Finite Group Databasebeta

← #27300973

Subgroups — #27300973

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287817C211yesmaximal, characteristic, complemented
2#20283409C211yesmaximal, characteristic
3#20276335C211yesmaximal, characteristic, complemented
4#20276326C211yesmaximal, characteristic, complemented
5#20272802C211yesmaximal, characteristic, complemented
6#20272800C211yesmaximal, characteristic, complemented
7C53×#2112C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9#16811134C411yescharacteristic
10#16811041—22no—
11#16811039—22no—
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14M32×D53—22no—
15#16810694—22no—
16C8×C22⋊D53—22no—
17#16810649C411yescharacteristic
18#16810589C411yescharacteristic
19#16810587C411yescharacteristic
20#16810578C2211yescharacteristic
21#16810404—22no—
22C22⋊#11719908—22no—
23C23×C8×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×C22⋊C16—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133223—21no—
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32C2×#11719917C2×C411yescharacteristic
33C2×#11719917C2×C411yescharacteristic
34C2×#11719912D412yes—
35C2×C8×D53D412yes—
36#13133200C812yes—
37#13133196—21no—
38#13133191—21no—
39#13133186C812yes—
40C4×C22⋊D53—22no—
41#13133176C2×C411yescharacteristic
42#13133157—21no—
43#13133157—21no—
44#13133157—21no—
45#13133157—21no—
46#13133156—24no—
47C2×#11719908—21no—
48C2×#11719908D412yescomplemented
49C16×D53—24no—
50#13133146—22no—
51C22⋊#88322—22no—
52#13133129—22no—
53#13133127—21no—
54C8×Dic53—21no—
55C23×C4×C53C2×C411yescharacteristic
56C22×C8×C53—21no—
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C22×C8×C53C2×C411yescharacteristic
63C22×C8×C53C2311yescharacteristic
64M32×C53—21no—
65M32×C53—21no—
66M32×C53—21no—
67M32×C53—21no—
68C2×C16×C53—21no—
69C2×C16×C53D412yescomplemented
70C24×C53C2×C811yescharacteristic
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53C2×C811yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C2×C22⋊D53—22no—
80C2×C22⋊D53C2×C812yes—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85C2×D212M1611yescharacteristic
86C2×C4×D53C22⋊C412yes—
87C2×Dic106M1611yescharacteristic
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C22⋊Dic53—22no—
102C22⋊Dic53C2×C812yes—
103#11719921—22no—
104C4⋊Dic53—21no—
105#11719919—22no—
106C4×Dic53—21no—
107#11719917—21no—
108#11719917—21no—
109#11719917—21no—
110#11719917—21no—
111C2×#88322—21no—
112C2×#88322C22⋊C412yes—
113#11719912—22no—
114#11719912—22no—
115C8×D53—22no—
116C8×D53—22no—
117C16×C53—22no—
118C16×C53—41no—
119#11719908—22no—
120#11719908—41no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—42no—
125C23×C53C22×C811yescharacteristic
126C22×D53C22⋊C812yes—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C22⋊D53—22no—
141C22⋊D53—22no—
142C22⋊D53—44no—
143C2×Dic53—22no—
144C2×Dic53C22⋊C812yes—
145D212—21no—
146D212—21no—
147C4×D53—22no—
148C4×D53—22no—
149Dic106—21no—
150Dic106—21no—
151C8×C53—41no—
152C8×C53#12011yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#88322—22no—
157#88322—41no—
158#12981—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—22no—
162C22×C53—41no—
163C22×C53—41no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53C2×C22⋊C811yescharacteristic
168C22×C53C8×D411yescharacteristic
169C22×C53C8×D411yescharacteristic
170C2×D53—22no—
171C2×D53—42no—
172C4×C53—41no—
173C4×C53#38611yescharacteristic
174C4×C53#40811yescharacteristic
175C4×C53#43511yescharacteristic
176C4×C53#43511yescharacteristic
177Dic53—22no—
178Dic53—42no—
179#3407—531no—
180#2929—531no—
181#2115—531no—
182#2112—531no—
183#2112D5311yescharacteristic, complemented
184#1378—531no—
185#1378—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—42no—
190C2×C53#186911yescharacteristic
191D53—42no—
192C2×#131—531no—
