Finite Group Databasebeta

← #20277573

Subgroups — #20277573

Order 6784.

532 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133173C211yesmaximal, characteristic
2#16810510C211yesmaximal, characteristic, complemented
3#16810510C211yesmaximal, characteristic, complemented
4#16810504C211yesmaximal, characteristic, complemented
5#16810504C211yesmaximal, characteristic, complemented
6#16810498C211yesmaximal, characteristic
7#16810490C211yesmaximal, characteristic, complemented
8#16810488C211yesmaximal, characteristic, complemented
9#16810484C211yesmaximal, characteristic
10C4⋊C4×Dic53C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic53C211yesmaximal, characteristic
12C4×C22⋊Dic53C211yesmaximal, characteristic
13#16810468C211yesmaximal, characteristic, complemented
14C2×#13133124C211yesmaximal, characteristic
15C53×#385C211yesmaximal, characteristic
16C23×Dic53C2211yescharacteristic
17C23×Dic53C2211yescharacteristic
18#13133173C418yescomplemented
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2211yescharacteristic
31#13133124C2211yescharacteristic
32#13133124C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#13133124C2211yescharacteristic
45C23×C4×C53C2211yescharacteristic
46C2×C4⋊C4×C53C2211yescharacteristic
47C2×C4⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51#95×C53C2211yescharacteristic
52C24×C53C2311yescharacteristic
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—41no—
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—41no—
67C22×Dic53C2×C414yes—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C22×Dic53C2311yescharacteristic
76C4⋊C4×C53—22no—
77C4⋊C4×C53—41no—
78C22⋊C4×C53—22no—
79C22⋊C4×C53C2×C414yes—
80C22⋊Dic53—22no—
81C22⋊Dic53C2×C414yes—
82C4⋊Dic53—22no—
83C4⋊Dic53—41no—
84#11719919C2×C414yescomplemented
85#11719919C2×C414yescomplemented
86C4×Dic53—22no—
87C4×Dic53—22no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53C2×C414yescomplemented
91C4×Dic53C2×C414yescomplemented
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53C22×C412yes—
97C23×C53#4112yes—
98C23×C53#4112yes—
99C23×C53C2411yescharacteristic
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—44no—
116C2×C4×C53C22×C412yes—
117C2×C4×C53C22×C412yes—
118C2×C4×C53#4112yes—
119C2×Dic53—28no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
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—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—44no—
140C2×Dic53C22×C412yes—
141C2×Dic53C22×C412yes—
142C2×Dic53C22×C412yes—
143C2×Dic53C22×C412yes—
144C2×Dic53#4112yes—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—22no—
151C22×C53#11714yes—
152C22×C53C23×C411yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53C2×#4111yescharacteristic
155C22×C53C2×#4111yescharacteristic
156C22×C53C2×#4111yescharacteristic
157C22×C53H(5,𝔽2)11yescharacteristic
158C22×C532−1+411yescharacteristic
159C4×C53—22no—
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165Dic53—22no—
166Dic53—22no—
167Dic53—22no—
168Dic53—22no—
169Dic53—22no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173#2343—531nomaximal, Sylow
174C2×C53—24no—
175C2×C53C2×#11711yescharacteristic
176C2×C53C4×#4111yescharacteristic
177C2×C53#52011yescharacteristic
178C2×C53#55111yescharacteristic
179C2×C53#55111yescharacteristic
180C2×C53#55211yescharacteristic
181C2×C53#55211yescharacteristic
182C2×#117—531no—
183#387—531no—
184#387—531no—
185#387—531no—
186#387—531no—
187#385—531no—
188#385D5311yescharacteristic
189#383—531no—
190#382—531no—
191#381—531no—
192C4×C4⋊C4—531no—
193C4×C22⋊C4—531no—
194C4×C22⋊C4—531no—
195#376—531no—
196C2×#95—531no—
197C53#234311yescharacteristic, complemented, Sylow
198C23×C4—531no—
199C23×C4—531no—
200C23×C4C2×D5311yescharacteristic
201#117—538no—
202C2×C4⋊C4—531no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4C2×D5311yescharacteristic
207C2×C4⋊C4C2×D5311yescharacteristic
208C2×C22⋊C4—531no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4C2×D5311yescharacteristic
211C2×C22⋊C4C2×D5311yescharacteristic
212C2×C42—531no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—531no—
218#95—531no—
219#95—531no—
220#95—531no—
221#95—531no—
222#95—531no—
223#95—531no—
224#95—531no—
225#95—531no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95—531no—
231#95—531no—
232#95C2×D5311yescharacteristic
233#95C2×D5311yescharacteristic
234C24C22×D5311yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—534no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—1062no—
250C22×C4—1062no—
251C22×C4—2121no—
252C22×C4C22×D5311yescharacteristic
253C22×C4C22×D5311yescharacteristic
254C22×C4C22×D5311yescharacteristic
255C22×C4C22×D5311yescharacteristic
256C22×C4C22×D5311yescharacteristic
257C22×C4C22×D5311yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—41no—
260C4⋊C4—534no—
261C4⋊C4—534no—
262C4⋊C4—1062no—
263C4⋊C4—2121no—
264C22⋊C4—22no—
265C22⋊C4—534no—
266C22⋊C4—1062no—
267C22⋊C4C4×D5314yes—
268C42—534no—
269C42—534no—
270C42—1062no—
271C42—1062no—
272C42—2121no—
273C42—2121no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C4×D5312yes—
279C23#1171994712yes—
280C23#1171994912yes—
281C23C23×D5311yescharacteristic
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—532no—
299C2×C4—532no—
300C2×C4—532no—
301C2×C4—532no—
302C2×C4—532no—
303C2×C4—1068no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—1061no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—2124no—
324C2×C4C2×C4×D5312yes—
325C2×C4C2×C4×D5312yes—
326C2×C4#1171994712yes—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22#1313317514yes—
334C22C22×C4×D5311yescharacteristic
335C22#41×D5311yescharacteristic
336C22C2×#1171994711yescharacteristic
337C22C2×#1171994711yescharacteristic
338C22C2×#1171994911yescharacteristic
339C22#1313326211yescharacteristic
340C22#1313326411yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—1062no—
348C4—1062no—
349C4—1062no—
350C4—1062no—
351C4—1062no—
352C4—2121no—
353C4—2121no—
354C4—2121no—
355C2—24no—
356C2C2×#1313317511yescharacteristic
357C2C4×#1171994711yescharacteristic
358C2#1681110211yescharacteristic
359C2#1681130011yescharacteristic
360C2#1681130411yescharacteristic
361C2#1681131011yescharacteristic
362C2#1681131511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutatorC22×C53C23×C4
FittingC53×#385C2
socleC23×C53C24
hypercenter#385D53
hypocenterC53#2343
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #16810510C2semidirect
3. #16810510C2semidirect
4. #16810504C2semidirect
5. #16810504C2semidirect
7. #16810490C2semidirect
8. #16810488C2semidirect
10. C4⋊C4×Dic53C2semidirect
13. #16810468C2semidirect
18. #13133173C4semidirect
84. #11719919C2×C4semidirect
85. #11719919C2×C4semidirect
90. C4×Dic53C2×C4semidirect
91. C4×Dic53C2×C4semidirect
197. C53#2343semidirect