Finite Group Databasebeta

← #20288332

Subgroups — #20288332

Order 6784.

1094 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#16811349C211yesmaximal, characteristic, complemented
2#16811345C213yesmaximal, complemented
3#16811314C213yesmaximal, complemented
4#16811308C213yesmaximal, complemented
5Q8×C22⋊D53C211yesmaximal, characteristic, complemented
6#16811285C211yesmaximal, characteristic, complemented
7#16811284C211yesmaximal, characteristic, complemented
8#16811274C213yesmaximal, complemented
9#16811177C213yesmaximal, complemented
10#16811160C213yesmaximal, complemented
11#16811157C213yesmaximal, complemented
12C4×#11719949C211yesmaximal, characteristic, complemented
13C4×#11719945C213yesmaximal, complemented
14C4×Q8×D53C211yesmaximal, characteristic, complemented
15C2×C4×Q8×C53C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719945C2213yes—
18C2×Q8×D53C2211yescharacteristic
19#13133216C2213yes—
20#13133215C2213yes—
21#13133212C2216yes—
22#13133211C2216yes—
23#13133209C2213yes—
24#13133208C2213yes—
25#13133207C2216yes—
26#13133206C2213yes—
27#13133205C2213yes—
28#13133202C2216yes—
29#13133200C2216yes—
30#13133197C2213yes—
31#13133196C2213yes—
32#13133195C2216yes—
33#13133193C2213yes—
34#13133193C2213yes—
35#13133192C2213yes—
36#13133191C2213yes—
37Q8×Dic53C2211yescharacteristic
38Q8×Dic53C2211yescharacteristic
39C4×Dic106C2213yes—
40C4×Dic106C2213yes—
41C4×Dic106C2213yes—
42#13133182C2213yes—
43C4×D212C2213yes—
44C4×D212C2213yes—
45C4×D212C2213yes—
46#13133179C2213yes—
47C4×C22⋊D53C2213yes—
48C4×C22⋊D53C2213yes—
49C4×C22⋊D53C2211yescharacteristic
50C4×C22⋊D53C2213yes—
51#13133176C2213yes—
52#13133175C2213yes—
53#13133174C2213yes—
54#13133174C2213yes—
55#13133174C2216yes—
56C4⋊C4×D53C2213yes—
57C42×D53C2213yes—
58C42×D53C2213yes—
59C22×Q8×C53C2211yescharacteristic
60C4×Q8×C53C2211yescharacteristic
61C4×Q8×C53C2213yes—
62C4×Q8×C53C2211yescharacteristic
63C4×Q8×C53C2213yes—
64C2×C4⋊C4×C53C2213yes—
65C2×C42×C53C2213yes—
66C2×Q8×C53—24no—
67C2×Q8×C53C2313yes—
68C2×Q8×C53C2311yescharacteristic
69C22×C4×C53C2311yescharacteristic
70C22×C4×C53C2313yes—
71C22×C4×C53C2313yes—
72C2×C22⋊D53C2313yes—
73C2×C22⋊D53C2311yescharacteristic
74#11719949—24no—
75Q8×D53—24no—
76#11719945—212no—
77C2×D212C2313yes—
78C2×D212C2313yes—
79C2×C4×D53C2311yescharacteristic
80C2×C4×D53C2313yes—
81C2×C4×D53C2313yes—
82C2×C4×D53C2313yes—
83C2×C4×D53C2313yes—
84C2×C4×D53C2313yes—
85C2×Dic106C2313yes—
86C2×Dic106C2313yes—
87C4⋊C4×C53C2313yes—
88C4⋊C4×C53C2313yes—
89C4⋊C4×C53C2316yes—
90C42×C53C2313yes—
91C42×C53C2313yes—
92C42×C53C2313yes—
93C42×C53C2313yes—
94C22⋊Dic53C2311yescharacteristic
95C22⋊Dic53C2313yes—
96#11719921C2311yescharacteristic
97#11719921C2313yes—
98#11719921C2313yes—
99#11719921C2313yes—
100#11719921C2316yes—
101#11719921C2316yes—
102#11719921C2316yes—
103C4⋊Dic53C2313yes—
