Finite Group Databasebeta

← #27279285

Subgroups — #27279285

Order 13568.

544 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284711C211yesmaximal, characteristic
2#20275513C211yesmaximal, characteristic, complemented
3#20273364C211yesmaximal, characteristic, complemented
4#20273139C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272560C211yesmaximal, characteristic, complemented
7C53×#2932C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810934C412yes—
10#16810900—21no—
11#16810815—21no—
12#16810756C412yes—
13C4×#11719922—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133129—21no—
18C2×#13133129—21no—
19#16810369—21no—
20#16810367—21no—
21#16810350—22no—
22C2×#131×C53—21no—
23C53×#532C2211yescharacteristic
24C53×#435—22no—
25C53×#435—22no—
26C53×#423C2211yescharacteristic
27C53×#408—21no—
28C2×#97×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719922—22no—
32C22×#88322—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—21no—
39#13133150—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44#13133138—22no—
45#13133137—22no—
46C22⋊#88322—21no—
47C22⋊#88322C2×C412yes—
48#13133129—42no—
49#13133129—42no—
50C4×#88322—21no—
51C4×#88322—21no—
52C2×#41×C53—21no—
53#131×C53—42no—
54#131×C53—42no—
55C2×M16×C53—22no—
56C2×M16×C53—21no—
57C22×C8×C53—22no—
58C22×C8×C53D411yescharacteristic
59C22×C8×C53D411yescharacteristic
60C4⋊Q8×C53D411yescharacteristic
61#127×C53D411yescharacteristic
62#124×C53—21no—
63#121×C53—21no—
64#121×C53C2×C412yes—
65C4×D4×C53—21no—
66C4×D4×C53D412yes—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53D412yes—
70C22⋊C8×C53—24no—
71#97×C53—21no—
72#97×C53—21no—
73C23×D53—21no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77#41×C53—41no—
78C2×Q8×C53—21no—
79C2×D4×C53—22no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53C22⋊C412yes—
83C22×C4×C53—22no—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×Dic53—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—42no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—41no—
97M16×C53—44no—
98M16×C53—41no—
99M16×C53—41no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—22no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—44no—
107C2×C8×C53D812yescomplemented
108C4⋊C4×C53—21no—
109C4⋊C4×C53C22⋊C412yes—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116#11719922—44no—
117#11719921—41no—
118#11719921—41no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53—22no—
127C23×C53C4×D411yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135Q8×C53—41no—
136Q8×C53—41no—
137D4×C53—42no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—81no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12311yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161D212—22no—
162D212—22no—
163D212—41no—
164D212—41no—
165C4×D53—81no—
166C4×D53—81no—
167C4×D53—81no—
168C4×D53—81no—
169C8×C53—22no—
170C8×C53—42no—
171C8×C53—41no—
172#88322—42no—
173#88322—81no—
174#88322—81no—
175#9049—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53#38611yescharacteristic
182C22×C53C4×D811yescharacteristic
183C22×C53#44211yescharacteristic
184C2×D53—42no—
185C2×D53—42no—
186C2×D53—42no—
187C2×D53—81no—
188C2×D53—81no—
189C2×D53—81no—
190C2×D53—81no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—42no—
196C4×C53#44711yescharacteristic
197C4×C53#44911yescharacteristic
198C4×C53#45911yescharacteristic
199C4×C53#46311yescharacteristic
200Dic53—81no—
201#3100—531no—
202#2932D5311yescharacteristic, complemented
203#1955—531no—
204#1540—531no—
205#1486—531no—
206#1301—531no—
207#1281—531no—
208C2×C53—21no—
209C2×C53—21no—
210C2×C53—82no—
211C2×C53#187911yescharacteristic
212C2×C53#188111yescharacteristic
213C2×C53#189111yescharacteristic
214D53—82no—
215C2×#131—21no—
216#532C2×D5311yescharacteristic
217C2×C4×D4—531no—
218#480—532no—
219#474—1061no—
220#459—1061no—
221#447—532no—
222C4×D8—1062no—
223#435—22no—
224#435—22no—
225#431—531no—
