Finite Group Databasebeta

← #20286271

Subgroups — #20286271

Order 6784.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133248C211yesmaximal, characteristic, complemented
2C2×#13133237C211yesmaximal, characteristic, complemented
3#16811278C211yesmaximal, characteristic, complemented
4#16811018C211yesmaximal, characteristic, complemented
5#16810867C211yesmaximal, characteristic, complemented
6#16810862C211yesmaximal, characteristic, complemented
7#16810835C211yesmaximal, characteristic, complemented
8#16810833C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810783C211yesmaximal, characteristic, complemented
11#16810779C211yesmaximal, characteristic, complemented
12#16810766C211yesmaximal, characteristic, complemented
13#16810721C211yesmaximal, characteristic, complemented
14#16810695C211yesmaximal, characteristic, complemented
15C53×#548C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18#13133248—24no—
19#13133237—24no—
20C2×#11719925C2211yescharacteristic
21C2×#11719925C2211yescharacteristic
22C2×#11719925C2211yescharacteristic
23C2×#11719925C2211yescharacteristic
24C2×#11719923C2211yescharacteristic
25C2×#11719923C2211yescharacteristic
26C2×#11719924C2211yescharacteristic
27C2×#11719924C2211yescharacteristic
28C2×#11719922C2211yescharacteristic
29C2×#11719917C2211yescharacteristic
30C22×#88322C2211yescharacteristic
31#13133217C2211yescharacteristic
32#13133199—21no—
33#13133199—21no—
34#13133196C2211yescharacteristic
35#13133196C2211yescharacteristic
36#13133188—21no—
37C4×D212C2211yescharacteristic
38C4⋊C4×D53—21no—
39#13133150C2211yescharacteristic
40C4⋊#88322C2211yescharacteristic
41#13133141C2211yescharacteristic
42#13133141C2211yescharacteristic
43#13133140C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133137C2211yescharacteristic
47C22⋊#88322C2211yescharacteristic
48C22⋊#88322C2211yescharacteristic
49#13133128C2211yescharacteristic
50C2×#41×C53C2211yescharacteristic
51C22×Q8×C53C2211yescharacteristic
52#124×C53—21no—
53#124×C53C2211yescharacteristic
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53C2211yescharacteristic
56#121×C53—21no—
57#121×C53C2211yescharacteristic
58C4×Q8×C53C2211yescharacteristic
59C4×D4×C53—21no—
60C4×D4×C53C2211yescharacteristic
61#41×C53—22no—
62#41×C53—41no—
63C2×Q8×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53C2311yescharacteristic
67C2×Q8×C53C2311yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53D411yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2311yescharacteristic
75C22×C4×C53C2311yescharacteristic
76C2×C22⋊D53—21no—
77C2×C22⋊D53—21no—
78#11719945—22no—
79#11719945—22no—
80#11719945—41no—
81#11719945—41no—
82C2×D212C2311yescharacteristic
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×Dic106C2311yescharacteristic
87C2×Dic106C2311yescharacteristic
88C4⋊C4×C53—21no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53D411yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C22⋊C4×C53—21no—
94C22⋊C4×C53—21no—
95C22⋊C4×C53—21no—
96C22⋊C4×C53D411yescharacteristic
97C22⋊C4×C53D411yescharacteristic
98C42×C53—21no—
99C42×C53C2311yescharacteristic
100C22⋊Dic53—21no—
101C22⋊Dic53—21no—
102#11719925—22no—
103#11719925—22no—
104#11719925—22no—
105#11719925—41no—
106#11719924—22no—
107#11719924D414yescomplemented
108#11719923—22no—
109#11719923—41no—
110#11719922—22no—
111#11719921—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53C2311yescharacteristic
115#11719919—21no—
116#11719919—21no—
117#11719917—22no—
118C2×#88322—22no—
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121C2×#88322C2311yescharacteristic
122C2×#88322C2311yescharacteristic
123C23×C53—21no—
124C23×C53C2×D411yescharacteristic
125C23×C53C2×D411yescharacteristic
126C22×D53—21no—
127Q8×C53—21no—
128Q8×C53—21no—
129Q8×C53—21no—
130Q8×C53—42no—
131Q8×C53C2×D412yes—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—42no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53C2×D411yescharacteristic
147C2×C4×C53C2×D411yescharacteristic
148C2×C4×C53C2×D411yescharacteristic
149C2×C4×C53C2×D411yescharacteristic
150C2×C4×C53C2411yescharacteristic
151C22⋊D53—42no—
152C22⋊D53—42no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156D212—21no—
157D212C2×D412yes—
158C4×D53—42no—
159C4×D53—42no—
160C4×D53—42no—
161Dic106—21no—
162Dic106—21no—
163Dic106—21no—
164Dic106—21no—
165#88322—21no—
166#88322—21no—
167#88322—21no—
168#88322C2×D412yes—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53C22×D411yescharacteristic
