Finite Group Databasebeta

← #36336251

Subgroups — #36336251

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#24923928C211yesmaximal, characteristic
3#24923629C211yesmaximal, characteristic, complemented
4#24923600C211yesmaximal, characteristic, complemented
5#24923588C211yesmaximal, characteristic, complemented
6#24923555C211yesmaximal, characteristic, complemented
7#24923527C211yesmaximal, characteristic, complemented
8#24923498C211yesmaximal, characteristic, complemented
9#24923492C211yesmaximal, characteristic, complemented
10#24923433C211yesmaximal, characteristic, complemented
11#24923427C211yesmaximal, characteristic, complemented
12#24923377C211yesmaximal, characteristic, complemented
13#24923271C211yesmaximal, characteristic, complemented
14C2×#19128757C211yesmaximal, characteristic, complemented
15#435×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C22×Dic334C2211yescharacteristic
18C2×Dic668C2211yescharacteristic
19C2×Dic668C2211yescharacteristic
20C2×#14581939C2211yescharacteristic
21#19128832C2211yescharacteristic
22#19128823—21no—
23#19128820C2211yescharacteristic
24#19128818—21no—
25#19128811C2211yescharacteristic
26#19128811C2211yescharacteristic
27#19128811C2211yescharacteristic
28#19128810—21no—
29#19128809—21no—
30#19128809—21no—
31#19128806C2211yescharacteristic
32#19128804—21no—
33C4×Dic334C2211yescharacteristic
34C4×Dic334C2211yescharacteristic
35D4×Dic167—21no—
36(C22⋊D167)⋊C4—21no—
37C2×C4⋊Dic167C2211yescharacteristic
38C2×C4⋊Dic167C2211yescharacteristic
39#19128764C2211yescharacteristic
40#19128764C2211yescharacteristic
41#19128764C2211yescharacteristic
42#19128757C2211yescharacteristic
43#19128757C2211yescharacteristic
44#19128757C2211yescharacteristic
45#19128757C2211yescharacteristic
46#19128757C2211yescharacteristic
47#19128757C2211yescharacteristic
48#19128757C2211yescharacteristic
49#19128754C2211yescharacteristic
50#19128754C2211yescharacteristic
51#19128754C2211yescharacteristic
52C2×M16×C167C2211yescharacteristic
53C22×C8×C167C2211yescharacteristic
54C4×D4×C167C2211yescharacteristic
55C4⋊C8×C167C2211yescharacteristic
56C22⋊C8×C167C2211yescharacteristic
57C22⋊C8×C167C2211yescharacteristic
58#97×C167C2211yescharacteristic
59#14581950—22no—
60#14581950—41no—
61C2×Dic334—22no—
62C2×Dic334—21no—
63C2×Dic334—21no—
64C2×Dic334—41no—
65C2×Dic334C2311yescharacteristic
66C2×Dic334C2311yescharacteristic
67C2×Dic334C2311yescharacteristic
68C2×D668C2311yescharacteristic
69C2×C22⋊D167—21no—
70C2×C22⋊D167—21no—
71C22×Dic167—21no—
72C22×Dic167—21no—
73C22×Dic167—21no—
74C2×C4×D167—21no—
75Dic668—22no—
76Dic668—41no—
77#14581939—41no—
78C4⋊Dic167—21no—
79C4⋊Dic167—21no—
80C4⋊Dic167—21no—
81C4⋊Dic167C2311yescharacteristic
82C4⋊Dic167C2311yescharacteristic
83C4⋊Dic167C2311yescharacteristic
84C4⋊Dic167C2311yescharacteristic
85Dic167⋊C4—21no—
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—
96C4×Dic167—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—
112Dic334—21no—
113Dic334—21no—
114Dic334—21no—
115Dic334—21no—
116Dic334—21no—
117Dic334—42no—
118Dic334—81no—
119Dic334#4112yes—
120D668—21no—
121D668—21no—
122C22⋊D167—42no—
123C22⋊D167—42no—
124C2×Dic167—21no—
125C2×Dic167—21no—
126C2×Dic167—21no—
127C2×Dic167—21no—
128C2×Dic167—21no—
129C2×Dic167—21no—
130C2×Dic167—21no—
131C2×Dic167—42no—
132C2×Dic167—81no—
133C2×Dic167—81no—
134C4×D167—42no—
135C23×C167C2×D411yescharacteristic
136C23×C167C2×D411yescharacteristic
137D4×C167—22no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—21no—
141C2×C4×C167—21no—
142C2×C4×C167C2×D411yescharacteristic
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2411yescharacteristic
147C8×C167—21no—
148C8×C167—21no—
149C8×C167—21no—
150C8×C167—21no—
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×C167C2×#4111yescharacteristic
166C4×C167H(5,𝔽2)11yescharacteristic
167Dic167—22no—
