Finite Group Databasebeta

← #27301520

Subgroups — #27301520

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284435C211yesmaximal, characteristic
2#20276452C211yesmaximal, characteristic, complemented
3C2×#16810431C211yesmaximal, characteristic, complemented
4#20274769C211yesmaximal, characteristic, complemented
5#20272827C211yesmaximal, characteristic, complemented
6#20272811C211yesmaximal, characteristic, complemented
7C2×C53×#504C211yesmaximal, characteristic, complemented
8#16811421—21no—
9C2×#13133239C2211yescharacteristic
10#16810898C412yescomplemented
11#16810893C412yescomplemented
12#16810886—22no—
13C2×#13133153C2211yescharacteristic
14C2×#13133153C2211yescharacteristic
15#16810661C2211yescharacteristic
16#16810640—21no—
17#16810605—21no—
18#16810434—22no—
19#16810431—21no—
20#16810431—21no—
21#16810395—21no—
22C2×#131×C53C2211yescharacteristic
23C53×#504—22no—
24C53×#504—22no—
25C2×M32×C53—21no—
26C2×M32×C53C2211yescharacteristic
27C22×C16×C53—21no—
28C22×C16×C53C2211yescharacteristic
29C2×#11719945—21no—
30#13133248—41no—
31#13133247—22no—
32#13133243—41no—
33#13133242—22no—
34#13133239—21no—
35#13133239—21no—
36#13133239—21no—
37#13133239—21no—
38#13133237—42no—
39C2×Dic212C2×C411yescharacteristic
40C2×#11719913—21no—
41C2×D424C2×C411yescharacteristic
42C2×#11719917—21no—
43C2×#11719917—21no—
44#13133176—21no—
45#13133153—21no—
46#13133153—21no—
47#13133153—21no—
48#13133153C2×C412yes—
49#13133151—21no—
50#13133149—21no—
51#13133144—42no—
52#13133135—22no—
53#13133133—22no—
54C2×#41×C53D411yescharacteristic
55#131×C53—21no—
56#131×C53—21no—
57#131×C53C2×C412yes—
58#131×C53D412yes—
59C2×M16×C53—21no—
60C2×M16×C53D411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53C2311yescharacteristic
63M32×C53—24no—
64M32×C53—21no—
65M32×C53D412yes—
66C2×C16×C53—24no—
67C2×C16×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53—21no—
70C2×C16×C53D411yescharacteristic
71C2×C16×C53D411yescharacteristic
72#41×C53—21no—
73#41×C53—21no—
74#41×C53C22⋊C412yes—
75#41×C53D812yes—
76C2×Q8×C53SD1611yescharacteristic
77C2×D4×C53—21no—
78C2×D4×C53SD1611yescharacteristic
79C22×C4×C53—21no—
80C22×C4×C53C2×D411yescharacteristic
81C2×C22⋊D53—41no—
82#11719945—41no—
83#11719945—41no—
84#11719945—41no—
85#11719945—41no—
86C2×D212—21no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89M16×C53—21no—
90M16×C53—22no—
91M16×C53—22no—
92M16×C53C22⋊C412yes—
93C2×C8×C53—21no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53C22×C411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C22⋊Dic53—41no—
103#11719925—42no—
104#11719924—42no—
105#11719923—42no—
106#11719922—42no—
107C4⋊Dic53—41no—
108C4×Dic53—41no—
109#11719917—22no—
110#11719917—41no—
111#11719917—41no—
112#11719917—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115Dic212—21no—
116Dic212—21no—
117D424—21no—
118D424—21no—
119#11719913—41no—
120#11719913—41no—
121C16×C53—21no—
122C16×C53—22no—
123C16×C53—21no—
124C23×C53—21no—
125C23×C53#13511yescharacteristic
126C22×D53—41no—
127Q8×C53—21no—
128Q8×C53#10212yes—
129D4×C53—22no—
130D4×C53—22no—
131D4×C53—21no—
132D4×C53#10212yes—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53C2×D811yescharacteristic
140C2×C4×C53C2×SD1611yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C22⋊D53—81no—
143C22⋊D53—81no—
144C2×Dic53—41no—
145C2×Dic53—81no—
146D212—41no—
147D212—41no—
148C4×D53—81no—
149C4×D53—81no—
150Dic106—41no—
151Dic106—41no—
152C8×C53—22no—
153C8×C53C4×D411yescharacteristic
154C8×C53C4×D411yescharacteristic
155C8×C53#12011yescharacteristic
156C8×C53#12311yescharacteristic
157#88322—42no—
158#88322—81no—
159#13071—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—41no—
162C22×C53—42no—
163C22×C53#41611yescharacteristic
164C22×C53#46511yescharacteristic
165C22×C53#46611yescharacteristic
166C2×D53—81no—
167C2×D53—161no—
168C4×C53—22no—
169C4×C53#38611yescharacteristic
170C4×C53C2×#10211yescharacteristic
171C4×C53#46511yescharacteristic
172C4×C53#46611yescharacteristic
173Dic53—81no—
174Dic53—161no—
175C2×#504D5311yescharacteristic, complemented
176#3067—531no—
177#2145—531no—
178C2×#359—531no—
