Finite Group Databasebeta

← #36336249

Subgroups — #36336249

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#24923976C211yesmaximal, characteristic, complemented
2#24923628C211yesmaximal, characteristic, complemented
3#24923597C211yesmaximal, characteristic, complemented
4#24923585C211yesmaximal, characteristic, complemented
5#24923554C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923496C211yesmaximal, characteristic, complemented
8#24923488C211yesmaximal, characteristic, complemented
9#24923431C211yesmaximal, characteristic, complemented
10#24923425C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923270C211yesmaximal, characteristic, complemented
14C2×#19128756C211yesmaximal, characteristic, complemented
15#548×C167C211yesmaximal, characteristic
16C2×#14581940C2211yescharacteristic
17C2×#14581940C2211yescharacteristic
18C2×#14581938C2211yescharacteristic
19C2×#14581934C2211yescharacteristic
20C22×C167⋊C8C2211yescharacteristic
21#19128832C2211yescharacteristic
22#19128823—21no—
23#19128811C2211yescharacteristic
24#19128811C2211yescharacteristic
25#19128809—21no—
26#19128809—21no—
27C4×Dic334C2211yescharacteristic
28D4×Dic167—21no—
29C2×C4⋊Dic167C2211yescharacteristic
30C2×C4⋊Dic167C2211yescharacteristic
31#19128763C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128756C2211yescharacteristic
39#19128756C2211yescharacteristic
40#19128756C2211yescharacteristic
41#19128755C2211yescharacteristic
42#19128753C2211yescharacteristic
43#19128753C2211yescharacteristic
44#19128753C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48C2×#41×C167C2211yescharacteristic
49C22×Q8×C167C2211yescharacteristic
50#124×C167—21no—
51#124×C167C2211yescharacteristic
52C22⋊Q8×C167—21no—
53C22⋊Q8×C167C2211yescharacteristic
54#121×C167—21no—
55#121×C167C2211yescharacteristic
56C4×Q8×C167C2211yescharacteristic
57C4×D4×C167—21no—
58C4×D4×C167C2211yescharacteristic
59C2×Dic334—21no—
60C2×Dic334C2311yescharacteristic
61C22×Dic167—21no—
62C22×Dic167—21no—
63#14581940—22no—
64#14581940—41no—
65#14581938—41no—
66#14581934—41no—
67C2×C167⋊C8—41no—
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C2×C167⋊C8C2311yescharacteristic
71C2×C167⋊C8C2311yescharacteristic
72C4⋊Dic167—21no—
73C4⋊Dic167—21no—
74C4⋊Dic167—21no—
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77C4⋊Dic167C2311yescharacteristic
78Dic167⋊C4—21no—
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81C22⋊Dic167—21no—
82C22⋊Dic167—21no—
83C4×Dic167—21no—
84#41×C167—22no—
85#41×C167—41no—
86C2×Q8×C167—21no—
87C2×Q8×C167—22no—
88C2×Q8×C167—41no—
89C2×Q8×C167C2311yescharacteristic
90C2×Q8×C167C2311yescharacteristic
91C2×D4×C167—21no—
92C2×D4×C167—21no—
93C2×D4×C167D411yescharacteristic
94C2×D4×C167C2311yescharacteristic
95C22×C4×C167—21no—
96C22×C4×C167—21no—
97C22×C4×C167C2311yescharacteristic
98C22×C4×C167C2311yescharacteristic
99C4⋊C4×C167—21no—
100C4⋊C4×C167—21no—
101C4⋊C4×C167D411yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167—21no—
107C22⋊C4×C167D411yescharacteristic
108C22⋊C4×C167D411yescharacteristic
109C42×C167—21no—
110C42×C167C2311yescharacteristic
111Dic334—21no—
112Dic334—21no—
113Dic334—81no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C2×Dic167—81no—
119C2×Dic167—81no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C167⋊C8—21no—
123C167⋊C8—21no—
124C23×C167—21no—
125C23×C167C2×D411yescharacteristic
126C23×C167C2×D411yescharacteristic
127Q8×C167—21no—
128Q8×C167—21no—
129Q8×C167—21no—
130Q8×C167—42no—
131Q8×C167#4112yes—
132D4×C167—22no—
133D4×C167—21no—
134D4×C167—21no—
135D4×C167—42no—
136D4×C167—42no—
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—42no—
