Finite Group Databasebeta

← #27279953

Subgroups — #27279953

Order 13568.

432 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20283380C211yesmaximal, characteristic
3#20273396C211yesmaximal, characteristic, complemented
4#20273100C211yesmaximal, characteristic
5#20272612C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810915C412yes—
10#16810910C412yes—
11#16810905—21no—
12#16810900—21no—
13#16810683—22no—
14#16810679—21no—
15#16810676C2211yescharacteristic
16#16810675—21no—
17#16810663—21no—
18#16810647C2211yescharacteristic
19#16810647C2211yescharacteristic
20#16810622—21no—
21#16810589—21no—
22#16810578—21no—
23#16810372—22no—
24#16810353—21no—
25#16810350—21no—
26C2×C4⋊Q8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×#332—21no—
30C53×#332—21no—
31C2×#11719945D411yescharacteristic
32C2×#11719917—21no—
33C2×#11719917—21no—
34C4×Dic106C2×C411yescharacteristic
35C4×Dic106D411yescharacteristic
36C4×D212C2×C411yescharacteristic
37C4×D212D411yescharacteristic
38C4×C22⋊D53—21no—
39C4×C22⋊D53—21no—
40#13133176D411yescharacteristic
41#13133174—21no—
42C42×D53—21no—
43#13133150—22no—
44#13133148—22no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yes—
49C4⋊#88322D412yes—
50#13133140—22no—
51#13133137—22no—
52C22⋊#88322—21no—
53#13133129—22no—
54#13133129—22no—
55#13133127—22no—
56C4×#88322—21no—
57C8×Dic53—22no—
58C22×Q8×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53C2×C412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D411yescharacteristic
72C4⋊C8×C53D411yescharacteristic
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—81no—
76C2×Q8×C53—81no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C2×C22⋊D53—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945—21no—
86C2×D212C22⋊C411yescharacteristic
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×Dic106C22⋊C411yescharacteristic
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—81no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C22⋊Dic53—21no—
106#11719921—21no—
107#11719921—21no—
108C4⋊Dic53C22⋊C411yescharacteristic
109C4⋊Dic53C22⋊C411yescharacteristic
110#11719919—21no—
111#11719919—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115#11719917—41no—
116#11719917—41no—
117#11719917—41no—
118#11719917—41no—
119C2×#88322—22no—
120C2×#88322—22no—
121C2×#88322—41no—
122C23×C53#12311yescharacteristic
123C22×D53—21no—
124Q8×C53—82no—
125Q8×C53—81no—
126Q8×C53—81no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—82no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53#12011yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C22⋊D53—41no—
143C22⋊D53—41no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147D212—21no—
148D212—21no—
149C4×D53—41no—
150C4×D53—41no—
151C4×D53—41no—
152C4×D53—41no—
153C4×D53—81no—
154Dic106—21no—
155Dic106—21no—
156C8×C53—41no—
157C8×C53—41no—
158C8×C53—81no—
159#88322—42no—
160#88322—42no—
161#88322—81no—
162#9168—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53#38611yescharacteristic
166C22×C53#48511yescharacteristic
167C22×C53#48511yescharacteristic
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—82no—
175C4×C53#41611yescharacteristic
176C4×C53#41911yescharacteristic
177C4×C53#45211yescharacteristic
178C4×C53#45211yescharacteristic
179C4×C53#47111yescharacteristic
180C4×C53#47211yescharacteristic
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184#3079—531no—
185#2925—531no—
186#1548—531no—
187C2×#332D5311yescharacteristic, complemented
188#1477—531no—
189#1301—531no—
190#1292—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53#187311yescharacteristic
194C2×C53#190211yescharacteristic
195C2×C53#190711yescharacteristic
196D53—81no—
197C2×C4⋊Q8C2×D5311yescharacteristic
198C4×#41—531no—
