Finite Group Databasebeta

← #27386195

Subgroups — #27386195

Order 13568.

906 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810649C211yesmaximal, characteristic
2C2×#16810578C211yesmaximal, characteristic
3#20277122C211yesmaximal, characteristic
4#20275280C214yesmaximal, complemented
5#20275273C214yesmaximal, complemented
6#20274411C211yesmaximal, characteristic
7#20274411C211yesmaximal, characteristic
8#20274409C211yesmaximal, characteristic
9C2×C53×#336C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810649—24no—
15C2×#13133146C2212yes—
16C2×#13133146C2212yes—
17#16810578—24no—
18C2×#13133127C2212yes—
19C2×C8×Dic53C2212yes—
20#16810493C411yescharacteristic
21#16810493C411yescharacteristic
22#16810489C411yescharacteristic
23#16810489C411yescharacteristic
24#16810483C414yes—
25C4×C4⋊Dic53C2214yes—
26#16810378C2214yes—
27#16810377C2214yes—
28#16810377C2214yes—
29C23×C8×C53C2211yescharacteristic
30C23×C8×C53C2211yescharacteristic
31C22×C42×C53C2211yescharacteristic
32C53×#336C2212yes—
33C53×#336C2212yes—
34C2×#11719917—22no—
35C2×#11719917—22no—
36C2×#11719917—22no—
37C2×#11719917—22no—
38C2×#11719917—22no—
39C2×#11719917Q814yes—
40C22×#88322C2312yes—
41C22×#88322C2312yes—
42#13133176—22no—
43#13133176—22no—
44#13133176—22no—
45#13133176—22no—
46C2×C4⋊Dic53C2×C411yescharacteristic
47C2×C4⋊Dic53C2×C411yescharacteristic
48C2×C4⋊Dic53C2×C411yescharacteristic
49C2×C4⋊Dic53C2×C411yescharacteristic
50C2×C4⋊Dic53C2×C414yes—
51C2×C4⋊Dic53C2×C414yes—
52C2×C22⋊Dic53C2×C412yes—
53C2×C22⋊Dic53C2×C412yes—
54C2×C4×Dic53C2312yes—
55C2×C4×Dic53C2312yes—
56#13133146—24no—
57#13133146—24no—
58#13133127—24no—
59C8×Dic53—24no—
60#13133124C2×C414yes—
61#13133124C2×C414yes—
62C23×C4×C53C2×C412yes—
63C23×C4×C53C2311yescharacteristic
64C22×C8×C53—22no—
65C22×C8×C53—22no—
66C22×C8×C53—22no—
67C22×C8×C53—22no—
68C22×C8×C53—22no—
69C22×C8×C53D414yes—
70C22×C8×C53C2312yes—
71C22×C8×C53C2311yescharacteristic
72C22×C8×C53C2311yescharacteristic
73C2×C42×C53—22no—
74C2×C42×C53—22no—
75C2×C42×C53C2×C411yescharacteristic
76C2×C42×C53C2×C411yescharacteristic
77C2×C42×C53C2312yes—
78C24×C53C22×C411yescharacteristic
79C22×C4×C53—24no—
80C22×C4×C53—24no—
81C22×C4×C53—42no—
82C22×C4×C53C22×C412yes—
83C22×C4×C53C22×C412yes—
84C22×C4×C53C22×C411yescharacteristic
85C22×C4×C53C2×D412yes—
86C22×C4×C53C2×Q812yes—
87C22×C4×C53#4112yes—
88C22×C4×C53#4112yes—
89C22×C4×C53#4112yes—
90C22×C4×C53#4112yes—
91C22×C4×C53C2411yescharacteristic
92C22×Dic53C22×C412yes—
93C22×Dic53C22×C412yes—
94C22×Dic53C22×C412yes—
95C22×Dic53C22×C412yes—
96C2×C8×C53—28no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—44no—
105C2×C8×C53C2×D412yes—
106C2×C8×C53C2×D412yes—
107C42×C53—22no—
108C42×C53—22no—
109C42×C53—22no—
110C42×C53—22no—
111C22⋊Dic53—24no—
112C22⋊Dic53—24no—
113C4⋊Dic53—22no—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117C4⋊Dic53—28no—
118C4⋊Dic53—44no—
119C4×Dic53—22no—
120C4×Dic53—22no—
121C4×Dic53—22no—
122C4×Dic53—22no—
