Finite Group Databasebeta

← #36336204

Subgroups — #36336204

Order 21376.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24923989C211yesmaximal, characteristic, complemented
2#24923628C211yesmaximal, characteristic, complemented
3#24923607C211yesmaximal, characteristic, complemented
4#24923590C211yesmaximal, characteristic, complemented
5#24923564C211yesmaximal, characteristic, complemented
6#24923513C211yesmaximal, characteristic, complemented
7#24923508C211yesmaximal, characteristic, complemented
8#24923488C211yesmaximal, characteristic, complemented
9#24923431C211yesmaximal, characteristic, complemented
10#24923425C211yesmaximal, characteristic, complemented
11#24923374C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923278C211yesmaximal, characteristic, complemented
14#24923265C211yesmaximal, characteristic, complemented
15#548×C167C211yesmaximal, characteristic
16C2×#14581940C2211yescharacteristic
17C2×#14581940C2211yescharacteristic
18C2×#14581938C2211yescharacteristic
19C2×#14581934C2211yescharacteristic
20C22×C167⋊C8C2211yescharacteristic
21#19128832C2211yescharacteristic
22#19128825—21no—
23#19128825—21no—
24#19128819—21no—
25#19128811C2211yescharacteristic
26#19128811C2211yescharacteristic
27C4×Dic334C2211yescharacteristic
28#19128797—21no—
29#19128791C2211yescharacteristic
30#19128791C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128765C2211yescharacteristic
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
61#14581940—22no—
62#14581940—41no—
63#14581938—41no—
64#14581934—41no—
65C2×C167⋊C8—41no—
66C2×C167⋊C8C2311yescharacteristic
67C2×C167⋊C8C2311yescharacteristic
68C2×C167⋊C8C2311yescharacteristic
69C2×C167⋊C8C2311yescharacteristic
70C4⋊Dic167—21no—
71C4⋊Dic167C2311yescharacteristic
72C4⋊Dic167C2311yescharacteristic
73C4⋊Dic167C2311yescharacteristic
74Dic167⋊C4—21no—
75Dic167⋊C4—21no—
76Dic167⋊C4—21no—
77C22⋊Dic167—21no—
78C22⋊Dic167—21no—
79C22⋊Dic167—21no—
80C22⋊Dic167—21no—
81C4×Dic167—21no—
82C4×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—42no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—21no—
118C167⋊C8—21no—
119C167⋊C8—21no—
120C167⋊C8—21no—
121C167⋊C8—21no—
122C23×C167—21no—
123C23×C167C2×D411yescharacteristic
124C23×C167C2×D411yescharacteristic
125Q8×C167—21no—
126Q8×C167—21no—
127Q8×C167—21no—
128Q8×C167—42no—
129Q8×C167#4112yes—
130D4×C167—21no—
131D4×C167—21no—
132D4×C167—41no—
133D4×C167—42no—
134D4×C167—42no—
135C2×C4×C167—21no—
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—42no—
143C2×C4×C167—42no—
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2×D411yescharacteristic
148C2×C4×C167C2411yescharacteristic
149C22×C167—21no—
150C22×C167—21no—
151C22×C167—21no—
152C22×C167—21no—
153C22×C167—21no—
154C22×C167—21no—
155C22×C167—41no—
156C22×C167—41no—
157C22×C167—41no—
158C22×C167C22×D411yescharacteristic
159C4×C167—21no—
160C4×C167—21no—
161C4×C167—22no—
162C4×C167—21no—
163C4×C167—21no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167C2×#4111yescharacteristic
167C4×C167H(5,𝔽2)11yescharacteristic
168Dic167—41no—
169Dic167—41no—
170Dic167—41no—
171Dic167—41no—
172C2×C167—41no—
173C2×C167—41no—
174C2×C167—81no—
175C2×C167#54611yescharacteristic
176C2×C167#57511yescharacteristic
177C2×C167#57811yescharacteristic
178C167#331311yescharacteristic, complemented, Sylow
179#3313—1671nomaximal, Sylow
180#556—1671no—
181#548D16711yescharacteristic
182#489—1671no—
183#485—1671no—
184#482—1671no—
185#477—1671no—
186#468—1671no—
187#467—1671no—
188#464—1671no—
189#452—1671no—
190#451—1671no—
191#441—1671no—
192#435—1671no—
193#419—1671no—
