Finite Group Databasebeta

← #36334331

Subgroups — #36334331

Order 21376.

396 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#24923749C211yesmaximal, characteristic, complemented
2#24923585C211yesmaximal, characteristic, complemented
3#24923580C211yesmaximal, characteristic, complemented
4#24923573C211yesmaximal, characteristic, complemented
5#24923566C211yesmaximal, characteristic, complemented
6#24923466C211yesmaximal, characteristic, complemented
7#24923461C211yesmaximal, characteristic, complemented
8#24923412C211yesmaximal, characteristic, complemented
9#24923412C211yesmaximal, characteristic, complemented
10#24923379C211yesmaximal, characteristic, complemented
11#24923369C211yesmaximal, characteristic, complemented
12C4×#14581938C211yesmaximal, characteristic, complemented
13C4×#14581935C211yesmaximal, characteristic, complemented
14#24923330C211yesmaximal, characteristic, complemented
15#540×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C2×#14581938C2211yescharacteristic
18C2×#14581938C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20C2×#14581935C2211yescharacteristic
21C4×Dic334C2211yescharacteristic
22C4×Dic334C2211yescharacteristic
23C4×D668C2211yescharacteristic
24C4×D668C2211yescharacteristic
25(C22⋊D167)⋊C4—21no—
26(C22⋊D167)⋊C4—21no—
27#19128790—21no—
28C2×C4⋊Dic167C2211yescharacteristic
29C4⋊C4×D167—21no—
30#19128765C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35C4⋊(C167⋊C8)C2211yescharacteristic
36#19128755C2211yescharacteristic
37#19128755C2211yescharacteristic
38#19128753C2211yescharacteristic
39#19128753C2211yescharacteristic
40#19128753C2211yescharacteristic
41#19128753C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44C22⋊(C167⋊C8)C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46#19128743C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48#127×C167C2211yescharacteristic
49#125×C167C2211yescharacteristic
50#123×C167—21no—
51#123×C167—21no—
52#121×C167—21no—
53#121×C167—21no—
54#121×C167C2211yescharacteristic
55#121×C167C2211yescharacteristic
56C4×D4×C167C2211yescharacteristic
57C4×D4×C167C2211yescharacteristic
58#117×C167C2211yescharacteristic
59#14581950—22no—
60#14581950—22no—
61C2×Dic334C2311yescharacteristic
62C2×D668C2311yescharacteristic
63C2×C22⋊D167—21no—
64C22×Dic167—21no—
65C2×C4×D167—21no—
66C2×C4×D167—21no—
67C2×C4×D167—21no—
68#14581938—22no—
69#14581938—22no—
70#14581935—22no—
71#14581935—22no—
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C2×C167⋊C8C2311yescharacteristic
75C2×C167⋊C8C2311yescharacteristic
76C4⋊Dic167—21no—
77C4⋊Dic167C2311yescharacteristic
78C4⋊Dic167C2311yescharacteristic
79Dic167⋊C4—21no—
80Dic167⋊C4—21no—
81(C2×D167)⋊C4—21no—
82(C2×D167)⋊C4—21no—
83C4×Dic167—21no—
84C4×Dic167—21no—
85C2×D4×C167—21no—
86C2×D4×C167—21no—
87C2×D4×C167—21no—
88C2×D4×C167—21no—
89C2×D4×C167C2311yescharacteristic
90C2×D4×C167C2311yescharacteristic
91C22×C4×C167—21no—
92C22×C4×C167—21no—
93C22×C4×C167C2311yescharacteristic
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167D411yescharacteristic
97C4⋊C4×C167D411yescharacteristic
98C4⋊C4×C167C2311yescharacteristic
99C4⋊C4×C167C2311yescharacteristic
100C22⋊C4×C167—21no—
101C22⋊C4×C167—21no—
102C22⋊C4×C167—21no—
103C22⋊C4×C167—21no—
104C22⋊C4×C167D411yescharacteristic
105C22⋊C4×C167D411yescharacteristic
106C42×C167C2311yescharacteristic
107C42×C167C2311yescharacteristic
108C22×D167—21no—
109Dic334—21no—
110Dic334#4112yes—
111D668—21no—
112D668#4112yes—
113C22⋊D167—24no—
114C2×Dic167—21no—
115C2×Dic167—21no—
116C2×Dic167—21no—
117C2×Dic167—81no—
118C4×D167—24no—
119C4×D167—81no—
120C4×D167—81no—
121C167⋊C8—21no—
122C167⋊C8—21no—
123C167⋊C8—21no—
124C167⋊C8—21no—
125C23×C167—21no—
126C23×C167—21no—
127C23×C167C2×D411yescharacteristic
128D4×C167—21no—
129D4×C167—21no—
130D4×C167—21no—
131D4×C167—21no—
132D4×C167—81no—
133D4×C167—81no—
134D4×C167—81no—
135D4×C167—81no—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—21no—
140C2×C4×C167—81no—
141C2×C4×C167—81no—
142C2×C4×C167C2×D411yescharacteristic
143C2×C4×C167C2×D411yescharacteristic
144C2×C4×C167C2×D411yescharacteristic
145C2×C4×C167C2×D411yescharacteristic
146C2×C4×C167C2×D411yescharacteristic
147C2×C4×C167C2411yescharacteristic
148C22×C167—21no—
149C22×C167—21no—
150C22×C167—21no—
151C22×C167—41no—
152C22×C167—41no—
153C22×C167—41no—
154C22×C167—41no—
155C22×C167—41no—
156C22×C167—41no—
