Finite Group Databasebeta

← #20286584

Subgroups — #20286584

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811331C211yesmaximal, characteristic
2Q8×C22⋊D53C211yesmaximal, characteristic
3#16810972C211yesmaximal, characteristic, complemented
4#16810959C211yesmaximal, characteristic, complemented
5#16810957C211yesmaximal, characteristic, complemented
6#16810916C211yesmaximal, characteristic, complemented
7#16810862C211yesmaximal, characteristic, complemented
8#16810861C211yesmaximal, characteristic, complemented
9#16810831C211yesmaximal, characteristic, complemented
10#16810777C211yesmaximal, characteristic, complemented
11Q16×Dic53C211yesmaximal, characteristic, complemented
12#16810686C211yesmaximal, characteristic
13C2×#13133141C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#451C211yesmaximal, characteristic, complemented
16C2×Q8×D53C2211yescharacteristic
17C2×#11719925C2211yescharacteristic
18C22×#88322C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20#13133216C2211yescharacteristic
21#13133215—21no—
22#13133215C2211yescharacteristic
23#13133207—21no—
24#13133197—21no—
25#13133196—21no—
26#13133196C2211yescharacteristic
27#13133194—21no—
28#13133193—21no—
29#13133193C2211yescharacteristic
30#13133193C2211yescharacteristic
31Q8×Dic53C2211yescharacteristic
32Q8×Dic53C2211yescharacteristic
33C4×C22⋊D53—21no—
34C4×C22⋊D53C2211yescharacteristic
35#13133177—21no—
36C2×C4⋊Dic53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133141C2211yescharacteristic
46#13133141C2211yescharacteristic
47#13133141C2211yescharacteristic
48#13133141C2211yescharacteristic
49#13133140C2211yescharacteristic
50#13133129C2211yescharacteristic
51C8×Dic53C2211yescharacteristic
52C22×Q8×C53C2211yescharacteristic
53C2×Q16×C53C2211yescharacteristic
54C2×Q16×C53C2211yescharacteristic
55C22⋊Q8×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53C2311yescharacteristic
63C2×Q8×C53C2311yescharacteristic
64C2×Q8×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53C2311yescharacteristic
67C2×C22⋊D53D411yescharacteristic
68C22×Dic53—21no—
69Q8×D53—22no—
70Q8×D53—41no—
71C2×C4×D53—21no—
72C2×C4×D53—21no—
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—21no—
76C2×Dic106C2311yescharacteristic
77Q16×C53—22no—
78Q16×C53—41no—
79C2×C8×C53C2311yescharacteristic
80C2×C8×C53C2311yescharacteristic
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53D411yescharacteristic
86#11719925—41no—
87#11719921—21no—
88#11719921—21no—
89#11719921D411yescharacteristic
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53C2311yescharacteristic
94C4⋊Dic53C2311yescharacteristic
95C4⋊Dic53C2311yescharacteristic
96#11719919—21no—
97#11719919—21no—
98#11719919—21no—
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53C2311yescharacteristic
107C2×#88322—41no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110C8×D53—41no—
111C23×C53C2×D411yescharacteristic
112C22×D53C2×D411yescharacteristic
113Q8×C53—21no—
114Q8×C53—21no—
115Q8×C53—21no—
116Q8×C53—21no—
117Q8×C53—21no—
118Q8×C53—42no—
119Q8×C53#4112yes—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—42no—
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53Q1614yescomplemented
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—81no—
135C2×Dic53C2×D411yescharacteristic
136C2×Dic53C2×D411yescharacteristic
137C2×Dic53C2×D411yescharacteristic
138C4×D53—21no—
139C4×D53—21no—
140C4×D53—42no—
141C4×D53—81no—
142Dic106—21no—
143Dic106—21no—
144Dic106—42no—
145Dic106—81no—
146C8×C53—21no—
147C8×C53—21no—
148#88322—21no—
149#88322—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53C2×Q1612yes—
153C22×C53C22×D411yescharacteristic
154C2×D53—21no—
155C2×D53—21no—
156C2×D53C2×Q1612yes—
