Finite Group Databasebeta

← #20286723

Subgroups — #20286723

Order 6784.

404 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811319C211yesmaximal, characteristic, complemented
2#16811294C211yesmaximal, characteristic, complemented
3#16810988C211yesmaximal, characteristic, complemented
4#16810948C211yesmaximal, characteristic, complemented
5#16810937C211yesmaximal, characteristic, complemented
6#16810931C211yesmaximal, characteristic, complemented
7#16810876C211yesmaximal, characteristic, complemented
8#16810859C211yesmaximal, characteristic, complemented
9#16810819C211yesmaximal, characteristic, complemented
10#16810770C211yesmaximal, characteristic, complemented
11#16810770C211yesmaximal, characteristic, complemented
12#16810725C211yesmaximal, characteristic, complemented
13#16810625C211yesmaximal, characteristic, complemented
14#16810620C211yesmaximal, characteristic, complemented
15C53×#435C211yesmaximal, characteristic, complemented
16C2×#11719945C2211yescharacteristic
17C2×#11719945C2211yescharacteristic
18C2×#11719913C2211yescharacteristic
19C2×D424C2211yescharacteristic
20C2×D424C2211yescharacteristic
21#13133214C2211yescharacteristic
22#13133213—21no—
23#13133212—21no—
24#13133209C2211yescharacteristic
25#13133199—21no—
26#13133191C2211yescharacteristic
27#13133191C2211yescharacteristic
28#13133191C2211yescharacteristic
29#13133191C2211yescharacteristic
30#13133190—21no—
31#13133190—21no—
32#13133189—21no—
33C4×Dic106C2211yescharacteristic
34C4×D212C2211yescharacteristic
35#13133179—21no—
36#13133177—21no—
37#13133176C2211yescharacteristic
38C2×C4⋊Dic53C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133151C2211yescharacteristic
41#13133149C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133142C2211yescharacteristic
44#13133142C2211yescharacteristic
45#13133139C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133139C2211yescharacteristic
48#13133139C2211yescharacteristic
49#13133139C2211yescharacteristic
50#13133139C2211yescharacteristic
51#13133139C2211yescharacteristic
52C2×M16×C53C2211yescharacteristic
53C22×C8×C53C2211yescharacteristic
54C4×D4×C53C2211yescharacteristic
55C4⋊C8×C53C2211yescharacteristic
56C22⋊C8×C53C2211yescharacteristic
57C22⋊C8×C53C2211yescharacteristic
58#97×C53C2211yescharacteristic
59C2×D4×C53D411yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C22×Dic53—21no—
69#11719945—22no—
70#11719945—22no—
71#11719945—41no—
72#11719945—41no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212C2311yescharacteristic
76C2×D212C2311yescharacteristic
77C2×D212C2311yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×Dic106C2311yescharacteristic
82M16×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C2×C8×C53C2311yescharacteristic
87C2×C8×C53C2311yescharacteristic
88C4⋊C4×C53D411yescharacteristic
89C22⋊C4×C53D411yescharacteristic
90C22⋊C4×C53D411yescharacteristic
91C42×C53C2311yescharacteristic
92C22⋊Dic53—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53—21no—
99C4⋊Dic53C2311yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108D424—41no—
109D424—41no—
110#11719913—22no—
111C23×C53C2×D411yescharacteristic
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116D4×C53—41no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53C2×D411yescharacteristic
122C2×C4×C53C2×D411yescharacteristic
123C2×C4×C53C2×D411yescharacteristic
124C2×C4×C53C2×D411yescharacteristic
125C2×C4×C53C2411yescharacteristic
126C22⋊D53—42no—
127C22⋊D53—42no—
128C22⋊D53—42no—
129C22⋊D53—81no—
130C22⋊D53—81no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—81no—
138D212—21no—
139D212—21no—
140D212—21no—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—42no—
145D212—81no—
146C4×D53—42no—
147C4×D53—42no—
148C4×D53—81no—
149Dic106—21no—
150Dic106#4112yes—
151C8×C53—21no—
152C8×C53—21no—
153C8×C53—21no—
154C8×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53C22×D411yescharacteristic
