Finite Group Databasebeta

← #27301256

Subgroups — #27301256

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#135×Dic53C211yesmaximal, characteristic, complemented
2#20276451C211yesmaximal, characteristic, complemented
3#20276443C211yesmaximal, characteristic, complemented
4#20275604C211yesmaximal, characteristic, complemented
5#20272830C211yesmaximal, characteristic, complemented
6#20272796C211yesmaximal, characteristic, complemented
7C53×#3414C211yesmaximal, characteristic
8#41×Dic53—21no—
9C2×#13133157C2211yescharacteristic
10Q16×Dic53—21no—
11SD16×Dic53—22no—
12D8×Dic53—21no—
13C2×#13133153C2211yescharacteristic
14#16810661C2211yescharacteristic
15C2×#13133144—21no—
16#16810619—21no—
17#16810590—21no—
18C2×C8×Dic53C2211yescharacteristic
19#16810433—22no—
20#16810430—22no—
21C2×#135×C53C2211yescharacteristic
22C2×#135×C53C2211yescharacteristic
23C53×#510—21no—
24C53×#510C412yescomplemented
25C53×#509—21no—
26C53×#509C412yescomplemented
27C53×#508—22no—
28C2×M32×C53C2211yescharacteristic
29C2×#11719917—21no—
30C22×#88322D411yescharacteristic
31Q8×Dic53—22no—
32D4×Dic53—22no—
33D4×Dic53—42no—
34#13133176—21no—
35#13133176—41no—
36C2×C4×Dic53—41no—
37C2×C4×Dic53D411yescharacteristic
38#13133157—21no—
39#13133157—21no—
40C2×#11719908—21no—
41#13133153—21no—
42#13133153—21no—
43#13133151—21no—
44#13133149D412yes—
45#13133144—41no—
46#13133144—41no—
47#13133141—22no—
48#13133138—22no—
49C22⋊#88322—42no—
50C8×Dic53—21no—
51C8×Dic53—21no—
52C2×#41×C53—21no—
53C2×#41×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53—21no—
57#135×C53—21no—
58#135×C53—21no—
59#135×C53—21no—
60#135×C53C2×C412yes—
61#135×C53D412yes—
62C2×Q16×C53C2×C411yescharacteristic, complemented
63C2×Q16×C53D411yescharacteristic
64C2×SD16×C53—21no—
65C2×SD16×C53—21no—
66C2×D8×C53C2×C411yescharacteristic, complemented
67C2×D8×C53D411yescharacteristic
68C22×C8×C53C2311yescharacteristic
69Q32×C53—22no—
70SD32×C53—22no—
71SD32×C53—22no—
72D16×C53—22no—
73M32×C53—21no—
74M32×C53C2×C412yescomplemented
75C2×C16×C53—21no—
76#41×C53—22no—
77#41×C53—22no—
78#41×C53—41no—
79#41×C53—41no—
80#41×C53—41no—
81#41×C53—41no—
82#41×C53—41no—
83#41×C53—41no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53—21no—
94C22×Dic53—42no—
95Q16×C53—21no—
96Q16×C53—21no—
97Q16×C53—21no—
98Q16×C53C22⋊C412yes—
99SD16×C53—22no—
100SD16×C53—22no—
101SD16×C53—41no—
102SD16×C53—41no—
103D8×C53—21no—
104D8×C53—21no—
105D8×C53—21no—
106D8×C53C22⋊C412yes—
107C2×C8×C53C22×C411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53#4111yescharacteristic
112C2×C8×C53#4111yescharacteristic
113C22⋊Dic53—42no—
114C22⋊Dic53—41no—
115C4⋊Dic53—22no—
116C4⋊Dic53—42no—
117C4×Dic53—21no—
118C4×Dic53—42no—
119C4×Dic53—41no—
120C4×Dic53D811yescharacteristic
121C4×Dic53D811yescharacteristic
122#11719917—41no—
123#11719917—41no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—41no—
128C16×C53—22no—
129#11719908—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53AGL(1,ℤ/8)11yescharacteristic
133Q8×C53—22no—
134Q8×C53—41no—
135Q8×C53—41no—
136Q8×C53—41no—
137D4×C53—22no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—81no—
144D4×C53—81no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—84no—
162C8×C53C2×C22⋊C411yescharacteristic
163C8×C53C4×D411yescharacteristic
164C8×C53#12111yescharacteristic
165C8×C53#12311yescharacteristic
166#88322—41no—
167#88322—81no—
168#13027—531nomaximal, Sylow
169C22×C53—41no—
170C22×C53—42no—
171C22×C53—82no—
172C22×C53—82no—
173C22×C53—81no—
174C22×C53#44211yescharacteristic
175C22×C53#44711yescharacteristic
176C22×C53#46311yescharacteristic
177C4×C53—42no—
178C4×C53—81no—
179C4×C53#38611yescharacteristic
180C4×C53C4×D811yescharacteristic
181C4×C53#44911yescharacteristic
182C4×C53#45911yescharacteristic
183Dic53—22no—
184Dic53—41no—
185Dic53—82no—
186#3414D5311yescharacteristic
187C4×#135—531no—
188#2145—531no—
189#2143—531no—
190#1974—531no—
191#1389—531no—
192#1376—531no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—21no—
196C2×C53—82no—
197C2×C53—161no—
198C2×C53#187911yescharacteristic
199C2×#135C2×D5311yescharacteristic
200C2×#135C2×D5311yescharacteristic
