Finite Group Databasebeta

← #27301212

Subgroups — #27301212

Order 13568.

462 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#20287849C211yesmaximal, characteristic
2C2×#16810429C211yesmaximal, characteristic, complemented
3#20276422C211yesmaximal, characteristic, complemented
4#20275613C211yesmaximal, characteristic, complemented
5#20272829C211yesmaximal, characteristic, complemented
6#20272804C211yesmaximal, characteristic, complemented
7C53×#2957C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9#16811064—21no—
10#16811063C412yescomplemented
11#16811059—21no—
12#16811058C412yescomplemented
13#16811053—22no—
14C2×#13133157C2211yescharacteristic
15C22×#11719908C2211yescharacteristic
16#16810661C2211yescharacteristic
17#16810639—21no—
18#16810596—21no—
19#16810429—22no—
20#16810427—21no—
21#16810424—21no—
22C2×#135×C53C2211yescharacteristic
23C53×#568—21no—
24C53×#444—22no—
25C53×#443—22no—
26C2×#108×C53C2211yescharacteristic
27C53×#421—21no—
28C53×#405C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239—21no—
33#13133239—21no—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
37#13133176—21no—
38#13133163—22no—
39#13133161—22no—
40#13133160—22no—
41#13133158—22no—
42#13133157—21no—
43#13133157C2×C412yes—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48#13133151—21no—
49#13133149—21no—
50#13133143—42no—
51#13133131—42no—
52C2×#41×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53C2×C412yes—
56#135×C53D412yes—
57C2×Q16×C53D411yescharacteristic
58C2×SD16×C53—21no—
59C2×D8×C53D411yescharacteristic
60#131×C53—24no—
61#131×C53—42no—
62C2×M16×C53—21no—
63C2×M16×C53—42no—
64C2×M16×C53D411yescharacteristic
65C22×C8×C53C2311yescharacteristic
66#117×C53D411yescharacteristic
67#108×C53—21no—
68#108×C53D412yes—
69#104×C53—24no—
70#97×C53—21no—
71C4×C8×C53—21no—
72#41×C53—22no—
73#41×C53—22no—
74#41×C53—41no—
75#41×C53—41no—
76C2×Q8×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—41no—
79C22×C4×C53—41no—
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—
89Q16×C53—21no—
90Q16×C53C22⋊C412yes—
91SD16×C53—22no—
92SD16×C53—22no—
93D8×C53—21no—
94D8×C53C22⋊C412yes—
95M16×C53—21no—
96M16×C53—22no—
97M16×C53—44no—
98M16×C53—41no—
99M16×C53D812yes—
100C2×C8×C53—21no—
101C2×C8×C53—44no—
102C2×C8×C53—41no—
103C2×C8×C53C22×C411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C4⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C42×C53—21no—
112C22⋊Dic53—41no—
113C4⋊Dic53—41no—
114C4×Dic53—41no—
115Dic212—21no—
116Dic212—21no—
117D424—21no—
118D424—21no—
119#11719913—41no—
120#11719913—41no—
121#11719908—22no—
122#11719908—41no—
123C23×C53—41no—
124C23×C53AGL(1,ℤ/8)11yescharacteristic
125C22×D53—41no—
126Q8×C53—22no—
127Q8×C53—41no—
128D4×C53—22no—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53C2×D811yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
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×C53—42no—
154C8×C53C2×C22⋊C411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158#13021—531nomaximal, Sylow
159C22×C53—41no—
160C22×C53—42no—
161C22×C53—161no—
162C22×C53#44211yescharacteristic
163C22×C53#44911yescharacteristic
164C22×C53#45911yescharacteristic
165C2×D53—81no—
166C2×D53—161no—
167C4×C53—42no—
168C4×C53—81no—
169C4×C53#38611yescharacteristic
170C4×C53C4×D811yescharacteristic
171C4×C53#44711yescharacteristic
172C4×C53#46311yescharacteristic
173Dic53—81no—
174Dic53—161no—
175#3414—531no—
176#2957D5311yescharacteristic, complemented
177C2×#359—531no—
178#2138—531no—
179#1976—531no—
180#1389—531no—
181#1379—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—82no—
186C2×C53#187911yescharacteristic
187D53—161no—
188C2×#135—531no—
189C2×#135C2×D5311yescharacteristic
190#568—21no—
191#510—532no—
192#510—1061no—
193#509—532no—
194#509—1061no—
195#508—1062no—
196C2×M32—531no—
197C22×C16—531no—
198#444—22no—
