Finite Group Databasebeta

← #20286768

Subgroups — #20286768

Order 6784.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811313C211yesmaximal, characteristic
2D4×Dic106C211yesmaximal, characteristic
3#16810992C211yesmaximal, characteristic, complemented
4#16810965C211yesmaximal, characteristic, complemented
5#16810954C211yesmaximal, characteristic, complemented
6#16810931C211yesmaximal, characteristic, complemented
7#16810881C211yesmaximal, characteristic, complemented
8#16810864C211yesmaximal, characteristic, complemented
9#16810834C211yesmaximal, characteristic, complemented
10#16810781C211yesmaximal, characteristic, complemented
11#16810781C211yesmaximal, characteristic, complemented
12#16810731C211yesmaximal, characteristic, complemented
13#16810633C211yesmaximal, characteristic, complemented
14C2×#13133142C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic
16C22×Dic106C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×Dic212C2211yescharacteristic
19C2×Dic212C2211yescharacteristic
20C2×Dic212C2211yescharacteristic
21#13133217C2211yescharacteristic
22#13133216—21no—
23#13133210—21no—
24#13133209C2211yescharacteristic
25#13133207—21no—
26#13133198—21no—
27#13133196C2211yescharacteristic
28#13133196C2211yescharacteristic
29#13133196C2211yescharacteristic
30#13133196C2211yescharacteristic
31#13133194—21no—
32#13133194—21no—
33C4×Dic106C2211yescharacteristic
34C4×Dic106C2211yescharacteristic
35D4×Dic53—21no—
36#13133176C2211yescharacteristic
37#13133173—21no—
38C2×C4⋊Dic53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133149C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133142C2211yescharacteristic
46#13133142C2211yescharacteristic
47#13133142C2211yescharacteristic
48#13133142C2211yescharacteristic
49#13133142C2211yescharacteristic
50#13133142C2211yescharacteristic
51#13133142C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53D411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×Dic106—22no—
66C2×Dic106—22no—
67C2×Dic106—21no—
68C2×Dic106—21no—
69C2×Dic106—41no—
70C2×Dic106—41no—
71C2×Dic106C2311yescharacteristic
72C2×Dic106C2311yescharacteristic
73C2×Dic106C2311yescharacteristic
74C2×Dic106C2311yescharacteristic
75M16×C53—41no—
76C2×C8×C53—41no—
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53D411yescharacteristic
82C22⋊C4×C53D411yescharacteristic
83C22⋊C4×C53D411yescharacteristic
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108Dic212—22no—
109Dic212—41no—
110Dic212—41no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113D4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53C2×D411yescharacteristic
119C2×C4×C53C2×D411yescharacteristic
120C2×C4×C53C2×D411yescharacteristic
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2411yescharacteristic
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134Dic106—21no—
135Dic106—21no—
136Dic106—21no—
137Dic106—21no—
138Dic106—21no—
139Dic106—21no—
140Dic106—21no—
141Dic106—42no—
142Dic106—42no—
143Dic106#4112yes—
144C8×C53—21no—
145C8×C53—21no—
146C8×C53—21no—
147C8×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53#13712yes—
154C22×C53C22×D411yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53C2×#4111yescharacteristic
160C4×C53H(5,𝔽2)11yescharacteristic
161Dic53—22no—
162Dic53—41no—
163Dic53—41no—
164Dic53—41no—
165Dic53—41no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169#3312—531nomaximal, Sylow
170C2×C53—22no—
171C2×C53—41no—
172C2×C53#54611yescharacteristic
173C2×C53C2×#13711yescharacteristic
