Finite Group Databasebeta

← #27292284

Subgroups — #27292284

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133149C211yesmaximal, characteristic
2C2×#16810386C212yesmaximal
3C2×#16810386C211yesmaximal, characteristic
4#20272945C212yesmaximal
5C2×C53×#336C211yesmaximal, characteristic
6C22×C4⋊Dic53C2212yes—
7C22×C4⋊Dic53C2212yes—
8C2×#13133149—24no—
9C2×#13133149—22no—
10C2×#13133124—22no—
11C2×#13133124—22no—
12#16810386—24no—
13#16810386C414yescomplemented
14C23×C8×C53C2211yescharacteristic
15C23×C8×C53C2211yescharacteristic
16C22×C42×C53C2211yescharacteristic
17C53×#336—22no—
18C23×Dic53—22no—
19C23×Dic53—22no—
20C2×C4⋊Dic53—22no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—22no—
23C2×C4⋊Dic53—22no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53—22no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53C2×C414yes—
31C2×C4⋊Dic53D414yes—
32#13133149—28no—
33#13133124—44no—
34#13133124—44no—
35C23×C4×C53D412yes—
36C23×C4×C53C2311yescharacteristic
37C22×C8×C53—22no—
38C22×C8×C53—21no—
39C22×C8×C53—22no—
40C22×C8×C53—22no—
41C22×C8×C53—22no—
42C22×C8×C53—21no—
43C22×C8×C53C2×C412yes—
44C22×C8×C53D412yes—
45C22×C8×C53Q812yes—
46C22×C8×C53Q812yes—
47C2×C42×C53—21no—
48C2×C42×C53—21no—
49C2×C42×C53—22no—
50C2×C42×C53—22no—
51C24×C53C2×D411yescharacteristic
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53SD1614yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D412yes—
62C22×C4×C53C2×Q811yescharacteristic
63C22×C4×C53C2×Q811yescharacteristic
64C22×C4×C53#4112yes—
65C22×C4×C53#4112yes—
66C22×C4×C53#4112yes—
67C22×C4×C53#4111yescharacteristic
68C22×C4×C53#4111yescharacteristic
69C22×Dic53—24no—
70C22×Dic53—24no—
71C22×Dic53—42no—
72C22×Dic53—42no—
73C22×Dic53—42no—
74C22×Dic53—42no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C2×C8×C53—24no—
84C2×C8×C53—22no—
85C2×C8×C53—24no—
86C2×C8×C53—22no—
87C2×C8×C53—24no—
88C2×C8×C53—22no—
89C2×C8×C53—41no—
90C2×C8×C53—42no—
91C2×C8×C53—41no—
92C2×C8×C53C4⋊C414yes—
93C42×C53—41no—
94C42×C53—41no—
95C42×C53—42no—
96C4⋊Dic53—24no—
97C4⋊Dic53—24no—
98C4⋊Dic53—24no—
99C4⋊Dic53—24no—
100C4⋊Dic53—42no—
101C4⋊Dic53—42no—
102C4⋊Dic53—42no—
103C4⋊Dic53—42no—
104C23×C53C4×D411yescharacteristic
105C23×C53#12111yescharacteristic
106C23×C53#12112yes—
107C23×C53#12312yes—
108C23×C53C4⋊Q811yescharacteristic
109C23×C53C2×SD1612yes—
110C23×C53#13512yes—
111C23×C53AGL(1,ℤ/8)12yes—
112C23×C53#13712yes—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—24no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53C2×C4⋊C411yescharacteristic
121C2×C4×C53#11712yes—
122C2×C4×C53C4×D412yes—
123C2×C4×C53C4×Q811yescharacteristic
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53C22⋊Q812yes—
126C2×C4×C53C22⋊Q812yes—
127C2×C4×C53C22⋊Q811yescharacteristic
128C2×C4×C53#12311yescharacteristic
129C2×C4×C53#12312yes—
130C2×C4×C53#12412yes—
131C2×C4×C53#12411yescharacteristic
132C2×C4×C53#12411yescharacteristic
133C2×C4×C53#12512yes—