123#11719917—28no—
124#11719917—44no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322C2×Q814yes—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53#11712yes—
134C23×C53C4×D412yes—
135C23×C53C4×Q812yes—
136C23×C53C23×C411yescharacteristic
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—48no—
143C2×C4×C53#11712yes—
144C2×C4×C53C4×D412yes—
145C2×C4×C53C4×Q812yes—
146C2×C4×C53C22⋊Q814yes—
147C2×C4×C53#12314yes—
148C2×C4×C53#12514yes—
149C2×C4×C53#12614yes—
150C2×C4×C53#13114yes—
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C22×Q811yescharacteristic
153C2×C4×C53C2×#4111yescharacteristic
154C2×C4×C53C2×#4111yescharacteristic
155C2×C4×C53C2×#4111yescharacteristic
156C2×C4×C53C2×#4111yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—22no—
161C2×Dic53—24no—
162C2×Dic53—24no—
163C2×Dic53—24no—
164C2×Dic53—24no—
165C8×C53—22no—
166C8×C53—22no—
167C8×C53—42no—
168#88322—24no—
169#88322—42no—
170#21025—531nomaximal, Sylow
171C22×C53—24no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—24no—
175C22×C53#38714yes—
176C22×C53C2×#11711yescharacteristic
177C22×C53C2×C4×D411yescharacteristic
178C22×C53C2×C4×Q811yescharacteristic
179C22×C53C2×#13112yes—
180C22×C53#56812yes—
181C4×C53—24no—
182C4×C53—42no—
183C4×C53#38714yes—
184C4×C53C2×C22⋊Q811yescharacteristic
185C4×C53C2×#12311yescharacteristic
186C4×C53C2×#12511yescharacteristic
187C4×C53C2×#12611yescharacteristic
188Dic53—42no—
189Dic53—42no—
190Dic53—42no—
191Dic53—42no—
192C2×#424—531no—
193C2×#405—531no—
194#2285—531no—
195#1917—534no—
196#1916—534no—
197#1729—531no—
198#1729—531no—
199#1728—531no—
200C2×#336D5311yescharacteristic
201C2×C53—24no—
202C2×C53C2×#38711yescharacteristic
203C2×C53#292412yes—
204C2×C53#293111yescharacteristic
205C2×C53#293211yescharacteristic
206C2×C53#296212yes—
207C22×M16—531no—
208C22×M16—531no—
209C23×C8C2×D5311yescharacteristic
210C23×C8C2×D5311yescharacteristic
211C2×#117—531no—
212C2×#117—531no—
213C22×C42C2×D5311yescharacteristic
214#424—1064no—
215C2×C4⋊C8—532no—
216C2×C4⋊C8—532no—
217#405—1064no—
218C2×#97—532no—
219C2×C4×C8—532no—
220#384—531no—
221#384—531no—
222#382—531no—
223#382—531no—
224#380—534no—
225C4×C4⋊C4—534no—
226#336—534no—
227#336—534no—
228#336—534no—
229#336C2×D5312yes—
230#336C2×D5312yes—
231C53#2102511yescharacteristic, complemented, Sylow
232C23×C4C4×D5312yes—
233C23×C4C22×D5311yescharacteristic
234C2×M16—534no—
235C2×M16—1062no—
236C2×M16—1062no—
237C2×M16—1062no—
238C2×M16—1062no—
239C2×M16—1062no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—532no—
246C22×C8—532no—
247C22×C8D21214yes—
248C22×C8C22×D5312yes—
249C22×C8C22×D5311yescharacteristic
250C22×C8C22×D5311yescharacteristic
251#117—1062no—
252#117—1062no—
253#117—1062no—
254#117—1062no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—534no—
260C2×C4⋊C4—534no—
261C2×C22⋊C4—532no—
262C2×C22⋊C4—532no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—532no—
266C2×C42—532no—
267C2×C42C4×D5311yescharacteristic
