Finite Group Databasebeta

← #27275934

Subgroups — #27275934

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274699C211yesmaximal, characteristic
2#20273372C212yesmaximal, complemented
3#20273118C212yesmaximal, complemented
4#20273029C211yesmaximal, characteristic
5C53×#3458C211yesmaximal, characteristic
6#16810676C2212yes—
7C2×C4⋊#88322C2211yescharacteristic
8C2×C4⋊#88322C2211yescharacteristic
9C2×C22⋊#88322—21no—
10#16810378—21no—
11#16810372—22no—
12#16810369—22no—
13Q8⋊#88322—24no—
14#16810349—21no—
15#16810347—21no—
16C53×#544—22no—
17C53×#544—22no—
18C53×#544—21no—
19C53×#544C412yes—
20C53×#543—21no—
21C53×#543C412yes—
22C53×#533—24no—
23C53×#533—21no—
24C53×#533C2212yes—
25C2×#117×C53—21no—
26C2×#117×C53C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C4⋊#88322—22no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322—42no—
34C4×#88322—22no—
35C23×C4×C53D411yescharacteristic
36C4⋊Q8×C53—21no—
37C4⋊Q8×C53—21no—
38C4⋊Q8×C53—24no—
39C4⋊Q8×C53—24no—
40C4⋊Q8×C53D412yes—
41C4⋊Q8×C53D412yes—
42#125×C53—24no—
43#125×C53—22no—
44#125×C53—24no—
45#125×C53—22no—
46C22⋊Q8×C53—24no—
47C22⋊Q8×C53—24no—
48C22⋊Q8×C53—24no—
49C22⋊Q8×C53—24no—
50C22⋊Q8×C53—22no—
51C22⋊Q8×C53—22no—
52C22⋊Q8×C53—22no—
53C22⋊Q8×C53C2×C414yes—
54C4×Q8×C53—22no—
55C4×Q8×C53—22no—
56C4×Q8×C53—22no—
57C4×Q8×C53D414yes—
58#117×C53—22no—
59#117×C53—22no—
60#117×C53—22no—
61#117×C53—22no—
62#117×C53—22no—
63#117×C53—22no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53C2×C411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53C2×C411yescharacteristic
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C42×C53—22no—
72C2×C42×C53D411yescharacteristic
73C2×C42×C53C2311yescharacteristic
74C24×C53C22⋊C411yescharacteristic
75C2×Q8×C53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—22no—
78C2×Q8×C53C22⋊C414yes—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53SD1612yes—
89C22×C4×C53Q1612yes—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—24no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53C22⋊C414yes—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—24no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C42×C53—24no—
112C42×C53—24no—
113C42×C53—21no—
114C42×C53—21no—
115C42×C53C2×D412yes—
116C42×C53C2×D412yes—
117C2×#88322—42no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53#10312yes—
129C23×C53C2×C22⋊C411yescharacteristic
130Q8×C53—44no—
131Q8×C53—82no—
132Q8×C53—82no—
133Q8×C53—82no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—24no—
141C2×C4×C53—24no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—82no—
149C2×C4×C53—82no—
150C2×C4×C53#10312yes—
151C2×C4×C53C2×C22⋊C412yes—
152C2×C4×C53#12012yes—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53C2×SD1611yescharacteristic
156C2×C4×C53C2×Q1611yescharacteristic
157C2×C4×C53#13711yescharacteristic
158C2×C4×C53#13711yescharacteristic
159#88322—82no—
160#88322—81no—
161#88322—81no—
162#8410—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—81no—
170C22×C53#37911yescharacteristic
171C22×C53C2×#10311yescharacteristic
172C22×C53#41911yescharacteristic
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—22no—
176C4×C53—41no—
177C4×C53—44no—
178C4×C53—41no—
179C4×C53—44no—
180C4×C53#44811yescharacteristic
181C4×C53#44811yescharacteristic
182C4×C53#45111yescharacteristic
183C4×C53#45111yescharacteristic
184#3458D5311yescharacteristic
185#1798—531no—
186#1542—532no—
187#1481—532no—
188#1458—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—42no—
192C2×C53#178711yescharacteristic
193C2×C53#179111yescharacteristic
