Finite Group Databasebeta

← #27277423

Subgroups — #27277423

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810474C211yesmaximal, characteristic
2#20274781C212yesmaximal
3#20272558C211yesmaximal, characteristic
4#20272557C212yesmaximal
5C53×#1280C211yesmaximal, characteristic
6C22×#11719919C411yescharacteristic
7C22×#11719919C411yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9C22×C4×Dic53C411yescharacteristic
10C22×C4×Dic53C2211yescharacteristic
11C2×C22⋊#88322C2212yes—
12#16810474—22no—
13#16810474—22no—
14C2×#13133124C2211yescharacteristic
15#16810377—22no—
16#16810377—22no—
17C2×C22⋊C8×C53C2212yes—
18C2×#95×C53C2211yescharacteristic
19C23×Dic53C2×C411yescharacteristic
20C23×Dic53C2×C411yescharacteristic
21C23×Dic53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C22×#88322—22no—
24C2×#11719919—22no—
25C2×#11719919—22no—
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×#11719919C814yescomplemented
30C2×C4×Dic53—22no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53D412yes—
40C2×C4×Dic53D412yes—
41C22⋊#88322—22no—
42C22⋊#88322—22no—
43#13133124—22no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47C22×C8×C53—22no—
48C22⋊C8×C53—22no—
49C22⋊C8×C53—22no—
50#95×C53—22no—
51C24×C53C22×C411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—41no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53C2×C812yes—
63C22×C4×C53M1612yes—
64C22×C4×C53C2×D412yes—
65C22×C4×C53#4112yes—
66C22×C4×C53#4111yescharacteristic
67C22×C4×C53#4111yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53C22⋊C412yes—
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C2×C812yes—
88C22×Dic53C2×C812yes—
89C22×Dic53M1612yes—
90C22×Dic53M1612yes—
91C2×C8×C53—42no—
92C2×C8×C53—42no—
93C2×C8×C53—42no—
94#11719919—28no—
95#11719919—28no—
96C4×Dic53—22no—
97C4×Dic53—22no—
98C4×Dic53—22no—
99C4×Dic53—22no—
100C4×Dic53—42no—
101C4×Dic53—42no—
102C4×Dic53—42no—
103C4×Dic53—42no—
104C2×#88322—42no—
105C2×#88322—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53#11711yescharacteristic
113C23×C53#11711yescharacteristic
114C23×C53C22×C811yescharacteristic
115C23×C53C2×M1611yescharacteristic
116C23×C53C2×M1611yescharacteristic
117C23×C53C2×M1611yescharacteristic
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53#12312yes—
134C2×C4×C53#12312yes—
135C2×Dic53—24no—
136C2×Dic53—24no—
137C2×Dic53—22no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—22no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53C22⋊C814yes—
158C2×Dic53#10012yes—
159C2×Dic53#10112yes—
160C8×C53—82no—
161#88322—82no—
162#8701—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53#38111yescharacteristic
176C22×C53C2×C22⋊C811yescharacteristic
177C22×C53#40711yescharacteristic
178C22×C53#41011yescharacteristic
179C22×C53#41011yescharacteristic
180C22×C53C2×#10011yescharacteristic
181C22×C53C2×#10111yescharacteristic
182C22×C53#43111yescharacteristic
183C22×C53#43111yescharacteristic
184C22×C53#43211yescharacteristic
185C22×C53#43211yescharacteristic
186C4×C53—22no—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—42no—
190Dic53—24no—
191Dic53—22no—
192Dic53—22no—
193Dic53—42no—
194Dic53—42no—
195C2×C4×C4⋊C4—531no—
196#1817—532no—
197#1280—532no—
198#1280—531no—
199#1280D5311yescharacteristic
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53#146011yescharacteristic
205C2×C53#146111yescharacteristic
206C2×C53#146611yescharacteristic
207C2×C53#146711yescharacteristic
208C2×C53#181711yescharacteristic
