Finite Group Databasebeta

← #27279181

Subgroups — #27279181

Order 13568.

528 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277983C211yesmaximal, characteristic
2#20273143C212yesmaximal, complemented
3#20273029C211yesmaximal, characteristic
4#20272616C212yesmaximal, complemented
5C53×#2905C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810676C2212yes—
8#16810562C411yescharacteristic
9#16810559C411yescharacteristic
10#16810556—21no—
11#16810546—21no—
12#16810543C411yescharacteristic
13#16810535C411yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810353—24no—
18#16810349—21no—
19#16810347—21no—
20C22×M16×C53—21no—
21C2×#117×C53C2211yescharacteristic
22C53×#424—24no—
23C53×#423—21no—
24C53×#423C2212yes—
25C22×Dic106C2×C411yescharacteristic
26C4×Dic106—22no—
27C4×Dic106—22no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C4⋊#88322—22no—
38C22⋊#88322—22no—
39C4×#88322—22no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—22no—
48C2×M16×C53—22no—
49C2×M16×C53—42no—
50C2×M16×C53—42no—
51C22×C8×C53—22no—
52#117×C53—22no—
53#117×C53—22no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—24no—
61C4⋊C8×C53—22no—
62C4⋊C8×C53D414yes—
63C24×C53C22⋊C411yescharacteristic
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53SD1612yes—
72C22×C4×C53Q1612yes—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C2×Dic106—22no—
81C2×Dic106—42no—
82M16×C53—44no—
83M16×C53—42no—
84M16×C53—42no—
85C2×C8×C53—24no—
86C2×C8×C53—44no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—22no—
89C22⋊C4×C53—22no—
90C22⋊C4×C53—22no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53C2×D412yes—
94C42×C53#4112yes—
95C22⋊Dic53—42no—
96C22⋊Dic53—42no—
97C4⋊Dic53—22no—
98C4⋊Dic53—42no—
99#11719919—42no—
100#11719919—42no—
101#11719919—42no—
102C4×Dic53—41no—
103C4×Dic53—41no—
104C4×Dic53—41no—
105C4×Dic53—41no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53#10312yes—
112C23×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#10312yes—
126C2×C4×C53C4×D412yes—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53#12112yes—
129C2×C4×C53#12311yescharacteristic
130C2×C4×C53C2×SD1611yescharacteristic
131C2×C4×C53C2×Q1611yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53#13711yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146Dic106—24no—
147Dic106—42no—
148C8×C53—44no—
149#88322—82no—
150#9033—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—22no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—81no—
158C22×C53#38611yescharacteristic
159C22×C53C2×#10311yescharacteristic
160C22×C53#41711yescharacteristic
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—41no—
165C4×C53—41no—
166C4×C53#45011yescharacteristic
167C4×C53#45111yescharacteristic
168C4×C53#48311yescharacteristic
169C4×C53#48411yescharacteristic
170Dic53—42no—
171Dic53—42no—
172Dic53—81no—
173Dic53—81no—
174#2905D5311yescharacteristic
175#2396—531no—
176#1487—532no—
177#1458—531no—
178#1303—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53#187211yescharacteristic
183C2×C53#188411yescharacteristic
184C2×C53#189111yescharacteristic
185C22×M16—21no—
186C2×C4×Q8—531no—
187C2×#117C2×D5311yescharacteristic
188#431—532no—
189#424—24no—
190#423—21no—
191#423C2×D5312yes—
192#400—531no—
193#399—531no—
194#398—1061no—
195#396—1061no—
196#395—531no—
197#393—531no—
198C4×C4⋊C4—531no—
199C4×C4⋊C4—1061no—
200C4×C22⋊C4—1061no—
201Q8⋊C8—1064no—
202#324—1061no—
203#323—1061no—
204C53#903311yescharacteristic, complemented, Sylow
205C22×Q8—531no—
206C23×C4C22⋊D5311yescharacteristic
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—42no—
210C2×M16—42no—
211C22×C8—22no—
212C4×Q8—1062no—
213C4×Q8—1062no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4C4×D5311yescharacteristic
222C2×C4⋊C4D21211yescharacteristic
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4C4×D5311yescharacteristic
226C2×C22⋊C4D21211yescharacteristic
227C2×C42—1061no—
228C2×C42—1061no—
229C2×C42C22⋊D5311yescharacteristic
230C2×C42C22×D5311yescharacteristic
231C4⋊C8—24no—
232C4⋊C8—22no—
233C4⋊C8—1062no—
234C4⋊C8C22⋊D5314yes—
235C22⋊C8—1062no—
236C4×C8—1062no—
237#95—531no—
238#95—531no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243C24#1171992111yescharacteristic
244C2×Q8—1062no—
245C2×Q8—2122no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4#1171992111yescharacteristic
257C22×C4#1171992312yes—
258C22×C4#1171992512yes—
259C22×C4C2×C4×D5311yescharacteristic
260C22×C4C2×D21211yescharacteristic
261C22×C4C2×C22⋊D5311yescharacteristic
262M16—44no—
263M16—42no—
264M16—42no—
265C2×C8—24no—
266C2×C8—44no—
267C2×C8—2122no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—1062no—
271C4⋊C4—2122no—
272C4⋊C4—2122no—
273C4⋊C4—2122no—
274C4⋊C4—2122no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—2122no—
278C22⋊C4—2122no—
279C42—21no—
280C42—21no—
281C42—2121no—
282C42—2121no—
283C42—2121no—
284C42—2121no—
285C42#1171994512yes—
286C42C2×C22⋊D5312yes—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1313314012yes—
292C23C2×#1171992111yescharacteristic
293Q8—1064no—
294Q8—2122no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4#1313314012yes—
320C2×C4C4×C22⋊D5312yes—
321C2×C4#1313318611yescharacteristic
322C2×C4#1313319112yes—
323C2×C4#1313320011yescharacteristic
324C2×C4C2×#1171992311yescharacteristic
325C2×C4C2×#1171992511yescharacteristic
326C2×C4#1313324311yescharacteristic
327C2×C4#1313324811yescharacteristic
328C8—44no—
329C8—4242no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22#1681050211yescharacteristic
338C22C2×#1313314011yescharacteristic
339C22#1681062211yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—4241no—
346C4—4241no—
347C4—2122no—
348C4—2122no—
349C4#1681077411yescharacteristic
350C4#1681077911yescharacteristic
351C4#1681095311yescharacteristic
352C4#1681095811yescharacteristic
353C2—21no—
354C2—21no—
355C2—42no—
356C2#2027502211yescharacteristic
357C2#2027509811yescharacteristic
358C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2905C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2905D53
hypocenterC53#9033
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2905

Complements

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

NH ≅ G/Ntype
2. #20273143C2semidirect
4. #20272616C2semidirect
204. C53#9033semidirect

Maximal subgroups

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

subgroupindex
#202779832
#202731432
#202730292
#202726162
C53×#29052
#903353