Finite Group Databasebeta

← #27272400

Subgroups — #27272400

Order 13568.

500 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
1C4×#13133196C211yesmaximal, characteristic
2#20274783C211yesmaximal, characteristic
3#20272540C212yesmaximal, complemented
4#20272537C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7C2×C4×Dic106C411yescharacteristic
8C2×C22⋊#88322—21no—
9#16810529C2211yescharacteristic
10#16810521C2211yescharacteristic
11#16810507—21no—
12#16810496—21no—
13#16810483C411yescharacteristic
14C4×#11719919—21no—
15C4×C4⋊Dic53C411yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810378—21no—
18#16810378C2212yes—
19C22×C42×C53C2211yescharacteristic
20C2×C4⋊C8×C53—21no—
21C2×C22⋊C8×C53—21no—
22C53×#336C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×#88322—22no—
25#13133196—22no—
26#13133196—22no—
27C4×Dic106—42no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C22⋊#88322—42no—
39#13133124—21no—
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—22no—
48C2×C42×C53—42no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53C2311yescharacteristic
53C4⋊C8×C53—42no—
54C22⋊C8×C53—42no—
55C24×C53C22⋊C411yescharacteristic
56C22×C4×C53—22no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53SD1612yes—
69C22×C4×C53Q1612yes—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×Dic106—21no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80C2×Dic106M1612yes—
81C2×C8×C53—42no—
82C2×C8×C53—42no—
83C2×C8×C53—42no—
84C42×C53—22no—
85C42×C53—22no—
86C42×C53—41no—
87C42×C53—41no—
88C22⋊Dic53—24no—
89C22⋊Dic53—42no—
90C4⋊Dic53—21no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53M1612yes—
94#11719919—24no—
95#11719919—42no—
96#11719919—42no—
97C4×Dic53—42no—
98C4×Dic53—42no—
99C2×#88322—42no—
100C2×#88322—42no—
101C2×#88322—42no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53#10312yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
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—42no—
124C2×C4×C53—42no—
125C2×C4×C53—44no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53#9912yes—
129C2×C4×C53#10312yes—
130C2×C4×C53C2×M1611yescharacteristic
131C2×C4×C53C2×M1611yescharacteristic
132C2×C4×C53C2×SD1611yescharacteristic
133C2×C4×C53C2×Q1611yescharacteristic
134C2×C4×C53#13512yes—
135C2×Dic53—22no—
136C2×Dic53—22no—
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—
146C2×Dic53—41no—
147Dic106—42no—
148Dic106—42no—
149C8×C53—82no—
150#88322—82no—
151#7648—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—22no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53#40711yescharacteristic
160C22×C53C2×#9911yescharacteristic
161C22×C53#41311yescharacteristic
162C22×C53C2×#10311yescharacteristic
163C22×C53#41611yescharacteristic
164C22×C53#42111yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—22no—
169C4×C53—22no—
170C4×C53—21no—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—41no—
174Dic53—42no—
175Dic53—42no—
176Dic53—81no—
177Dic53—81no—
178C4×C22⋊Q8—531no—
179#1817—531no—
180#1797D5311yescharacteristic
181#1272—532no—
182#1271—532no—
183C2×C53—42no—
184C2×C53#146511yescharacteristic
185C2×C53#148711yescharacteristic
186C2×C53#148811yescharacteristic
187C2×C53#151611yescharacteristic
188C2×C53#152011yescharacteristic
189C2×C53#153511yescharacteristic
190C2×C53#154211yescharacteristic
191C2×C22⋊Q8—531no—
192C2×C4×Q8—531no—
193C22×C42C2×D5311yescharacteristic
194C2×C4⋊C8—21no—
195C2×C22⋊C8—21no—
196C2×C22⋊C8—1061no—
197#391—531no—
198#389—531no—
199#387—1061no—
200#385—1061no—
201#380—531no—
202C4×C4⋊C4—531no—
203C4×C4⋊C4—1061no—
204C4×C22⋊C4—1061no—
205#336—532no—
206#336—1061no—
207#336C2×D5312yes—
208C53#764811yescharacteristic, complemented, Sylow
209C22×Q8—531no—
210C23×C4C4×D5311yescharacteristic
211C23×C4D21211yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C22×C8—22no—
214C22×C8—1062no—
215C22⋊Q8—1062no—
216C22⋊Q8—1062no—
217C4×Q8—2122no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—531no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C42—42no—
227C2×C42—1061no—
228C2×C42—1061no—
229C2×C42C4×D5311yescharacteristic
230C2×C42D21211yescharacteristic
231C2×C42C22⋊D5311yescharacteristic
232C2×C42C22×D5311yescharacteristic
233C4⋊C8—42no—
234C22⋊C8—42no—
235C22⋊C8—2122no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241C24#1171992111yescharacteristic
242C2×Q8—532no—
243C2×Q8—1061no—
244C2×Q8—2121no—
245C2×Q8—2121no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4—1061no—
259C22×C4#1171992111yescharacteristic
260C22×C4#1171992111yescharacteristic
261C22×C4#1171992111yescharacteristic
262C22×C4#1171992312yes—
263C22×C4#1171992512yes—
264C22×C4C2×C4×D5311yescharacteristic
265C22×C4C2×D21211yescharacteristic
266C22×C4C2×C22⋊D5311yescharacteristic
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—2122no—
271C2×C8—2122no—
272C2×C8—2122no—
273C4⋊C4—532no—
274C4⋊C4—1064no—
275C4⋊C4—1061no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C4⋊C4—2122no—
279C4⋊C4—2122no—
280C22⋊C4—1064no—
281C22⋊C4—2122no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—2122no—
287C42—2122no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#1313314012yes—
295C23C2×#1171992111yescharacteristic
296Q8—2122no—
297Q8—2122no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
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#1313313112yes—
329C2×C4#1313314012yes—
330C2×C4M16×D5311yescharacteristic
331C2×C4M16×D5311yescharacteristic
332C2×C4C2×#1171992311yescharacteristic
333C2×C4C2×#1171992511yescharacteristic
334C2×C4#1313323912yes—
335C8—82no—
336C8—4242no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#1681058611yescharacteristic
345C22C2×#1313313111yescharacteristic
346C22#1681060511yescharacteristic
347C22C2×#1313314011yescharacteristic
348C22#1681062011yescharacteristic
349C22#1681063911yescharacteristic
350C22#1681064111yescharacteristic
351C4—22no—
352C4—22no—
353C4—21no—
354C4—22no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—41no—
359C4—4241no—
360C4—4241no—
361C4—2122no—
362C4—2122no—
363C2—42no—
364C2#2027305111yescharacteristic
365C2#2027314311yescharacteristic
366C2#2027315011yescharacteristic
367C2#2027327211yescharacteristic
368C2#2027328611yescharacteristic
369C2#2027334411yescharacteristic
370C2#2027337311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1797D53
hypocenterC53#7648
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272540C2semidirect
4. #20272537C2semidirect
208. C53#7648semidirect

Maximal subgroups

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

subgroupindex
C4×#131331962
#202747832
#202725402
#202725372
C53×#17972
#764853