Finite Group Databasebeta

← #27412207

Subgroups — #27412207

Order 13568.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283304C212yesmaximal
2#20277339C211yesmaximal, characteristic
3#20276062C214yesmaximal, complemented
4#20276057C214yesmaximal, complemented
5C2×#16810392C211yesmaximal, characteristic
6#20274424C212yesmaximal
7C2×C53×#544C211yesmaximal, characteristic
8C22×#11719917C2212yes—
9C2×#13133176C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11#16810666C2214yes—
12#16810666C2214yes—
13#16810663—22no—
14#16810663—22no—
15#16810659—24no—
16#16810519—22no—
17#16810496—22no—
18#16810492C2212yes—
19#16810488C2212yes—
20#16810392C2214yes—
21#16810382C2218yes—
22C53×#544—24no—
23C53×#544—22no—
24C53×#544C2214yes—
25C2×C4⋊Q8×C53—22no—
26C2×#125×C53—22no—
27C2×C22⋊Q8×C53—22no—
28C2×#117×C53C2212yes—
29C22×C4⋊C4×C53C2211yescharacteristic
30C2×#11719917—22no—
31C2×#11719917—24no—
32C2×#11719917Q818yescomplemented
33C2×#11719917C2314yes—
34C22×#88322—22no—
35#13133176—22no—
36#13133176—22no—
37#13133176—22no—
38#13133176Q814yescomplemented
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2311yescharacteristic
41C2×C4⋊Dic53C2311yescharacteristic
42C2×#11719919—24no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53C2312yes—
47#13133150—24no—
48#13133150—24no—
49#13133148—24no—
50#13133148—24no—
51#13133124—24no—
52C22×Q8×C53—22no—
53C23×C4×C53D412yes—
54C23×C4×C53C2311yescharacteristic
55C4⋊Q8×C53—28no—
56C4⋊Q8×C53—28no—
57#125×C53—28no—
58#125×C53—28no—
59C22⋊Q8×C53—28no—
60C22⋊Q8×C53—28no—
61#117×C53—22no—
62#117×C53—22no—
63#117×C53D414yes—
64#117×C53C2314yes—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53—24no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53D412yes—
70C2×C4⋊C4×C53D412yes—
71C2×C4⋊C4×C53D412yes—
72C2×C4⋊C4×C53C2312yes—
73C2×C22⋊C4×C53—22no—
74C2×C42×C53—22no—
75C24×C53C2×D411yescharacteristic
76C2×Q8×C53—28no—
77C2×Q8×C53—84no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—82no—
83C22×C4×C53C2×D414yes—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D412yes—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×Q814yes—
88C22×C4×C53C2×Q812yes—
89C22×C4×C53#4112yes—
90C22×C4×C53C2411yescharacteristic
91C22×Dic53—22no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C4⋊C4×C53—28no—
95C4⋊C4×C53—216no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53C2×D414yes—
101C4⋊C4×C53C2×D414yes—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—24no—
104C42×C53—24no—
105C42×C53—24no—
106C22⋊Dic53—24no—
107C22⋊Dic53—41no—
108C22⋊Dic53—41no—
109C4⋊Dic53—24no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112#11719919—84no—
113C4×Dic53—22no—
114C4×Dic53—42no—
115C4×Dic53C2×Q814yescomplemented
116#11719917—28no—
117#11719917—24no—
118#11719917C2×Q818yescomplemented
119C2×#88322—28no—
120C2×#88322—24no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53#12011yescharacteristic
124C23×C53#12011yescharacteristic
125C23×C53C22⋊Q812yes—
126C23×C53C4⋊Q812yes—
127C23×C53C22×D411yescharacteristic
128Q8×C53—88no—
129C2×C4×C53—24no—
130C2×C4×C53—24no—
131C2×C4×C53—24no—
132C2×C4×C53—44no—
133C2×C4×C53—44no—
134C2×C4×C53—44no—
135C2×C4×C53—44no—
136C2×C4×C53—82no—
137C2×C4×C53#12012yes—
138C2×C4×C53C22⋊Q814yes—
139C2×C4×C53C22⋊Q814yes—
140C2×C4×C53C22⋊Q812yes—
141C2×C4×C53C4⋊Q814yes—
142C2×C4×C53C4⋊Q814yes—
143C2×C4×C53C4⋊Q812yes—
144C2×C4×C53C22×D412yes—
145C2×C4×C53C22×Q812yes—
146C2×C4×C53C22×Q811yescharacteristic
147C2×C4×C53C2×#4111yescharacteristic
148C2×Dic53—24no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155#88322—28no—
156#23561—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53#39512yes—
166C22×C53#39512yes—
167C22×C53C2×#12011yescharacteristic
168C22×C53C2×C22⋊Q811yescharacteristic
169C22×C53C2×C4⋊Q811yescharacteristic
170C4×C53—44no—
171C4×C53—44no—
172C4×C53—44no—
173C4×C53#39514yes—
174C4×C53C2×C22⋊Q812yes—
175C4×C53C2×C4⋊Q812yes—
176Dic53—44no—
177Dic53—81no—
178Dic53—81no—
