Finite Group Databasebeta

← #36336243

Subgroups — #36336243

Order 21376.

388 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#24923928C211yesmaximal, characteristic
2#24923631C211yesmaximal, characteristic, complemented
3#24923599C211yesmaximal, characteristic, complemented
4#24923587C211yesmaximal, characteristic, complemented
5#24923551C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923500C211yesmaximal, characteristic, complemented
8#24923491C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923424C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923268C211yesmaximal, characteristic, complemented
14C2×#19128755C211yesmaximal, characteristic, complemented
15#554×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×Dic334C2211yescharacteristic
18C2×#14581940C2211yescharacteristic
19C2×#14581940C2211yescharacteristic
20C2×#14581937C2211yescharacteristic
21C2×#14581934C2211yescharacteristic
22C22×C167⋊C8C2211yescharacteristic
23#19128820C2211yescharacteristic
24#19128818—21no—
25#19128811C2211yescharacteristic
26#19128810—21no—
27#19128806C2211yescharacteristic
28#19128804—21no—
29C4×Dic334C2211yescharacteristic
30(C22⋊D167)⋊C4—21no—
31#19128763C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128755C2211yescharacteristic
37#19128755C2211yescharacteristic
38#19128755C2211yescharacteristic
39#19128755C2211yescharacteristic
40#19128755C2211yescharacteristic
41#19128755C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44#19128752C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48C4⋊Q8×C167C2211yescharacteristic
49#125×C167—21no—
50C22⋊Q8×C167—21no—
51C22⋊Q8×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53C22⋊Q8×C167C2211yescharacteristic
54C4×Q8×C167C2211yescharacteristic
55C4×D4×C167—21no—
56C4×D4×C167C2211yescharacteristic
57C2×C4⋊C4×C167C2211yescharacteristic
58C2×C4⋊C4×C167C2211yescharacteristic
59#14581950—22no—
60#14581950—41no—
61C2×Dic334—22no—
62C2×Dic334—21no—
63C2×Dic334—41no—
64C2×Dic334C2311yescharacteristic
65C2×Dic334C2311yescharacteristic
66C2×D668C2311yescharacteristic
67C2×C22⋊D167—21no—
68C2×C22⋊D167—21no—
69C22×Dic167—21no—
70C2×C4×D167—21no—
71#14581940—22no—
72#14581940—41no—
73#14581937—41no—
74#14581934—41no—
75C2×C167⋊C8—41no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C2×C167⋊C8C2311yescharacteristic
80C4⋊Dic167C2311yescharacteristic
81Dic167⋊C4—21no—
82Dic167⋊C4—21no—
83C22⋊Dic167—21no—
84(C2×D167)⋊C4—21no—
85(C2×D167)⋊C4—21no—
86(C2×D167)⋊C4—21no—
87C4×Dic167—21no—
88C2×Q8×C167—21no—
89C2×Q8×C167C2311yescharacteristic
90C2×D4×C167D411yescharacteristic
91C22×C4×C167—21no—
92C22×C4×C167—21no—
93C22×C4×C167C2311yescharacteristic
94C22×C4×C167C2311yescharacteristic
95C4⋊C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167—21no—
98C4⋊C4×C167—21no—
99C4⋊C4×C167—21no—
100C4⋊C4×C167—21no—
101C4⋊C4×C167D411yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C4⋊C4×C167C2311yescharacteristic
104C4⋊C4×C167C2311yescharacteristic
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167D411yescharacteristic
108C22⋊C4×C167D411yescharacteristic
109C42×C167—21no—
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112Dic334—21no—
113Dic334—21no—
114Dic334—21no—
115Dic334—42no—
116Dic334#4112yes—
117D668—21no—
118D668—21no—
119C22⋊D167—42no—
120C22⋊D167—42no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C2×Dic167—21no—
124C2×Dic167—42no—
125C4×D167—42no—
126C167⋊C8—21no—
127C167⋊C8—21no—
128C167⋊C8—21no—
129C167⋊C8—21no—
130C23×C167C2×D411yescharacteristic
131C23×C167C2×D411yescharacteristic
132Q8×C167—21no—
133Q8×C167—21no—
134Q8×C167—81no—
135D4×C167—22no—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167—21no—
143C2×C4×C167—21no—
144C2×C4×C167—81no—
145C2×C4×C167—81no—
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2×D411yescharacteristic
149C2×C4×C167C2×D411yescharacteristic
150C2×C4×C167C2411yescharacteristic
151C22×C167—21no—
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—21no—
155C22×C167—21no—
156C22×C167#13712yes—
157C22×C167C22×D411yescharacteristic
158C2×D167—41no—
159C2×D167—41no—
160C2×D167—41no—
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—21no—
164C4×C167—21no—
165C4×C167—41no—
166C4×C167—41no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167C2×#4111yescharacteristic
