Finite Group Databasebeta

← #20288791

Subgroups — #20288791

Order 6784.

1104 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#16811355C213yesmaximal, complemented
2Q8×Dic106C211yesmaximal, characteristic, complemented
3#16811331C213yesmaximal, complemented
4#16811292C211yesmaximal, characteristic, complemented
5#16811284C211yesmaximal, characteristic, complemented
6#16811277C213yesmaximal, complemented
7#16811273C213yesmaximal, complemented
8#16811246C213yesmaximal, complemented
9#16811242C213yesmaximal, complemented
10#16811168C213yesmaximal, complemented
11#16811153C213yesmaximal, complemented
12#16811106C211yesmaximal, characteristic, complemented
13#16811105C211yesmaximal, characteristic, complemented
14C2×Q8×Dic53C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16C2×#11719949C2211yescharacteristic
17C2×#11719947C2216yes—
18#13133217C2213yes—
19#13133216C2213yes—
20#13133216C2213yes—
21#13133213C2213yes—
22#13133212C2213yes—
23#13133210C2213yes—
24#13133207C2213yes—
25#13133206C2213yes—
26#13133203C2213yes—
27#13133203C2216yes—
28#13133201C2213yes—
29#13133201C2216yes—
30#13133199C2216yes—
31#13133198C2213yes—
32#13133197C2216yes—
33#13133194C2216yes—
34#13133194C2211yescharacteristic
35#13133193C2213yes—
36#13133191C2213yes—
37#13133191C2213yes—
38#13133189C2213yes—
39#13133188C2213yes—
40Q8×Dic53C2211yescharacteristic
41Q8×Dic53C2211yescharacteristic
42Q8×Dic53C2213yes—
43Q8×Dic53C2211yescharacteristic
44Q8×Dic53C2213yes—
45C4×Dic106C2213yes—
46C4×D212C2213yes—
47D4×Dic53C2213yes—
48D4×Dic53C2213yes—
49#13133179C2213yes—
50C4×C22⋊D53C2213yes—
51C4×C22⋊D53C2213yes—
52#13133177C2216yes—
53#13133176C2213yes—
54#13133174C2213yes—
55C2×C4⋊Dic53C2213yes—
56C2×C4⋊Dic53C2213yes—
57C4⋊C4×D53C2213yes—
58C2×C4×Dic53C2213yes—
59C2×#41×C53C2211yescharacteristic
60C22×Q8×C53C2211yescharacteristic
61#124×C53C2213yes—
62C22⋊Q8×C53C2213yes—
63#121×C53C2213yes—
64C4×Q8×C53C2211yescharacteristic
65C4×D4×C53C2213yes—
66#41×C53—24no—
67C2×Q8×C53—24no—
68C2×Q8×C53C2313yes—
69C2×Q8×C53C2311yescharacteristic
70C2×D4×C53C2313yes—
71C2×D4×C53C2313yes—
72C22×C4×C53C2313yes—
73C22×C4×C53C2313yes—
74C2×C22⋊D53C2313yes—
75C2×C22⋊D53C2313yes—
76C2×C22⋊D53C2316yes—
77C22×Dic53C2311yescharacteristic
78C22×Dic53C2313yes—
79C22×Dic53C2316yes—
80#11719949—24no—
81#11719947—412no—
82C2×D212C2313yes—
83C2×C4×D53C2313yes—
84C2×C4×D53C2316yes—
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2316yes—
87C4⋊C4×C53C2313yes—
88C4⋊C4×C53C2313yes—
89C22⋊C4×C53C2311yescharacteristic
90C22⋊C4×C53C2313yes—
91C22⋊C4×C53C2316yes—
92C42×C53C2313yes—
93C22⋊Dic53C2313yes—
94C22⋊Dic53C2313yes—
95C22⋊Dic53C2311yescharacteristic
96C22⋊Dic53C2313yes—
97#11719921C2313yes—
98#11719921C2313yes—
99#11719921C2316yes—
100#11719921C2313yes—