194#416—1671no—
195C2×#41C2×D16711yescharacteristic
196C22×Q8C2×D16711yescharacteristic
197C2×Q16—1671no—
198C2×Q16—1671no—
199C2×SD16—1671no—
200C2×M16—1671no—
201C22×C8—1671no—
202C4⋊Q8—1671no—
203#126—3341no—
204#126—3341no—
205#124—21no—
206#124—3341no—
207#124C2×D16711yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—1671no—
210C22⋊Q8—1671no—
211C22⋊Q8C2×D16711yescharacteristic
212#121—21no—
213#121C2×D16711yescharacteristic
214C4×Q8—3341no—
215C4×Q8—1671no—
216C4×Q8C2×D16711yescharacteristic
217C4×D4—21no—
218C4×D4C2×D16711yescharacteristic
219#117—1671no—
220#117—1671no—
221#107—1671no—
222#107—1671no—
223#106—1671no—
224C4⋊C8—1671no—
225#103—1671no—
226#103—1671no—
227#103—1671no—
228#103—1671no—
229#103—1671no—
230#103—1671no—
231#103—1671no—
232#102—1671no—
233#102—1671no—
234#102—1671no—
235C22⋊C8—1671no—
236C22⋊C8—1671no—
237#97—1671no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—3341no—
244C2×Q8—1671no—
245C2×Q8C22×D16711yescharacteristic
246C2×Q8C22×D16711yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4C22⋊D16711yescharacteristic
250C2×D4C22×D16711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4C22×D16711yescharacteristic
254C22×C4C22×D16711yescharacteristic
255Q16—3342no—
256Q16—6681no—
257SD16—6681no—
258M16—6681no—
259C2×C8—6681no—
260C2×C8—1671no—
261C2×C8—1671no—
262C2×C8—1671no—
263C2×C8—1671no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—3341no—
267C4⋊C4—3341no—
268C4⋊C4—3341no—
269C4⋊C4—3341no—
270C4⋊C4—1671no—
271C4⋊C4—1671no—
272C4⋊C4—1671no—
273C4⋊C4C22⋊D16711yescharacteristic
274C4⋊C4C22×D16711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4C22⋊D16711yescharacteristic
284C22⋊C4C22⋊D16711yescharacteristic
285C42—21no—
286C42—3341no—
287C42—3341no—
288C42—3341no—
289C42C22×D16711yescharacteristic
290C23—21no—
291C23C2×C22⋊D16711yescharacteristic
292C23C2×C22⋊D16711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—3341no—
298Q8—3341no—
299Q8—6682no—
300Q8#1458194912yes—
301D4—21no—
302D4—21no—
303D4—41no—
304D4—42no—
305D4—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3341no—
316C2×C4—3341no—
317C2×C4—3341no—
318C2×C4—3341no—
319C2×C4C2×C22⋊D16711yescharacteristic
320C2×C4C2×C22⋊D16711yescharacteristic
321C2×C4C2×C22⋊D16711yescharacteristic
322C2×C4C2×C22⋊D16711yescharacteristic
323C2×C4C23×D16711yescharacteristic
324C8—3341no—
325C8—3341no—
326C8—3341no—
327C8—3341no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C22×C22⋊D16711yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—6681no—
346C4—6681no—
347C4—6681no—
348C4—6681no—
349C4C2×#1458194911yescharacteristic
350C4#1912887711yescharacteristic
351C2—41no—
352C2—41no—
353C2—81no—
354C2#2492391311yescharacteristic
355C2#2492406711yescharacteristic
356C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting, prime radical#548×C167C2
socleC22×C167C22×D4
hypercenter#548D167
hypocenterC167#3313
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. #24923989C2semidirect
2. #24923628C2semidirect
3. #24923607C2semidirect
4. #24923590C2semidirect
5. #24923564C2semidirect
6. #24923513C2semidirect
7. #24923508C2semidirect
8. #24923488C2semidirect
9. #24923431C2semidirect
10. #24923425C2semidirect
11. #24923374C2semidirect
12. #24923341C2semidirect
13. #24923278C2semidirect
14. #24923265C2semidirect
178. C167#3313semidirect

Maximal subgroups

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

subgroupindex
#249239892
#249236282
#249236072
#249235902
#249235642
#249235132
#249235082
#249234882
#249234312
#249234252
#249233742
#249233412
#249232782
#249232652
#548×C1672
#3313167