170C4×C167H(5,𝔽2)11yescharacteristic
171Dic167—22no—
172Dic167—41no—
173Dic167—41no—
174C2×C167—22no—
175C2×C167—41no—
176C2×C167#54611yescharacteristic
177C2×C167C2×#13711yescharacteristic
178C2×C167#57711yescharacteristic
179D167—81no—
180C167#331611yescharacteristic, complemented, Sylow
181#3316—1671nomaximal, Sylow
182#554D16711yescharacteristic, complemented
183#548—1671no—
184#489—1671no—
185#483—1671no—
186#481—1671no—
187#475—1671no—
188#470—1671no—
189#465—1671no—
190#464—1671no—
191#452—1671no—
192#451—1671no—
193#441—1671no—
194#435—1671no—
195#417—1671no—
196C2×#103—1671no—
197C2×#41—1671no—
198C22×Q8—1671no—
199C2×Q16—1671no—
200C2×Q16—1671no—
201C2×SD16—1671no—
202C2×M16—1671no—
203C22×C8—1671no—
204C4⋊Q8C2×D16711yescharacteristic
205#125—21no—
206#124—3341no—
207#124—1671no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—3341no—
211C22⋊Q8—1671no—
212C22⋊Q8C2×D16711yescharacteristic
213C22⋊Q8C2×D16711yescharacteristic
214#121—3341no—
215#121—1671no—
216C4×Q8—1671no—
217C4×Q8C2×D16711yescharacteristic
218C4×D4—21no—
219C4×D4—3341no—
220C4×D4C2×D16711yescharacteristic
221C2×C4⋊C4C2×D16711yescharacteristic
222C2×C4⋊C4C2×D16711yescharacteristic
223#106—1671no—
224#106—1671no—
225#106—1671no—
226C4⋊C8—1671no—
227#103—1671no—
228#103—1671no—
229#103—1671no—
230#103—1671no—
231#103—1671no—
232#103—1671no—
233#103—1671no—
234#102—1671no—
235#102—1671no—
236#102—1671no—
237C22⋊C8—1671no—
238C22⋊C8—1671no—
239#97—1671no—
240#41—3342no—
241#41—6681no—
242C2×Q8—21no—
243C2×Q8—3342no—
244C2×Q8—3341no—
245C2×Q8—6681no—
246C2×Q8—1671no—
247C2×Q8—1671no—
248C2×Q8C22×D16711yescharacteristic
249C2×D4—3341no—
250C2×D4—3341no—
251C2×D4—1671no—
252C2×D4C22⋊D16711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3341no—
256C22×C4—3341no—
257C22×C4C22×D16711yescharacteristic
258C22×C4C22×D16711yescharacteristic
259Q16—3342no—
260Q16—6681no—
261SD16—6681no—
262M16—6681no—
263C2×C8—6681no—
264C2×C8—1671no—
265C2×C8—1671no—
266C2×C8—1671no—
267C2×C8—1671no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—1671no—
277C4⋊C4C22⋊D16711yescharacteristic
278C4⋊C4C22×D16711yescharacteristic
279C4⋊C4C22×D16711yescharacteristic
280C4⋊C4C22×D16711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4C22⋊D16711yescharacteristic
288C22⋊C4C22⋊D16711yescharacteristic
289C42—21no—
290C42—3341no—
291C42C22×D16711yescharacteristic
292C23—3341no—
293C23C2×C22⋊D16711yescharacteristic
294C23C2×C22⋊D16711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—3341no—
299Q8—3341no—
300Q8—3341no—
301Q8—6682no—
302Q8—1672no—
303D4—22no—
304D4—3341no—
305D4—3341no—
306D4—6682no—
307D4—6682no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4—3341no—
321C2×C4—6682no—
322C2×C4—6682no—
323C2×C4C2×C22⋊D16711yescharacteristic
324C2×C4C2×C22⋊D16711yescharacteristic
325C2×C4C2×C22⋊D16711yescharacteristic
326C2×C4C2×C22⋊D16711yescharacteristic
327C2×C4C23×D16711yescharacteristic
328C8—3341no—
329C8—3341no—
330C8—3341no—
331C8—3341no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—6681no—
338C22—6681no—
339C22—6681no—
340C22#1912886312yes—
341C22C22×C22⋊D16711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3342no—
351C4—6681no—
352C4—6681no—
353C4#41×D16711yescharacteristic
354C4#1912887811yescharacteristic
355C2—22no—
356C2—41no—
357C2—13361no—
358C2#2492390711yescharacteristic
359C2C2×#1912886311yescharacteristic
360C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#554×C167C2
socleC22×C167C22×D4
prime radical#24923928C2
hypercenter#554D167
hypocenterC167#3316
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #24923631C2semidirect
3. #24923599C2semidirect
4. #24923587C2semidirect
5. #24923551C2semidirect
6. #24923525C2semidirect
7. #24923500C2semidirect
8. #24923491C2semidirect
9. #24923429C2semidirect
10. #24923424C2semidirect
11. #24923374C2semidirect
12. #24923341C2semidirect
13. #24923268C2semidirect
14. C2×#19128755C2semidirect
15. #554×C167C2semidirect
180. C167#3316semidirect
182. #554D167semidirect

Maximal subgroups

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

subgroupindex
#249239282
#249236312
#249235992
#249235872
#249235512
#249235252
#249235002
#249234912
#249234292
#249234242
#249233742
#249233412
#249232682
C2×#191287552
#554×C1672
#3316167