168Dic167—41no—
169Dic167—41no—
170Dic167—41no—
171Dic167—41no—
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#554—1671no—
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#435D16711yescharacteristic, complemented
195#417—1671no—
196C2×#103—1671no—
197C2×#41—1671no—
198C22×Q8—1671no—
199C2×Q16—1671no—
200C2×Q16—1671no—
201C2×SD16—1671no—
202C2×M16C2×D16711yescharacteristic
203C22×C8C2×D16711yescharacteristic
204C4⋊Q8—1671no—
205#125—3341no—
206#124—3341no—
207#124—1671no—
208C22⋊Q8—3341no—
209C22⋊Q8—3341no—
210C22⋊Q8—3341no—
211C22⋊Q8—1671no—
212C22⋊Q8—1671no—
213C22⋊Q8—1671no—
214#121—3341no—
215#121—1671no—
216C4×Q8—1671no—
217C4×Q8—1671no—
218C4×D4—3341no—
219C4×D4—3341no—
220C4×D4C2×D16711yescharacteristic
221C2×C4⋊C4—1671no—
222C2×C4⋊C4—1671no—
223#106—1671no—
224#106—1671no—
225#106—1671no—
226C4⋊C8C2×D16711yescharacteristic
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⋊C8C2×D16711yescharacteristic
238C22⋊C8C2×D16711yescharacteristic
239#97C2×D16711yescharacteristic
240#41—3342no—
241#41—6681no—
242C2×Q8—3342no—
243C2×Q8—3341no—
244C2×Q8—3341no—
245C2×Q8—6681no—
246C2×Q8—1671no—
247C2×Q8—1671no—
248C2×Q8—1671no—
249C2×D4—3341no—
250C2×D4—3341no—
251C2×D4—1671no—
252C2×D4D66811yescharacteristic
253C22×C4—3341no—
254C22×C4—3341no—
255C22×C4—3341no—
256C22×C4—3341no—
257C22×C4C22×D16711yescharacteristic
258C22×C4C22×D16711yescharacteristic
259Q16—3342no—
260Q16—6681no—
261SD16—6681no—
262M16—41no—
263C2×C8—41no—
264C2×C8C22×D16711yescharacteristic
265C2×C8C22×D16711yescharacteristic
266C2×C8C22×D16711yescharacteristic
267C2×C8C22×D16711yescharacteristic
268C4⋊C4—3341no—
269C4⋊C4—3341no—
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⋊C4—1671no—
280C4⋊C4D66811yescharacteristic
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4—3341no—
286C22⋊C4—3341no—
287C22⋊C4D66811yescharacteristic
288C22⋊C4D66811yescharacteristic
289C42—3341no—
290C42—3341no—
291C42C22×D16711yescharacteristic
292C23—3341no—
293C23C2×D66811yescharacteristic
294C23C2×D66811yescharacteristic
295Q8—13361no—
296Q8—3341no—
297Q8—3341no—
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—13361no—
313C2×C4—13361no—
314C2×C4—3341no—
315C2×C4—3341no—
316C2×C4—3341no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4—3341no—
320C2×C4—3341no—
321C2×C4—6682no—
322C2×C4—6682no—
323C2×C4C2×D66811yescharacteristic
324C2×C4C2×D66811yescharacteristic
325C2×C4C2×D66811yescharacteristic
326C2×C4C2×D66811yescharacteristic
327C2×C4C23×D16711yescharacteristic
328C8—21no—
329C8—21no—
330C8—21no—
331C8—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—6681no—
338C22—6681no—
339C22—6681no—
340C22#1912886512yes—
341C22C22×D66811yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—3342no—
347C4—6681no—
348C4—6681no—
349C4—6681no—
350C4—6681no—
351C4—6681no—
352C4—6681no—
353C4#41×D16711yescharacteristic
354C4#1912887711yescharacteristic
355C2—22no—
356C2—41no—
357C2—13361no—
358C2#2492391411yescharacteristic
359C2C2×#1912886511yescharacteristic
360C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#435×C167C2
socleC22×C167C22×D4
prime radical#24923928C2
hypercenter#435D167
hypocenterC167#3316
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. #24923600C2semidirect
5. #24923588C2semidirect
6. #24923555C2semidirect
7. #24923527C2semidirect
8. #24923498C2semidirect
9. #24923492C2semidirect
10. #24923433C2semidirect
11. #24923427C2semidirect
12. #24923377C2semidirect
13. #24923271C2semidirect
14. C2×#19128757C2semidirect
15. #435×C167C2semidirect
180. C167#3316semidirect
194. #435D167semidirect

Maximal subgroups

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

subgroupindex
#249239762
#249239282
#249236292
#249236002
#249235882
#249235552
#249235272
#249234982
#249234922
#249234332
#249234272
#249233772
#249232712
C2×#191287572
#435×C1672
#3316167