179C2×#544D5311yescharacteristic
180#2911—532no—
181#2317—531no—
182#2061—534no—
183#2060—534no—
184C2×#343—531no—
185#1733—532no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—22no—
189C2×C53—22no—
190C2×C53C2×#39511yescharacteristic
191C2×C53#301911yescharacteristic
192C2×C53#302011yescharacteristic
193C22×M16—532no—
194#544—24no—
195#544—22no—
196#544C2×D5314yes—
197C2×C4⋊Q8—22no—
198C2×#125—22no—
199C2×C22⋊Q8—22no—
200C2×#117—531no—
201C2×#117—531no—
202C2×#117C2×D5312yes—
203C22×C4⋊C4C2×D5311yescharacteristic
204#428—534no—
205#428—534no—
206#428—1062no—
207#428—1062no—
208#427—1064no—
209#389—1062no—
210#385—1062no—
211#384—532no—
212#382—532no—
213#343—534no—
214#339—538no—
215C53#2356111yescharacteristic, complemented, Sylow
216C22×Q8—22no—
217C23×C4C22⋊D5312yes—
218C23×C4C22×D5311yescharacteristic
219C2×M16—538no—
220C2×M16—534no—
221C2×M16—1064no—
222C2×M16—1062no—
223C22×C8—1062no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230#117—22no—
231#117—22no—
232#117—534no—
233#117—1062no—
234#117—1062no—
235#117—1062no—
236#117C22⋊D5314yes—
237#117C22×D5314yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1064no—
246C2×C4⋊C4C22⋊D5312yes—
247C2×C4⋊C4C22⋊D5312yes—
248C2×C4⋊C4C22⋊D5312yes—
249C2×C4⋊C4C22×D5312yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—1061no—
252C2×C22⋊C4—1061no—
253C2×C42—22no—
254C2×C42—532no—
255C2×C42—1061no—
256#107—1064no—
257#107—1064no—
258#106—1064no—
259#106—1064no—
260#95—1064no—
261C24C2×C22⋊D5311yescharacteristic
262C2×Q8—28no—
263C2×Q8—84no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—82no—
269C22×C4—1062no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4Q8×D5314yes—
273C22×C4Q8×D5312yes—
274C22×C4#1171994912yes—
275C22×C4C2×C22⋊D5314yes—
276C22×C4C2×C22⋊D5311yescharacteristic
277C22×C4C2×C22⋊D5312yes—
278C22×C4C2×C22⋊D5312yes—
279C22×C4C23×D5311yescharacteristic
280M16—538no—
281M16—1068no—
282M16—1064no—
283C2×C8—1068no—
284C2×C8—1064no—
285C4⋊C4—28no—
286C4⋊C4—216no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—1064no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—4244no—
295C4⋊C4C2×C22⋊D5314yes—
296C4⋊C4C2×C22⋊D5314yes—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—1064no—
300C22⋊C4—2121no—
301C22⋊C4—2121no—
302C42—24no—
303C42—24no—
304C42—534no—
305C42—1062no—
306C42—2122no—
307C23—41no—
308C23—41no—
309C23#1313318611yescharacteristic
310C23#1313318611yescharacteristic
311C23#1313319312yes—
312C23#1313321512yes—
313C23C22×C22⋊D5311yescharacteristic
314Q8—88no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—1064no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4#1313318612yes—
331C2×C4#1313319312yes—
332C2×C4#1313319314yes—
333C2×C4#1313319314yes—
334C2×C4#1313321512yes—
335C2×C4#1313321514yes—
336C2×C4#1313321514yes—
337C2×C4C22×C22⋊D5312yes—
338C2×C4C2×Q8×D5312yes—
339C2×C4C2×Q8×D5311yescharacteristic
340C2×C4C2×#1171994911yescharacteristic
341C8—1068no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22#1681054212yes—
351C22#1681054212yes—
352C22C2×#1313318611yescharacteristic
353C22C2×#1313319311yescharacteristic
354C22C2×#1313321511yescharacteristic
355C4—44no—
356C4—44no—
357C4—44no—
358C4—4241no—
359C4—4241no—
360C4—2124no—
361C4#1681054214yes—
362C4C2×#1313319312yes—
363C4C2×#1313321512yes—
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#1681054211yescharacteristic
369C2#2028407211yescharacteristic
370C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#544C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenterC2×#544D53
hypocenterC53#23561
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20276062C2semidirect
4. #20276057C2semidirect
32. C2×#11719917Q8semidirect
38. #13133176Q8semidirect
115. C4×Dic53C2×Q8semidirect
118. #11719917C2×Q8semidirect
215. C53#23561semidirect

Maximal subgroups

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

subgroupindex
#202833042
#202773392
#202760622
#202760572
C2×#168103922
#202744242
C2×C53×#5442
#2356153