Finite Group Databasebeta

← #28306454

Subgroups — #28306454

Order 13568.

1646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811565C212yesmaximal
2C2×#16811497C214yesmaximal, complemented
3C23×C8×D53C211yesmaximal, characteristic
4C22×C4⋊#6157C2214yes—
5C22×C22⋊#6157—22no—
6#16811565—24no—
7#16811565C418yescomplemented
8C2×(C2×#6157)⋊C4—22no—
9#16811497—28no—
10C23×C4×D53C2211yescharacteristic
11C23×#88322C411yescharacteristic, complemented
12C22×C8×D53—24no—
13C22×C8×D53—28no—
14C22×C8×D53C412yescomplemented
15C22×C8×D53C2212yes—
16C23×C8×C53C411yescharacteristic, complemented
17C23×#6157—24no—
18C2×C4⋊#6157—28no—
19C2×C4⋊#6157—28no—
20C2×C4⋊#6157—24no—
21C2×C4⋊#6157C2×C418yes—
22C2×C22⋊#6157—28no—
23C2×C22⋊#6157—44no—
24C2×C22⋊#6157—44no—
25(C2×#6157)⋊C4—28no—
26C24×D53D411yescharacteristic
27C23×Dic53Q811yescharacteristic
28C22×C4×D53—24no—
29C22×C4×D53—24no—
30C22×C4×D53C2×C411yescharacteristic
31C22×C4×D53C2×C414yes—
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D414yes—
34C22×C4×D53Q811yescharacteristic
35C22×C4×D53C2311yescharacteristic
36C22×#88322—24no—
37C22×#88322—22no—
38C22×#88322C2×C412yes—
39C2×C8×D53—232no—
40C2×C8×D53—24no—
41C2×C8×D53—24no—
42C2×C8×D53—24no—
43C2×C8×D53—24no—
44C2×C8×D53—24no—
45C2×C8×D53D418yes—
46C23×C4×C53C2×C411yescharacteristic
47C22×C8×C53—24no—
48C22×C8×C53—22no—
49C22×C8×C53C2×C412yes—
50C24×C53C4⋊C411yescharacteristic
51C23×D53—24no—
52C23×D53—24no—
53C23×D53C22⋊C414yes—
54C23×D53C4⋊C411yescharacteristic
55C23×D53D812yes—
56C23×D53SD1614yes—
57C23×D53Q1612yes—
58C23×D53C2×D411yescharacteristic
59C22×#6157—28no—
60C22×#6157—48no—
61C22×#6157—44no—
62C22×#6157—44no—
63C22×#6157—48no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53C4212yes—
67C22×C4×C53C22⋊C412yes—
68C22×C4×C53C4⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×Dic53—24no—
71C22×Dic53C4⋊C411yescharacteristic
72C22×Dic53C4⋊C414yes—
73C22×Dic53C2×Q811yescharacteristic
74C2×C4×D53—24no—
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—24no—
78C2×C4×D53—216no—
79C2×C4×D53—28no—
80C2×C4×D53—28no—
81C2×C4×D53C4212yes—
82C2×C4×D53C22⋊C414yes—
83C2×C4×D53C22⋊C412yes—
84C2×C4×D53C4⋊C414yes—
85C2×C4×D53C22×C412yes—
86C2×C4×D53C2×D414yes—
87C2×C4×D53C2×D412yes—
88C2×C4×D53#4114yes—
89C22⋊#6157—416no—
90C4⋊#6157—216no—
91C4⋊#6157—44no—
92C4⋊#6157—44no—
93C2×C8×C53—28no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53C22⋊C414yescomplemented
97C2×#88322—28no—
98C2×#88322—22no—
99C2×#88322—22no—
100C2×#88322C22⋊C414yescomplemented
101C8×D53—216no—
102C8×D53—216no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53#9512yes—
106C23×C53#10612yes—
107C23×C53#10712yes—
108C23×C53C2×C4⋊C411yescharacteristic
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—28no—
112C22×D53—28no—
113C22×D53—216no—
114C22×D53—28no—
115C22×D53—28no—
116C22×D53#9512yes—
117C22×D53#10218yes—
118C22×D53#10318yes—
119C22×D53#10612yes—
120C22×D53#10712yes—
121C22×D53C2×C22⋊C412yes—
122C22×D53#12114yes—
123C22×D53C2×D811yescharacteristic
124C22×D53C2×SD1612yes—
125C22×D53C2×Q1611yescharacteristic