199#477—532no—
200#476—532no—
201#475—1061no—
202#474—1061no—
203#433—1062no—
204#432—1061no—
205#431—531no—
206#431—1061no—
207#428—1061no—
208#423—531no—
209#423—531no—
210C2×C4⋊C8C2×D5311yescharacteristic
211C2×C4⋊C8C2×D5311yescharacteristic
212#417—1061no—
213#408—1061no—
214#405—1061no—
215#333—1062no—
216#332—21no—
217#332—21no—
218Q8⋊C8—1061no—
219D4⋊C8—1061no—
220C53#916811yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C22×Q8—21no—
223C2×M16—1061no—
224C2×M16—1061no—
225C22×C8—21no—
226C22×C8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8C4×D5312yes—
231C4×Q8—531no—
232C4×Q8—531no—
233C4×D4—531no—
234C4×D4—531no—
235C4×D4—1061no—
236C4×D4—1061no—
237#117—531no—
238#117—1061no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C42—1061no—
242C2×C42C22×D5311yescharacteristic
243#107—1062no—
244#106—1062no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—532no—
249C4⋊C8—532no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253C4⋊C8C22⋊D5311yescharacteristic
254C4⋊C8C22⋊D5311yescharacteristic
255#103—1062no—
256#102—1062no—
257C22⋊C8—1062no—
258C22⋊C8—1062no—
259C22⋊C8—1061no—
260#97—1062no—
261C4×C8—1062no—
262C4×C8—1061no—
263#41—1061no—
264#41—1061no—
265#41—1061no—
266#41—1061no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8—531no—
272C2×D4—531no—
273C2×D4—1061no—
274C22×C4—41no—
275C22×C4—1061no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4#1171994511yescharacteristic
279C22×C4D4×D5311yescharacteristic
280C22×C4#1171994711yescharacteristic
281M16—2121no—
282M16—2121no—
283M16—2121no—
284M16—2121no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—81no—
291C2×C8—1062no—
292C2×C8—1062no—
293C2×C8—2121no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—531no—
299C4⋊C4—531no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C22⋊C4—1061no—
303C22⋊C4—1061no—
304C22⋊C4—1061no—
305C42—1061no—
306C42—1061no—
307C42—1061no—
308C42C2×C4×D5311yescharacteristic
309C42D4×D5311yescharacteristic
310C42D4×D5311yescharacteristic
311C42C2×C22⋊D5311yescharacteristic
312C23—1061no—
313C23#1313319911yescharacteristic
314Q8—82no—
315Q8—81no—
316Q8—81no—
317Q8—1061no—
318Q8—1061no—
319D4—1061no—
320D4—1061no—
321D4—2121no—
322D4—2121no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—4241no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4C22⋊C4×D5311yescharacteristic
337C2×C4#1313317711yescharacteristic
338C2×C4C4×C22⋊D5311yescharacteristic
339C2×C4#1313318511yescharacteristic
340C2×C4#1313318811yescharacteristic
341C2×C4#1313319011yescharacteristic
342C2×C4#1313323811yescharacteristic
343C2×C4#1313324011yescharacteristic
344C2×C4#1313324611yescharacteristic
345C2×C4#1313324911yescharacteristic
346C8—41no—
347C8—41no—
348C8—81no—
349C8—4241no—
350C8—2122no—
351C8—2122no—
352C22—21no—
353C22—21no—
354C22—2121no—
355C22—2121no—
356C22—2121no—
357C22#1681049911yescharacteristic
358C22#1681096211yescharacteristic
359C22#1681096311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—2121no—
365C4—2121no—
366C4—2121no—
367C4#1681061811yescharacteristic
368C4#1681063011yescharacteristic
369C4#1681078211yescharacteristic
370C4#1681078411yescharacteristic
371C4#1681088311yescharacteristic
372C4#1681089011yescharacteristic
373C2—21no—
374C2—21no—
375C2—4241no—
376C2#2027502411yescharacteristic
377C2#2027519111yescharacteristic
378C2#2027522611yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9168
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #20273396C2semidirect
5. #20272612C2semidirect
6. #20272590C2semidirect
7. C2×C53×#332C2semidirect
187. C2×#332D53semidirect
220. C53#9168semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202833802
#202733962
#202731002
#202726122
#202725902
C2×C53×#3322
#916853