101#11719921C2313yes—
102C4⋊Dic53C2313yes—
103C4⋊Dic53C2313yes—
104C4⋊Dic53C2313yes—
105C4⋊Dic53C2313yes—
106C4⋊Dic53C2316yes—
107C4⋊Dic53C2311yescharacteristic
108C4⋊Dic53C2313yes—
109#11719919C2313yes—
110#11719919C2311yescharacteristic
111#11719919C2313yes—
112#11719919C2311yescharacteristic
113#11719919C2313yes—
114#11719919C2313yes—
115C4×Dic53C2313yes—
116C4×Dic53C2313yes—
117C4×Dic53C2313yes—
118C4×Dic53C2313yes—
119C4×Dic53C2313yes—
120C23×C53C2413yes—
121C23×C53C2411yescharacteristic
122C22×D53C2413yes—
123Q8×C53—26no—
124Q8×C53#4114yescomplemented
125D4×C53—26no—
126D4×C53—26no—
127C2×C4×C53—26no—
128C2×C4×C53—26no—
129C2×C4×C53C2413yes—
130C2×C4×C53C2411yescharacteristic
131C2×C4×C53C2411yescharacteristic
132C2×C4×C53C2413yes—
133C2×C4×C53C2413yes—
134C22⋊D53—26no—
135C22⋊D53—26no—
136C22⋊D53—46no—
137C2×Dic53—26no—
138C2×Dic53—22no—
139C2×Dic53—46no—
140C2×Dic53C2413yes—
141C2×Dic53C2413yes—
142C2×Dic53C2411yescharacteristic
143C2×Dic53C2411yescharacteristic
144C2×Dic53C2413yes—
145C2×Dic53C2411yescharacteristic
146C2×Dic53C2411yescharacteristic
147D212—26no—
148C4×D53—26no—
149C4×D53—46no—
150Dic106—22no—
151Dic106—46no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—23no—
155C22×C53—23no—
156C22×C53—23no—
157C22×C53H(5,𝔽2)12yes—
158C22×C53C2511yescharacteristic
159C2×D53—23no—
160C2×D53—23no—
161C2×D53—23no—
162C4×C53—21no—
163C4×C53—23no—
164C4×C53—23no—
165C4×C53—21no—
166C4×C53C2×#4116yes—
167Dic53—21no—
168Dic53—23no—
169Dic53—23no—
170Dic53—23no—
171Dic53—21no—
172Dic53—21no—
173Dic532−1+412yes—
174#3522—531nomaximal, Sylow
175C2×C53—22no—
176C2×C53—43no—
177C2×C53C22×#4111yescharacteristic
178C2×C53C2×H(5,𝔽2)11yescharacteristic
179C2×C53C2×2−1+411yescharacteristic
180D53—43no—
181#559—533no—
182Q82—531no—
183#554—533no—
184#550—531no—
185#548—531no—
186#548D5311yescharacteristic, complemented
187#547—533no—
188#547—533no—
189#542—533no—
190#542—533no—
191#532—533no—
192#529—533no—
193#520—531no—
194#520—531no—
195C2×C4×Q8—531no—
196C53#352211yescharacteristic, complemented, Sylow
197C2×#41—531no—
198C2×#41—536no—
199C2×#41C2×D5311yescharacteristic
200C22×Q8C2×D5311yescharacteristic
201C4⋊Q8—533no—
202C4⋊Q8—533no—
203C4⋊Q8—533no—
204#127—533no—
205#126—533no—
206#126—533no—
207#125—533no—
208#124—533no—
209#124—533no—
210#124—536no—
211#124C2×D5313yes—
212#123—533no—
213#123—533no—
214#123—536no—
215#123—536no—
216C22⋊Q8—533no—
217C22⋊Q8—536no—
218C22⋊Q8—531no—
219C22⋊Q8—536no—
220C22⋊Q8C2×D5313yes—
221#121—533no—
222#121—533no—
223#121—533no—
224#121—533no—
225#121C2×D5313yes—
226C4×Q8—531no—
227C4×Q8—533no—
228C4×Q8—531no—
229C4×Q8—533no—
230C4×Q8—531no—
231C4×Q8—533no—
232C4×Q8C2×D5311yescharacteristic
233C4×D4—533no—
234C4×D4—533no—
235C4×D4—533no—
236C4×D4—533no—
237C4×D4—533no—
