Finite Group Databasebeta

← #20289124

Subgroups — #20289124

Order 6784.

1002 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811376C212yesmaximal, complemented
2#16811372C212yesmaximal, complemented
3#16811357C212yesmaximal, complemented
4#16811353C212yesmaximal, complemented
5#16811344C212yesmaximal, complemented
6#16811341C212yesmaximal, complemented
7Q8×C22⋊D53C212yesmaximal, complemented
8#16811286C212yesmaximal, complemented
9#16811241C214yesmaximal, complemented
10#16811237C214yesmaximal, complemented
11#16811196C212yesmaximal, complemented
12#16811194C211yesmaximal, characteristic, complemented
13#16811175C212yesmaximal, complemented
14#16811169C211yesmaximal, characteristic
15C53×#536C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C22×Dic106C2211yescharacteristic
18C2×Q8×D53C2212yes—
19#13133216C2214yes—
20#13133216C2212yes—
21#13133216C2214yes—
22#13133215C2212yes—
23#13133215C2214yes—
24#13133215C2211yescharacteristic
25#13133215C2212yes—
26#13133213C2211yescharacteristic
27#13133212C2214yes—
28#13133211C2214yes—
29#13133210C2214yes—
30#13133210C2214yes—
31#13133207C2214yes—
32#13133206C2211yescharacteristic
33#13133204C2211yescharacteristic
34#13133204C2212yes—
35#13133203C2214yes—
36#13133203C2214yes—
37#13133203C2214yes—
38#13133202C2214yes—
39#13133199C2214yes—
40#13133198C2212yes—
41#13133198C2212yes—
42#13133197C2212yes—
43#13133196C2212yes—
44#13133196C2214yes—
45#13133196C2212yes—
46#13133195C2214yes—
47#13133194C2214yes—
48#13133194C2214yes—
49#13133193C2212yes—
50#13133193C2214yes—
51#13133189C2212yes—
52#13133189C2212yes—
53Q8×Dic53C2212yes—
54C4×Dic106C2214yes—
55#13133182C2212yes—
56#13133182C2214yes—
57#13133182C2214yes—
58C4×C22⋊D53C2212yes—
59C4×C22⋊D53C2212yes—
60C4×C22⋊D53C2214yes—
61#13133177C2214yes—
62#13133176C2211yescharacteristic
63#13133175C2212yes—
64#13133173C2214yes—
65C2×C4×Dic53C2211yescharacteristic
66C2×#41×C53C2211yescharacteristic
67C22×Q8×C53C2211yescharacteristic
68C4⋊Q8×C53C2212yes—
69#124×C53C2212yes—
70#123×C53C2212yes—
71#123×C53C2212yes—
72C22⋊Q8×C53C2212yes—
73C22⋊Q8×C53C2212yes—
74#117×C53C2211yescharacteristic
75#41×C53—42no—
76C2×Q8×C53—42no—
77C2×Q8×C53C2312yes—
78C2×Q8×C53C2312yes—
79C2×Q8×C53C2311yescharacteristic
80C2×D4×C53C2311yescharacteristic
81C2×D4×C53C2311yescharacteristic
82C2×D4×C53C2311yescharacteristic
83C22×C4×C53C2311yescharacteristic
84C22×C4×C53C2311yescharacteristic
85C22×C4×C53C2311yescharacteristic
86C22×C4×C53C2312yes—
87C2×C22⋊D53C2312yes—
88C2×C22⋊D53C2312yes—
89C2×C22⋊D53C2312yes—
90C22×Dic53C2312yes—
91C22×Dic53C2312yes—
92Q8×D53—44no—
93#11719945—42no—
94C2×D212C2311yescharacteristic
95C2×C4×D53C2312yes—
96C2×C4×D53C2314yes—
97C2×Dic106—42no—
98C2×Dic106C2314yes—
99C2×Dic106C2312yes—
100C2×Dic106C2314yes—
101C2×Dic106C2312yes—
102C2×Dic106C2311yescharacteristic
103C4⋊C4×C53C2312yes—
104C4⋊C4×C53C2314yes—
105C4⋊C4×C53C2314yes—
106C22⋊C4×C53C2312yes—
107C22⋊C4×C53C2314yes—
108C22⋊C4×C53C2314yes—
109C42×C53C2312yes—
110C22⋊Dic53C2314yes—
111C22⋊Dic53C2312yes—
112C22⋊Dic53C2312yes—
