Finite Group Databasebeta

← #27277552

Subgroups — #27277552

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277072C211yesmaximal, characteristic
2C2×#16810349C211yesmaximal, characteristic
3#20272562C212yesmaximal
4#20272561C212yesmaximal
5C2×C53×#324C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic
7C2×#13133176C411yescharacteristic
8C2×C22⋊#88322C2212yes—
9C4⋊C4×Dic53—21no—
10C4⋊C4×Dic53—21no—
11C4⋊C4×Dic53—21no—
12C4⋊C4×Dic53—21no—
13C4×C22⋊Dic53C411yescharacteristic
14C4×C22⋊Dic53C411yescharacteristic
15C4×C22⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810349—21no—
18#16810349—21no—
19C22×C4⋊C4×C53C2211yescharacteristic
20C2×C22⋊C8×C53C2212yes—
21C53×#324—21no—
22C53×#324—21no—
23C22×#88322—22no—
24#13133176—22no—
25#13133176—22no—
26#13133176—22no—
27#13133176C814yescomplemented
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C22⋊Dic53C2×C411yescharacteristic
31C2×C22⋊Dic53C2×C411yescharacteristic
32C2×C22⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C4×Dic53—21no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53—21no—
40C22⋊#88322—22no—
41C22⋊#88322—22no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C23×C4×C53C2×C411yescharacteristic
47C23×C4×C53C2×C411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49C22×C8×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53—22no—
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C22⋊C8×C53—22no—
57C22⋊C8×C53—22no—
58C24×C53C22×C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—81no—
66C22×C4×C53—81no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C2×C812yes—
72C22×C4×C53M1612yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4112yes—
76C22×C4×C53#4112yes—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C2×C8×C53—42no—
82C2×C8×C53—82no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53—21no—
91C22⋊Dic53C2×C812yes—
92C22⋊Dic53C2×C812yes—
93C22⋊Dic53M1612yes—
94C22⋊Dic53M1612yes—
95C4⋊Dic53—24no—
96C4⋊Dic53—24no—
97C4×Dic53—24no—
98C4×Dic53—24no—
99C4×Dic53—81no—
100C4×Dic53—81no—
101C4×Dic53—81no—
102C4×Dic53—81no—
103C2×#88322—42no—
104C2×#88322—82no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53C2×C22⋊C411yescharacteristic
108C23×C53#11711yescharacteristic
109C23×C53#11711yescharacteristic
110C23×C53C22×C811yescharacteristic
111C23×C53C2×M1611yescharacteristic
112C23×C53C2×M1611yescharacteristic
113C23×C53C2×M1611yescharacteristic
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—81no—
131C2×C4×C53—81no—
132C2×C4×C53C22⋊C812yes—
133C2×C4×C53C22⋊C812yes—
134C2×C4×C53#12312yes—
135C2×C4×C53#12312yes—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—22no—
139C2×Dic53—22no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C8×C53—82no—
149#88322—82no—
150#8735—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—21no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53#38111yescharacteristic
162C22×C53C2×C22⋊C811yescharacteristic
163C22×C53#40711yescharacteristic
164C22×C53#43111yescharacteristic
165C22×C53#43111yescharacteristic
166C22×C53#43211yescharacteristic
167C22×C53#43211yescharacteristic
168C4×C53—22no—
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—22no—
172C4×C53—41no—
173C4×C53—41no—
174Dic53—42no—
175Dic53—42no—
176Dic53—42no—
177Dic53—42no—
178#2274—531no—
179C2×#324—531no—
180C2×#324D5311yescharacteristic
181#1282—532no—
182#1282—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53#181711yescharacteristic
190C2×C53#182811yescharacteristic
191C2×C53#183011yescharacteristic
192C2×#117—531no—
193C2×#117—531no—
194C22×C4⋊C4C2×D5311yescharacteristic
195C2×C22⋊C8—532no—
196C2×C22⋊C8C2×D5312yes—
197C4×C4⋊C4—1061no—
198C4×C4⋊C4—1061no—
199C4×C4⋊C4—1061no—
200C4×C4⋊C4—1061no—
201C4×C22⋊C4—531no—
202C4×C22⋊C4—531no—
203C4×C22⋊C4—531no—
204C4×C22⋊C4—531no—
205#324—21no—
206#324—21no—
207#324—1061no—
208#324—1061no—
209C53#873511yescharacteristic, complemented, Sylow
210C23×C4C4×D5311yescharacteristic
211C23×C4C4×D5311yescharacteristic
212C23×C4C22×D5311yescharacteristic
213C22×C8—22no—
214C22×C8—1062no—
215#117—534no—
216#117—1062no—
217#117—1062no—
218#117—1062no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4D21211yescharacteristic
224C2×C4⋊C4D21211yescharacteristic
225C2×C4⋊C4C22⋊D5311yescharacteristic
226C2×C4⋊C4C22⋊D5311yescharacteristic
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C42—1061no—
232C2×C42—1061no—
233C2×C42—1061no—
234C2×C42—1061no—
235C2×C42—1061no—
236C2×C42—1061no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—1062no—
240C22⋊C8—1062no—
241#95—1061no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245C24C2×C4×D5311yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—1061no—
258C22×C4C8×D5312yes—
259C22×C4#1171991212yes—
260C22×C4#1171992111yescharacteristic
261C22×C4#1171992111yescharacteristic
262C22×C4#1171992111yescharacteristic
263C22×C4#1171992111yescharacteristic
264C22×C4C2×D21211yescharacteristic
265C22×C4#1171994712yes—
266C22×C4#1171994712yes—
267C22×C4C2×C22⋊D5311yescharacteristic
268C2×C8—42no—
269C2×C8—82no—
270C2×C8—4242no—
271C2×C8—2122no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—1064no—
277C4⋊C4—1064no—
278C22⋊C4—532no—
279C22⋊C4—532no—
280C22⋊C4—532no—
281C22⋊C4—532no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—1064no—
287C42—1064no—
288C42—4241no—
289C42—4241no—
290C42—4241no—
291C42—4241no—
292C23—41no—
293C23—41no—
294C23C2×#1171992111yescharacteristic
295C23#1313317311yescharacteristic
296C23#1313317311yescharacteristic
297C23C2×C8×D5311yescharacteristic
298C23M16×D5311yescharacteristic
299C23M16×D5311yescharacteristic
300C23C2×#1171991211yescharacteristic
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4#1313312912yes—
332C2×C4#1313312912yes—
333C2×C4#1313319812yes—
334C2×C4#1313320112yes—
335C8—82no—
336C8—4242no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22#1681048411yescharacteristic
348C22C2×#1313312911yescharacteristic
349C22#1681058611yescharacteristic
350C22#1681067311yescharacteristic
351C22#1681067311yescharacteristic
352C22#1681067711yescharacteristic
353C22#1681067711yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—2122no—
361C4—2122no—
362C4—2122no—
363C4—2122no—
364C2—21no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2027478111yescharacteristic
371C2#2027482111yescharacteristic
372C2#2027482711yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC2×C53×#324C2
socleC22×C53#381
prime radicalC24×C53C22×C4
cosocleC23×C4×C53C23
hypercenterC2×#324D53
hypocenterC53#8735
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
27. #13133176C8semidirect
209. C53#8735semidirect

Maximal subgroups

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

subgroupindex
#202770722
C2×#168103492
#202725622
#202725612
C2×C53×#3242
#873553