Finite Group Databasebeta

← #27279287

Subgroups — #27279287

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#20283433C211yesmaximal, characteristic, complemented
2#20275514C211yesmaximal, characteristic, complemented
3#20273364C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272559C211yesmaximal, characteristic, complemented
7C53×#3100C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810695—22no—
10#16810695—22no—
11#16810676C2211yescharacteristic
12#16810647C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14#16810590—21no—
15C2×C22⋊#88322—21no—
16C2×C22⋊#88322—21no—
17C2×#13133128C2211yescharacteristic
18#16810369—21no—
19#16810367—21no—
20D4⋊#88322—22no—
21C53×#532C2211yescharacteristic
22C2×C4×D4×C53C2211yescharacteristic
23C53×#480C412yes—
24C53×#474—21no—
25C53×#459—21no—
26C53×#447C412yes—
27C4×D8×C53—22no—
28C53×#431C2211yescharacteristic
29#13133224—42no—
30#13133224—42no—
31C2×#11719917—22no—
32C2×#11719917—21no—
33C22×#88322—22no—
34C22×#88322—21no—
35C22×#88322D411yescharacteristic
36C22×#88322D411yescharacteristic
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322D412yes—
42C22⋊#88322—24no—
43C22⋊#88322—21no—
44C22⋊#88322—42no—
45C22⋊#88322—42no—
46#13133128—21no—
47#13133128—21no—
48C4×#88322—21no—
49C2×#41×C53—21no—
50C22×D4×C53C2×C411yescharacteristic
51C23×C4×C53—21no—
52C2×D8×C53—22no—
53C4⋊Q8×C53D411yescharacteristic
54#127×C53D411yescharacteristic
55#124×C53—21no—
56#121×C53—21no—
57#121×C53C2×C412yes—
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53D412yes—
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C42×C53C2311yescharacteristic
67#107×C53—22no—
68C4⋊C8×C53—21no—
69#102×C53—22no—
70#102×C53—22no—
71C22⋊C8×C53C2×C412yes—
72C4×C8×C53—21no—
73C24×C53—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×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—42no—
85C2×D4×C53C22⋊C412yes—
86C22×C4×C53—22no—
87C22×C4×C53—21no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97D8×C53—44no—
98C2×C8×C53—22no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53C22⋊C412yes—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53#4111yescharacteristic
111#11719917—44no—
112#11719917—41no—
113#11719917—41no—
114C2×#88322—21no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—22no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—44no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322D812yescomplemented
126C23×C53—22no—
127C23×C53—22no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C4×D411yescharacteristic
135Q8×C53—41no—
136Q8×C53—41no—
137D4×C53—22no—
138D4×C53—22no—
139D4×C53—42no—
140D4×C53—41no—
141D4×C53—41no—
142D4×C53—42no—
143D4×C53—42no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—81no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—81no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12311yescharacteristic
165C2×C4×C53C2×D811yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
169C8×C53—42no—
170#88322—22no—
171#88322—42no—
172#88322—41no—
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—42no—
182C22×C53—42no—
183C22×C53—42no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53#38611yescharacteristic
189C22×C53C4×D811yescharacteristic
190C22×C53#44211yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53#44711yescharacteristic
198C4×C53#44911yescharacteristic
199C4×C53#45911yescharacteristic
200C4×C53#46311yescharacteristic
201#3100D5311yescharacteristic
202#2932—531no—
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—82no—
212C2×C53#187911yescharacteristic
213C2×C53#188111yescharacteristic
214C2×C53#189111yescharacteristic
215C2×#131—1061no—
216#532C2×D5311yescharacteristic
217C2×C4×D4C2×D5311yescharacteristic
218#480Dic5312yes—
219#474—21no—
220#459—21no—
221#447Dic5312yes—
222C4×D8—22no—
223#435—1062no—
224#435—1062no—
225#431—531no—
226#431C2×D5311yescharacteristic
227#423—531no—
228C2×C4⋊C8—531no—
229#408—1061no—
230C2×C22⋊C8—1061no—
231C2×C22⋊C8—1061no—
232C2×#97—531no—
233#332—1061no—
234#331—1061no—
235D4⋊C8—1062no—
236C53#904911yescharacteristic, complemented, Sylow
237C2×#41—21no—
238C22×D4C2×Dic5311yescharacteristic
239C23×C4—21no—
240C2×D8—22no—
241#131—2122no—
242#131—2122no—
243C2×M16—1062no—
244C2×M16—1061no—
245C22×C8—531no—
246C22×C8—531no—
247C22×C8—1062no—
248C22×C8—1061no—
249C4⋊Q8C22⋊D5311yescharacteristic
250#127C22⋊D5311yescharacteristic
251#124—21no—
252#121—21no—
253#121C2×Dic5312yes—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C22⋊D5312yes—
260C2×C4⋊C4C2×Dic5311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C42C22×D5311yescharacteristic
263#107—22no—
264C4⋊C8—21no—
265C4⋊C8—532no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270#102—22no—
271#102—22no—
272C22⋊C8—1064no—
273C22⋊C8—1061no—
274C22⋊C8—2122no—
275C22⋊C8—2122no—
276C22⋊C8C2×Dic5312yes—
277#97—1061no—
278#97—1061no—
279C4×C8—21no—
280C4×C8—1061no—
281C24—21no—
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—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4C22⋊Dic5312yes—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4D4×D5311yescharacteristic
303C22×C4#1171994711yescharacteristic
304C22×C4C22×Dic5311yescharacteristic
305D8—44no—
306M16—2124no—
307M16—2121no—
308M16—2121no—
309C2×C8—22no—
310C2×C8—532no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—1062no—
315C2×C8—1061no—
316C2×C8—1061no—
317C2×C8—2124no—
318C2×C8—2121no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8—2121no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4C22⋊Dic5312yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42D4×D5311yescharacteristic
331C42#1171994711yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23D4×Dic5311yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—41no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4C2×C22⋊Dic5311yescharacteristic
368C2×C4D4×Dic5311yescharacteristic
369C2×C4#1313318511yescharacteristic
370C2×C4#1313318911yescharacteristic
371C2×C4#1313318911yescharacteristic
372C2×C4#1313319811yescharacteristic
373C2×C4D8×D5311yescharacteristic
374C2×C4#1313323611yescharacteristic
375C2×C4#1313324111yescharacteristic
376C2×C4#1313324111yescharacteristic
377C8—42no—
378C8—1062no—
379C8—4241no—
380C8—4241no—
381C8—2122no—
382C8—2121no—
383C22—21no—
384C22—21no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22#1681050311yescharacteristic
396C22D8×Dic5311yescharacteristic
397C22#1681073511yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—42no—
403C4—81no—
404C4#1681075511yescharacteristic
405C4#1681076511yescharacteristic
406C4#1681081611yescharacteristic
407C4#1681083611yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2—82no—
412C2#2027506511yescharacteristic
413C2#2027507911yescharacteristic
414C2#2027513411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3100C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3100D53
hypocenterC53#9049
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #20283433C2semidirect
2. #20275514C2semidirect
3. #20273364C2semidirect
4. #20273140C2semidirect
5. #20272611C2semidirect
6. #20272559C2semidirect
125. C2×#88322D8semidirect
236. C53#9049semidirect

Maximal subgroups

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

subgroupindex
#202834332
#202755142
#202733642
#202731402
#202726112
#202725592
C53×#31002
#904953