113C22⋊Dic53C2312yes—
114C22⋊Dic53C2314yes—
115#11719921C2314yes—
116#11719921C2314yes—
117#11719921C2312yes—
118#11719921C2312yes—
119#11719921C2314yes—
120C4⋊Dic53C2312yes—
121C4⋊Dic53C2314yes—
122C4⋊Dic53C2314yes—
123#11719919C2314yes—
124#11719919C2314yes—
125#11719919C2314yes—
126#11719919C2312yes—
127#11719919C2312yes—
128#11719919C2314yes—
129#11719919C2314yes—
130#11719919C2314yes—
131#11719919C2314yes—
132C4×Dic53C2312yes—
133C4×Dic53C2311yescharacteristic
134C4×Dic53C2311yescharacteristic
135C4×Dic53C2314yes—
136C4×Dic53C2314yes—
137C4×Dic53C2314yes—
138C4×Dic53C2312yes—
139C23×C53C2411yescharacteristic
140C23×C53C2411yescharacteristic
141C23×C53C2411yescharacteristic
142C22×D53C2412yes—
143Q8×C53—24no—
144Q8×C53—42no—
145Q8×C53—41no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—41no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—42no—
153C2×C4×C53C2411yescharacteristic
154C2×C4×C53C2414yes—
155C2×C4×C53C2411yescharacteristic
156C2×C4×C53C2414yes—
157C2×C4×C53C2411yescharacteristic
158C2×C4×C53C2411yescharacteristic
159C22⋊D53—24no—
160C22⋊D53—24no—
161C22⋊D53—42no—
162C2×Dic53—24no—
163C2×Dic53—42no—
164C2×Dic53C2414yes—
165C2×Dic53C2412yes—
166C2×Dic53C2414yes—
167C2×Dic53C2412yes—
168C2×Dic53C2412yes—
169D212—41no—
170C4×D53—24no—
171C4×D53—44no—
172Dic106—24no—
173Dic106—28no—
174Dic106—42no—
175Dic106—41no—
176Dic106—44no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—22no—
182C22×C53—22no—
183C22×C53C2511yescharacteristic
184C2×D53—22no—
185C2×D53—22no—
186C2×D53—22no—
187C4×C53—21no—
188C4×C53—24no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—24no—
192C4×C53H(5,𝔽2)12yes—
193Dic53—24no—
194Dic53—22no—
195Dic53—24no—
196Dic53—22no—
197Dic532−1+414yes—
198#3550—531nomaximal, Sylow
199C2×C53—41no—
200C2×C53—41no—
201C2×C53—41no—
202C2×C53C2×H(5,𝔽2)11yescharacteristic
203C2×C53C2×2−1+412yes—
204D53—42no—
205#563—532no—
206#563—532no—
207#559—532no—
208Q82—532no—
209#556—532no—
210#556—532no—
211D4×Q8—532no—
212D4×Q8—532no—
213#542—534no—
214#542—534no—
215#536—531no—
216#536—532no—
217#536D5311yescharacteristic, complemented
218#533—532no—
219#532—531no—
220C53#355011yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C2×#41C2×D5311yescharacteristic
223C22×Q8—532no—
224C22×Q8—531no—
225C22×Q8C2×D5311yescharacteristic
226C4⋊Q8—531no—
227C4⋊Q8—532no—
228C4⋊Q8—532no—
229C4⋊Q8—534no—
230C4⋊Q8—532no—
231C4⋊Q8—534no—
232C4⋊Q8—534no—
233C4⋊Q8C2×D5312yes—
234#127—531no—
235#126—534no—
236#126—534no—
237#126—534no—
238#126—534no—
239#125—534no—
240#124—531no—
241#124—531no—
242#124—532no—
243#124—534no—
244#124—534no—
245#124—534no—
246#124C2×D5312yes—
247#123—532no—
248#123—532no—
249#123—534no—
250#123—534no—
251#123C2×D5312yes—
252#123C2×D5312yes—
253C22⋊Q8—532no—
254C22⋊Q8—534no—
255C22⋊Q8—532no—
256C22⋊Q8—532no—
257C22⋊Q8—534no—
258C22⋊Q8—532no—
259C22⋊Q8—534no—
260C22⋊Q8—534no—
261C22⋊Q8—534no—
262C22⋊Q8C2×D5312yes—
263C22⋊Q8C2×D5312yes—
264#121—532no—
265#121—532no—
266C4×Q8—532no—
267C4×Q8—532no—
268C4×Q8—534no—
269C4×Q8—534no—