238C4×D4—536no—
239C4×D4—533no—
240C4×D4C2×D5313yes—
241#117—533no—
242#117—533no—
243C2×C4⋊C4—533no—
244C2×C4⋊C4—533no—
245C2×C4⋊C4—533no—
246C2×C42—533no—
247#41—24no—
248#41—1064no—
249#41—21212no—
250C2×Q8—24no—
251C2×Q8—531no—
252C2×Q8—536no—
253C2×Q8C22×D5313yes—
254C2×Q8C22×D5311yescharacteristic
255C2×D4—533no—
256C2×D4—533no—
257C2×D4—536no—
258C2×D4—533no—
259C2×D4C22×D5313yes—
260C2×D4C22×D5313yes—
261C22×C4—531no—
262C22×C4—533no—
263C22×C4—536no—
264C22×C4—533no—
265C22×C4—536no—
266C22×C4C22×D5313yes—
267C22×C4C22×D5313yes—
268C4⋊C4—533no—
269C4⋊C4—533no—
270C4⋊C4—533no—
271C4⋊C4—533no—
272C4⋊C4—533no—
273C4⋊C4—536no—
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—533no—
278C4⋊C4—533no—
279C4⋊C4—533no—
280C4⋊C4—533no—
281C4⋊C4C22×D5313yes—
282C4⋊C4C22×D5313yes—
283C22⋊C4—533no—
284C22⋊C4—533no—
285C22⋊C4—531no—
286C22⋊C4—533no—
287C22⋊C4—533no—
288C22⋊C4—533no—
289C22⋊C4—533no—
290C22⋊C4—536no—
291C22⋊C4—533no—
292C22⋊C4C22×D5316yes—
293C22⋊C4C22×D5311yescharacteristic
294C22⋊C4C22×D5313yes—
295C42—533no—
296C42—533no—
297C42—533no—
298C42—533no—
299C42—533no—
300C42C22×D5313yes—
301C23—533no—
302C23C23×D5313yes—
303C23C23×D5311yescharacteristic
304Q8—26no—
305Q8—1062no—
306Q8—2126no—
307Q8#1171994714yescomplemented
308D4—26no—
309D4—26no—
310D4—1066no—
311D4—1066no—
312D4—1066no—
313D4—2126no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—531no—
319C2×C4—533no—
320C2×C4—533no—
321C2×C4—533no—
322C2×C4—531no—
323C2×C4—1062no—
324C2×C4—1066no—
325C2×C4—1066no—
326C2×C4—2126no—
327C2×C4—2126no—
328C2×C4C23×D5313yes—
329C2×C4C23×D5311yescharacteristic
330C2×C4C23×D5313yes—
331C2×C4C23×D5311yescharacteristic
332C2×C4C23×D5313yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1063no—
339C22—1063no—
340C22—1063no—
341C22#1313326312yes—
342C22C24×D5311yescharacteristic
343C4—21no—
344C4—23no—
345C4—23no—
346C4—21no—
347C4—532no—
348C4—1061no—
349C4—1063no—
350C4—1063no—
351C4—1063no—
352C4—1061no—
353C4—1061no—
354C4C2×#1171994716yes—
355C2—22no—
356C2—43no—
357C2—2123no—
358C2C22×#1171994711yescharacteristic
359C2C2×#1313326311yescharacteristic
360C22−1+4×D5311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#548C2
hypercenter#548D53
hypocenterC53#3522
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16811355C2semidirect
2. Q8×Dic106C2semidirect
3. #16811331C2semidirect
4. #16811292C2semidirect
5. #16811284C2semidirect
6. #16811277C2semidirect
7. #16811273C2semidirect
8. #16811246C2semidirect
9. #16811242C2semidirect
10. #16811168C2semidirect
11. #16811153C2semidirect
12. #16811106C2semidirect
13. #16811105C2semidirect
14. C2×Q8×Dic53C2semidirect
15. C53×#548C2semidirect
124. Q8×C53#41semidirect
186. #548D53semidirect
196. C53#3522semidirect
307. Q8#11719947semidirect