Finite Group Databasebeta

← #36336228

Subgroups — #36336228

Order 21376.

382 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#24923924C211yesmaximal, characteristic
2#24923639C211yesmaximal, characteristic, complemented
3#24923610C211yesmaximal, characteristic, complemented
4#24923604C211yesmaximal, characteristic, complemented
5#24923566C211yesmaximal, characteristic, complemented
6#24923525C211yesmaximal, characteristic, complemented
7#24923508C211yesmaximal, characteristic, complemented
8#24923479C211yesmaximal, characteristic, complemented
9#24923429C211yesmaximal, characteristic, complemented
10#24923429C211yesmaximal, characteristic, complemented
11#24923367C211yesmaximal, characteristic, complemented
12#24923341C211yesmaximal, characteristic, complemented
13#24923268C211yesmaximal, characteristic, complemented
14#24923263C211yesmaximal, characteristic, complemented
15#552×C167C211yesmaximal, characteristic, complemented
16C2×#14581950C2211yescharacteristic
17C2×#14581950C2211yescharacteristic
18C2×#14581940C2211yescharacteristic
19C2×#14581940C2211yescharacteristic
20C2×#14581937C2211yescharacteristic
21C2×#14581934C2211yescharacteristic
22C22×C167⋊C8C2211yescharacteristic
23#19128832C2211yescharacteristic
24#19128814—21no—
25#19128814—21no—
26#19128811C2211yescharacteristic
27#19128811C2211yescharacteristic
28#19128803—21no—
29C4×D668C2211yescharacteristic
30C4⋊C4×D167—21no—
31#19128765C2211yescharacteristic
32#19128765C2211yescharacteristic
33#19128763C2211yescharacteristic
34C4⋊(C167⋊C8)C2211yescharacteristic
35#19128756C2211yescharacteristic
36#19128755C2211yescharacteristic
37#19128755C2211yescharacteristic
38#19128755C2211yescharacteristic
39#19128755C2211yescharacteristic
40#19128755C2211yescharacteristic
41#19128755C2211yescharacteristic
42#19128752C2211yescharacteristic
43#19128752C2211yescharacteristic
44#19128752C2211yescharacteristic
45C22⋊(C167⋊C8)C2211yescharacteristic
46C22⋊(C167⋊C8)C2211yescharacteristic
47#19128743C2211yescharacteristic
48#126×C167—21no—
49#125×C167—21no—
50#125×C167C2211yescharacteristic
51#123×C167—21no—
52C22⋊Q8×C167C2211yescharacteristic
53C22⋊Q8×C167C2211yescharacteristic
54C4×Q8×C167C2211yescharacteristic
55C4×D4×C167C2211yescharacteristic
56#117×C167—21no—
57#117×C167C2211yescharacteristic
58C2×C4⋊C4×C167C2211yescharacteristic
59#14581950—22no—
60#14581950—22no—
61#14581950—41no—
62#14581950—41no—
63C2×Dic334C2311yescharacteristic
64C2×Dic334C2311yescharacteristic
65C2×D668C2311yescharacteristic
66C2×C22⋊D167—21no—
67C2×C22⋊D167—21no—
68C2×C4×D167—21no—
69C2×C4×D167—21no—
70C2×C4×D167—21no—
71#14581940—41no—
72#14581940—41no—
73#14581937—22no—
74#14581934—41no—
75C2×C167⋊C8—41no—
76C2×C167⋊C8C2311yescharacteristic
77C2×C167⋊C8C2311yescharacteristic
78C2×C167⋊C8C2311yescharacteristic
79C2×C167⋊C8C2311yescharacteristic
80C4⋊Dic167—21no—
81C4⋊Dic167—21no—
82C4⋊Dic167C2311yescharacteristic
83Dic167⋊C4—21no—
84Dic167⋊C4—21no—
85C22⋊Dic167—21no—
86C22⋊Dic167—21no—
87(C2×D167)⋊C4—21no—
88C2×Q8×C167C2311yescharacteristic
89C2×D4×C167D411yescharacteristic
90C22×C4×C167—21no—
91C22×C4×C167C2311yescharacteristic
92C22×C4×C167C2311yescharacteristic
93C4⋊C4×C167—21no—
94C4⋊C4×C167—21no—
95C4⋊C4×C167—21no—
96C4⋊C4×C167—21no—
97C4⋊C4×C167—21no—
98C4⋊C4×C167—21no—
99C4⋊C4×C167D411yescharacteristic
100C4⋊C4×C167C2311yescharacteristic
101C4⋊C4×C167C2311yescharacteristic
102C4⋊C4×C167C2311yescharacteristic
103C22⋊C4×C167—21no—
104C22⋊C4×C167—21no—
105C22⋊C4×C167—21no—
106C22⋊C4×C167D411yescharacteristic
107C22⋊C4×C167D411yescharacteristic
108C42×C167—21no—
109C42×C167—21no—
110C42×C167C2311yescharacteristic
111C22×D167—21no—
112Dic334—21no—
113Dic334—21no—
114Dic334—21no—
115Dic334—21no—
116D668—21no—
117D668#4112yes—
118C22⋊D167—42no—
119C22⋊D167—42no—
120C2×Dic167—21no—
121C2×Dic167—21no—
122C2×Dic167—21no—
123C4×D167—42no—
124C4×D167—42no—
125C4×D167—42no—
126C167⋊C8—21no—
127C167⋊C8—21no—
128C167⋊C8—21no—
129C167⋊C8—21no—
130C23×C167C2×D411yescharacteristic
131C23×C167C2×D411yescharacteristic
132Q8×C167—21no—
133Q8×C167—21no—
134D4×C167—41no—
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—21no—
