Finite Group Databasebeta

← #27283111

Subgroups — #27283111

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274568C211yesmaximal, characteristic, complemented
2#20274464C211yesmaximal, characteristic, complemented
3C2×#16810365C212yesmaximal, complemented
4C2×#16810363C212yesmaximal
5C53×#2861C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×#13133128C2212yes—
8C2×#13133128C2212yes—
9#16810378—21no—
10#16810378—21no—
11#16810378—21no—
12#16810365—24no—
13#16810363—24no—
14C2×#125×C53C2211yescharacteristic
15C2×#124×C53C2211yescharacteristic
16C53×#391C411yescharacteristic
17C53×#391C411yescharacteristic
18C53×#390C411yescharacteristic
19C53×#390C411yescharacteristic
20C53×#388—21no—
21C53×#388—21no—
22C53×#388—21no—
23C53×#388—21no—
24C43×C53C2211yescharacteristic
25C22×#88322—22no—
26C22×#88322—22no—
27C4⋊#88322—24no—
28#13133128—22no—
29#13133128—22no—
30#13133128—22no—
31#13133128D414yes—
32C22×Q8×C53C2×C411yescharacteristic
33C22×D4×C53C2×C411yescharacteristic
34#125×C53—22no—
35#125×C53—22no—
36#124×C53—22no—
37#124×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53C2311yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59C24×C53—21no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53—41no—
63C2×D4×C53—22no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C42×C53—22no—
89C42×C53—41no—
90C42×C53—41no—
91C42×C53—41no—
92C42×C53—41no—
93C42×C53—41no—
94C42×C53C2×D412yes—
95C42×C53#4112yes—
96C2×#88322—24no—
97C2×#88322—24no—
98C2×#88322—42no—
99C2×#88322—42no—
100C2×#88322—42no—
101C2×#88322—42no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110Q8×C53—82no—
111D4×C53—82no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53#10412yes—
139C2×C4×C53C4×D412yes—
140C2×C4×C53#12112yes—
141C2×C4×C53#12411yescharacteristic
142C2×C4×C53#12711yescharacteristic
143#88322—44no—
144#88322—44no—
145#9688—531nomaximal, Sylow
146C22×C53—81no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53#39011yescharacteristic
154C22×C53#41311yescharacteristic
155C22×C53#41311yescharacteristic
156C22×C53C2×#10411yescharacteristic
157C22×C53#42111yescharacteristic
158C22×C53#42111yescharacteristic
159C22×C53#42111yescharacteristic
160C4×C53—22no—
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—22no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—81no—
169C4×C53—81no—
170C4×C53—81no—
171#2861D5311yescharacteristic
172#1767—531no—
173#1745—531no—
174C2×#330—532no—
175C2×#329—532no—
176C2×C53—161no—
177C2×C53#152911yescharacteristic
178C2×C53#153211yescharacteristic
179C2×C53#153511yescharacteristic
180C2×C53#153511yescharacteristic
181C2×C53#196311yescharacteristic
182C2×C53#196711yescharacteristic
183C2×C53#198011yescharacteristic
184C2×#125C2×D5311yescharacteristic
185C2×#124C2×D5311yescharacteristic
186C2×C4⋊C8—1061no—
187C2×#97—532no—
188C2×#97—532no—
189#391Dic5311yescharacteristic
190#391Dic5311yescharacteristic
191#390Dic5311yescharacteristic
192#390Dic5311yescharacteristic
193#388—21no—
194#388—21no—
195#388—21no—
196#388—21no—
197C43C2×D5311yescharacteristic
198#336—1061no—
199#336—1061no—
200#336—1061no—
201#330—1064no—
202#329—1064no—
203C53#968811yescharacteristic, complemented, Sylow
204C22×Q8C2×Dic5311yescharacteristic
205C22×D4C2×Dic5311yescharacteristic
206C22×C8—1062no—
207C22×C8—1062no—
208#125—22no—
209#125—22no—
210#124—22no—
211#124—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5311yescharacteristic
215C2×C4⋊C4C2×Dic5311yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42C2×Dic5311yescharacteristic
223C2×C42C2×Dic5311yescharacteristic
224C2×C42C22⋊D5311yescharacteristic
225C2×C42C22⋊D5311yescharacteristic
226C2×C42C22⋊D5311yescharacteristic
227C2×C42C22⋊D5311yescharacteristic
228C2×C42C22×D5311yescharacteristic
229C4⋊C8—1064no—
230#97—534no—
231#97—1062no—
232#97—1062no—
233#97—1062no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238C24—21no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—41no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C22×Dic5311yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255C2×C8—1064no—
256C2×C8—1064no—
257C2×C8—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C42—22no—
274C42—41no—
275C42—41no—
276C42—41no—
277C42—41no—
278C42—41no—
279C42D4×D5312yes—
280C42#1171994712yes—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×C22⋊Dic5311yescharacteristic
289Q8—82no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4#1313314412yes—
318C2×C4D4×Dic5312yes—
319C2×C4#1313318812yes—
320C2×C4#1313320411yescharacteristic
321C2×C4#1313321311yescharacteristic
322C8—2124no—
323C8—2124no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#1681052611yescharacteristic
332C22#1681060711yescharacteristic
333C22#1681060711yescharacteristic
334C22C2×#1313314411yescharacteristic
335C22#1681064011yescharacteristic
336C22#1681064011yescharacteristic
337C22#1681064011yescharacteristic
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—161no—
350C2#2027332211yescharacteristic
351C2#2027333511yescharacteristic
352C2#2027334611yescharacteristic
353C2#2027334611yescharacteristic
354C2#2027554811yescharacteristic
355C2#2027556511yescharacteristic
356C2#2027563511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2861C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2861D53
hypocenterC53#9688
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2861

Complements

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

NH ≅ G/Ntype
1. #20274568C2semidirect
2. #20274464C2semidirect
3. C2×#16810365C2semidirect
203. C53#9688semidirect

Maximal subgroups

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

subgroupindex
#202745682
#202744642
C2×#168103652
C2×#168103632
C53×#28612
#968853