Finite Group Databasebeta

← #27279544

Subgroups — #27279544

Order 13568.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283755C211yesmaximal, characteristic, complemented
2C2×#16810736C211yesmaximal, characteristic
3#20273107C211yesmaximal, characteristic, complemented
4#20273105C211yesmaximal, characteristic, complemented
5#20273035C211yesmaximal, characteristic, complemented
6#20272546C211yesmaximal, characteristic, complemented
7C53×#1281C211yesmaximal, characteristic, complemented
8C22×D424C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810736—22no—
12#16810736—22no—
13#16810702—22no—
14#16810700—22no—
15#16810676C2211yescharacteristic
16#16810676C2211yescharacteristic
17C2×#13133149C411yescharacteristic
18C2×#13133139C411yescharacteristic
19C2×#13133139C411yescharacteristic
20#16810585—22no—
21#16810352—22no—
22#16810350—22no—
23#16810346—22no—
24C53×#431C2211yescharacteristic
25C53×#431C2211yescharacteristic
26C2×#97×C53C2211yescharacteristic
27C22×D212C2×C411yescharacteristic
28C22×D212C2×C411yescharacteristic
29C22×C4×D53—21no—
30C22×C4×D53—21no—
31C2×D424—22no—
32C2×D424—22no—
33C2×D424—22no—
34C2×#11719912—24no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212D412yes—
42C4×D212D412yes—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×C4⋊Dic53C2×C411yescharacteristic
45C2×#11719921—21no—
46C2×#11719921—21no—
47#13133149—22no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133139—22no—
51#13133139—22no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133129—24no—
55C4×#88322—21no—
56C4×#88322D412yes—
57C22×C8×C53C2×C411yescharacteristic
58C22×C8×C53C2×C411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C22⋊C8×C53—21no—
63C22⋊C8×C53D412yes—
64#97×C53—21no—
65#97×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C23×D53—21no—
69C23×D53—21no—
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53#4111yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212C22⋊C412yes—
86C2×D212M1612yescomplemented
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—82no—
92C2×C4×D53—82no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—21no—
96C2×C8×C53—22no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—41no—
100C2×C8×C53M1612yescomplemented
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105#11719921—22no—
106#11719921—22no—
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53C22⋊C412yes—
113C2×#88322—22no—
114C2×#88322—41no—
115D424—28no—
116#11719912—84no—
117C23×C53C4×D411yescharacteristic
118C22×D53—22no—
119C22×D53—22no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—81no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53C2×C22⋊C411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12111yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12311yescharacteristic
136C2×C4×C53C2×M1611yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C2×Dic53—22no—
141C2×Dic53—41no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144D212—22no—
145D212—22no—
146D212—22no—
147D212—22no—
148D212—41no—
149D212—41no—
150D212—41no—
151D212—41no—
152C4×D53—44no—
153C4×D53—82no—
154C4×D53—82no—
155C4×D53—82no—
156C8×C53—22no—
157C8×C53—42no—
158C8×C53—41no—
159C8×C53—81no—
160#88322—42no—
161#88322—81no—
162#9093—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53#38611yescharacteristic
166C22×C53#43511yescharacteristic
167C22×C53#43611yescharacteristic
168C2×D53—42no—
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—42no—
172C2×D53—82no—
173C2×D53—82no—
174C2×D53—82no—
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—82no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53#40711yescharacteristic
183C4×C53#40811yescharacteristic
184C4×C53#43511yescharacteristic
185C4×C53#43611yescharacteristic
186Dic53—42no—
187Dic53—81no—
188#2975—531no—
189C2×#442—531no—
190#1478—531no—
191#1478—531no—
192#1460—531no—
193#1281D5311yescharacteristic, complemented
194#1275—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53#187011yescharacteristic
198C2×C53#190311yescharacteristic
199C2×C53#190411yescharacteristic
200D53—82no—
201D53—82no—
202C22×D8—531no—
203C2×C4×D4—531no—
204C2×C4×D4—531no—