179#1814—531no—
180#1388—531no—
181#1382—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—42no—
186C2×C53#189311yescharacteristic
187D53—161no—
188#575—1061no—
189C2×#135—531no—
190C2×#131C2×D5311yescharacteristic
191#504—22no—
192#504—22no—
193C2×M32—21no—
194C2×M32C2×D5311yescharacteristic
195C22×C16—21no—
196C22×C16C2×D5311yescharacteristic
197#473—532no—
198#472—532no—
199#471—1062no—
200C2×#108—531no—
201C2×#108—531no—
202#427—531no—
203#421—1061no—
204#413—1061no—
205#360—1062no—
206#359—1061no—
207#359—1061no—
208#344—1061no—
209C53#1307111yescharacteristic, complemented, Sylow
210C2×#41—1061no—
211C2×#41D21211yescharacteristic
212#137—1062no—
213#137—2121no—
214AGL(1,ℤ/8)—1062no—
215AGL(1,ℤ/8)—2121no—
216#135—1061no—
217#135—1061no—
218#135—1061no—
219#135—1061no—
220#135—2122no—
221C2×Q16—531no—
222C2×SD16—1061no—
223C2×D8—531no—
224#131—21no—
225#131—21no—
226#131C4×D5312yes—
227#131C22⋊D5312yes—
228C2×M16—21no—
229C2×M16—1061no—
230C2×M16—1061no—
231C2×M16D21211yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5311yescharacteristic
234#117—1061no—
235M32—24no—
236M32—21no—
237M32C22⋊D5312yes—
238C2×C16—24no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16C22⋊D5311yescharacteristic
243C2×C16C22⋊D5311yescharacteristic
244#108—532no—
245#108—1061no—
246#108—1061no—
247#108—1061no—
248#107—1061no—
249#106—1061no—
250#104—2122no—
251#101—1062no—
252#100—1062no—
253#41—21no—
254#41—21no—
255#41—2121no—
256#41—2121no—
257#41—2121no—
258#41—2121no—
259#41D42412yes—
260#41#1171992112yes—
261C2×Q8—1061no—
262C2×Q8#1171991311yescharacteristic
263C2×D4—21no—
264C2×D4—1061no—
265C2×D4—2121no—
266C2×D4#1171991311yescharacteristic
267C22×C4—21no—
268C22×C4—2121no—
269C22×C4C2×D21211yescharacteristic
270Q16—1061no—
271Q16—1061no—
272Q16—2122no—
273SD16—2122no—
274SD16—2122no—
275SD16—2121no—
276SD16—2121no—
277D8—1061no—
278D8—1061no—
279D8—2122no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—1062no—
284M16—2121no—
285M16—2121no—
286M16—2121no—
287M16#1171992112yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8#1171994511yescharacteristic
295C2×C8#1171994511yescharacteristic
296C2×C8C2×C22⋊D5311yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—2121no—
300C22⋊C4—2121no—
301C42—2121no—
302C16—21no—
303C16—22no—
304C16—21no—
305C23—21no—
306C23—2121no—
307C23#1313323911yescharacteristic
308Q8—21no—
309Q8—2121no—
310Q8—2121no—
311Q8#1313313912yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—4241no—
316D4—4241no—
317D4—2121no—
318D4—2121no—
319D4#1313313912yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4C2×#1171992111yescharacteristic
328C2×C4#1313319111yescharacteristic
329C2×C4#1313319111yescharacteristic
330C2×C4C2×D42411yescharacteristic
331C2×C4C2×#1171991311yescharacteristic
332C2×C4#1313323911yescharacteristic
333C8—22no—
334C8—4241no—
335C8—2122no—
336C8C4×C22⋊D5311yescharacteristic
337C8C4×C22⋊D5311yescharacteristic
338C8#1313318611yescharacteristic
339C8#1313320011yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—8481no—
344C22—4241no—
345C22#1681062011yescharacteristic
346C22#1681085211yescharacteristic
347C22#1681085911yescharacteristic
348C4—22no—
349C4—8481no—
350C4—4241no—
351C4#1681050211yescharacteristic
352C4C2×#1313313911yescharacteristic
353C4#1681085211yescharacteristic
354C4#1681085911yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—8481no—
360C2#2027514311yescharacteristic
subgroupG/H
centerC4#16810859
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#504C2
socleC2×C53#1893
prime radical#20284435C2
cosocleC22×C8×C53C23
hypercenterC2×#504D53
hypocenterC53#13071
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. #20276452C2semidirect
3. C2×#16810431C2semidirect
4. #20274769C2semidirect
5. #20272827C2semidirect
6. #20272811C2semidirect
7. C2×C53×#504C2semidirect
10. #16810898C4semidirect
11. #16810893C4semidirect
175. C2×#504D53semidirect
209. C53#13071semidirect

Maximal subgroups

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

subgroupindex
#202844352
#202764522
C2×#168104312
#202747692
#202728272
#202728112
C2×C53×#5042
#1307153