Finite Group Databasebeta

← #324×Dic53 #27274600

Subgroups — #324×Dic53 #27274600

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×Dic53C211yesmaximal, characteristic
2C22⋊C8×Dic53C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272557C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C22×C4⋊Dic53C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10C2×C22⋊#88322C2212yes—
11C2×C8×Dic53—22no—
12C4⋊C4×Dic53—22no—
13C4⋊C4×Dic53—22no—
14C2×#13133124C411yescharacteristic
15C2×#13133124C2211yescharacteristic
16#16810377—22no—
17#16810349C414yescomplemented
18C22×C4⋊C4×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#324C414yescomplemented
21C23×Dic53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C22×#88322—22no—
26C2×C4⋊Dic53—22no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C814yescomplemented
29C2×C4×Dic53—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—22no—
34C2×C4×Dic53—42no—
35C2×C4×Dic53—42no—
36C2×C4×Dic53C814yescomplemented
37C2×C4×Dic53D412yes—
38C2×C4×Dic53D412yes—
39C22⋊#88322—22no—
40C22⋊#88322C2×C414yes—
41C8×Dic53—44no—
42#13133124—22no—
43#13133124—22no—
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47C22×C8×C53—22no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53C2×C412yes—
51C2×C4⋊C4×C53C2×C412yes—
52C22⋊C8×C53—22no—
53C22⋊C8×C53C2×C414yes—
54C24×C53C22×C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—41no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53C4212yes—
66C22×C4×C53C4212yes—
67C22×C4×C53C2×C812yes—
68C22×C4×C53C2×C812yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53#4112yes—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—41no—
80C22×Dic53—41no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53C22⋊C412yes—
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C2×C812yes—
89C22×Dic53C2×C812yes—
90C22×Dic53C2×C812yes—
91C22×Dic53C2×C812yes—
92C22×Dic53M1612yes—
93C22×Dic53M1612yes—
94C2×C8×C53—24no—
95C2×C8×C53—42no—
96C2×C8×C53—42no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—24no—
99C4⋊Dic53—28no—
100C4×Dic53—24no—
101C4×Dic53—24no—
102C4×Dic53—24no—
103C4×Dic53—42no—
104C4×Dic53—42no—
105C4×Dic53—42no—
106C2×#88322—24no—
107C2×#88322—42no—
108C2×#88322—42no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53C2×C4211yescharacteristic
114C23×C53C2×C22⋊C411yescharacteristic
115C23×C53#11711yescharacteristic
116C23×C53C22×C811yescharacteristic
117C23×C53C22×C811yescharacteristic
118C23×C53C2×M1611yescharacteristic
119C23×C53C2×M1611yescharacteristic
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53C4×C814yescomplemented
137C2×C4×C53C4×D412yes—
138C2×C4×C53C4×D412yes—
139C2×Dic53—21no—
140C2×Dic53—22no—
141C2×Dic53—21no—
142C2×Dic53—22no—
143C2×Dic53—24no—
144C2×Dic53—24no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—42no—
148C2×Dic53—42no—
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⋊C812yes—
158C2×Dic53C22⋊C814yes—
159C2×Dic53C22⋊C812yes—
160C2×Dic53#9912yes—
161C2×Dic53#10112yes—
162C8×C53—44no—
163#88322—44no—
164C4×#324—531nomaximal, Sylow
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—21no—
176C22×C53—21no—
177C22×C53C4×C22⋊C411yescharacteristic
178C22×C53C2×C4×C811yescharacteristic
179C22×C53C4×M1611yescharacteristic
180C22×C53C2×C22⋊C811yescharacteristic
181C22×C53C2×C22⋊C811yescharacteristic
182C22×C53C2×#9911yescharacteristic
183C22×C53#41011yescharacteristic
184C22×C53C2×#10111yescharacteristic
185C22×C53#41311yescharacteristic
186C22×C53#43111yescharacteristic
187C22×C53#43111yescharacteristic
188C4×C53—24no—
189C4×C53—22no—
190C4×C53—22no—
191C4×C53—42no—
192Dic53—22no—
193Dic53—24no—
194Dic53—42no—
195Dic53—42no—
196Dic53#32414yesdirect product factor
197C2×C4×C4⋊C4—531no—
198C4×C22⋊C8—532no—
199C2×#324—531no—
200C2×#324D5311yescharacteristic
201#1280—532no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53C2×#32411yescharacteristic
207C2×C53#145811yescharacteristic