199#443—22no—
200C2×#108C2×D5311yescharacteristic
201#427—531no—
202#421—21no—
203#421—1061no—
204#410—1061no—
205#405C2×D5311yescharacteristic
206#359—1062no—
207#358—1061no—
208#357—1061no—
209C53#1302111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×#41—1061no—
212#135—21no—
213#135—21no—
214#135—1061no—
215#135—1061no—
216#135—1061no—
217#135—1061no—
218#135C4×D5312yes—
219#135C22⋊D5312yes—
220C2×Q16—531no—
221C2×Q16D21211yescharacteristic
222C2×SD16—21no—
223C2×SD16—1061no—
224C2×D8—531no—
225C2×D8D21211yescharacteristic
226#131—24no—
227#131—42no—
228C2×M16—21no—
229C2×M16—42no—
230C2×M16C22⋊D5311yescharacteristic
231C22×C8C22×D5311yescharacteristic
232#117—1061no—
233#117C22⋊D5311yescharacteristic
234Q32—1062no—
235SD32—1062no—
236SD32—1062no—
237D16—1062no—
238M32—532no—
239M32—1061no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16—1061no—
244#108—21no—
245#108C22⋊D5312yes—
246#107—1061no—
247#106—1061no—
248#104—24no—
249#104—2122no—
250#99—2122no—
251#97—21no—
252C4×C8—21no—
253#41—22no—
254#41—22no—
255#41—41no—
256#41—41no—
257#41—2121no—
258#41—2121no—
259#41—2121no—
260#41—2121no—
261C2×Q8—21no—
262C2×Q8—1061no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—1061no—
266C2×D4—2121no—
267C22×C4—41no—
268C22×C4—2121no—
269C22×C4C2×C22⋊D5311yescharacteristic
270Q16—21no—
271Q16—1061no—
272Q16—1061no—
273Q16#1171992112yes—
274SD16—22no—
275SD16—22no—
276SD16—2121no—
277SD16—2121no—
278D8—21no—
279D8—1061no—
280D8—1061no—
281D8#1171992112yes—
282M16—21no—
283M16—22no—
284M16—44no—
285M16—41no—
286M16#1171992212yes—
287C2×C8—21no—
288C2×C8—44no—
289C2×C8—41no—
290C2×C8C2×C4×D5311yescharacteristic
291C2×C8C2×D21211yescharacteristic
292C2×C8#1171994511yescharacteristic
293C2×C8#1171994511yescharacteristic
294C2×C8C2×C22⋊D5311yescharacteristic
295C2×C8C2×C22⋊D5311yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—2121no—
298C22⋊C4—21no—
299C22⋊C4—2121no—
300C42—21no—
301C42—2121no—
302C16—1062no—
303C16—2121no—
304C23—41no—
305C23—2121no—
306C23#1313324211yescharacteristic
307Q8—22no—
308Q8—41no—
309Q8—2121no—
310Q8—2121no—
311D4—22no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—4241no—
316D4—4241no—
317D4—2121no—
318D4—2121no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—4241no—
324C2×C4—4241no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4C4×C22⋊D5311yescharacteristic
328C2×C4#1313318611yescharacteristic
329C2×C4#1313319111yescharacteristic
330C2×C4C2×#1171992211yescharacteristic
331C2×C4#1313323711yescharacteristic
332C2×C4#1313324311yescharacteristic
333C8—22no—
334C8—42no—
335C8C2×#1171992111yescharacteristic
336C8C4×C22⋊D5311yescharacteristic
337C8#1313319111yescharacteristic
338C8#1313320011yescharacteristic
339C22—41no—
340C22—42no—
341C22—161no—
342C22—8481no—
343C22—4241no—
344C22#1681073311yescharacteristic
345C22#1681076811yescharacteristic
346C22#1681081811yescharacteristic
347C4—42no—
348C4—81no—
349C4—8481no—
350C4—4241no—
351C4#1681050211yescharacteristic
352C4C4×#1171992211yescharacteristic
353C4#1681075711yescharacteristic
354C4#1681083811yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—82no—
359C2—8481no—
360C2#2027506311yescharacteristic
subgroupG/H
centerC4#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2957C2
socleC2×C53#1879
prime radical#20287849C2
cosocleC22×C8×C53C23
hypercenter#2957D53
hypocenterC53#13021
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. C2×#16810429C2semidirect
3. #20276422C2semidirect
4. #20275613C2semidirect
5. #20272829C2semidirect
6. #20272804C2semidirect
7. C53×#2957C2semidirect
10. #16811063C4semidirect
12. #16811058C4semidirect
176. #2957D53semidirect
209. C53#13021semidirect

Maximal subgroups

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

subgroupindex
#202878492
C2×#168104292
#202764222
#202756132
#202728292
#202728042
C53×#29572
#1302153