Finite Group Databasebeta

← #36334337

Subgroups — #36334337

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#24923874C211yesmaximal, characteristic, complemented
2#24923587C211yesmaximal, characteristic, complemented
3#24923582C211yesmaximal, characteristic, complemented
4#24923576C211yesmaximal, characteristic, complemented
5#24923569C211yesmaximal, characteristic, complemented
6#24923469C211yesmaximal, characteristic, complemented
7#24923464C211yesmaximal, characteristic, complemented
8#24923414C211yesmaximal, characteristic, complemented
9#24923414C211yesmaximal, characteristic, complemented
10#24923379C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12C4×#14581937C211yesmaximal, characteristic, complemented
13C4×#14581935C211yesmaximal, characteristic, complemented
14#24923330C211yesmaximal, characteristic, complemented
15#520×C167C211yesmaximal, characteristic, complemented
16C2×#14581937C2211yescharacteristic
17C2×#14581937C2211yescharacteristic
18C2×#14581935C2211yescharacteristic
19C2×#14581935C2211yescharacteristic
20#19128829C2211yescharacteristic
21#19128824C2211yescharacteristic
22#19128817—21no—
23#19128817—21no—
24#19128806C2211yescharacteristic
25#19128806C2211yescharacteristic
26#19128805—21no—
27#19128805—21no—
28C4×D668C2211yescharacteristic
29C4×D668C2211yescharacteristic
30#19128765C2211yescharacteristic
31#19128765C2211yescharacteristic
32#19128763C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35C4⋊(C167⋊C8)C2211yescharacteristic
36#19128756C2211yescharacteristic
37#19128756C2211yescharacteristic
38#19128753C2211yescharacteristic
39#19128753C2211yescharacteristic
40#19128752C2211yescharacteristic
41#19128752C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44C22⋊(C167⋊C8)C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46#19128743C2211yescharacteristic
47C4×C167⋊C8C2211yescharacteristic
48C2×#41×C167C2211yescharacteristic
49C4×Q8×C167C2211yescharacteristic
50C4×Q8×C167C2211yescharacteristic
51C4×D4×C167—21no—
52C4×D4×C167—21no—
53C4×D4×C167C2211yescharacteristic
54C4×D4×C167C2211yescharacteristic
55#117×C167—21no—
56#117×C167C2211yescharacteristic
57C2×C4⋊C4×C167—21no—
58C2×C4⋊C4×C167C2211yescharacteristic
59C2×D668—21no—
60C2×D668—21no—
61C2×D668C2311yescharacteristic
62C2×D668C2311yescharacteristic
63C2×C22⋊D167—21no—
64C2×C22⋊D167—21no—
65C2×C4×D167—21no—
66C2×C4×D167—21no—
67#14581937—22no—
68#14581937—22no—
69#14581935—22no—
70#14581935—22no—
71C2×C167⋊C8C2311yescharacteristic
72C2×C167⋊C8C2311yescharacteristic
73C2×C167⋊C8C2311yescharacteristic
74C2×C167⋊C8C2311yescharacteristic
75C4⋊Dic167C2311yescharacteristic
76C4⋊Dic167C2311yescharacteristic
77Dic167⋊C4—21no—
78Dic167⋊C4—21no—
79(C2×D167)⋊C4—21no—
80(C2×D167)⋊C4—21no—
81(C2×D167)⋊C4—21no—
82(C2×D167)⋊C4—21no—
83#41×C167—22no—
84#41×C167—22no—
85C2×Q8×C167C2311yescharacteristic
86C2×D4×C167—21no—
87C2×D4×C167C2311yescharacteristic
88C22×C4×C167—21no—
89C22×C4×C167—21no—
90C22×C4×C167—21no—
91C22×C4×C167—21no—
92C22×C4×C167C2311yescharacteristic
93C4⋊C4×C167—21no—
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×C167D411yescharacteristic
103C22⋊C4×C167D411yescharacteristic
104C42×C167—21no—
105C42×C167—21no—
106C42×C167C2311yescharacteristic
107C42×C167C2311yescharacteristic
108C22×D167—21no—
109C22×D167—21no—
110D668—21no—
111D668—21no—
112D668—21no—
113D668—21no—
114D668—81no—
115D668—81no—
116C22⋊D167—81no—
117C22⋊D167—81no—
118C2×Dic167—21no—
119C2×Dic167—21no—
120C4×D167—81no—
121C4×D167—81no—
122C167⋊C8—21no—
123C167⋊C8—21no—
124C167⋊C8—21no—
125C167⋊C8—21no—
126C23×C167—21no—
127C23×C167C2×D411yescharacteristic
128Q8×C167—21no—
129Q8×C167#4112yes—
130D4×C167—24no—
131D4×C167—21no—
132D4×C167#4112yes—
133C2×C4×C167—24no—
134C2×C4×C167—21no—
135C2×C4×C167—21no—
136C2×C4×C167—21no—
137C2×C4×C167—21no—
138C2×C4×C167—21no—
