Finite Group Databasebeta

← #27491272

Subgroups — #27491272

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284655C212yesmaximal, complemented
2C2×#16810876C211yesmaximal, characteristic
3#20276225C212yesmaximal, complemented
4#20276074C214yesmaximal, complemented
5#20275971C212yesmaximal, complemented
6#20275178C212yesmaximal, complemented
7#20274757C211yesmaximal, characteristic, complemented
8C53×#2905C211yesmaximal, characteristic, complemented
9C22×D424C2211yescharacteristic
10C2×#13133191C2212yes—
11#16810947—24no—
12#16810941—22no—
13#16810936—22no—
14#16810876—22no—
15#16810876—22no—
16C2×#13133153C2212yes—
17#16810666C2214yes—
18C2×#13133149C2211yescharacteristic
19#16810627C2214yes—
20C2×#13133139C2212yes—
21C2×#13133133C2214yes—
22#16810587C2212yes—
23#16810391C2214yes—
24#16810388C2212yes—
25C22×M16×C53C2211yescharacteristic
26C22×M16×C53C2211yescharacteristic
27C2×#117×C53C2211yescharacteristic
28C53×#424—22no—
29C53×#424C2212yes—
30C53×#424C2212yes—
31C53×#423—22no—
32C22×D212C2312yes—
33C22×C22⋊D53—22no—
34C2×D424—22no—
35C2×D424—42no—
36C2×#11719917C2314yes—
37#13133191—22no—
38#13133191—22no—
39#13133191—22no—
40#13133191—22no—
41C2×C4⋊Dic53C2312yes—
42C2×#11719921—22no—
43#13133153—42no—
44#13133150—24no—
45#13133149—22no—
46#13133148—24no—
47#13133139—24no—
48#13133137—24no—
49#13133137—24no—
50#13133133—44no—
51C22⋊#88322—22no—
52C22⋊#88322—22no—
53C23×C4×C53C2311yescharacteristic
54C2×M16×C53—21no—
55C2×M16×C53—42no—
56C2×M16×C53—42no—
57C2×M16×C53D412yes—
58C2×M16×C53D412yes—
59C2×M16×C53C2312yes—
60C22×C8×C53—21no—
61C22×C8×C53C2311yescharacteristic
62C22×C8×C53C2311yescharacteristic
63#117×C53—21no—
64#117×C53—21no—
65#117×C53D412yes—
66#117×C53C2312yes—
67C2×C4⋊C4×C53D412yes—
68C2×C22⋊C4×C53—21no—
69C2×C42×C53—21no—
70C4⋊C8×C53—24no—
71C4⋊C8×C53—24no—
72C24×C53#4111yescharacteristic
73C23×D53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53C2411yescharacteristic
83C2×C22⋊D53—42no—
84C2×C22⋊D53—42no—
85C2×C22⋊D53—84no—
86C22×Dic53—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—24no—
90C2×D212—24no—
91M16×C53—24no—
92M16×C53—42no—
93M16×C53—42no—
94M16×C53—42no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53—22no—
98C2×C8×C53—42no—
99C2×C8×C53C2×D412yes—
100C2×C8×C53C2×D412yes—
101C4⋊C4×C53—22no—
102C4⋊C4×C53C2×D414yes—
103C22⋊C4×C53—22no—
104C22⋊C4×C53—22no—
105C42×C53—22no—
106C42×C53—22no—
107#11719921—42no—
108#11719921—42no—
109C4⋊Dic53—22no—
110C4⋊Dic53—22no—
111#11719917—24no—
112#11719917—24no—
113C2×#88322—24no—
114D424—44no—
115C23×C53—22no—
116C23×C53—42no—
117C23×C53#12111yescharacteristic
118C23×C53#12111yescharacteristic
119C23×C53C2×#4111yescharacteristic
120C22×D53—42no—
121C22×D53—44no—
122C22×D53—82no—
123C22×D53—82no—
124C2×C4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—24no—
132C2×C4×C53—82no—
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53C22×D411yescharacteristic
136C2×C4×C53C22×D411yescharacteristic
137C2×C4×C53H(5,𝔽2)12yes—
138C2×C4×C532−1+412yes—
139C22⋊D53—84no—
140C22⋊D53—84no—
141C2×Dic53—42no—
142C2×Dic53—82no—
143D212—42no—
144D212—42no—
145D212—44no—
146C8×C53—22no—
147C8×C53—22no—
148C8×C53—42no—
149#88322—44no—
150#29791—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—82no—
157C22×C53C2×#12111yescharacteristic
158C22×C53#53912yes—
159C2×D53—82no—
160C2×D53—82no—
161C2×D53—82no—
162C2×D53—164no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—44no—
166C4×C53#53411yescharacteristic
167C4×C53#53611yescharacteristic
168C4×C53#53712yes—
169Dic53—82no—
170#3094—532no—
171C2×#469—531no—
172#2905D5311yescharacteristic, complemented
173#2090—532no—
174#2063—534no—
175#2045—532no—
176#1900—532no—
177#1811—531no—
178C2×C53—22no—
179C2×C53—41no—
180C2×C53—41no—
181C2×C53#261811yescharacteristic
182C2×C53#306812yes—