134C2×C4×C53#12512yes—
135C2×C4×C53#12511yescharacteristic
136C2×C4×C53#12612yes—
137C2×C4×C53#12612yes—
138C2×Dic53—44no—
139C2×Dic53—44no—
140C2×Dic53—44no—
141C2×Dic53—44no—
142C2×Dic53—44no—
143C2×Dic53—44no—
144C2×Dic53—82no—
145C2×Dic53—82no—
146C2×Dic53—82no—
147C2×Dic53—82no—
148C2×Dic53—82no—
149C2×Dic53—82no—
150C2×Dic53—82no—
151C2×Dic53—82no—
152C8×C53—24no—
153C8×C53—42no—
154#11257—531nomaximal, Sylow
155C22×C53#38812yes—
156C22×C53#38911yescharacteristic
157C22×C53#39411yescharacteristic
158C22×C53#39711yescharacteristic
159C22×C53#39812yes—
160C22×C53C4×SD1612yes—
161C22×C53#44111yescharacteristic
162C22×C53#44211yescharacteristic
163C22×C53#46012yes—
164C22×C53#46112yes—
165C22×C53#46312yes—
166C22×C53#46412yes—
167C22×C53#46512yes—
168C22×C53#46812yes—
169C22×C53#48112yes—
170C22×C53#48212yes—
171C22×C53#48312yes—
172C22×C53#48512yes—
173C22×C53#49811yescharacteristic
174C22×C53#49911yescharacteristic
175C22×C53#50112yes—
176C4×C53—44no—
177C4×C53#38211yescharacteristic
178C4×C53#38511yescharacteristic
179C4×C53#38712yes—
180C4×C53#39311yescharacteristic
181C4×C53#40011yescharacteristic
182C4×C53#40012yes—
183Dic53—84no—
184Dic53—84no—
185C22×#106—531no—
186C2×#340—531no—
187C2×#340—532no—
188C2×#336D5311yescharacteristic
189C4⋊#95—532no—
190C2×C53#144211yescharacteristic
191C2×C53#192612yes—
192C2×C53#192712yes—
193C2×C53#194111yescharacteristic
194C2×C53#194311yescharacteristic
195C2×C53#205512yes—
196C2×C53#207712yes—
197C2×C53#208412yes—
198C2×C53#208512yes—
199C23×C8C2×D5311yescharacteristic
200C23×C8C2×D5311yescharacteristic
201C22×C4⋊C4—532no—
202C22×C4⋊C4—532no—
203C22×C42C2×D5311yescharacteristic
204C2×#106—1064no—
205C2×#106—1062no—
206C2×#95—1062no—
207C2×#95—1062no—
208#340—534no—
209#340—1064no—
210#336—22no—
211C53#1125711yescharacteristic, complemented, Sylow
212C23×C4—1062no—
213C23×C4—1062no—
214C23×C4D21212yes—
215C23×C4C22×D5311yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—21no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8Dic10612yes—
223C22×C8Dic10612yes—
224C22×C8C4×D5312yes—
225C22×C8C22⋊D5312yes—
226C2×C4⋊C4—534no—
227C2×C4⋊C4—534no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—1062no—
231C2×C4⋊C4—1062no—
232C2×C4⋊C4—1062no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—1062no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—1062no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—21no—
242#106—1068no—
243#95—2124no—
244#95—2124no—
245C24C2×D21211yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—1064no—
253C22×C4—1064no—
254C22×C4—2122no—
255C22×C4—2122no—
256C22×C4—2122no—
257C22×C4—2122no—
258C22×C4—2122no—
259C22×C4—2122no—
260C22×C4—2122no—
261C22×C4—2122no—
262C22×C4—2122no—
263C22×C4—2122no—
264C22×C4—2122no—
265C22×C4—2122no—
266C22×C4#1171991314yes—
267C22×C4C2×Dic10611yescharacteristic
268C22×C4C2×Dic10611yescharacteristic
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4#1171994511yescharacteristic