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×C167C22×D411yescharacteristic
156C2×D167—22no—
157C2×D167—41no—
158C2×D167—41no—
159C4×C167—21no—
160C4×C167—21no—
161C4×C167—21no—
162C4×C167—21no—
163C4×C167—41no—
164C4×C167—41no—
165C4×C167—41no—
166C4×C167—41no—
167C4×C167C2×#4111yescharacteristic
168C4×C167H(5,𝔽2)11yescharacteristic
169Dic167—41no—
170Dic167—41no—
171Dic167—41no—
172C2×C167—41no—
173C2×C167—41no—
174C2×C167#54611yescharacteristic
175C2×C167#57511yescharacteristic
176C2×C167#57811yescharacteristic
177D167—42no—
178C167#331511yescharacteristic, complemented, Sylow
179#3315—1671nomaximal, Sylow
180#552D16711yescharacteristic, complemented
181#547—1671no—
182#491—1671no—
183#485—1671no—
184#484—1671no—
185#478—1671no—
186#470—1671no—
187#467—1671no—
188#462—1671no—
189#452—1671no—
190#452—1671no—
191#440—1671no—
192#435—1671no—
193#417—1671no—
194#416—1671no—
195C2×#41—1671no—
196C2×#41—1671no—
197C2×Q16—1671no—
198C2×Q16—1671no—
199C2×SD16—1671no—
200C2×M16—1671no—
201C22×C8—1671no—
202C4⋊Q8—1671no—
203#126—21no—
204#125—21no—
205#125C2×D16711yescharacteristic
206#123—21no—
207#123—3341no—
208#123—3341no—
209C22⋊Q8—1671no—
210C22⋊Q8—1671no—
211C22⋊Q8C2×D16711yescharacteristic
212C22⋊Q8C2×D16711yescharacteristic
213#121—3341no—
214C4×Q8C2×D16711yescharacteristic
215C4×D4—1671no—
216C4×D4C2×D16711yescharacteristic
217#117—21no—
218#117C2×D16711yescharacteristic
219C2×C4⋊C4—3341no—
220C2×C4⋊C4C2×D16711yescharacteristic
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—3342no—
239#41—3342no—
240#41—6681no—
241#41—6681no—
242C2×Q8—1671no—
243C2×Q8—1671no—
244C2×Q8C22×D16711yescharacteristic
245C2×D4—3341no—
246C2×D4—3341no—
247C2×D4—1671no—
248C2×D4C22⋊D16711yescharacteristic
249C22×C4—21no—
250C22×C4—3341no—
251C22×C4—3341no—
252C22×C4—3341no—
253C22×C4C22×D16711yescharacteristic
254C22×C4C22×D16711yescharacteristic
255Q16—6681no—
256Q16—6681no—
257SD16—3342no—
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—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—3341no—
271C4⋊C4—3341no—
272C4⋊C4—3341no—
273C4⋊C4—3341no—
274C4⋊C4—1671no—
275C4⋊C4C22⋊D16711yescharacteristic
276C4⋊C4C22×D16711yescharacteristic
277C4⋊C4C22×D16711yescharacteristic
278C4⋊C4C22×D16711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—3341no—
283C22⋊C4—3341no—
284C22⋊C4—3341no—
285C22⋊C4C22⋊D16711yescharacteristic
286C22⋊C4C22⋊D16711yescharacteristic
287C42—21no—
288C42—21no—
289C42C22×D16711yescharacteristic
290C23—3341no—
291C23C2×C22⋊D16711yescharacteristic
292C23C2×C22⋊D16711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3341no—
296Q8—3341no—
297Q8—3341no—
298Q8—3341no—
299D4—41no—
300D4—3341no—
301D4—6682no—
302D4—6682no—
303D4—1672no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—3341no—
314C2×C4—3341no—
315C2×C4—3341no—
316C2×C4—6682no—
317C2×C4—6682no—
318C2×C4—6682no—
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—3342no—
335C22—6681no—
336C22—6681no—
337C22C22×C22⋊D16711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—6681no—
347C4—6681no—
348C4—6681no—
349C4#41×D16711yescharacteristic
350C4#1912887811yescharacteristic
351C2—41no—
352C2—41no—
353C2—6682no—
354C2#2492390711yescharacteristic
355C2#2492406711yescharacteristic
356C2#2492408411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
Fitting#552×C167C2
socleC22×C167C22×D4
prime radical#24923924C2
hypercenter#552D167
hypocenterC167#3315
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #24923639C2semidirect
3. #24923610C2semidirect
4. #24923604C2semidirect
5. #24923566C2semidirect
6. #24923525C2semidirect
7. #24923508C2semidirect
8. #24923479C2semidirect
9. #24923429C2semidirect
10. #24923429C2semidirect
11. #24923367C2semidirect
12. #24923341C2semidirect
13. #24923268C2semidirect
14. #24923263C2semidirect
15. #552×C167C2semidirect
178. C167#3315semidirect
180. #552D167semidirect

Maximal subgroups

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

subgroupindex
#249239242
#249236392
#249236102
#249236042
#249235662
#249235252
#249235082
#249234792
#249234292
#249233672
#249233412
#249232682
#249232632
#552×C1672
#3315167