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2313yes—
106C4⋊Dic53C2313yes—
107#11719919C2316yes—
108#11719919C2311yescharacteristic
109#11719919C2313yes—
110#11719919C2313yes—
111#11719919C2313yes—
112#11719919C2316yes—
113#11719919C2316yes—
114C4×Dic53C2311yescharacteristic
115C4×Dic53C2313yes—
116C4×Dic53C2313yes—
117C4×Dic53C2313yes—
118C4×Dic53C2313yes—
119C4×Dic53C2313yes—
120C23×C53C2411yescharacteristic
121C22×D53C2413yes—
122C22×D53C2411yescharacteristic
123Q8×C53—26no—
124Q8×C53#4114yescomplemented
125C2×C4×C53—26no—
126C2×C4×C53—26no—
127C2×C4×C53—22no—
128C2×C4×C53C2413yes—
129C2×C4×C53C2413yes—
130C2×C4×C53C2413yes—
131C2×C4×C53C2411yescharacteristic
132C2×C4×C53C2413yes—
133C2×C4×C53C2411yescharacteristic
134C22⋊D53—26no—
135C22⋊D53#4114yescomplemented
136C2×Dic53C2413yes—
137C2×Dic53C2413yes—
138C2×Dic53C2411yescharacteristic
139C2×Dic53C2411yescharacteristic
140C2×Dic53C2413yes—
141C2×Dic53C2411yescharacteristic
142D212—26no—
143D212—26no—
144C4×D53—26no—
145C4×D53—26no—
146C4×D53—26no—
147C4×D53—22no—
148C4×D53—26no—
149C4×D53—26no—
150Dic106—26no—
151Dic106—26no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53C2×#4112yes—
155C22×C53C2511yescharacteristic
156C2×D53—21no—
157C2×D53—21no—
158C2×D53—23no—
159C2×D53—23no—
160C2×D53—23no—
161C2×D53C2×#4112yes—
162C4×C53—23no—
163C4×C53—23no—
164C4×C53—23no—
165C4×C53—21no—
166C4×C53C2×#4116yes—
167C4×C532−1+412yes—
168Dic53—23no—
169Dic53—23no—
170Dic53—21no—
171Dic53—21no—
172Dic53—23no—
173Dic53C2×#4112yes—
174#3482—531nomaximal, Sylow
175C2×C53—22no—
176C2×C53C22×#4111yescharacteristic
177C2×C53C22×#4111yescharacteristic
178C2×C53C2×2−1+411yescharacteristic
179D53—22no—
180D53—43no—
181#557—531no—
182#556—533no—
183#552—533no—
184#552—533no—
185D4×Q8—531no—
186#548—531no—
187#548—531no—
188#547—533no—
189#533—533no—
190#529—533no—
191#529—533no—
192C4×#41—531no—
193C4×#41—533no—
194C2×C4×Q8—531no—
195C2×C4×Q8D5311yescharacteristic, complemented
196C53#348211yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×#41—533no—
199C22×Q8—531no—
200C22×Q8C2×D5311yescharacteristic
201C4⋊Q8—533no—
202C4⋊Q8—533no—
203#126—536no—
204#126—536no—
205#125—533no—
206#125—536no—
207#125—533no—
208#124—533no—
209#124—533no—
210#123—536no—
211#123—536no—
212C22⋊Q8—533no—
213C22⋊Q8—533no—
214C22⋊Q8—533no—
215C22⋊Q8—536no—
216C22⋊Q8—533no—
217#121—533no—
218#121—533no—
219C4×Q8—531no—
220C4×Q8—531no—
221C4×Q8—533no—
222C4×Q8—533no—
223C4×Q8—533no—
224C4×Q8—533no—
225C4×Q8C2×D5311yescharacteristic
226C4×Q8C2×D5313yes—
227C4×Q8C2×D5311yescharacteristic
228C4×Q8C2×D5313yes—
229C4×D4—531no—
230C4×D4—533no—
231C4×D4—533no—
232C4×D4—533no—
233C4×D4—533no—
234C4×D4—533no—
235C4×D4—533no—
236C4×D4—533no—
237#117—533no—
238#117—533no—
239#117—533no—
240#117—533no—
241#117—536no—
242C2×C4⋊C4—533no—
243C2×C4⋊C4C2×D5313yes—
244C2×C42—533no—