183D53—162no—
184C22×D8—531no—
185C22×M16C2×D5311yescharacteristic
186C22×M16C2×D5311yescharacteristic
187C2×#121—532no—
188C2×#117C2×D5311yescharacteristic
189#482—1064no—
190#481—1062no—
191#480—1062no—
192#469—1062no—
193#469—1062no—
194C2×#108—532no—
195#428—534no—
196C2×#106—531no—
197#424—22no—
198#424C2×D5312yes—
199#424C2×D5312yes—
200#423—22no—
201#418—534no—
202C2×#102—532no—
203C2×#100—534no—
204#407—532no—
205#343—534no—
206#341—532no—
207C53#2979111yescharacteristic, complemented, Sylow
208C22×D4—532no—
209C22×D4—1062no—
210C23×C4C22×D5311yescharacteristic
211C2×D8—1062no—
212C2×D8—2122no—
213C2×M16—21no—
214C2×M16—42no—
215C2×M16—42no—
216C2×M16—534no—
217C2×M16C22⋊D5312yes—
218C2×M16C22⋊D5312yes—
219C2×M16C22×D5312yes—
220C22×C8—21no—
221C22×C8C22×D5311yescharacteristic
222C22×C8C22×D5311yescharacteristic
223#121—1062no—
224#121—1062no—
225#121—1062no—
226#121—1062no—
227#117—21no—
228#117—21no—
229#117D21212yes—
230#117C22×D5312yes—
231C2×C4⋊C4—532no—
232C2×C4⋊C4D21212yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—1062no—
235C2×C42—21no—
236#108—2122no—
237#107—1064no—
238#106—1062no—
239#106—1064no—
240C4⋊C8—24no—
241C4⋊C8—24no—
242#102—1064no—
243#102—1064no—
244#102—1064no—
245#100—2124no—
246C22⋊C8—1062no—
247C22⋊C8—1062no—
248C24—1062no—
249C24#1171994511yescharacteristic
250C2×D4—1062no—
251C2×D4—1062no—
252C2×D4—1064no—
253C2×D4—1064no—
254C2×D4—4244no—
255C2×D4—2122no—
256C2×D4—2122no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—1062no—
261C22×C4C2×D21211yescharacteristic
262C22×C4C2×D21211yescharacteristic
263C22×C4#1171994511yescharacteristic
264C22×C4C2×C22⋊D5311yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266C22×C4C23×D5311yescharacteristic
267D8—2124no—
268M16—24no—
269M16—42no—
270M16—42no—
271M16—42no—
272M16—1064no—
273M16—1064no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—42no—
278C2×C8—1064no—
279C2×C8C2×C22⋊D5312yes—
280C2×C8C2×C22⋊D5312yes—
281C4⋊C4—22no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4C2×D21214yes—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—2122no—
288C22⋊C4—2122no—
289C42—22no—
290C42—22no—
291C23—22no—
292C23—42no—
293C23—4242no—
294C23—4242no—
295C23—2124no—
296C23—2122no—
297C23#1313319111yescharacteristic
298C23#1313319111yescharacteristic
299C23C2×#1171994511yescharacteristic
300D4—4244no—
301D4—4244no—
302D4—2122no—
303D4—2122no—
304D4—2124no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—82no—
314C2×C4—4242no—
315C2×C4—2122no—
316C2×C4#1313319111yescharacteristic
317C2×C4#1313319111yescharacteristic
318C2×C4C22×C22⋊D5311yescharacteristic
319C2×C4C22×D21211yescharacteristic
320C2×C4#1313326212yes—
321C2×C4#1313326512yes—
322C8—22no—
323C8—22no—
324C8—42no—
325C8—2124no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—82no—
332C22—8484no—
333C22—4242no—
334C22—4242no—
335C22—4242no—
336C22C2×#1313319111yescharacteristic
337C22#1681122012yes—
338C4—21no—
339C4—21no—
340C4—44no—
341C4—4242no—
342C4#1681118211yescharacteristic
343C4#1681119711yescharacteristic
344C4#1681120412yes—
345C2—22no—
346C2—41no—
347C2—41no—
348C2—8482no—
349C2#2028045711yescharacteristic
350C2#2028444312yes—
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2905C2
socleC22×C53C2×#121
prime radicalC2×#16810876C2
hypercenter#2905D53
hypocenterC53#29791
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. #20284655C2semidirect
3. #20276225C2semidirect
4. #20276074C2semidirect
5. #20275971C2semidirect
6. #20275178C2semidirect
7. #20274757C2semidirect
8. C53×#2905C2semidirect
172. #2905D53semidirect
207. C53#29791semidirect

Maximal subgroups

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

subgroupindex
#202846552
C2×#168108762
#202762252
#202760742
#202759712
#202751782
#202747572
C53×#29052
#2979153