Finite Group Databasebeta

← #27300326

Subgroups — #27300326

Order 13568.

416 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283131C211yesmaximal, characteristic
2C2×#16810404C211yesmaximal, characteristic
3C2×#16810404C211yesmaximal, characteristic
4#20272802C211yesmaximal, characteristic, complemented
5#20272801C211yesmaximal, characteristic, complemented
6#20272800C211yesmaximal, characteristic, complemented
7C53×#1378C211yesmaximal, characteristic, complemented
8C22×#11719912C2211yescharacteristic
9C2×M16×D53C411yescharacteristic
10#117×D53C411yescharacteristic
11C2×#13133157C2211yescharacteristic
12C2×#13133157C2211yescharacteristic
13#16810581C411yescharacteristic
14#16810580—21no—
15#16810579C411yescharacteristic
16#16810578C2211yescharacteristic
17#16810577—21no—
18#16810575—21no—
19C4×#11719912—21no—
20#16810404—21no—
21#16810404—21no—
22#16810404—21no—
23#16810404—21no—
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×#405C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C2×#11719917C2×C411yescharacteristic
29C2×#11719912—21no—
30C2×#11719912—21no—
31C2×#11719912—21no—
32C2×#11719912—21no—
33C2×#11719912—21no—
34C2×#11719912D411yescharacteristic
35C2×#11719912D411yescharacteristic
36C2×#11719912D411yescharacteristic
37C2×#11719912D411yescharacteristic
38M16×D53—22no—
39M16×D53—22no—
40C22×#88322C2×C411yescharacteristic
41C2×C8×D53—21no—
42#13133176C2×C411yescharacteristic
43#13133175—21no—
44#13133174—21no—
45#13133173C812yes—
46C4⋊C4×D53—21no—
47C22⋊C4×D53C812yes—
48C42×D53—21no—
49#13133157—21no—
50#13133157—21no—
51#13133157—21no—
52#13133157—21no—
53C2×#11719908—21no—
54C2×#11719908—21no—
55#13133128—21no—
56#13133127—21no—
57#13133127—21no—
58C4×#88322—21no—
59C8×Dic53—21no—
60C8×Dic53—21no—
61C2×M16×C53C2×C411yescharacteristic
62C22×C8×C53C2311yescharacteristic
63#117×C53C2×C411yescharacteristic
64M32×C53—21no—
65M32×C53—21no—
66M32×C53—21no—
67M32×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53—21no—
70#97×C53—21no—
71C4×C8×C53—21no—
72C23×D53C2×C811yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×Dic53C2×C811yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53—21no—
79C2×C4×D53—22no—
80C2×C4×D53C22⋊C411yescharacteristic
81C2×C4×D53C22⋊C411yescharacteristic
82C2×C4×D53M1611yescharacteristic
83C2×C4×D53M1611yescharacteristic
84M16×C53—21no—
85M16×C53M1612yes—
86C2×C8×C53—21no—
87C2×C8×C53C2×D411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53#4111yescharacteristic
90C2×C8×C53#4111yescharacteristic
91C2×C8×C53#4111yescharacteristic
92C2×C8×C53#4111yescharacteristic
93C4⋊C4×C53—21no—
94C22⋊C4×C53C2×C812yes—
95C42×C53—21no—
96C22⋊Dic53C2×C812yes—
97#11719921—22no—
98C4⋊Dic53—21no—
99#11719919—22no—
100C4×Dic53—21no—
101C4×Dic53—22no—
102#11719917—21no—
103#11719917M1612yes—
104C2×#88322—21no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322C22⋊C411yescharacteristic
108C2×#88322C22⋊C411yescharacteristic
109#11719912—24no—
110#11719912—21no—
111#11719912—21no—
112#11719912—21no—
113#11719912—21no—
114#11719912—21no—
115#11719912—21no—
116#11719912—21no—
117#11719912—21no—
118C8×D53—24no—
119C16×C53—41no—
120C16×C53—41no—
121#11719908—41no—
122#11719908—41no—
123C23×C53C22×C811yescharacteristic
124C22×D53—21no—
125C22×D53—21no—
126C22×D53—81no—
127C22×D53C22⋊C811yescharacteristic
128C22×D53C22⋊C811yescharacteristic
129C2×C4×C53—22no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53#11711yescharacteristic
132C2×C4×C53#11711yescharacteristic
133C2×C4×C53C2×M1611yescharacteristic
134C2×C4×C53C2×M1611yescharacteristic
135C2×C4×C53C2×M1611yescharacteristic
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—22no—
139C2×Dic53C22⋊C811yescharacteristic
140C2×Dic53C22⋊C811yescharacteristic
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—21no—
146C4×D53—21no—
147C4×D53—21no—
148C4×D53—21no—
149C4×D53—82no—
150C8×C53—22no—
151C8×C53#12311yescharacteristic
152C8×C53#12311yescharacteristic
153C8×C53#12311yescharacteristic
154C8×C53#12311yescharacteristic
155#88322—21no—
156#88322—21no—
157#88322—22no—
158#12851—531nomaximal, Sylow
159C22×C53—41no—
160C22×C53C2×C22⋊C811yescharacteristic
161C22×C53#43111yescharacteristic
162C22×C53#43111yescharacteristic
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—81no—