126C22×D53#13514yes—
127C2×#6157—48no—
128C2×#6157—48no—
129C2×#6157—48no—
130C2×#6157—88no—
131C2×#6157—88no—
132C2×C4×C53—22no—
133C2×C4×C53—24no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53#9512yes—
137C2×C4×C53C2×C4211yescharacteristic
138C2×C4×C53C2×C22⋊C412yes—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53#11712yes—
141C2×Dic53—22no—
142C2×Dic53—28no—
143C2×Dic53#9512yes—
144C2×Dic53C2×C4⋊C412yes—
145C2×Dic53C22⋊Q814yes—
146C4×D53—28no—
147C4×D53—24no—
148C4×D53—22no—
149C4×D53—22no—
150C4×D53—28no—
151C4×D53C4×D418yes—
152C4×D53#12014yes—
153C4×D53#12314yes—
154C8×C53—24no—
155#88322—24no—
156#10721—531nomaximal, Sylow
157C22×C53—24no—
158C22×C53—24no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53#34014yescomplemented
162C22×C53C2×#9511yescharacteristic
163C22×C53#38012yes—
164C22×C53C2×#10611yescharacteristic
165C22×C53C2×#10711yescharacteristic
166C22×C53#42712yes—
167C2×D53—24no—
168C2×D53—24no—
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—216no—
172C2×D53—216no—
173C2×D53#34014yescomplemented
174C2×D53#38614yes—
175C2×D53C2×#10212yes—
176C2×D53C2×#10312yes—
177C2×D53#41614yes—
178C2×D53#46518yes—
179C2×D53#46614yes—
180C2×D53#46714yes—
181#6157—88no—
182C4×C53—22no—
183C4×C53C4×C22⋊C412yes—
184C4×C53#37911yescharacteristic
185C4×C53#38111yescharacteristic
186Dic53—22no—
187Dic53#38514yes—
188C24×C8—531no—
189C2×#380—532no—
190C2×#340—534no—
191C2×C53—24no—
192C2×C53C22⋊#9511yescharacteristic
193C2×C53C2×#34011yescharacteristic
194C2×C53#173511yescharacteristic
195C2×C53#180712yes—
196C2×C53#180812yes—
197D53—24no—
198D53#189214yes—
199D53#189314yes—
200C24×C4—531no—
201C23×C8—532no—
202C23×C8—531no—
203C23×C8—532no—
204C23×C8—1068no—
205C23×C8—1064no—
206C23×C8#615711yescharacteristic, complemented
207C22×C4⋊C4—534no—
208C22×C22⋊C4—1062no—
209#380—538no—
210#380—1064no—
211C2×#95—1062no—
212#340—1068no—
213C53#1072111yescharacteristic, complemented, Sylow
214C25—531no—
215C23×C4—531no—
216C23×C4—531no—
217C23×C4—534no—
218C23×C4—531no—
219C23×C4—531no—
220C23×C4—534no—
221C23×C4—531no—
222C23×C4—1064no—
223C23×C4—1064no—
224C23×C4—1064no—
225C23×C4C2×#615711yescharacteristic
226C22×C8—22no—
227C22×C8—24no—
228C22×C8—532no—
229C22×C8—538no—
230C22×C8—1064no—
231C22×C8—1064no—
232C22×C8—1064no—
233C22×C8—1064no—
234C22×C8—10632no—
235C22×C8—1064no—
236C22×C8—1062no—
237C22×C8—1064no—
238C22×C8C2×#615712yes—
239C2×C4⋊C4—538no—
240C2×C4⋊C4—1068no—
241C2×C4⋊C4—1068no—
242C2×C4⋊C4—1064no—
243C2×C22⋊C4—1068no—
244C2×C22⋊C4—2124no—
245C2×C22⋊C4—2124no—
246#95—1068no—
247C24—531no—
248C24—534no—
249C24—534no—
250C24—532no—
251C24—532no—
252C24—531no—
253C24—1064no—
254C24—1064no—
255C24C4⋊#615711yescharacteristic
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—531no—
259C22×C4—532no—
260C22×C4—534no—
261C22×C4—532no—
262C22×C4—534no—
263C22×C4—534no—
264C22×C4—531no—
265C22×C4—534no—
266C22×C4—534no—
267C22×C4—532no—
268C22×C4—532no—
269C22×C4—1068no—
270C22×C4—1064no—
271C22×C4—10616no—
272C22×C4—1068no—
273C22×C4—1064no—
274C22×C4—1064no—
275C22×C4—1062no—
