Finite Group Databasebeta

← #27301108

Subgroups — #27301108

Order 13568.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283310C211yesmaximal, characteristic, complemented
2#20276451C211yesmaximal, characteristic, complemented
3C2×#16810433C211yesmaximal, characteristic, complemented
4#20276343C211yesmaximal, characteristic, complemented
5#20272829C211yesmaximal, characteristic, complemented
6#20272815C211yesmaximal, characteristic, complemented
7C53×#3067C211yesmaximal, characteristic
8#16811106—21no—
9C2×#13133157C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11#16810665—22no—
12#16810665—21no—
13#16810661—21no—
14#16810661C2211yescharacteristic
15C2×#13133149—21no—
16C2×#13133149C2211yescharacteristic
17#16810433—22no—
18#16810433—21no—
19#16810433—21no—
20#16810407—22no—
21C53×#575—21no—
22C2×#135×C53C2211yescharacteristic
23C2×#131×C53C2211yescharacteristic
24C53×#473C412yescomplemented
25C53×#472C412yescomplemented
26C53×#471—22no—
27C2×#108×C53C2211yescharacteristic
28C53×#413—21no—
29Q8×Dic53—41no—
30D4×Dic53—42no—
31D4×Dic53—41no—
32#13133176—22no—
33#13133176—21no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53—21no—
36#13133157—21no—
37#13133157—21no—
38#13133157—21no—
39#13133157—21no—
40C2×#11719908—21no—
41C2×#11719908—21no—
42#13133151—22no—
43#13133151—21no—
44#13133149—22no—
45#13133149—21no—
46#13133149D411yescharacteristic
47#13133149D412yes—
48#13133149D411yescharacteristic
49C2×#41×C53—21no—
50C2×#41×C53D411yescharacteristic
51#137×C53—22no—
52#137×C53—41no—
53AGL(1,ℤ/8)×C53—22no—
54AGL(1,ℤ/8)×C53—41no—
55#135×C53—21no—
56#135×C53—21no—
57#135×C53—21no—
58#135×C53—21no—
59#135×C53—42no—
60C2×Q16×C53C2×C411yescharacteristic, complemented
61C2×SD16×C53—21no—
62C2×D8×C53C2×C411yescharacteristic, complemented
63#131×C53—21no—
64#131×C53—21no—
65#131×C53C2×C412yes—
66#131×C53D412yes—
67C2×M16×C53—21no—
68C2×M16×C53—21no—
69C2×M16×C53D411yescharacteristic
70C22×C8×C53—21no—
71C22×C8×C53C2311yescharacteristic
72#108×C53—21no—
73#108×C53C2×C412yescomplemented
74#101×C53—22no—
75#100×C53—22no—
76#41×C53—21no—
77#41×C53—21no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53C22⋊C412yes—
83#41×C53SD1612yes—
84C2×Q8×C53—21no—
85C2×Q8×C53D811yescharacteristic
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—41no—
89C2×D4×C53D811yescharacteristic
90C22×C4×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53—42no—
94C22×Dic53—41no—
95Q16×C53—21no—
96Q16×C53—21no—
97Q16×C53—42no—
98SD16×C53—42no—
99SD16×C53—42no—
100SD16×C53—41no—
101SD16×C53—41no—
102D8×C53—21no—
103D8×C53—21no—
104D8×C53—42no—
105M16×C53—21no—
106M16×C53—22no—
107M16×C53—22no—
108M16×C53—22no—
109M16×C53—41no—
110M16×C53C22⋊C412yes—
111C2×C8×C53—21no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—41no—
115C2×C8×C53C22×C411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53#4111yescharacteristic
120C2×C8×C53#4111yescharacteristic
121C22⋊Dic53—42no—
122C22⋊Dic53—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—22no—
125C4⋊Dic53—21no—
126C4⋊Dic53—42no—
127C4⋊Dic53—41no—
128C4×Dic53—42no—
129C4×Dic53—41no—
130#11719908—41no—
131#11719908—41no—
132C23×C53—21no—
133C23×C53—41no—
134C23×C53AGL(1,ℤ/8)11yescharacteristic
135Q8×C53—21no—
136Q8×C53—41no—
137Q8×C53—41no—
138Q8×C53#10212yes—
139D4×C53—22no—
140D4×C53—22no—
141D4×C53—21no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53#10212yes—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×SD1611yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×Dic53—41no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—161no—
163C2×Dic53—162no—
164C8×C53—22no—
165C8×C53—42no—
166C8×C53C4×D411yescharacteristic
167C8×C53C4×D411yescharacteristic
168C8×C53#12111yescharacteristic
169C8×C53#12111yescharacteristic
170#13005—531nomaximal, Sylow
171C22×C53—22no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53—81no—
175C22×C53—161no—
176C22×C53#41711yescharacteristic
177C22×C53#44711yescharacteristic
178C22×C53#48111yescharacteristic
179C4×C53—22no—
180C4×C53—81no—
181C4×C53#38611yescharacteristic
182C4×C53C2×#10211yescharacteristic
183C4×C53#44811yescharacteristic
184C4×C53#48011yescharacteristic
185Dic53—81no—
186Dic53—81no—
187Dic53—82no—
188#3067D5311yescharacteristic
189#2912—531no—
190#2145—531no—
191C2×#360—531no—
192#2117—531no—
193#1389—531no—
194#1383—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—42no—
199C2×C53—161no—
200C2×C53#187411yescharacteristic
