Finite Group Databasebeta

← #27276759

Subgroups — #27276759

Order 13568.

682 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
2#20274693C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272557C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C22×C4⋊Dic53C411yescharacteristic
7C22×C4×Dic53C411yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9C22×C4×Dic53C2211yescharacteristic
10C2×#13133146—22no—
11C2×C22⋊#88322C2212yes—
12C4⋊C4×Dic53—22no—
13C4⋊C4×Dic53—22no—
14C2×#13133124C411yescharacteristic
15C2×#13133124C2211yescharacteristic
16#16810377—22no—
17#16810349—22no—
18C22×C4⋊C4×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2212yes—
20C53×#324—22no—
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—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—42no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53D412yes—
38C2×C4×Dic53D412yes—
39#13133146—44no—
40C22⋊#88322—22no—
41C22⋊#88322—22no—
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×C53D412yes—
51C2×C4⋊C4×C53Q812yes—
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
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×C53C4⋊C412yes—
66C22×C4×C53C4⋊C412yes—
67C22×C4×C53C2×C812yes—
68C22×C4×C53M1612yes—
69C22×C4×C53C2×D412yes—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×Q811yescharacteristic
72C22×C4×C53#4112yes—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—41no—
82C22×Dic53—41no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53—41no—
86C22×Dic53—41no—
87C22×Dic53C22⋊C412yes—
88C22×Dic53C22⋊C411yescharacteristic
89C22×Dic53C22⋊C411yescharacteristic
90C22×Dic53C2×C812yes—
91C22×Dic53C2×C812yes—
92C22×Dic53M1612yes—
93C22×Dic53M1612yes—
94C2×C8×C53—42no—
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—42no—
107C2×#88322—42no—
108C2×#88322—42no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53C2×C22⋊C411yescharacteristic
114C23×C53C2×C4⋊C411yescharacteristic
115C23×C53#11711yescharacteristic
116C23×C53C22×C811yescharacteristic
117C23×C53C2×M1611yescharacteristic
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×C53#12112yes—
138C2×C4×C53C22⋊Q812yes—
139C2×Dic53—24no—
140C2×Dic53—24no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—22no—
145C2×Dic53—21no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53C22⋊C814yes—
160C2×Dic53#9912yes—
161C2×Dic53#10112yes—
162C8×C53—82no—
163#88322—82no—
164#8560—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×C53#38011yescharacteristic
178C22×C53C2×C22⋊C811yescharacteristic
179C22×C53#40711yescharacteristic
180C22×C53C2×#9911yescharacteristic
181C22×C53#41011yescharacteristic
182C22×C53C2×#10111yescharacteristic
183C22×C53#41311yescharacteristic
184C22×C53C2×C4⋊C811yescharacteristic
185C22×C53#42311yescharacteristic
186C22×C53#43111yescharacteristic
187C22×C53#43211yescharacteristic
188C4×C53—24no—
189C4×C53—22no—
190C4×C53—22no—
191C4×C53—42no—
192Dic53—24no—
193Dic53—22no—
194Dic53—22no—
195Dic53—42no—
196Dic53—42no—
197C2×C4×C4⋊C4—531no—
198#1797—532no—
199C2×#324—531no—
200C2×#324D5311yescharacteristic
201#1280—532no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53#145911yescharacteristic
207C2×C53#146111yescharacteristic
208C2×C53#146311yescharacteristic
209C2×C53#146411yescharacteristic
210C2×C53#179711yescharacteristic
211C2×C53#179911yescharacteristic
212C2×C53#180111yescharacteristic
213C22×C4⋊C4—531no—
214C22×C4⋊C4C2×D5311yescharacteristic
215C22×C42—531no—
216C22×C42—531no—
217C22×C42—531no—
218C2×C4⋊C8—1062no—
219C2×C22⋊C8—532no—
220C2×C22⋊C8C2×D5312yes—
221C4×C4⋊C4—1062no—
222C4×C4⋊C4—1062no—
223C2×#95—531no—
224C2×#95—531no—
225#336—1062no—
226#324—22no—
227#324—1062no—
228C53#856011yescharacteristic, 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⋊C4Dic10612yes—
244C2×C4⋊C4C22⋊D5312yes—
245C2×C42—532no—
246C2×C42—532no—
247C2×C42—1061no—
248C2×C42—1061no—
249C2×C42—1062no—
250C2×C42—1062no—
251C2×C42—1061no—
252C2×C42—1061no—
253C2×C42—2122no—
254C2×C42—2122no—
255C4⋊C8—2124no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—1062no—
259C22⋊C8—1062no—
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—532no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—532no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1061no—
283C22×C4—1061no—
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×C4#1171991212yes—
296C22×C4#1171991912yes—
297C22×C4#1171991912yes—
298C22×C4C2×Dic10611yescharacteristic
299C22×C4D4×D5312yes—
300C22×C4#1171994712yes—
301C22×C4C2×C22⋊D5311yescharacteristic
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—2122no—
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—
321C23C22⋊C4×D5311yescharacteristic
322C23C2×#1171991911yescharacteristic
323C23#1313317311yescharacteristic
324C23C2×C8×D5311yescharacteristic
325C23M16×D5311yescharacteristic
326C23M16×D5311yescharacteristic
327C23C2×#1171991211yescharacteristic
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—534no—
345C2×C4—532no—
346C2×C4—532no—
347C2×C4—1064no—
348C2×C4—1064no—
349C2×C4—1061no—
350C2×C4—1061no—
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×C4#1313314614yescomplemented
368C2×C4#1313318912yes—
369C2×C4#1313319412yes—
370C8—82no—
371C8—4242no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#1681048111yescharacteristic
385C22C22⋊C8×D5311yescharacteristic
386C22#1681058511yescharacteristic
387C22#99×D5311yescharacteristic
388C22#1681059511yescharacteristic
389C22#101×D5311yescharacteristic
390C22#1681060611yescharacteristic
391C22C2×#1313314611yescharacteristic
392C22#1681064611yescharacteristic
393C22#1681067311yescharacteristic
394C22#1681067711yescharacteristic
395C4—24no—
396C4—22no—
397C4—22no—
398C4—42no—
399C4—1064no—
400C4—1062no—
401C4—1062no—
402C4—2122no—
403C4—2122no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2#2027303211yescharacteristic
409C2#2027303711yescharacteristic
410C2#2027304311yescharacteristic
411C2#2027304811yescharacteristic
412C2#2027469311yescharacteristic
413C2#2027470111yescharacteristic
414C2#2027470911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22×C4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8560
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
28. C2×C4⋊Dic53C8semidirect
136. C2×C4×C53C4⋊C8semidirect
228. C53#8560semidirect
367. C2×C4#13133146semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×Dic532
#202746932
C2×#168103492
#202725572
C2×C53×#3242
#856053