168C2×D53—81no—
169C4×C53—42no—
170C4×C53#38111yescharacteristic
171C4×C53#40711yescharacteristic
172C4×C53#43211yescharacteristic
173C4×C53#43211yescharacteristic
174Dic53—21no—
175Dic53—21no—
176Dic53—42no—
177#2873—531no—
178C2×#349—531no—
179C2×#349—531no—
180#1378—531no—
181#1378—531no—
182#1378—531no—
183#1378D5311yescharacteristic, complemented
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#181711yescharacteristic
188D53—41no—
189D53—41no—
190C22×M16—531no—
191C22×M16—531no—
192C2×#117—531no—
193C2×M32—531no—
194C2×M32—531no—
195C2×M32C2×D5311yescharacteristic
196C2×M32C2×D5311yescharacteristic
197#405—531no—
198#405—531no—
199#405—531no—
200#405—1061no—
201#405—1061no—
202#405C2×D5311yescharacteristic
203C4×M16—1061no—
204C4×M16—1061no—
205#349—1061no—
206#349—1061no—
207#349—1061no—
208#349—1061no—
209C53#1285111yescharacteristic, complemented, Sylow
210C23×C4—531no—
211C2×M16—531no—
212C2×M16—531no—
213C2×M16—531no—
214C2×M16—531no—
215C2×M16—531no—
216C2×M16—1061no—
217C2×M16—1062no—
218C2×M16—1062no—
219C2×M16—1061no—
220C2×M16—1061no—
221C2×M16—1061no—
222C2×M16—1061no—
223C2×M16C4×D5311yescharacteristic
224C22×C8—531no—
225C22×C8—1061no—
226C22×C8C22×D5311yescharacteristic
227#117—531no—
228#117—532no—
229#117—1061no—
230#117—1061no—
231#117C4×D5311yescharacteristic
232C2×C4⋊C4—1061no—
233C2×C22⋊C4—532no—
234C2×C42—1061no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242M32—1061no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1061no—
246C2×C16—1061no—
247#97—21no—
248#97—1061no—
249#97—1061no—
250#97—1061no—
251C4×C8—21no—
252C4×C8—1061no—
253C4×C8—1061no—
254C4×C8—1061no—
255C24—531no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—531no—
261C22×C4—1062no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4C2×C4×D5311yescharacteristic
267M16—21no—
268M16—532no—
269M16—1064no—
270M16—1061no—
271M16—1061no—
272M16—1061no—
273M16—1061no—
274M16—1061no—
275M16—1061no—
276M16—1061no—
277M16—1061no—
278M16—1061no—
279M16#1171991212yes—
280C2×C8—21no—
281C2×C8—531no—
282C2×C8—531no—
283C2×C8—1064no—
284C2×C8—1061no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8#1171994511yescharacteristic
288C2×C8#1171994511yescharacteristic
289C2×C8D4×D5311yescharacteristic
290C2×C8D4×D5311yescharacteristic
291C2×C8#1171994711yescharacteristic
292C2×C8#1171994911yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—1062no—
295C4⋊C4—1061no—
296C22⋊C4—532no—
297C22⋊C4—1062no—
298C22⋊C4C8×D5312yes—
299C42—21no—
300C42—1062no—
301C42—1061no—
302C16—41no—
303C16—41no—
304C16—2121no—
305C16—2121no—
306C23—531no—
307C23—531no—
308C23—1061no—
309C23—1061no—
310C23—4241no—
311C23C2×C8×D5311yescharacteristic
312C2×C4—22no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1062no—
326C2×C4—4242no—
327C2×C4C22⋊C4×D5311yescharacteristic
328C2×C4#1313317411yescharacteristic
329C2×C4#1313317511yescharacteristic
330C2×C4M16×D5311yescharacteristic
331C2×C4M16×D5311yescharacteristic
332C2×C4C2×#1171991211yescharacteristic
333C8—22no—
334C8—1062no—
335C8—1061no—
336C8—1061no—
337C8#1313319911yescharacteristic
338C8#1313319911yescharacteristic
339C8#1313320211yescharacteristic
340C8#1313320211yescharacteristic
341C22—41no—
342C22—1061no—
343C22—1061no—
344C22—4241no—
345C22—4241no—
346C22—2121no—
347C22—2121no—
348C22C22⋊C8×D5311yescharacteristic
349C22#1681067411yescharacteristic
350C22#1681067511yescharacteristic
351C4—42no—
352C4—1061no—
353C4—1061no—
354C4—2122no—
355C4#1681048511yescharacteristic
356C4#1681058511yescharacteristic
357C4#1681067811yescharacteristic
358C4#1681067911yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—2121no—
363C2—2121no—
364C2#2027478211yescharacteristic
subgroupG/H
centerC4#16810485
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1378C2
socleC2×C53#1817
prime radicalC23×D53C2×C8
cosocleC22×C8×C53C23
hypercenter#1378D53
hypocenterC53#12851
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
4. #20272802C2semidirect
5. #20272801C2semidirect
6. #20272800C2semidirect
7. C53×#1378C2semidirect
183. #1378D53semidirect
209. C53#12851semidirect

Maximal subgroups

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

subgroupindex
#202831312
C2×#168104042
#202728022
#202728012
#202728002
C53×#13782
#1285153