270C4×Q8—534no—
271C4×D4—532no—
272C4×D4—532no—
273C4×D4—534no—
274C4×D4—534no—
275#117—531no—
276#117—532no—
277#117—534no—
278#117C2×D5311yescharacteristic
279C2×C42—531no—
280#41—42no—
281#41—2122no—
282C2×Q8—42no—
283C2×Q8—534no—
284C2×Q8—532no—
285C2×Q8—534no—
286C2×Q8—532no—
287C2×Q8—531no—
288C2×Q8—2122no—
289C2×Q8—2124no—
290C2×Q8C22×D5312yes—
291C2×Q8C22×D5312yes—
292C2×Q8C22×D5311yescharacteristic
293C2×D4—532no—
294C2×D4—532no—
295C2×D4—532no—
296C2×D4—531no—
297C2×D4C22×D5311yescharacteristic
298C2×D4C22×D5311yescharacteristic
299C2×D4C22×D5311yescharacteristic
300C22×C4—532no—
301C22×C4—532no—
302C22×C4—532no—
303C22×C4—534no—
304C22×C4C22×D5311yescharacteristic
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307C22×C4C22×D5312yes—
308C4⋊C4—532no—
309C4⋊C4—534no—
310C4⋊C4—534no—
311C4⋊C4—534no—
312C4⋊C4—532no—
313C4⋊C4—532no—
314C4⋊C4—534no—
315C4⋊C4—534no—
316C4⋊C4—534no—
317C4⋊C4—534no—
318C4⋊C4—534no—
319C4⋊C4—534no—
320C4⋊C4C22×D5312yes—
321C4⋊C4C22×D5314yes—
322C4⋊C4C22×D5314yes—
323C22⋊C4—534no—
324C22⋊C4—532no—
325C22⋊C4—532no—
326C22⋊C4—532no—
327C22⋊C4—534no—
328C22⋊C4—534no—
329C22⋊C4—532no—
330C22⋊C4—532no—
331C22⋊C4—534no—
332C22⋊C4—534no—
333C22⋊C4C22×D5314yes—
334C22⋊C4C22×D5312yes—
335C22⋊C4C22×D5314yes—
336C42—532no—
337C42—531no—
338C42—531no—
339C42—534no—
340C42—534no—
341C42—532no—
342C42—534no—
343C42C22×D5312yes—
344C23—532no—
345C23C23×D5311yescharacteristic
346C23C23×D5311yescharacteristic
347C23C23×D5311yescharacteristic
348Q8—24no—
349Q8—42no—
350Q8—41no—
351Q8—1068no—
352Q8—1064no—
353Q8—2122no—
354Q8—2124no—
355Q8—2121no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—1064no—
360D4—1064no—
361D4—2121no—
362D4—2122no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—42no—
367C2×C4—532no—
368C2×C4—534no—
369C2×C4—532no—
370C2×C4—532no—
371C2×C4—534no—
372C2×C4—1064no—
373C2×C4—1064no—
374C2×C4—2124no—
375C2×C4—2122no—
376C2×C4C23×D5314yes—
377C2×C4C23×D5311yescharacteristic
378C2×C4C23×D5311yescharacteristic
379C2×C4C23×D5314yes—
380C2×C4C23×D5311yescharacteristic
381C2×C4C23×D5311yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—1062no—
389C22—1062no—
390C22—1062no—
391C22C24×D5311yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—24no—
396C4—24no—
397C4—534no—
398C4—1062no—
399C4—1064no—
400C4—1062no—
401C4—1064no—
402C4#1313326212yes—
403C2—41no—
404C2—41no—
405C2—41no—
406C2—2122no—
407C2C2×#1313326211yescharacteristic
408C22−1+4×D5312yes—
subgroupG/H
center, FrattiniC22C24×D53
commutator, socleC22×C53C25
FittingC53×#536C2
hypercenter#536D53
hypocenterC53#3550
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #16811376C2semidirect
2. #16811372C2semidirect
3. #16811357C2semidirect
4. #16811353C2semidirect
5. #16811344C2semidirect
6. #16811341C2semidirect
7. Q8×C22⋊D53C2semidirect
8. #16811286C2semidirect
9. #16811241C2semidirect
10. #16811237C2semidirect
11. #16811196C2semidirect
12. #16811194C2semidirect
13. #16811175C2semidirect
15. C53×#536C2semidirect
217. #536D53semidirect
220. C53#3550semidirect