194C2×C53#179411yescharacteristic
195#544—22no—
196#544—22no—
197#544—21no—
198#544Dic5312yes—
199#543—21no—
200#543Dic5312yes—
201#533—24no—
202#533—21no—
203#533C2×D5312yes—
204C2×#117—21no—
205C2×#117C2×D5311yescharacteristic
206#431—532no—
207C2×C4⋊C8—531no—
208C2×C4⋊C8—531no—
209C2×C22⋊C8—1061no—
210#336—1061no—
211#333—1062no—
212#332—1062no—
213Q8⋊C8—1064no—
214#324—1061no—
215#323—1061no—
216C53#841011yescharacteristic, complemented, Sylow
217C23×C4C22⋊D5311yescharacteristic
218C22×C8—1061no—
219C22×C8—1061no—
220C4⋊Q8—24no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—24no—
224C4⋊Q8C22⋊D5312yes—
225C4⋊Q8C22⋊D5312yes—
226#125—24no—
227#125—22no—
228#125—24no—
229#125—22no—
230C22⋊Q8—24no—
231C22⋊Q8—24no—
232C22⋊Q8—24no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—24no—
237C22⋊Q8C2×Dic5314yes—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8C22⋊D5314yes—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4C2×Dic5311yescharacteristic
251C2×C4⋊C4C22⋊D5311yescharacteristic
252C2×C22⋊C4—22no—
253C2×C22⋊C4C2×Dic5311yescharacteristic
254C2×C22⋊C4C22⋊D5311yescharacteristic
255C2×C42—22no—
256C2×C42C22⋊D5311yescharacteristic
257C2×C42C22×D5311yescharacteristic
258C4⋊C8—1062no—
259C4⋊C8—1062no—
260C4⋊C8—1062no—
261C22⋊C8—1062no—
262C22⋊C8—2122no—
263C4×C8—1062no—
264C24C22⋊Dic5311yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8C22⋊Dic5314yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4#1171992412yes—
276C22×C4#1171992512yes—
277C22×C4C22⋊Dic5311yescharacteristic
278C22×C4C22⋊Dic5311yescharacteristic
279C22×C4C22⋊Dic5311yescharacteristic
280C22×C4C22×Dic5311yescharacteristic
281C22×C4C2×C22⋊D5311yescharacteristic
282C22×C4C2×C22⋊D5311yescharacteristic
283C2×C8—2122no—
284C2×C8—2121no—
285C2×C8—2121no—
286C2×C8—2121no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8—2121no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4C22⋊Dic5314yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C42—24no—
309C42—21no—
310C42—24no—
311C42—21no—
312C42C2×C22⋊D5312yes—
313C42C2×C22⋊D5312yes—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23#1313314112yes—
319C23C2×C22⋊Dic5311yescharacteristic
320Q8—44no—
321Q8—82no—
322Q8—82no—
323Q8—82no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—24no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#1313314112yes—
341C2×C4C2×C22⋊Dic5312yes—
342C2×C4#1313318612yes—
343C2×C4#1313318611yescharacteristic
344C2×C4#1313318611yescharacteristic
345C2×C4C2×#1171992411yescharacteristic
346C2×C4C2×#1171992511yescharacteristic
347C2×C4#1313324811yescharacteristic
348C2×C4#1313324811yescharacteristic
349C8—4242no—
350C8—4241no—
351C8—4241no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22#1681047911yescharacteristic
360C22C2×#1313314111yescharacteristic
361C22#1681063211yescharacteristic
362C4—22no—
363C4—22no—
364C4—22no—
365C4—41no—
366C4—44no—
367C4—41no—
368C4—44no—
369C4#1681076211yescharacteristic
370C4#1681076211yescharacteristic
371C4#1681077911yescharacteristic
372C4#1681077911yescharacteristic
373C2—21no—
374C2—21no—
375C2—42no—
376C2#2027465011yescharacteristic
377C2#2027466611yescharacteristic
378C2#2027467911yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3458C2
socleC22×C53#379
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#3458D53
hypocenterC53#8410
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #20273372C2semidirect
3. #20273118C2semidirect
216. C53#8410semidirect

Maximal subgroups

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

subgroupindex
#202746992
#202733722
#202731182
#202730292
C53×#34582
#841053