193C23×C8C2×D5311yescharacteristic
194#525—531no—
195#504—1062no—
196#504—1062no—
197C2×M32—531no—
198C2×M32—531no—
199C2×M32—1062no—
200C2×M32C2×D5311yescharacteristic
201C2×M32C2×D5311yescharacteristic
202#435—1062no—
203C8×D4—1062no—
204#424—531no—
205#408—531no—
206#407—531no—
207#405—531no—
208#349—1062no—
209C22⋊C16—22no—
210C22⋊C16—1062no—
211C53#1298111yescharacteristic, complemented, Sylow
212C2×#41—531no—
213C23×C4C4×D5311yescharacteristic
214#131—1061no—
215#131—1061no—
216#131—1061no—
217#131—1061no—
218C2×M16—531no—
219C2×M16—532no—
220C2×M16—531no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—532no—
228C22×C8C4×D5311yescharacteristic
229C22×C8C22×D5311yescharacteristic
230#123—532no—
231C22⋊Q8—1061no—
232#121—1061no—
233#120—532no—
234C4×D4—1062no—
235#117—531no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—1064no—
241M32—1061no—
242M32—1061no—
243M32—1061no—
244M32—1061no—
245C2×C16—21no—
246C2×C16—532no—
247C2×C16—1064no—
248C2×C16—1061no—
249C2×C16C22⋊D5312yescomplemented
250C4⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253#97—1061no—
254C4×C8—1061no—
255C24C8×D5311yescharacteristic
256#41—1061no—
257#41—1061no—
258#41—1061no—
259#41—1061no—
260C2×Q8—531no—
261C2×D4—532no—
262C2×D4—531no—
263C2×D4—1062no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—532no—
271C22×C4C8×D5311yescharacteristic
272C22×C4C2×C4×D5311yescharacteristic
273M16—1061no—
274M16—1061no—
275M16—1062no—
276M16—1062no—
277M16—1061no—
278M16—1061no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—532no—
287C2×C8—1062no—
288C2×C8—1062no—
289C2×C8—1061no—
290C2×C8#1171994511yescharacteristic
291C2×C8D4×D5311yescharacteristic
292C2×C8D4×D5311yescharacteristic
293C2×C8D4×D5311yescharacteristic
294C2×C8#1171994711yescharacteristic
295C2×C8C2×C22⋊D5311yescharacteristic
296C4⋊C4—1062no—
297C4⋊C4—1061no—
298C22⋊C4—532no—
299C22⋊C4—1062no—
300C22⋊C4—1062no—
301C42—1061no—
302C16—22no—
303C16—41no—
304C16—1062no—
305C16—2121no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—42no—
310C23—532no—
311C23C2×C8×D5311yescharacteristic
312Q8—1061no—
313Q8—1061no—
314D4—1062no—
315D4—1062no—
316D4—1061no—
317D4—1061no—
318D4—2124no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—532no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—1062no—
330C2×C4C22⋊C4×D5311yescharacteristic
331C2×C4#1313317711yescharacteristic
332C2×C4C4×C22⋊D5311yescharacteristic
333C2×C4M16×D5311yescharacteristic
334C2×C4#1313322311yescharacteristic
335C2×C4#1313322511yescharacteristic
336C8—41no—
337C8—1062no—
338C8—2121no—
339C8#1313318511yescharacteristic
340C8#1313318811yescharacteristic
341C8#1313319011yescharacteristic
342C8#1313319911yescharacteristic
343C22—22no—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—1062no—
352C22—2122no—
353C22C22⋊C8×D5311yescharacteristic
354C22#1681068611yescharacteristic
355C22C8×C22⋊D5311yescharacteristic
356C4—41no—
357C4—1062no—
358C4—2122no—
359C4#1681049911yescharacteristic
360C4#1681058811yescharacteristic
361C4#1681069211yescharacteristic
362C4#1681069711yescharacteristic
363C2—21no—
364C2—21no—
365C2—21no—
366C2—42no—
367C2—2122no—
368C2#2027500411yescharacteristic
subgroupG/H
centerC8#13133185
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2112C2
socleC2×C53#1869
prime radical#16811134C4
cosocleC22×C8×C53C23
hypercenter#2112D53
hypocenterC53#12981
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
1. #20287817C2semidirect
3. #20276335C2semidirect
4. #20276326C2semidirect
5. #20272802C2semidirect
6. #20272800C2semidirect
7. C53×#2112C2semidirect
48. C2×#11719908D4semidirect
69. C2×C16×C53D4semidirect
183. #2112D53semidirect
211. C53#12981semidirect
249. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202878172
#202834092
#202763352
#202763262
#202728022
#202728002
C53×#21122
#1298153