276C22×C4—1062no—
277C22×C4—1068no—
278C22×C4—2124no—
279C22×C4—2124no—
280C22×C4—2128no—
281C22×C4—2128no—
282C22×C4C4×#615712yes—
283C22×C4C4⋊#615711yescharacteristic
284C22×C4C22⋊#615712yes—
285C22×C4C22×#615711yescharacteristic
286C2×C8—28no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—534no—
290C2×C8—10616no—
291C2×C8—10616no—
292C2×C8—1068no—
293C2×C8—1062no—
294C2×C8—1062no—
295C2×C8C22⋊#615714yescomplemented
296C4⋊C4—10616no—
297C4⋊C4—2124no—
298C4⋊C4—2124no—
299C22⋊C4—21216no—
300C23—22no—
301C23—22no—
302C23—532no—
303C23—534no—
304C23—532no—
305C23—531no—
306C23—531no—
307C23—534no—
308C23—532no—
309C23—532no—
310C23—532no—
311C23—538no—
312C23—538no—
313C23—1068no—
314C23—10616no—
315C23—1068no—
316C23—1068no—
317C23—1068no—
318C23—1062no—
319C23—1062no—
320C23(C2×#6157)⋊C412yes—
321C23#1313328512yes—
322C23#1313328612yes—
323C23C2×C4⋊#615711yescharacteristic
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—532no—
329C2×C4—534no—
330C2×C4—538no—
331C2×C4—534no—
332C2×C4—534no—
333C2×C4—532no—
334C2×C4—1068no—
335C2×C4—1068no—
336C2×C4—1068no—
337C2×C4—1062no—
338C2×C4—1064no—
339C2×C4—1062no—
340C2×C4—1062no—
341C2×C4—4248no—
342C2×C4—4248no—
343C2×C4—2128no—
344C2×C4—2128no—
345C2×C4—2128no—
346C2×C4(C2×#6157)⋊C412yes—
347C2×C4C2×C4×#615711yescharacteristic
348C2×C4C2×C22⋊#615712yes—
349C2×C4C2×C22⋊#615711yescharacteristic
350C2×C4#1313329612yes—
351C8—24no—
352C8—1064no—
353C22—24no—
354C22—24no—
355C22—22no—
356C22—22no—
357C22—538no—
358C22—534no—
359C22—534no—
360C22—532no—
361C22—532no—
362C22—534no—
363C22—534no—
364C22—534no—
365C22—10616no—
366C22—10616no—
367C22—1064no—
368C22—1064no—
369C22—1062no—
370C22—1062no—
371C22#1681149714yescomplemented
372C22C2×(C2×#6157)⋊C411yescharacteristic
373C22#1681156512yes—
374C22C2×#1313328511yescharacteristic
375C22C2×#1313328611yescharacteristic
376C22#1681163312yes—
377C4—22no—
378C4—534no—
379C4—1062no—
380C4—4248no—
381C4C4×C22⋊#615712yes—
382C4#1681156411yescharacteristic
383C4#1681156611yescharacteristic
384C2—24no—
385C2—534no—
386C2—534no—
387C2—1064no—
388C2#2028988911yescharacteristic
389C2C2×#1681149711yescharacteristic
390C2#2029040111yescharacteristic
391C2#2029059112yes—
392C2#2029059212yes—
subgroupG/H
centerC23C2×C4⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC23×C8×C53C4
socleC23×C53C2×C4⋊C4
prime radicalC24×D53D4
cosocleC22×C4×D53C23
hypercenterC23×C8#6157
hypocenterC53#10721
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8

Complements

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

NH ≅ G/Ntype
2. C2×#16811497C2semidirect
7. #16811565C4semidirect
11. C23×#88322C4semidirect
14. C22×C8×D53C4semidirect
16. C23×C8×C53C4semidirect
96. C2×C8×C53C22⋊C4semidirect
100. C2×#88322C22⋊C4semidirect
161. C22×C53#340semidirect
173. C2×D53#340semidirect
206. C23×C8#6157semidirect
213. C53#10721semidirect
295. C2×C8C22⋊#6157semidirect
371. C22#16811497semidirect

Maximal subgroups

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

subgroupindex
C2×#168115652
C2×#168114972
C23×C8×D532
#1072153