Finite Group Databasebeta

← #36336221

Subgroups — #36336221

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#24923976C211yesmaximal, characteristic, complemented
2#24923915C211yesmaximal, characteristic
3#24923629C211yesmaximal, characteristic, complemented
4#24923605C211yesmaximal, characteristic, complemented
5#24923583C211yesmaximal, characteristic, complemented
6#24923550C211yesmaximal, characteristic, complemented
7#24923515C211yesmaximal, characteristic, complemented
8#24923510C211yesmaximal, characteristic, complemented
9#24923492C211yesmaximal, characteristic, complemented
10#24923433C211yesmaximal, characteristic, complemented
11#24923422C211yesmaximal, characteristic, complemented
12#24923371C211yesmaximal, characteristic, complemented
13#24923271C211yesmaximal, characteristic, complemented
14C2×#19128754C211yesmaximal, characteristic, complemented
15#435×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×D668C2211yescharacteristic
18C2×Dic668C2211yescharacteristic, complemented
19C2×#14581939C2211yescharacteristic
20C2×#14581939C2211yescharacteristic
21#19128832C2211yescharacteristic, complemented
22#19128823—21no—
23#19128820C2211yescharacteristic
24#19128817—21no—
25#19128811C2211yescharacteristic
26#19128811C2211yescharacteristic
27#19128811C2211yescharacteristic, complemented
28#19128809—21no—
29#19128809—21no—
30#19128806C2211yescharacteristic
31#19128805—21no—
32#19128800—21no—
33C4×Dic334C2211yescharacteristic, complemented
34C4×D668C2211yescharacteristic
35D4×Dic167—21no—
36C2×C4⋊Dic167C2211yescharacteristic
37C2×C4⋊Dic167C2211yescharacteristic, complemented
38C22⋊C4×D167—21no—
39#19128766C2211yescharacteristic, complemented
40#19128766C2211yescharacteristic, complemented
41#19128766C2211yescharacteristic, 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, complemented
53C22×C8×C167C2211yescharacteristic
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—
72C22×Dic167—21no—
73C22×Dic167—21no—
74C2×C4×D167—21no—
75C2×C4×D167—21no—
76Dic668—41no—
77#14581939—22no—
78#14581939—41no—
79C4⋊Dic167—21no—
80C4⋊Dic167—21no—
81C4⋊Dic167—21no—
82C4⋊Dic167C2311yescharacteristic
83C4⋊Dic167C2311yescharacteristic
84C4⋊Dic167C2311yescharacteristic
85C4⋊Dic167C2311yescharacteristic
86Dic167⋊C4—21no—
87Dic167⋊C4—21no—
88Dic167⋊C4—21no—
89Dic167⋊C4—21no—
90C22⋊Dic167—21no—
91C22⋊Dic167—21no—
92C22⋊Dic167—21no—
93(C2×D167)⋊C4—21no—
94(C2×D167)⋊C4—21no—
95(C2×D167)⋊C4—21no—
96(C2×D167)⋊C4—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—81no—
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—
134C2×Dic167—81no—
135C2×Dic167—81no—
136C4×D167—42no—
137C4×D167—81no—
138C23×C167C2×D411yescharacteristic
139C23×C167C2×D411yescharacteristic
140D4×C167—22no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—21no—
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2411yescharacteristic
150C8×C167—21no—
151C8×C167—21no—
152C8×C167—21no—
153C8×C167—21no—
154C22×C167—21no—
155C22×C167—21no—
156C22×C167—21no—
157C22×C167—21no—
158C22×C167—21no—
159C22×C167#13712yes—
160C22×C167C22×D411yescharacteristic
161C2×D167—22no—
162C2×D167—41no—
163C2×D167—41no—
164C2×D167—41no—
165C2×D167—41no—
166C2×D167—41no—
167C2×D167—84no—
168C4×C167—21no—
169C4×C167—21no—
170C4×C167—21no—
171C4×C167—21no—
172C4×C167C2×#4111yescharacteristic
173C4×C167H(5,𝔽2)11yescharacteristic
174Dic167—41no—
175Dic167—41no—
176Dic167—41no—
177Dic167—41no—
178Dic167—41no—
179Dic167—41no—
180C2×C167—22no—
181C2×C167—41no—
182C2×C167#54611yescharacteristic
183C2×C167C2×#13711yescharacteristic
184C2×C167#57611yescharacteristic
185D167—42no—
186D167—81no—
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#450—1671no—
200#440—1671no—
201#435D16711yescharacteristic, complemented
202#417—1671no—
203C2×#102—1671no—
204C2×#41—1671no—
205C22×D4—1671no—
206C2×Q16—1671no—
207C2×SD16—1671no—
208C2×SD16—1671no—
209C2×M16C2×D16711yescharacteristic, complemented
210C22×C8C2×D16711yescharacteristic