145C2×C4×C167—42no—
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—41no—
157C22×C167—41no—
158C22×C167—41no—
159C22×C167#13712yes—
160C22×C167C22×D411yescharacteristic
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—22no—
164C4×C167—21no—
165C4×C167—21no—
166C4×C167—41no—
167C4×C167—41no—
168C4×C167C2×#4111yescharacteristic
169C4×C167H(5,𝔽2)11yescharacteristic
170Dic167—41no—
171Dic167—41no—
172Dic167—41no—
173Dic167—41no—
174C2×C167—22no—
175C2×C167—41no—
176C2×C167—81no—
177C2×C167#54611yescharacteristic
178C2×C167C2×#13711yescharacteristic
179C2×C167#57711yescharacteristic
180C167#331611yescharacteristic, complemented, Sylow
181#3316—1671nomaximal, Sylow
182#554—1671no—
183#548D16711yescharacteristic
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×#41C2×D16711yescharacteristic
198C22×Q8C2×D16711yescharacteristic
199C2×Q16—1671no—
200C2×Q16—1671no—
201C2×SD16—1671no—
202C2×M16—1671no—
203C22×C8—1671no—
204C4⋊Q8—1671no—
205#125—3341no—
206#124—21no—
207#124C2×D16711yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—3341no—
210C22⋊Q8—3341no—
211C22⋊Q8—1671no—
212C22⋊Q8—1671no—
213C22⋊Q8C2×D16711yescharacteristic
214#121—21no—
215#121C2×D16711yescharacteristic
216C4×Q8—1671no—
217C4×Q8C2×D16711yescharacteristic
218C4×D4—21no—
219C4×D4—3341no—
220C4×D4C2×D16711yescharacteristic
221C2×C4⋊C4—1671no—
222C2×C4⋊C4—1671no—
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—22no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—3341no—
246C2×Q8—1671no—
247C2×Q8C22×D16711yescharacteristic
248C2×Q8C22×D16711yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D16711yescharacteristic
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—3341no—
271C4⋊C4—3341no—
272C4⋊C4—3341no—
273C4⋊C4—3341no—
274C4⋊C4—3341no—
275C4⋊C4—3341no—
276C4⋊C4—1671no—
277C4⋊C4—1671no—
278C4⋊C4—1671no—
279C4⋊C4C22⋊D16711yescharacteristic
280C4⋊C4C22×D16711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4C22⋊D16711yescharacteristic
288C22⋊C4C22⋊D16711yescharacteristic
289C42—21no—
290C42—3341no—
291C42C22×D16711yescharacteristic
292C23—21no—
293C23C2×C22⋊D16711yescharacteristic
294C23C2×C22⋊D16711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—13361no—
300Q8—3341no—
301Q8—3341no—
302Q8#1458194912yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—13361no—
318C2×C4—13361no—
319C2×C4—3341no—
320C2×C4—3341no—
321C2×C4—3341no—
322C2×C4—3341no—
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—41no—
338C22—41no—
339C22—41no—
340C22#1912886212yes—
341C22C22×C22⋊D16711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—6681no—
350C4—6681no—
351C4—6681no—
352C4—6681no—
353C4C2×#1458194911yescharacteristic
354C4#1912887711yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#2492391311yescharacteristic
359C2C2×#1912886211yescharacteristic
360C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radical#548×C167C2
socleC22×C167C22×D4
hypercenter#548D167
hypocenterC167#3316
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #24923976C2semidirect
2. #24923628C2semidirect
3. #24923597C2semidirect
4. #24923585C2semidirect
5. #24923554C2semidirect
6. #24923525C2semidirect
7. #24923496C2semidirect
8. #24923488C2semidirect
9. #24923431C2semidirect
10. #24923425C2semidirect
11. #24923374C2semidirect
12. #24923341C2semidirect
13. #24923270C2semidirect
14. C2×#19128756C2semidirect
180. C167#3316semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249236282
#249235972
#249235852
#249235542
#249235252
#249234962
#249234882
#249234312
#249234252
#249233742
#249233412
#249232702
C2×#191287562
#548×C1672
#3316167