Finite Group Databasebeta

← #27272107

Subgroups — #27272107

Order 13568.

568 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#20277198C211yesmaximal, characteristic
2#20274783C211yesmaximal, characteristic
3#20272539C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C2×C53×#336C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C2×#13133176C411yescharacteristic
8C2×C22⋊#88322—21no—
9#16810521C411yescharacteristic
10#16810521C411yescharacteristic
11#16810521C2211yescharacteristic
12#16810521C2211yescharacteristic
13#16810496—21no—
14#16810496—21no—
15C4×#11719919—21no—
16C4×#11719919—21no—
17#16810378—21no—
18#16810378C2212yes—
19C23×C8×C53—21no—
20C22×C42×C53C2211yescharacteristic
21C53×#336—21no—
22C53×#336C2212yes—
23C22×#88322—22no—
24#13133176—22no—
25#13133176—42no—
26#13133176C814yescomplemented
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C22⋊#88322—42no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C22×C8×C53—24no—
48C22×C8×C53—22no—
49C22×C8×C53—42no—
50C22×C8×C53—42no—
51C2×C42×C53—42no—
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—22no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C2×C812yes—
70C22×C4×C53M1612yes—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×C8×C53—24no—
79C2×C8×C53—48no—
80C2×C8×C53—42no—
81C2×C8×C53—42no—
82C42×C53—41no—
83C42×C53—41no—
84C42×C53—41no—
85C42×C53—41no—
86C22⋊Dic53—24no—
87C22⋊Dic53—42no—
88C4⋊Dic53—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53C2×C812yescomplemented
93C4⋊Dic53C2×C812yescomplemented
94#11719919—42no—
95#11719919—42no—
96#11719919—42no—
97#11719919—41no—
98#11719919—41no—
99C4×Dic53—24no—
100C4×Dic53—42no—
101C2×#88322—42no—
102C2×#88322—42no—
103C2×#88322—42no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53C22⋊C812yes—
109C23×C53#9912yes—
110C23×C53#10112yes—
111C23×C53C2×C22⋊C411yescharacteristic
112C2×C4×C53—24no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—24no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—44no—
130C2×C4×C53C22⋊C812yes—
131C2×C4×C53#10412yes—
132C2×C4×C53C22×C811yescharacteristic
133C2×C4×C53C2×M1611yescharacteristic
134C2×C4×C53#13512yes—
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53#13711yescharacteristic
137C2×Dic53—22no—
138C2×Dic53—22no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C8×C53—44no—
150#88322—82no—
151#7579—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53#32414yes—
159C22×C53C2×C22⋊C811yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53C2×#10111yescharacteristic
162C22×C53#41611yescharacteristic
163C22×C53#41711yescharacteristic
164C22×C53C2×#10411yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—21no—
172C4×C53—22no—
173C4×C53—41no—
174Dic53—42no—
175Dic53—42no—
176Dic53—81no—
177Dic53—81no—
178#2296—531no—
179#1817—531no—
180C2×#336D5311yescharacteristic
181#1271—532no—
182#1271—532no—
183C2×C53—24no—
184C2×C53C2×#32411yescharacteristic
185C2×C53#147611yescharacteristic
186C2×C53#147711yescharacteristic
187C2×C53#150411yescharacteristic
188C2×C53#151611yescharacteristic
189C2×C53#153111yescharacteristic
190C2×C53#155211yescharacteristic
191C23×C8—21no—
192C2×#117—531no—
193C2×#117—531no—
194C22×C42C2×D5311yescharacteristic
195C2×C22⋊C8—1061no—
196#389—531no—
197#389—531no—
198#389—531no—
199#389—531no—
200#385—1061no—
201#385—1061no—
202C4×C4⋊C4—1061no—
203C4×C4⋊C4—1061no—
204#336—21no—
205#336—532no—
206#336—1061no—
207#336C2×D5312yes—
208C53#757911yescharacteristic, complemented, Sylow
209C23×C4C4×D5311yescharacteristic
210C23×C4D21211yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C22×C8—24no—
213C22×C8—22no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—1062no—
217#117—534no—
218#117—1062no—
219#117—2122no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1061no—
230C2×C42—42no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42C4×D5311yescharacteristic
234C2×C42D21211yescharacteristic
235C2×C42C22⋊D5311yescharacteristic
236C2×C42C22×D5311yescharacteristic
237C22⋊C8—2122no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242C24#1171992111yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4C8×D5312yes—
257C22×C4#1171991212yes—
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4#1171992111yescharacteristic
261C22×C4C2×C4×D5311yescharacteristic
262C22×C4C2×D21211yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264C2×C8—24no—
265C2×C8—48no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—2122no—
269C2×C8—2122no—
270C2×C8—2122no—
271C4⋊C4—532no—
272C4⋊C4—532no—
273C4⋊C4—1061no—
274C4⋊C4—1061no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C4⋊C4—2122no—
278C4⋊C4—2122no—
279C4⋊C4—2122no—
280C4⋊C4—2121no—
281C4⋊C4—2121no—
282C22⋊C4—1064no—
283C22⋊C4—2122no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—1064no—
289C42—2122no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1313312912yes—
295C23#1313313112yes—
296C23#1313313512yes—
297C23C2×#1171992111yescharacteristic
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—44no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4#1313312912yes—
329C2×C4#1313314512yes—
330C2×C4C2×C8×D5311yescharacteristic
331C2×C4C2×#1171991211yescharacteristic
332C2×C4#1313323912yes—
333C2×C4#1313324311yescharacteristic
334C2×C4#1313324811yescharacteristic
335C8—44no—
336C8—4242no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#1681034814yes—
344C22C2×#1313312911yescharacteristic
345C22C2×#1313313111yescharacteristic
346C22C2×#1313313511yescharacteristic
347C22#1681062011yescharacteristic
348C22#1681062211yescharacteristic
349C22C2×#1313314511yescharacteristic
350C22#1681063911yescharacteristic
351C4—21no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—4241no—
360C4—4241no—
361C4—2122no—
362C4—2122no—
363C2—24no—
364C2C2×#1681034811yescharacteristic
365C2#2027309811yescharacteristic
366C2#2027310011yescharacteristic
367C2#2027321711yescharacteristic
368C2#2027327211yescharacteristic
369C2#2027332911yescharacteristic
370C2#2027341311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×#336C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC2×#336D53
hypocenterC53#7579
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
3. #20272539C2semidirect
4. #20272537C2semidirect
26. #13133176C8semidirect
92. C4⋊Dic53C2×C8semidirect
93. C4⋊Dic53C2×C8semidirect
208. C53#7579semidirect

Maximal subgroups

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

subgroupindex
#202771982
#202747832
#202725392
#202725372
C2×C53×#3362
#757953