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⋊Q8—1671no—
220#121—3341no—
221#121—1671no—
222#120—3341no—
223C4×Q8—1671no—
224C4×D4—3341no—
225C4×D4—1671no—
226C4×D4C2×D16711yescharacteristic
227C2×C4⋊C4—1671no—
228C2×C4⋊C4—1671no—
229C2×C22⋊C4—3341no—
230#107—1671no—
231#107—1671no—
232#107—1671no—
233C4⋊C8C2×D16711yescharacteristic, complemented
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#102—1671no—
243#102—1671no—
244C22⋊C8C2×D16711yescharacteristic
245C22⋊C8C2×D16711yescharacteristic, complemented
246#97C2×D16711yescharacteristic, complemented
247C24—3341no—
248#41—3342no—
249#41—6681no—
250C2×Q8—3341no—
251C2×Q8—1671no—
252C2×Q8—1671no—
253C2×D4—3342no—
254C2×D4—3341no—
255C2×D4—3341no—
256C2×D4—3341no—
257C2×D4—6681no—
258C2×D4—1671no—
259C2×D4—1671no—
260C2×D4D66811yescharacteristic
261C22×C4—3341no—
262C22×C4—3341no—
263C22×C4—3341no—
264C22×C4—3341no—
265C22×C4C22×D16711yescharacteristic
266C22×C4C22×D16711yescharacteristic
267Q16—6681no—
268SD16—3342no—
269SD16—6681no—
270M16—41no—
271C2×C8—41no—
272C2×C8C22×D16711yescharacteristic
273C2×C8C22×D16711yescharacteristic
274C2×C8C22×D16711yescharacteristic
275C2×C8C22×D16711yescharacteristic
276C4⋊C4—3341no—
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⋊C4D66811yescharacteristic
288C22⋊C4—3341no—
289C22⋊C4—3341no—
290C22⋊C4—3341no—
291C22⋊C4—3341no—
292C22⋊C4—3341no—
293C22⋊C4—3341no—
294C22⋊C4—3341no—
295C22⋊C4D66811yescharacteristic
296C22⋊C4D66811yescharacteristic
297C42—3341no—
298C42C22×D16711yescharacteristic
299C23—13361no—
300C23—3341no—
301C23—3341no—
302C23—6682no—
303C23—6682no—
304C23C2×D66811yescharacteristic
305C23C2×D66811yescharacteristic
306Q8—13361no—
307Q8—3341no—
308Q8—3341no—
309Q8—3341no—
310Q8—3341no—
311D4—22no—
312D4—13361no—
313D4—3341no—
314D4—3341no—
315D4—3341no—
316D4—6682no—
317D4—6682no—
318D4—1672no—
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×C4C2×D66811yescharacteristic
334C2×C4C2×D66811yescharacteristic
335C2×C4C2×D66811yescharacteristic
336C2×C4C2×D66811yescharacteristic
337C2×C4C23×D16711yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—13364no—
348C22—3342no—
349C22—6681no—
350C22—6681no—
351C22—6681no—
352C22—6681no—
353C22—6681no—
354C22#1912886512yes—
355C22C22×D66811yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—6681no—
361C4—6681no—
362C4—6681no—
363C4—6681no—
364C4—6681no—
365C4—6681no—
366C4#41×D16711yescharacteristic
367C4#1912887711yescharacteristic
368C2—22no—
369C2—41no—
370C2—13361no—
371C2—6682no—
372C2#2492391411yescharacteristic
373C2C2×#1912886511yescharacteristic
374C2#2492407411yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#435×C167C2
socleC22×C167C22×D4
prime radical#24923915C2
hypercenter#435D167
hypocenterC167#3314
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. #24923976C2semidirect
3. #24923629C2semidirect
4. #24923605C2semidirect
5. #24923583C2semidirect
6. #24923550C2semidirect
7. #24923515C2semidirect
8. #24923510C2semidirect
9. #24923492C2semidirect
10. #24923433C2semidirect
11. #24923422C2semidirect
12. #24923371C2semidirect
13. #24923271C2semidirect
14. C2×#19128754C2semidirect
15. #435×C167C2semidirect
18. C2×Dic668C22semidirect
21. #19128832C22semidirect
27. #19128811C22semidirect
33. C4×Dic334C22semidirect
37. C2×C4⋊Dic167C22semidirect
39. #19128766C22semidirect
40. #19128766C22semidirect
41. #19128766C22semidirect
42. #19128757C22semidirect
43. #19128757C22semidirect
44. #19128757C22semidirect
45. #19128757C22semidirect
52. C2×M16×C167C22semidirect
55. C4⋊C8×C167C22semidirect
57. C22⋊C8×C167C22semidirect
58. #97×C167C22semidirect
187. C167#3314semidirect
201. #435D167semidirect
209. C2×M16C2×D167semidirect
233. C4⋊C8C2×D167semidirect
245. C22⋊C8C2×D167semidirect
246. #97C2×D167semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249239152
#249236292
#249236052
#249235832
#249235502
#249235152
#249235102
#249234922
#249234332
#249234222
#249233712
#249232712
C2×#191287542
#435×C1672
#3314167