201#575—21no—
202C2×#135C2×D5311yescharacteristic
203C2×#131C2×D5311yescharacteristic
204#520—1061no—
205C2×M32—531no—
206C2×M32—531no—
207#473Dic5312yescomplemented
208#472Dic5312yescomplemented
209#471—22no—
210C2×#108C2×D5311yescharacteristic
211#428—1062no—
212#428—1061no—
213#427—531no—
214#427—1061no—
215C2×#106—531no—
216C2×#106—1061no—
217#413—21no—
218#360—1062no—
219#360—1061no—
220#360—1061no—
221#350—1062no—
222C53#1300511yescharacteristic, complemented, Sylow
223C2×#41—21no—
224C2×#41C22⋊D5311yescharacteristic
225#137—22no—
226#137—41no—
227AGL(1,ℤ/8)—22no—
228AGL(1,ℤ/8)—41no—
229#135—21no—
230#135—21no—
231#135—21no—
232#135—21no—
233#135—42no—
234C2×Q16C2×Dic5311yescharacteristic, complemented
235C2×SD16—21no—
236C2×D8C2×Dic5311yescharacteristic, complemented
237#131—21no—
238#131—21no—
239#131C2×Dic5312yes—
240#131C22⋊D5312yes—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16C22⋊D5311yescharacteristic
244C22×C8—21no—
245C22×C8C22×D5311yescharacteristic
246C4×Q8—2121no—
247C4×D4—2122no—
248C4×D4—2121no—
249#117—1062no—
250#117—1061no—
251C2×C4⋊C4—1062no—
252C2×C4⋊C4—1061no—
253M32—1061no—
254M32—1061no—
255M32—1061no—
256M32—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259#108—21no—
260#108C2×Dic5312yescomplemented
261#107—1062no—
262#107—1061no—
263#106—532no—
264#106—531no—
265#106—531no—
266#106—1062no—
267#106—1061no—
268#101—22no—
269#100—22no—
270#41—21no—
271#41—21no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41#1171992312yes—
277#41C22⋊Dic5312yes—
278C2×Q8—21no—
279C2×Q8#1171992211yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4#1171992211yescharacteristic
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—2122no—
287C22×C4—2121no—
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—21no—
290Q16—21no—
291Q16—42no—
292SD16—42no—
293SD16—42no—
294SD16—41no—
295SD16—41no—
296D8—21no—
297D8—21no—
298D8—42no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—22no—
303M16—41no—
304M16C22⋊Dic5312yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8D4×D5311yescharacteristic
310C2×C8D4×D5311yescharacteristic
311C2×C8#1171994711yescharacteristic
312C2×C8#1171994711yescharacteristic
313C2×C8C22×Dic5311yescharacteristic
314C2×C8C2×C22⋊D5311yescharacteristic
315C4⋊C4—1062no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—2122no—
319C4⋊C4—2121no—
320C22⋊C4—2122no—
321C22⋊C4—2121no—
322C42—2122no—
323C42—2121no—
324C16—2121no—
325C16—2121no—
326C23—21no—
327C23—41no—
328C23#1313324211yescharacteristic
329Q8—21no—
330Q8—41no—
331Q8—41no—
332Q8#1313313812yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4#1313313812yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—8481no—
348C2×C4—8482no—
349C2×C4—2121no—
350C2×C4—2122no—
351C2×C4—2121no—
352C2×C4C2×C22⋊Dic5311yescharacteristic
353C2×C4#1313318511yescharacteristic
354C2×C4#1313319811yescharacteristic
355C2×C4C2×#1171992211yescharacteristic
356C2×C4C2×#1171992311yescharacteristic
357C2×C4#1313324711yescharacteristic
358C8—22no—
359C8—42no—
360C8D4×Dic5311yescharacteristic
361C8D4×Dic5311yescharacteristic
362C8#1313318911yescharacteristic
363C8#1313318911yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—81no—
368C22—161no—
369C22#1681062411yescharacteristic
370C22#1681075611yescharacteristic
371C22#1681093911yescharacteristic
372C4—22no—
373C4—81no—
374C4—4242no—
375C4—4241no—
376C4—4241no—
377C4#1681050311yescharacteristic
378C4C2×#1313313811yescharacteristic
379C4#1681076111yescharacteristic
380C4#1681093411yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—161no—
386C2#2027503311yescharacteristic
subgroupG/H
centerC2#20275033
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3067C2
socleC2×C53#1874
cosocleC22×C8×C53C23
hypercenter#3067D53
hypocenterC53#13005
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. #20283310C2semidirect
2. #20276451C2semidirect
3. C2×#16810433C2semidirect
4. #20276343C2semidirect
5. #20272829C2semidirect
6. #20272815C2semidirect
24. C53×#473C4semidirect
25. C53×#472C4semidirect
60. C2×Q16×C53C2×C4semidirect
62. C2×D8×C53C2×C4semidirect
73. #108×C53C2×C4semidirect
207. #473Dic53semidirect
208. #472Dic53semidirect
222. C53#13005semidirect
234. C2×Q16C2×Dic53semidirect
236. C2×D8C2×Dic53semidirect
260. #108C2×Dic53semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202833102
#202764512
C2×#168104332
#202763432
#202728292
#202728152
C53×#30672
#1300553