157C22×C167C22×D411yescharacteristic
158C2×D167—22no—
159C2×D167—41no—
160C2×D167—41no—
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—21no—
164C4×C167—21no—
165C4×C167—21no—
166C4×C167—41no—
167C4×C167—41no—
168C4×C167C2×#4111yescharacteristic
169C4×C167C2×#4111yescharacteristic
170Dic167—22no—
171Dic167—41no—
172Dic167—41no—
173C2×C167—41no—
174C2×C167—81no—
175C2×C167—81no—
176C2×C167#52511yescharacteristic
177C2×C167#57611yescharacteristic
178C2×C167#57711yescharacteristic
179D167—42no—
180C167#311911yescharacteristic, complemented, Sylow
181#3119—1671nomaximal, Sylow
182#540D16711yescharacteristic, complemented
183#520—1671no—
184#481—1671no—
185#480—1671no—
186#479—1671no—
187#478—1671no—
188#460—1671no—
189#459—1671no—
190#449—1671no—
191#449—1671no—
192#442—1671no—
193#440—1671no—
194C4×SD16—1671no—
195C4×D8—1671no—
196#433—1671no—
197C2×#41—1671no—
198C2×SD16—1671no—
199C2×SD16—1671no—
200C2×D8—1671no—
201C2×D8—1671no—
202#127C2×D16711yescharacteristic
203#125C2×D16711yescharacteristic
204#123—21no—
205#123—21no—
206#121—21no—
207#121—21no—
208#121C2×D16711yescharacteristic
209#121C2×D16711yescharacteristic
210C4×Q8—1671no—
211C4×Q8—1671no—
212C4×D4—3341no—
213C4×D4—3341no—
214C4×D4—1671no—
215C4×D4—1671no—
216C4×D4C2×D16711yescharacteristic
217C4×D4C2×D16711yescharacteristic
218#117—3341no—
219#117C2×D16711yescharacteristic
220C2×C4⋊C4—3341no—
221C2×C4⋊C4—1671no—
222#107—1671no—
223#107—1671no—
224#106—1671no—
225#106—1671no—
226C4⋊C8—1671no—
227C4⋊C8—1671no—
228#103—1671no—
229#103—1671no—
230#102—1671no—
231#102—1671no—
232#102—1671no—
233#102—1671no—
234#102—1671no—
235#102—1671no—
236C22⋊C8—1671no—
237C22⋊C8—1671no—
238#97—1671no—
239C4×C8—1671no—
240#41—3342no—
241#41—3342no—
242C2×Q8—1671no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—3341no—
248C2×D4—1671no—
249C2×D4C22×D16711yescharacteristic
250C2×D4C22×D16711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—3341no—
254C22×C4—3341no—
255C22×C4—3341no—
256C22×C4—3341no—
257C22×C4C22×D16711yescharacteristic
258SD16—3342no—
259SD16—3342no—
260D8—3342no—
261D8—3342no—
262C2×C8—1671no—
263C2×C8—1671no—
264C2×C8—1671no—
265C2×C8—1671no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—3341no—
269C4⋊C4—3341no—
270C4⋊C4—3341no—
271C4⋊C4—1671no—
272C4⋊C4—1671no—
273C4⋊C4C22⋊D16711yescharacteristic
274C4⋊C4C22⋊D16711yescharacteristic
275C4⋊C4C22×D16711yescharacteristic
276C4⋊C4C22×D16711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4C22⋊D16711yescharacteristic
284C22⋊C4C22⋊D16711yescharacteristic
285C42—3341no—
286C42—3341no—
287C42C22×D16711yescharacteristic
288C42C22×D16711yescharacteristic
289C23—21no—
290C23—21no—
291C23—3341no—
292C23C2×C22⋊D16711yescharacteristic
293Q8—3341no—
294Q8—1672no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—3344no—
304D4—3341no—
305D4—1672no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—13361no—
313C2×C4—13361no—
314C2×C4—13361no—
315C2×C4—3344no—
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×C4C2×C22⋊D16711yescharacteristic
324C2×C4C23×D16711yescharacteristic
325C8—3341no—
326C8—3341no—
327C8—3341no—
328C8—3341no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3342no—
339C22—6681no—
340C22—6681no—
341C22C22×C22⋊D16711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—3342no—
350C4—6681no—
351C4—6681no—
352C4#41×D16711yescharacteristic
353C4#41×D16711yescharacteristic
354C2—41no—
355C2—81no—
356C2—81no—
357C2—6682no—
358C2#2492377611yescharacteristic
359C2#2492407211yescharacteristic
360C2#2492408011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#540×C167C2
socleC22×C167C22×D4
hypercenter#540D167
hypocenterC167#3119
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #24923749C2semidirect
2. #24923585C2semidirect
3. #24923580C2semidirect
4. #24923573C2semidirect
5. #24923566C2semidirect
6. #24923466C2semidirect
7. #24923461C2semidirect
8. #24923412C2semidirect
9. #24923412C2semidirect
10. #24923379C2semidirect
11. #24923369C2semidirect
12. C4×#14581938C2semidirect
13. C4×#14581935C2semidirect
14. #24923330C2semidirect
15. #540×C167C2semidirect
180. C167#3119semidirect
182. #540D167semidirect

Maximal subgroups

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

subgroupindex
#249237492
#249235852
#249235802
#249235732
#249235662
#249234662
#249234612
#249234122
#249233792
#249233692
C4×#145819382
C4×#145819352
#249233302
#540×C1672
#3119167