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53C2×#4111yescharacteristic
163C4×C53H(5,𝔽2)11yescharacteristic
164Dic53—21no—
165Dic53—21no—
166Dic53—41no—
167Dic53—41no—
168Dic53—41no—
169Dic53—41no—
170Dic53C2×Q1612yes—
171#3299—531nomaximal, Sylow
172C2×C53—22no—
173C2×C53#54611yescharacteristic
174C2×C53C22×Q1611yescharacteristic
175C2×C53#57711yescharacteristic
176D53—22no—
177#554—531no—
178D4×Q8—531no—
179#487—531no—
180#484—531no—
181#484—531no—
182#477—531no—
183#467—531no—
184#467—531no—
185#462—531no—
186#451—531no—
187#451D5311yescharacteristic, complemented
188C4×Q16—531no—
189C8×D4—531no—
190C2×#103—531no—
191C2×#103—531no—
192C53#329911yescharacteristic, complemented, Sylow
193C22×Q8—531no—
194C22×Q8C2×D5311yescharacteristic
195C2×Q16—531no—
196C2×Q16C2×D5311yescharacteristic
197C2×Q16C2×D5311yescharacteristic
198C22×C8—531no—
199C22×C8—531no—
200C4⋊Q8—531no—
201C4⋊Q8—531no—
202C4⋊Q8—1061no—
203#125—1061no—
204C22⋊Q8—531no—
205C22⋊Q8—531no—
206C22⋊Q8—531no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209C22⋊Q8—1061no—
210C22⋊Q8—1061no—
211C22⋊Q8C2×D5311yescharacteristic
212C4×Q8—531no—
213C4×Q8—531no—
214C4×D4—531no—
215C4×D4—1061no—
216C4×D4—1061no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219#107—531no—
220#107—531no—
221#107—531no—
222C4⋊C8—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—
231#103C2×D5311yescharacteristic
232#103C2×D5311yescharacteristic
233C22⋊C8—531no—
234C22⋊C8C2×D5311yescharacteristic
235C4×C8—531no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—531no—
240C2×Q8—1061no—
241C2×Q8—1061no—
242C2×Q8—1062no—
243C2×Q8—2121no—
244C2×Q8C22×D5311yescharacteristic
245C2×Q8C22×D5311yescharacteristic
246C2×Q8C22×D5311yescharacteristic
247C2×D4—531no—
248C22×C4—21no—
249C22×C4—531no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4C22×D5311yescharacteristic
254Q16—22no—
255Q16—41no—
256Q16—2121no—
257C2×C8—531no—
258C2×C8—531no—
259C2×C8—2121no—
260C2×C8—2121no—
261C2×C8C22×D5311yescharacteristic
262C2×C8C22×D5311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—531no—
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—1061no—
276C4⋊C4—1061no—
277C4⋊C4—1061no—
278C4⋊C4C22×D5311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—531no—
281C22⋊C4—531no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C42—531no—
286C42—1061no—
287C42—1061no—
288C23—531no—
289C23D4×D5311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—42no—
296Q8—1061no—
297Q8—1061no—
298Q8—4241no—
299Q8—2122no—
300Q8#1171994712yes—
301D4—534no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—531no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—2122no—
321C2×C4D4×D5311yescharacteristic
322C2×C4C23×D5311yescharacteristic
323C8—21no—
324C8—21no—
325C8—1061no—
326C8—1061no—
327C22—21no—
328C22—21no—
329C22—532no—
330C22—1061no—
331C22—1061no—
332C22Q16×D5312yes—
333C22C2×D4×D5311yescharacteristic
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—532no—
340C4—1061no—
341C4—1061no—
342C4—2121no—
343C4—2121no—
344C4—2121no—
345C4—2121no—
346C4C2×#1171994711yescharacteristic
347C4#1313326211yescharacteristic
348C2—22no—
349C2—1062no—
350C2#1681126011yescharacteristic
351C2C2×Q16×D5311yescharacteristic
352C2#1681143211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53#3299
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #16810972C2semidirect
4. #16810959C2semidirect
5. #16810957C2semidirect
6. #16810916C2semidirect
7. #16810862C2semidirect
8. #16810861C2semidirect
9. #16810831C2semidirect
10. #16810777C2semidirect
11. Q16×Dic53C2semidirect
13. C2×#13133141C2semidirect
14. #103×D53C2semidirect
15. C53×#451C2semidirect
129. C22⋊D53Q16semidirect
187. #451D53semidirect
192. C53#3299semidirect