208C2×C53#145911yescharacteristic
209C2×C53#146111yescharacteristic
210C2×C53C4×C22⋊C811yescharacteristic
211C2×C53C4×#9911yescharacteristic
212C2×C53C4×#10111yescharacteristic
213C22×C4⋊C4—531no—
214C22×C4⋊C4C2×D5311yescharacteristic
215C22×C42—531no—
216C22×C42—531no—
217C22×C42—531no—
218C2×C22⋊C8—532no—
219C2×C22⋊C8C2×D5312yes—
220C2×C4×C8—1062no—
221C4×C4⋊C4—1062no—
222C4×C4⋊C4—1062no—
223C2×#95—531no—
224C2×#95—531no—
225#336—1062no—
226#324—534no—
227#324Dic5314yesdirect product factor
228C53C4×#32411yescharacteristic, complemented, Sylow
229C23×C4—531no—
230C23×C4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C23×C4C4×D5311yescharacteristic
234C23×C4C4×D5311yescharacteristic
235C23×C4C22×D5311yescharacteristic
236C22×C8—22no—
237C22×C8—1062no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—534no—
241C2×C4⋊C4—1062no—
242C2×C4⋊C4—1062no—
243C2×C4⋊C4C4×D5312yes—
244C2×C4⋊C4C2×Dic5312yes—
245C2×C42—532no—
246C2×C42—534no—
247C2×C42—532no—
248C2×C42—1061no—
249C2×C42—1061no—
250C2×C42—1062no—
251C2×C42—1061no—
252C2×C42—1061no—
253C2×C42—2122no—
254C2×C42—2122no—
255C22⋊C8—22no—
256C22⋊C8—534no—
257C22⋊C8—1062no—
258C22⋊C8C2×Dic5314yes—
259C4×C8—2124no—
260#95—1062no—
261#95—1062no—
262C24C2×C4×D5311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—531no—
274C22×C4—532no—
275C22×C4—531no—
276C22×C4—532no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4C8×D5312yes—
295C22×C4C8×D5312yes—
296C22×C4C4×Dic5312yes—
297C22×C4C4×Dic5312yes—
298C22×C4C2×C4×D5311yescharacteristic
299C22×C4D4×D5312yes—
300C22×C4#1171994712yes—
301C22×C4C22×Dic5311yescharacteristic
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—1064no—
306C2×C8—2122no—
307C2×C8—2122no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—1068no—
311C42—1064no—
312C42—1064no—
313C42—1064no—
314C42—2122no—
315C42—2122no—
316C42—2122no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C4×Dic5311yescharacteristic
322C23C22⋊C4×D5311yescharacteristic
323C23#1313317311yescharacteristic
324C23C2×C8×D5311yescharacteristic
325C23C2×C8×D5311yescharacteristic
326C23M16×D5311yescharacteristic
327C23M16×D5311yescharacteristic
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—532no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—532no—
348C2×C4—534no—
349C2×C4—1064no—
350C2×C4—1064no—
351C2×C4—1062no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1062no—
355C2×C4—1062no—
356C2×C4—1062no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4C8×Dic5314yescomplemented
368C2×C4#1313317712yes—
369C2×C4D4×Dic5312yes—
370C8—44no—
371C8—2124no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22C22⋊C4×Dic5311yescharacteristic
385C22C2×C8×Dic5311yescharacteristic
386C22M16×Dic5311yescharacteristic
387C22C22⋊C8×D5311yescharacteristic
388C22C22⋊C8×D5311yescharacteristic
389C22#99×D5311yescharacteristic
390C22#1681059511yescharacteristic
391C22#101×D5311yescharacteristic
392C22#1681060611yescharacteristic
393C22#1681067311yescharacteristic
394C22#1681067311yescharacteristic
395C4—24no—
396C4—22no—
397C4—22no—
398C4—42no—
399C4—534no—
400C4—1062no—
401C4—1064no—
402C4—2122no—
403C4—2122no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#324×D5311yescharacteristic
409C2#2027302811yescharacteristic
410C2#2027303211yescharacteristic
411C2#2027303711yescharacteristic
412C2C22⋊C8×Dic5311yescharacteristic
413C2#99×Dic5311yescharacteristic
414C2#101×Dic5311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C53C2×C4×C8
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53C4×#324
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
17. #16810349C4semidirect
20. C53×#324C4semidirect
28. C2×C4⋊Dic53C8semidirect
36. C2×C4×Dic53C8semidirect
136. C2×C4×C53C4×C8semidirect
196. Dic53#324direct (N × H)
227. #324Dic53direct (N × H)
228. C53C4×#324semidirect
367. C2×C4C8×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic532
C22⋊C8×Dic532
C2×#168103492
#202725572
C2×C53×#3242
C4×#32453