174C2×C53#57811yescharacteristic
175#552—531no—
176D4×Q8—531no—
177#491—531no—
178#485—531no—
179#483—531no—
180#479—531no—
181#470—531no—
182#467—531no—
183#462—531no—
184#451—531no—
185#451—531no—
186#441—531no—
187#435D5311yescharacteristic
188#419—531no—
189C2×#103—531no—
190C53#331211yescharacteristic, complemented, Sylow
191C22×Q8—531no—
192C22×Q8—531no—
193C2×Q16—531no—
194C2×Q16—531no—
195C2×Q16—531no—
196C2×M16C2×D5311yescharacteristic
197C22×C8C2×D5311yescharacteristic
198C4⋊Q8—531no—
199C4⋊Q8—1061no—
200#126—1061no—
201#125—531no—
202#125—1061no—
203#123—1061no—
204C22⋊Q8—531no—
205C22⋊Q8—531no—
206C22⋊Q8—531no—
207C22⋊Q8—531no—
208C22⋊Q8—1061no—
209C22⋊Q8—1061no—
210C4×Q8—531no—
211C4×Q8—531no—
212C4×D4—1061no—
213C4×D4C2×D5311yescharacteristic
214#117—531no—
215#117—1061no—
216C2×C4⋊C4—531no—
217#107—531no—
218#106—531no—
219#106—531no—
220C4⋊C8C2×D5311yescharacteristic
221#103—531no—
222#103—531no—
223#103—531no—
224#103—531no—
225#103—531no—
226#103—531no—
227#103—531no—
228#103—531no—
229#103—531no—
230#103—531no—
231C22⋊C8C2×D5311yescharacteristic
232C22⋊C8C2×D5311yescharacteristic
233#97C2×D5311yescharacteristic
234C2×Q8—531no—
235C2×Q8—531no—
236C2×Q8—531no—
237C2×Q8—531no—
238C2×Q8—1062no—
239C2×Q8—1062no—
240C2×Q8—1061no—
241C2×Q8—1061no—
242C2×Q8—2121no—
243C2×Q8—2121no—
244C2×D4D21211yescharacteristic
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4C22×D5311yescharacteristic
249C22×C4C22×D5311yescharacteristic
250Q16—1062no—
251Q16—2121no—
252Q16—2121no—
253M16—41no—
254C2×C8—41no—
255C2×C8C22×D5311yescharacteristic
256C2×C8C22×D5311yescharacteristic
257C2×C8C22×D5311yescharacteristic
258C2×C8C22×D5311yescharacteristic
259C4⋊C4—531no—
260C4⋊C4—531no—
261C4⋊C4—531no—
262C4⋊C4—531no—
263C4⋊C4—1061no—
264C4⋊C4—1061no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4D21211yescharacteristic
275C22⋊C4—1061no—
276C22⋊C4—1061no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4D21211yescharacteristic
281C22⋊C4D21211yescharacteristic
282C42—1061no—
283C42—1061no—
284C42—1061no—
285C42C22×D5311yescharacteristic
286C23C2×D21211yescharacteristic
287C23C2×D21211yescharacteristic
288Q8—532no—
289Q8—1061no—
290Q8—1061no—
291Q8—1061no—
292Q8—1061no—
293Q8—1061no—
294Q8—1061no—
295Q8—1061no—
296Q8—2122no—
297Q8—2122no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—1061no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4C2×D21211yescharacteristic
315C2×C4C2×D21211yescharacteristic
316C2×C4C2×D21211yescharacteristic
317C2×C4C2×D21211yescharacteristic
318C2×C4C23×D5311yescharacteristic
319C8—21no—
320C8—21no—
321C8—21no—
322C8—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1313325012yes—
329C22C22×D21211yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—1062no—
335C4—2121no—
336C4—2121no—
337C4—2121no—
338C4—2121no—
339C4—2121no—
340C4—2121no—
341C4—2121no—
342C4#41×D5311yescharacteristic
343C4#1313326211yescharacteristic
344C2—22no—
345C2—41no—
346C2#1681126811yescharacteristic
347C2C2×#1313325011yescharacteristic
348C2#1681144011yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3312
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
3. #16810992C2semidirect
4. #16810965C2semidirect
5. #16810954C2semidirect
6. #16810931C2semidirect
7. #16810881C2semidirect
8. #16810864C2semidirect
9. #16810834C2semidirect
10. #16810781C2semidirect
11. #16810781C2semidirect
12. #16810731C2semidirect
13. #16810633C2semidirect
14. C2×#13133142C2semidirect
190. C53#3312semidirect