226#431—531no—
227#423C2×D5311yescharacteristic
228C2×C4⋊C8—531no—
229#408—21no—
230C2×C22⋊C8—1061no—
231C2×C22⋊C8—1061no—
232C2×#97C2×D5311yescharacteristic
233#332—1061no—
234#331—1061no—
235D4⋊C8—1062no—
236C53#904911yescharacteristic, complemented, Sylow
237C2×#41—21no—
238C22×D4—531no—
239C23×C4—1061no—
240C2×D8—1062no—
241#131—42no—
242#131—42no—
243C2×M16—22no—
244C2×M16—21no—
245C22×C8—22no—
246C22×C8—1061no—
247C22×C8C22⋊D5311yescharacteristic
248C22×C8C22⋊D5311yescharacteristic
249C4⋊Q8D21211yescharacteristic
250#127D21211yescharacteristic
251#124—21no—
252#121—21no—
253#121C4×D5312yes—
254C4×D4—21no—
255C4×D4—1061no—
256C4×D4—1061no—
257C4×D4—1061no—
258C4×D4—1061no—
259C4×D4C22⋊D5312yes—
260C2×C4⋊C4—531no—
261C2×C22⋊C4—1061no—
262C2×C42C22×D5311yescharacteristic
263#107—1062no—
264C4⋊C8—21no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8C22⋊D5312yes—
270#102—1062no—
271#102—1062no—
272C22⋊C8—24no—
273C22⋊C8—532no—
274C22⋊C8—1061no—
275C22⋊C8—2122no—
276C22⋊C8—2122no—
277#97—21no—
278#97—21no—
279C4×C8—1061no—
280C4×C8—1061no—
281C24—1061no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—2122no—
293C2×D4#1171992112yes—
294C22×C4—22no—
295C22×C4—1061no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4C2×C4×D5311yescharacteristic
303C22×C4#1171994511yescharacteristic
304C22×C4C2×C22⋊D5311yescharacteristic
305D8—2124no—
306M16—44no—
307M16—41no—
308M16—41no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—44no—
316C2×C8—1062no—
317C2×C8—2121no—
318C2×C8—2121no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8#1171992212yescomplemented
322C4⋊C4—21no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4#1171992112yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42C2×D21211yescharacteristic
331C42#1171994511yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C23—22no—
335C23—1062no—
336C23—2121no—
337C23—2121no—
338C23—2121no—
339C23—2121no—
340C23—2121no—
341C23—2121no—
342C23C4×C22⋊D5311yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—42no—
346D4—42no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—81no—
351D4—1062no—
352D4—1062no—
353D4—2121no—
354D4—2121no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—4241no—
361C2×C4—4241no—
362C2×C4—4241no—
363C2×C4—4241no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4C2×#1171992111yescharacteristic
368C2×C4C4×C22⋊D5311yescharacteristic
369C2×C4#1313318611yescharacteristic
370C2×C4#1313319111yescharacteristic
371C2×C4#1313319111yescharacteristic
372C2×C4#1313320011yescharacteristic
373C2×C4C2×#1171992211yescharacteristic
374C2×C4#1313323711yescharacteristic
375C2×C4#1313324211yescharacteristic
376C2×C4#1313324311yescharacteristic
377C8—22no—
378C8—42no—
379C8—41no—
380C8—4241no—
381C8—4241no—
382C8—2122no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—4241no—
389C22—4241no—
390C22—4241no—
391C22—4241no—
392C22—2122no—
393C22—2122no—
394C22—2122no—
395C22#1681050211yescharacteristic
396C22C4×#1171992211yescharacteristic
397C22#1681073311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—4241no—
404C4#1681075711yescharacteristic
405C4#1681076811yescharacteristic
406C4#1681081811yescharacteristic
407C4#1681083811yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2—4242no—
412C2#2027506311yescharacteristic
413C2#2027507711yescharacteristic
414C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2932C2
socleC22×C53#386
prime radical#20284711C2
cosocleC2×C42×C53C23
hypercenter#2932D53
hypocenterC53#9049
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. #20275513C2semidirect
3. #20273364C2semidirect
4. #20273139C2semidirect
5. #20272610C2semidirect
6. #20272560C2semidirect
7. C53×#2932C2semidirect
107. C2×C8×C53D8semidirect
202. #2932D53semidirect
236. C53#9049semidirect
321. C2×C8#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847112
#202755132
#202733642
#202731392
#202726102
#202725602
C53×#29322
#904953