
hypercenter#450D167
hypocenterC167#3314
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #24923977C2semidirect
3. #24923627C2semidirect
4. #24923605C2semidirect
5. #24923581C2semidirect
6. #24923547C2semidirect
7. #24923512C2semidirect
8. #24923508C2semidirect
9. #24923489C2semidirect
10. #24923428C2semidirect
11. #24923370C2semidirect
12. #24923337C2semidirect
13. #24923267C2semidirect
14. C2×#19128753C2semidirect
15. #450×C167C2semidirect
17. C2×#14581940C22semidirect
18. C2×#14581933C22semidirect
20. #19128831C22semidirect
27. #19128808C22semidirect
31. Q8×Dic167C22semidirect
36. C4⋊C4×D167C22semidirect
38. #19128766C22semidirect
39. #19128766C22semidirect
40. #19128765C22semidirect
41. Dic167⋊C8C22semidirect
42. #19128757C22semidirect
43. #19128757C22semidirect
44. #19128756C22semidirect
45. #19128755C22semidirect
50. (C2×D167)⋊C8C22semidirect
51. #19128742C22semidirect
187. C167#3314semidirect
199. #450D167semidirect

Maximal subgroups

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

subgroupindex
#249239772
#249239132
#249236272
#249236052
#249235812
#249235472
#249235122
#249235082
#249234892
#249234282
#249233702
#249233372
#249232672
C2×#191287532
#450×C1672
#3314167