205#442—1062no—
206#442—1062no—
207#436—1062no—
208#436—1062no—
209#431—531no—
210#431—531no—
211#431C2×D5311yescharacteristic
212#431C2×D5311yescharacteristic
213C2×#106—531no—
214C2×#102—531no—
215C2×#102—531no—
216#407—1062no—
217C2×#97C2×D5311yescharacteristic
218D4⋊C8—1062no—
219D4⋊C8—1062no—
220#323—1062no—
221C53#909311yescharacteristic, complemented, Sylow
222C22×D4—531no—
223C22×D4—531no—
224C23×C4—1061no—
225C23×C4—1061no—
226C2×D8—1062no—
227C2×D8—1062no—
228C2×D8—1062no—
229C2×M16—1064no—
230C22×C8C4×D5311yescharacteristic
231C22×C8C4×D5311yescharacteristic
232C4×D4—532no—
233C4×D4—532no—
234C4×D4—1061no—
235C4×D4—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C4×D4—1061no—
239C4×D4—1061no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42C22×D5311yescharacteristic
245#106—1062no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—1061no—
249C4⋊C8—1061no—
250#102—1062no—
251#102—1062no—
252C22⋊C8—21no—
253C22⋊C8—1064no—
254C22⋊C8—1061no—
255C22⋊C8—1061no—
256C22⋊C8D21212yes—
257#97—21no—
258#97—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—532no—
262C4×C8—1061no—
263C24—1061no—
264C24—1061no—
265C2×D4—532no—
266C2×D4—532no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C22×C4—1061no—
278C22×C4—1062no—
279C22×C4—1062no—
280C22×C4—1061no—
281C22×C4—4242no—
282C22×C4—4242no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4C2×C4×D5311yescharacteristic
286C22×C4C2×D21211yescharacteristic
287C22×C4#1171994511yescharacteristic
288D8—1068no—
289M16—4244no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—1062no—
298C2×C8—2121no—
299C2×C8#1171991212yescomplemented
300C4⋊C4—532no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C22⋊C4—1062no—
305C22⋊C4—1062no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C42D4×D5311yescharacteristic
309C42D4×D5311yescharacteristic
310C42D4×D5311yescharacteristic
311C42#1171994911yescharacteristic
312C23—1062no—
313C23—1062no—
314C23—4241no—
315C23—4241no—
316C23—4241no—
317C23—4241no—
318C23—2121no—
319C23—2121no—
320C23—2121no—
321C23—2121no—
322C23C4×D21211yescharacteristic
323D4—1062no—
324D4—1062no—
325D4—1062no—
326D4—1062no—
327D4—2121no—
328D4—2121no—
329D4—2121no—
330D4—2121no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—1062no—
334C2×C4—4242no—
335C2×C4—4242no—
336C2×C4—4242no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—2124no—
340C2×C4—2121no—
341C2×C4C22⋊C4×D5311yescharacteristic
342C2×C4#1313317911yescharacteristic
343C2×C4#1313318711yescharacteristic
344C2×C4#1313319011yescharacteristic
345C2×C4#1313319211yescharacteristic
346C2×C4#1313320211yescharacteristic
347C2×C4M16×D5311yescharacteristic
348C2×C4C2×#1171991211yescharacteristic
349C2×C4#1313322311yescharacteristic
350C2×C4#1313322511yescharacteristic
351C8—22no—
352C8—42no—
353C8—41no—
354C8—81no—
355C8—4241no—
356C8—2122no—
357C22—21no—
358C22—21no—
359C22—4242no—
360C22—4242no—
361C22—4242no—
362C22—4242no—
363C22—4241no—
364C22—4241no—
365C22—2122no—
366C22—2122no—
367C22—2122no—
368C22—2122no—
369C22#1681050111yescharacteristic
370C22#1681069611yescharacteristic
371C22#1681070211yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—4241no—
377C4—2122no—
378C4#1681058511yescharacteristic
379C4#1681058811yescharacteristic
380C4#1681069311yescharacteristic
381C4#1681070011yescharacteristic
382C2—21no—
383C2—21no—
384C2—4242no—
385C2—4242no—
386C2#2027501111yescharacteristic
387C2#2027519411yescharacteristic
388C2#2027520111yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1281C2
socleC22×C53#386
prime radicalC22×D424C4
cosocleC2×C42×C53C23
hypercenter#1281D53
hypocenterC53#9093
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
1. #20283755C2semidirect
3. #20273107C2semidirect
4. #20273105C2semidirect
5. #20273035C2semidirect
6. #20272546C2semidirect
7. C53×#1281C2semidirect
86. C2×D212M16semidirect
100. C2×C8×C53M16semidirect
193. #1281D53semidirect
221. C53#9093semidirect
299. C2×C8#11719912semidirect

Maximal subgroups

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

subgroupindex
#202837552
C2×#168107362
#202731072
#202731052
#202730352
#202725462
C53×#12812
#909353