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53C2×#4111yescharacteristic
176C4×C53H(5,𝔽2)11yescharacteristic
177Dic53—22no—
178Dic53—41no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#3309—531nomaximal, Sylow
183C2×C53—41no—
184C2×C53—41no—
185C2×C53#54611yescharacteristic
186C2×C53#57511yescharacteristic
187C2×C53#57611yescharacteristic
188D53—81no—
189D53—81no—
190D53—81no—
191#553—531no—
192#550—531no—
193#490—531no—
194#482—531no—
195#480—531no—
196#479—531no—
197#469—531no—
198#466—531no—
199#459—531no—
200#449—531no—
201#449—531no—
202#440—531no—
203#435D5311yescharacteristic, complemented
204#417—531no—
205#416—531no—
206C53#330911yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41—531no—
209C2×SD16—531no—
210C2×D8—531no—
211C2×D8—531no—
212C2×M16C2×D5311yescharacteristic
213C22×C8C2×D5311yescharacteristic
214#127—531no—
215#127—1061no—
216#126—1061no—
217#125—531no—
218#123—1061no—
219#121—531no—
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224#121—1061no—
225#121—1061no—
226C4×Q8—531no—
227C4×D4—531no—
228C4×D4—1061no—
229C4×D4—1061no—
230C4×D4C2×D5311yescharacteristic
231#117—531no—
232C2×C4⋊C4—531no—
233#107—531no—
234#107—531no—
235#106—531no—
236C4⋊C8C2×D5311yescharacteristic
237#103—531no—
238#103—531no—
239#103—531no—
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244#102—531no—
245#102—531no—
246#102—531no—
247C22⋊C8C2×D5311yescharacteristic
248C22⋊C8C2×D5311yescharacteristic
249#97C2×D5311yescharacteristic
250#41—1062no—
251#41—1062no—
252#41—2121no—
253#41—2121no—
254C2×Q8—531no—
255C2×D4—531no—
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C2×D4D21211yescharacteristic
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4C22×D5311yescharacteristic
272C22×C4C22×D5311yescharacteristic
273SD16—1062no—
274D8—2121no—
275D8—2121no—
276M16—41no—
277C2×C8—41no—
278C2×C8C22×D5311yescharacteristic
279C2×C8C22×D5311yescharacteristic
280C2×C8C22×D5311yescharacteristic
281C2×C8C22×D5311yescharacteristic
282C4⋊C4—531no—
283C4⋊C4—531no—
284C4⋊C4—531no—
285C4⋊C4—531no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4D21211yescharacteristic
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4D21211yescharacteristic
298C22⋊C4D21211yescharacteristic
299C42—1061no—
300C42—1061no—
301C42C22×D5311yescharacteristic
302C23—1061no—
303C23—1061no—
304C23—1061no—
305C23C2×D21211yescharacteristic
306C23C2×D21211yescharacteristic
307Q8—532no—
308Q8—1061no—
309D4—41no—
310D4—1061no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—1061no—
316D4—4241no—
317D4—4241no—
318D4—4241no—
319D4—4241no—
320D4—2122no—
321D4—2122no—
322D4—2122no—
323D4—2122no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4C2×D21211yescharacteristic
338C2×C4C2×D21211yescharacteristic
339C2×C4C2×D21211yescharacteristic
340C2×C4C2×D21211yescharacteristic
341C2×C4C23×D5311yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—2121no—
353C22—2121no—
354C22—2121no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22—2121no—
359C22—2121no—
360C22—2121no—
361C22C22×D21211yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—1062no—
367C4—2121no—
368C4—2121no—
369C4—2121no—
370C4—2121no—
371C4#41×D5311yescharacteristic
372C4#1313326211yescharacteristic
373C2—41no—
374C2—41no—
375C2—4241no—
376C2—4241no—
377C2—4241no—
378C2#1681126811yescharacteristic
379C2#1681142311yescharacteristic
380C2#1681142811yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#435C2
socleC22×C53C22×D4
hypercenter#435D53
hypocenterC53#3309
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #16811319C2semidirect
2. #16811294C2semidirect
3. #16810988C2semidirect
4. #16810948C2semidirect
5. #16810937C2semidirect
6. #16810931C2semidirect
7. #16810876C2semidirect
8. #16810859C2semidirect
9. #16810819C2semidirect
10. #16810770C2semidirect
11. #16810770C2semidirect
12. #16810725C2semidirect
13. #16810625C2semidirect
14. #16810620C2semidirect
15. C53×#435C2semidirect
203. #435D53semidirect
206. C53#3309semidirect