209C2×C53#182011yescharacteristic
210C2×C53#182111yescharacteristic
211C22×C4⋊C4—531no—
212C22×C4⋊C4—531no—
213C22×C42—531no—
214C22×C42—531no—
215C22×C42—531no—
216C2×C22⋊C8—532no—
217C2×C22⋊C8C2×D5312yes—
218C4×C4⋊C4—1062no—
219C4×C4⋊C4—1062no—
220C2×#95—531no—
221C2×#95C2×D5311yescharacteristic
222#336—1062no—
223#336—1062no—
224C53#870111yescharacteristic, complemented, Sylow
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—531no—
228C23×C4—531no—
229C23×C4C4×D5311yescharacteristic
230C23×C4C4×D5311yescharacteristic
231C23×C4C22×D5311yescharacteristic
232C22×C8—22no—
233C22×C8—1062no—
234C2×C4⋊C4—534no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—1062no—
238C2×C4⋊C4—1062no—
239C2×C4⋊C4—1062no—
240C2×C42—532no—
241C2×C42—532no—
242C2×C42—1062no—
243C2×C42—1061no—
244C2×C42—1061no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42—1061no—
248C2×C42—1061no—
249C2×C42—2122no—
250C2×C42—2122no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—1062no—
254C22⋊C8—1062no—
255#95—22no—
256#95—1062no—
257C24C2×C4×D5311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—532no—
271C22×C4—532no—
272C22×C4—532no—
273C22×C4—532no—
274C22×C4—532no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4C8×D5312yes—
292C22×C4#1171991212yes—
293C22×C4#1171994512yes—
294C22×C4D4×D5312yes—
295C22×C4#1171994711yescharacteristic
296C22×C4#1171994911yescharacteristic
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—2122no—
301C2×C8—2122no—
302C2×C8—2122no—
303C4⋊C4—1068no—
304C4⋊C4—1068no—
305C42—1062no—
306C42—1062no—
307C42—1062no—
308C42—1062no—
309C42—2122no—
310C42—2122no—
311C42—2122no—
312C42—2122no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23C22⋊C4×D5311yescharacteristic
318C23#1313317411yescharacteristic
319C23#1313317511yescharacteristic
320C23C2×C8×D5311yescharacteristic
321C23M16×D5311yescharacteristic
322C23M16×D5311yescharacteristic
323C23C2×#1171991211yescharacteristic
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—534no—
340C2×C4—532no—
341C2×C4—532no—
342C2×C4—1064no—
343C2×C4—1064no—
344C2×C4—1062no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1062no—
355C2×C4—1062no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4#1313319912yes—
365C2×C4#1313320212yes—
366C8—82no—
367C8—4242no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22#1681048511yescharacteristic
381C22C22⋊C8×D5311yescharacteristic
382C22#1681058511yescharacteristic
383C22#1681059511yescharacteristic
384C22#1681059511yescharacteristic
385C22#100×D5311yescharacteristic
386C22#101×D5311yescharacteristic
387C22#1681067411yescharacteristic
388C22#1681067511yescharacteristic
389C22#1681067811yescharacteristic
390C22#1681067911yescharacteristic
391C4—22no—
392C4—22no—
393C4—42no—
394C4—42no—
395C4—1064no—
396C4—1062no—
397C4—1062no—
398C4—2122no—
399C4—2122no—
400C2—21no—
401C2—21no—
402C2—21no—
403C2—21no—
404C2#2027303411yescharacteristic
405C2#2027303711yescharacteristic
406C2#2027305411yescharacteristic
407C2#2027305711yescharacteristic
408C2#2027478211yescharacteristic
409C2#2027479111yescharacteristic
410C2#2027479411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×C4×C53C23
hypercenter#1280D53
hypocenterC53#8701
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
29. C2×#11719919C8semidirect
224. C53#8701semidirect

Maximal subgroups

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

subgroupindex
C2×#168104742
#202747812
#202725582
#202725572
C53×#12802
#870153