268C2×C42C4×D5311yescharacteristic
269C2×C42C22×D5312yes—
270C4⋊C8—1064no—
271C4⋊C8—1064no—
272#97—1064no—
273C4×C8—1064no—
274#95—534no—
275#95—534no—
276C24C2×C4×D5311yescharacteristic
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—532no—
284C22×C4C2×C4×D5312yes—
285C22×C4C2×C4×D5312yes—
286C22×C4C2×C4×D5311yescharacteristic
287C22×C4C2×D21212yes—
288C22×C4#1171994512yes—
289C22×C4#1171994712yes—
290C22×C4#1171994712yes—
291C22×C4#1171994712yes—
292C22×C4Q8×D5312yes—
293C22×C4C23×D5311yescharacteristic
294M16—1068no—
295M16—2124no—
296C2×C8—28no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—44no—
305C2×C8—534no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8—1062no—
309C2×C8—1062no—
310C2×C8—1062no—
311C2×C8C2×D21212yes—
312C2×C8C2×D21212yes—
313C4⋊C4—1062no—
314C4⋊C4—1062no—
315C4⋊C4—1062no—
316C4⋊C4—1062no—
317C4⋊C4—1068no—
318C4⋊C4—2124no—
319C22⋊C4—1064no—
320C22⋊C4—1064no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—1062no—
326C42—1062no—
327C42—1062no—
328C42—1062no—
329C23—22no—
330C23—22no—
331C23#1313317312yes—
332C23C4×D21212yes—
333C23#1313318212yes—
334C23C22×C4×D5311yescharacteristic
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—48no—
341C2×C4—1062no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—1062no—
345C2×C4—1064no—
346C2×C4—1064no—
347C2×C4—1064no—
348C2×C4—1064no—
349C2×C4#1313317312yes—
350C2×C4C4×D21212yes—
351C2×C4#1313318212yes—
352C2×C4#1313319714yes—
353C2×C4#1313320114yes—
354C2×C4#1313320714yes—
355C2×C4#1313321014yes—
356C2×C4#1313322314yes—
357C2×C4C22×D21211yescharacteristic
358C2×C4C2×Q8×D5311yescharacteristic
359C2×C4C2×#1171994711yescharacteristic
360C2×C4C2×#1171994711yescharacteristic
361C2×C4C2×#1171994711yescharacteristic
362C2×C4C2×#1171994511yescharacteristic
363C8—22no—
364C8—22no—
365C8—42no—
366C8—1064no—
367C8—2122no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—24no—
372C22#1681050814yes—
373C22C2×#1313317311yescharacteristic
374C22C2×C4×D21211yescharacteristic
375C22C2×#1313318211yescharacteristic
376C22C2×#1313322312yes—
377C22#1681138912yes—
378C4—24no—
379C4—42no—
380C4—2122no—
381C4—2122no—
382C4—2122no—
383C4—2122no—
384C4#1681050814yes—
385C4C2×#1313319711yescharacteristic
386C4C2×#1313320111yescharacteristic
387C4C2×#1313320711yescharacteristic
388C4C2×#1313321011yescharacteristic
389C2—24no—
390C2C2×#1681050811yescharacteristic
391C2#2028336912yes—
392C2#2028342811yescharacteristic
393C2#2028343411yescharacteristic
394C2#2028365812yes—
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC2×C53×#336C2
prime radicalC24×C53C22×C4
cosocleC22×C4×C53C24
hypercenterC2×#336D53
hypocenterC53#21025
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
4. #20275280C2semidirect
5. #20275273C2semidirect
231. C53#21025semidirect

Maximal subgroups

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

subgroupindex
C2×#168106492
C2×#168105782
#202771222
#202752802
#202752732
#202744112
#202744092
C2×C53×#3362
#2102553