179C2×D53—22no—
180C2×D53—41no—
181C2×D53—41no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—22no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53C22×D411yescharacteristic
190C4×C53H(5,𝔽2)11yescharacteristic
191Dic53—41no—
192Dic53—41no—
193Dic53—41no—
194#3269—531nomaximal, Sylow
195C2×C53—41no—
196C2×C53—41no—
197C2×C53—81no—
198C2×C53D4211yescharacteristic
199C2×C53#57511yescharacteristic
200C2×C53#57811yescharacteristic
201D53—42no—
202C2×#137—531no—
203C2×#135—531no—
204#548D5311yescharacteristic, complemented
205#547—531no—
206#497—531no—
207#468—531no—
208#467—531no—
209#463—531no—
210#462—531no—
211#452—531no—
212#452—531no—
213#451—531no—
214#449—531no—
215#440—531no—
216#435—531no—
217C53#326911yescharacteristic, complemented, Sylow
218C2×#41—531no—
219C2×#41—531no—
220C2×#41C2×D5311yescharacteristic
221C22×Q8C2×D5311yescharacteristic
222#137—1064no—
223#135—1064no—
224C2×Q16—531no—
225C2×Q16—531no—
226C2×Q16—531no—
227C2×Q16—531no—
228C2×SD16—531no—
229C2×SD16—531no—
230C2×SD16—531no—
231C2×SD16—531no—
232C2×D8—531no—
233C2×M16—531no—
234C22×C8—531no—
235C4⋊Q8—531no—
236#124—21no—
237#124C2×D5311yescharacteristic
238#123—1061no—
239#123—1061no—
240C22⋊Q8—21no—
241C22⋊Q8—531no—
242C22⋊Q8—531no—
243C22⋊Q8C2×D5311yescharacteristic
244#121—21no—
245#121—1061no—
246#121C2×D5311yescharacteristic
247C4×Q8C2×D5311yescharacteristic
248C4×D4—21no—
249C4×D4—531no—
250C4×D4C2×D5311yescharacteristic
251C2×C4⋊C4—1061no—
252#107—531no—
253C4⋊C8—531no—
254#103—531no—
255#103—531no—
256#103—531no—
257#103—531no—
258#102—531no—
259#102—531no—
260C22⋊C8—531no—
261C22⋊C8—531no—
262#97—531no—
263#41—22no—
264#41—41no—
265#41—1062no—
266#41—1062no—
267#41—2121no—
268#41—2121no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—531no—
273C2×Q8—531no—
274C2×Q8C22×D5311yescharacteristic
275C2×Q8C22×D5311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—531no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4C22⋊D5311yescharacteristic
282C2×D4C22×D5311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4C22×D5311yescharacteristic
289C22×C4C22×D5311yescharacteristic
290Q16—1062no—
291Q16—1062no—
292Q16—1062no—
293Q16—2121no—
294SD16—534no—
295SD16—1062no—
296SD16—1062no—
297SD16—2121no—
298D8—1062no—
299M16—1062no—
300C2×C8—531no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8—531no—
304C2×C8—1062no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—531no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4C22⋊D5311yescharacteristic
313C4⋊C4C22×D5311yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—21no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C22⋊C4C22⋊D5311yescharacteristic
322C22⋊C4C22⋊D5311yescharacteristic
323C42—21no—
324C42C22×D5311yescharacteristic
325C23—21no—
326C23—1061no—
327C23C2×C22⋊D5311yescharacteristic
328C23C2×C22⋊D5311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—42no—
333Q8—1061no—
334Q8—1061no—
335Q8—1061no—
336Q8—1061no—
337Q8D4×D5312yes—
338D4—21no—
339D4—21no—
340D4—41no—
341D4—42no—
342D4—42no—
343D4—532no—
344D4—1061no—
345D4—2122no—
346D4—2122no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4C2×C22⋊D5311yescharacteristic
363C2×C4C2×C22⋊D5311yescharacteristic
364C2×C4C2×C22⋊D5311yescharacteristic
365C2×C4C2×C22⋊D5311yescharacteristic
366C2×C4C23×D5311yescharacteristic
367C8—532no—
368C8—1061no—
369C8—1061no—
370C8—1061no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—1062no—
381C22—2121no—
382C22—2121no—
383C22C22×C22⋊D5311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—2121no—
392C4—2121no—
393C4—2121no—
394C4C2×D4×D5311yescharacteristic
395C4#1313326211yescharacteristic
396C2—41no—
397C2—41no—
398C2—81no—
399C2—2122no—
400C2D4×C22⋊D5311yescharacteristic
401C2#1681142111yescharacteristic
402C2#1681143811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#548C2
socleC22×C53C22×D4
hypercenter#548D53
hypocenterC53#3269
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#13133248C2semidirect
2. C2×#13133237C2semidirect
3. #16811278C2semidirect
4. #16811018C2semidirect
5. #16810867C2semidirect
6. #16810862C2semidirect
7. #16810835C2semidirect
8. #16810833C2semidirect
9. #16810785C2semidirect
10. #16810783C2semidirect
11. #16810779C2semidirect
12. #16810766C2semidirect
13. #16810721C2semidirect
14. #16810695C2semidirect
15. C53×#548C2semidirect
107. #11719924D4semidirect
204. #548D53semidirect
217. C53#3269semidirect