245C2×C42—533no—
246C2×C42C2×D5313yes—
247#41—10612no—
248#41—1064no—
249C2×Q8—24no—
250C2×Q8—533no—
251C2×Q8—533no—
252C2×Q8—1064no—
253C2×Q8C22×D5313yes—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—531no—
256C2×D4—533no—
257C2×D4—533no—
258C2×D4—533no—
259C22×C4—533no—
260C22×C4—533no—
261C22×C4—533no—
262C22×C4—533no—
263C22×C4—531no—
264C22×C4—533no—
265C22×C4C22×D5313yes—
266C22×C4C22×D5313yes—
267C22×C4C22×D5311yescharacteristic
268C4⋊C4—533no—
269C4⋊C4—536no—
270C4⋊C4—531no—
271C4⋊C4—533no—
272C4⋊C4—533no—
273C4⋊C4—533no—
274C4⋊C4—533no—
275C4⋊C4—533no—
276C4⋊C4—533no—
277C4⋊C4—536no—
278C4⋊C4—536no—
279C4⋊C4C22×D5313yes—
280C4⋊C4C22×D5313yes—
281C4⋊C4C22×D5316yes—
282C22⋊C4—531no—
283C22⋊C4—533no—
284C22⋊C4—531no—
285C22⋊C4—533no—
286C22⋊C4—536no—
287C22⋊C4—533no—
288C22⋊C4—533no—
289C22⋊C4—536no—
290C22⋊C4—536no—
291C42—533no—
292C42—533no—
293C42—533no—
294C42—533no—
295C42—531no—
296C42—533no—
297C42C22×D5313yes—
298C42C22×D5313yes—
299C42C22×D5313yes—
300C42C22×D5313yes—
301C23—531no—
302C23—533no—
303C23C23×D5311yescharacteristic
304Q8—26no—
305Q8—1066no—
306Q8—1066no—
307Q8#1171994514yescomplemented
308D4—534no—
309D4—1066no—
310D4—1066no—
311D4—1066no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—22no—
315C2×C4—531no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—533no—
319C2×C4—533no—
320C2×C4—533no—
321C2×C4—1066no—
322C2×C4—1066no—
323C2×C4—1066no—
324C2×C4—1066no—
325C2×C4—1062no—
326C2×C4—1066no—
327C2×C4C23×D5313yes—
328C2×C4C23×D5311yescharacteristic
329C2×C4C23×D5313yes—
330C2×C4C23×D5313yes—
331C2×C4C23×D5313yes—
332C2×C4C23×D5311yescharacteristic
333C22—21no—
334C22—21no—
335C22—532no—
336C22—1061no—
337C22—1061no—
338C22—1063no—
339C22—1063no—
340C22—1063no—
341C22#41×D5312yes—
342C22C24×D5311yescharacteristic
343C4—23no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—532no—
348C4—1063no—
349C4—1061no—
350C4—1063no—
351C4—1061no—
352C4—1063no—
353C4C2×#1171994516yes—
354C4#1313326512yes—
355C2—22no—
356C2—1062no—
357C2—2123no—
358C2C2×#41×D5311yescharacteristic
359C2C22×#1171994511yescharacteristic
360C2C2×#1313326511yescharacteristic
subgroupG/H
centerC2×C4C23×D53
commutator, socleC22×C53C25
FrattiniC22C24×D53
FittingC2×C4×Q8×C53C2
hypercenterC2×C4×Q8D53
hypocenterC53#3482
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #16811349C2semidirect
2. #16811345C2semidirect
3. #16811314C2semidirect
4. #16811308C2semidirect
5. Q8×C22⋊D53C2semidirect
6. #16811285C2semidirect
7. #16811284C2semidirect
8. #16811274C2semidirect
9. #16811177C2semidirect
10. #16811160C2semidirect
11. #16811157C2semidirect
12. C4×#11719949C2semidirect
13. C4×#11719945C2semidirect
14. C4×Q8×D53C2semidirect
15. C2×C4×Q8×C53C2semidirect
124. Q8×C53#41semidirect
135. C22⋊D53#41semidirect
195. C2×C4×Q8D53semidirect
196. C53#3482semidirect
307. Q8#11719945semidirect