139C2×C4×C167—81no—
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—22no—
150C22×C167—21no—
151C22×C167—21no—
152C22×C167—41no—
153C22×C167—41no—
154C22×C167C22×D411yescharacteristic
155C2×D167—41no—
156C2×D167—41no—
157C2×D167—41no—
158C2×D167—41no—
159C2×D167—41no—
160C2×D167—41no—
161C4×C167—21no—
162C4×C167—22no—
163C4×C167—21no—
164C4×C167—21no—
165C4×C167—21no—
166C4×C167—21no—
167C4×C167—41no—
168C4×C167—41no—
169C4×C167C2×#4111yescharacteristic
170C4×C167C2×#4111yescharacteristic
171Dic167—41no—
172Dic167—41no—
173C2×C167—41no—
174C2×C167—42no—
175C2×C167#52511yescharacteristic
176C2×C167#57611yescharacteristic
177C2×C167#57711yescharacteristic
178D167—81no—
179D167—81no—
180C167#311911yescharacteristic, complemented, Sylow
181#3119—1671nomaximal, Sylow
182#540—1671no—
183#520D16711yescharacteristic, complemented
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×#41C2×D16711yescharacteristic
198C2×SD16—1671no—
199C2×SD16—1671no—
200C2×D8—1671no—
201C2×D8—1671no—
202#127—1671no—
203#125—1671no—
204#123—3341no—
205#123—3341no—
206#121—3341no—
207#121—3341no—
208#121—1671no—
209#121—1671no—
210C4×Q8C2×D16711yescharacteristic
211C4×Q8C2×D16711yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1671no—
215C4×D4—1671no—
216C4×D4C2×D16711yescharacteristic
217C4×D4C2×D16711yescharacteristic
218#117—21no—
219#117C2×D16711yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D16711yescharacteristic
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—22no—
241#41—22no—
242C2×Q8C22×D16711yescharacteristic
243C2×D4—21no—
244C2×D4—3341no—
245C2×D4—3341no—
246C2×D4—3341no—
247C2×D4—3341no—
248C2×D4—1671no—
249C2×D4—1671no—
250C2×D4C22×D16711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
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—21no—
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—3341no—
280C22⋊C4—3341no—
281C22⋊C4—3341no—
282C22⋊C4—3341no—
283C22⋊C4C22⋊D16711yescharacteristic
284C22⋊C4C22⋊D16711yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D16711yescharacteristic
288C42C22×D16711yescharacteristic
289C23—21no—
290C23—3341no—
291C23—3341no—
292C23C2×C22⋊D16711yescharacteristic
293Q8—21no—
294Q8#1458195012yes—
295D4—24no—
296D4—21no—
297D4—13361no—
298D4—13361no—
299D4—13361no—
300D4—13361no—
301D4—3341no—
302D4—3341no—
303D4—3341no—
304D4—3341no—
305D4#1458195012yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—13361no—
316C2×C4—13361no—
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—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—6681no—
336C22—6681no—
337C22—6681no—
338C22—6681no—
339C22—6681no—
340C22—6681no—
341C22C22×C22⋊D16711yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—6681no—
351C4—6681no—
352C4C2×#1458195011yescharacteristic
353C4C2×#1458195011yescharacteristic
354C2—41no—
355C2—42no—
356C2—13361no—
357C2—13361no—
358C2#2492378011yescharacteristic
359C2#2492407211yescharacteristic
360C2#2492407711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#520×C167C2
socleC22×C167C22×D4
hypercenter#520D167
hypocenterC167#3119
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #24923874C2semidirect
2. #24923587C2semidirect
3. #24923582C2semidirect
4. #24923576C2semidirect
5. #24923569C2semidirect
6. #24923469C2semidirect
7. #24923464C2semidirect
8. #24923414C2semidirect
9. #24923414C2semidirect
10. #24923379C2semidirect
11. #24923367C2semidirect
12. C4×#14581937C2semidirect
13. C4×#14581935C2semidirect
14. #24923330C2semidirect
15. #520×C167C2semidirect
180. C167#3119semidirect
183. #520D167semidirect

Maximal subgroups

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

subgroupindex
#249238742
#249235872
#249235822
#249235762
#249235692
#249234692
#249234642
#249234142
#249233792
#249233672
C4×#145819372
C4×#145819352
#249233302
#520×C1672
#3119167