201C4×#41—1061no—
202#510—21no—
203#510Dic5312yescomplemented
204#509—21no—
205#509Dic5312yescomplemented
206#508—22no—
207C2×M32—531no—
208C2×M32C2×D5311yescharacteristic
209C4×Q16—1061no—
210C4×SD16—1062no—
211C4×D8—1061no—
212C2×#108—531no—
213#427—531no—
214C2×#104—1061no—
215#416—1061no—
216#408—1061no—
217C2×C4×C8—531no—
218#360—1062no—
219#359—1062no—
220C53#1302711yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—21no—
223#135—21no—
224#135—21no—
225#135—21no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135C2×Dic5312yes—
230#135C22⋊D5312yes—
231C2×Q16C2×Dic5311yescharacteristic, complemented
232C2×Q16C22⋊D5311yescharacteristic
233C2×SD16—21no—
234C2×SD16—21no—
235C2×D8C2×Dic5311yescharacteristic, complemented
236C2×D8C22⋊D5311yescharacteristic
237C2×M16—1061no—
238C22×C8—531no—
239C22×C8C22×D5311yescharacteristic
240C4×Q8—1062no—
241C4×D4—1062no—
242C4×D4—2122no—
243#117—1061no—
244#117—2121no—
245C2×C42—531no—
246C2×C42—2121no—
247Q32—22no—
248SD32—22no—
249SD32—22no—
250D16—22no—
251M32—21no—
252M32—1061no—
253M32—1061no—
254M32C2×Dic5312yescomplemented
255C2×C16—21no—
256C2×C16—1061no—
257#108—1061no—
258#108—1061no—
259#107—1061no—
260#106—532no—
261#104—2121no—
262#104—2121no—
263#103—1062no—
264#102—1062no—
265C22⋊C8—2122no—
266C4×C8—1061no—
267C4×C8—1061no—
268#41—22no—
269#41—22no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—1061no—
285C22×C4—2122no—
286C22×C4D4×D5311yescharacteristic
287Q16—21no—
288Q16—21no—
289Q16—21no—
290Q16C22⋊Dic5312yes—
291SD16—22no—
292SD16—22no—
293SD16—41no—
294SD16—41no—
295D8—21no—
296D8—21no—
297D8—21no—
298D8C22⋊Dic5312yes—
299M16—2121no—
300M16—2121no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—1061no—
304C2×C8—2121no—
305C2×C8D4×D5311yescharacteristic
306C2×C8#1171994711yescharacteristic
307C2×C8#1171994711yescharacteristic
308C2×C8C22×Dic5311yescharacteristic
309C2×C8C2×C22⋊D5311yescharacteristic
310C2×C8C2×C22⋊D5311yescharacteristic
311C4⋊C4—1062no—
312C4⋊C4—2122no—
313C22⋊C4—2122no—
314C22⋊C4—2121no—
315C42—531no—
316C42—531no—
317C42—1061no—
318C42—2122no—
319C42—2121no—
320C16—22no—
321C16—2121no—
322C23—41no—
323C23—41no—
324C23#1313324111yescharacteristic
325Q8—22no—
326Q8—41no—
327Q8—41no—
328Q8—41no—
329D4—22no—
330D4—42no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—81no—
336D4—81no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—4244no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4D4×Dic5311yescharacteristic
349C2×C4#1313318511yescharacteristic
350C2×C4#1313318911yescharacteristic
351C2×C4D8×D5311yescharacteristic
352C2×C4#1313323611yescharacteristic
353C2×C4#1313324111yescharacteristic
354C8—4241no—
355C8—2121no—
356C8C2×C22⋊Dic5311yescharacteristic
357C8D4×Dic5311yescharacteristic
358C8#1313318911yescharacteristic
359C8#1313319811yescharacteristic
360C22—41no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—81no—
365C22#1681073511yescharacteristic
366C22#1681075511yescharacteristic
367C22#1681083611yescharacteristic
368C4—42no—
369C4—81no—
370C4—1062no—
371C4—4242no—
372C4—2121no—
373C4#1681050311yescharacteristic
374C4D8×Dic5311yescharacteristic
375C4#1681076511yescharacteristic
376C4#1681081611yescharacteristic
377C2—21no—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—161no—
382C2#2027506511yescharacteristic
subgroupG/H
centerC4#16810816
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3414C2
socleC2×C53#1879
cosocleC22×C8×C53C23
hypercenter#3414D53
hypocenterC53#13027
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #135×Dic53C2semidirect
2. #20276451C2semidirect
3. #20276443C2semidirect
4. #20275604C2semidirect
5. #20272830C2semidirect
6. #20272796C2semidirect
24. C53×#510C4semidirect
26. C53×#509C4semidirect
62. C2×Q16×C53C2×C4semidirect
66. C2×D8×C53C2×C4semidirect
74. M32×C53C2×C4semidirect
203. #510Dic53semidirect
205. #509Dic53semidirect
220. C53#13027semidirect
231. C2×Q16C2×Dic53semidirect
235. C2×D8C2×Dic53semidirect
254. M32C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#135×Dic532
#202764512
#202764432
#202756042
#202728302
#202727962
C53×#34142
#1302753