271C22×C4#1171994511yescharacteristic
272C22×C4D4×D5312yes—
273C22×C4#1171994712yes—
274C22×C4#1171994712yes—
275C22×C4#1171994912yes—
276C22×C4C2×C22⋊D5311yescharacteristic
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8#1171991914yes—
287C4⋊C4—1064no—
288C4⋊C4—1064no—
289C4⋊C4—1064no—
290C4⋊C4—1064no—
291C4⋊C4—2122no—
292C4⋊C4—2122no—
293C4⋊C4—2122no—
294C4⋊C4—2122no—
295C42—41no—
296C42—41no—
297C42—42no—
298C23C4×D21211yescharacteristic
299C23#1313319111yescharacteristic
300C23#1313319212yes—
301C23#1313320112yes—
302C23#1313321711yescharacteristic
303C23C2×#1171991312yes—
304C23#1313323912yes—
305C23#1313324512yes—
306C23#1313325012yes—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—4242no—
315C2×C4—4242no—
316C2×C4—4242no—
317C2×C4—4242no—
318C2×C4—4242no—
319C2×C4—4242no—
320C2×C4—4242no—
321C2×C4—4242no—
322C2×C4—2124no—
323C2×C4—2124no—
324C2×C4—2124no—
325C2×C4—2124no—
326C2×C4—2124no—
327C2×C4—2124no—
328C2×C4C2×#1171991911yescharacteristic
329C2×C4#1313317512yes—
330C2×C4#1313317712yes—
331C2×C4C4×Dic10611yescharacteristic
332C2×C4#1313318511yescharacteristic
333C2×C4#1313319412yes—
334C2×C4#1313319412yes—
335C2×C4#1313319611yescharacteristic
336C2×C4#1313319811yescharacteristic
337C2×C4#1313320212yes—
338C2×C4#1313320312yes—
339C2×C4#1313320411yescharacteristic
340C2×C4#1313320611yescharacteristic
341C2×C4#1313320812yes—
342C2×C4#1313320812yes—
343C2×C4#1313320911yescharacteristic
344C2×C4#1313321012yes—
345C2×C4#1313321212yes—
346C8—24no—
347C8—42no—
348C22#1681051512yes—
349C22#1681052111yescharacteristic
350C22#1681054011yescharacteristic
351C22#1681054811yescharacteristic
352C22#1681055512yes—
353C22C4×#1171991312yes—
354C22#1681073111yescharacteristic
355C22#1681073611yescharacteristic
356C22#1681082412yes—
357C22#1681082912yes—
358C22#1681083912yes—
359C22#1681084612yes—
360C22#1681085212yes—
361C22#1681086912yes—
362C22#1681094212yes—
363C22#1681094812yes—
364C22#1681095412yes—
365C22#1681096512yes—
366C22#1681102111yescharacteristic
367C22#1681102511yescharacteristic
368C22#1681103212yes—
369C4—44no—
370C4—4244no—
371C4—4244no—
372C4#1681048811yescharacteristic
373C4#1681049511yescharacteristic
374C4#1681051012yes—
375C4#1681053511yescharacteristic
376C4#1681056211yescharacteristic
377C4#1681056512yes—
378C2#2027297111yescharacteristic
379C2#2027534312yes—
380C2#2027535012yes—
381C2#2027544311yescharacteristic
382C2#2027545311yescharacteristic
383C2#2027603312yes—
384C2#2027614812yes—
385C2#2027619312yes—
386C2#2027619812yes—
subgroupG/H
centerC24C2×D212
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#336C2
socle, prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#336D53
hypocenterC53#11257
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
13. #16810386C4semidirect
211. C53#11257semidirect

Maximal subgroups

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

subgroupindex
C22×#131331492
C2×#168103862
#202729452
C2×C53×#3362
#1125753