Finite Group Databasebeta

← #27295137

Subgroups — #27295137

Order 13568.

764 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278071C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic, complemented
3#20272717C214yesmaximal, complemented
4C53×#1800C211yesmaximal, characteristic, complemented
5C23×D212C411yescharacteristic
6C2×#13133191—21no—
7C2×#13133185—21no—
8C2×C22⋊#88322—21no—
9#16810540C2212yes—
10#16810532C412yes—
11#16810525—21no—
12#16810483C411yescharacteristic
13#16810478—21no—
14#16810349C2212yescomplemented
15#16810347—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#423—21no—
18C2×#100×C53—21no—
19C53×#343C2212yescomplemented
20C24×D53—21no—
21C22×D212—21no—
22C22×D212—44no—
23C22×D212—41no—
24C22×D212—41no—
25C22×D212C2×C412yes—
26C22×C22⋊D53—22no—
27C22×#88322—41no—
28#13133191—44no—
29#13133185—42no—
30#13133185—42no—
31C2×C4⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×#11719921—22no—
34C2×#11719921—22no—
35C2×#11719921—22no—
36C22⋊#88322—22no—
37C22⋊#88322—41no—
38#13133124C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53C2×C411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C2×M16×C53—22no—
43C4×D4×C53—41no—
44C4×D4×C53—41no—
45C2×C4⋊C4×C53C2311yescharacteristic
46C2×C22⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C4⋊C8×C53—41no—
50C4⋊C8×C53—41no—
51#100×C53—42no—
52C24×C53C22⋊C411yescharacteristic
53C24×C53C22⋊C411yescharacteristic
54C23×D53—22no—
55C23×D53—44no—
56C23×D53—42no—
57C23×D53—42no—
58C23×D53—42no—
59C23×D53—82no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53D812yes—
63C2×D4×C53SD1612yes—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—44no—
76C2×C22⋊D53—44no—
77C2×C22⋊D53—82no—
78C22×Dic53—22no—
79C2×D212—24no—
80C2×D212—42no—
81C2×D212—42no—
82C2×D212—48no—
83C2×D212—44no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—44no—
87C2×D212—44no—
88M16×C53—42no—
89M16×C53—42no—
90C2×C8×C53—42no—
91C4⋊C4×C53—41no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C42×C53—21no—
95C42×C53—21no—
96C22⋊Dic53—42no—
97#11719921—44no—
98#11719921—42no—
99#11719921—42no—
100#11719921—82no—
101C4⋊Dic53—42no—
102C2×#88322—42no—
103C2×#88322—82no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53#10212yes—
113C23×C53#10212yes—
114C23×C53C2×C22⋊C411yescharacteristic
115C22×D53—24no—
116C22×D53—42no—
117C22×D53—42no—
118C22×D53—42no—
119C22×D53—48no—
120C22×D53—42no—
121C22×D53—42no—
122C22×D53—42no—
123C22×D53—88no—
124C22×D53—88no—
125C22×D53—84no—
126C22×D53—84no—
127D4×C53—42no—
128D4×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—22no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—81no—
142C2×C4×C53#9911yescharacteristic
143C2×C4×C53#9911yescharacteristic
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53C2×SD1611yescharacteristic
146C22⋊D53—88no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150D212—44no—
151D212—44no—
152D212—44no—
153D212—44no—
154C8×C53—82no—
155#88322—82no—
156#11880—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—42no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—81no—
169C22×C53C2×#9911yescharacteristic
170C22×C53C2×#10211yescharacteristic
171C22×C53#42111yescharacteristic
172C2×D53—44no—
173C2×D53—44no—
174C2×D53—44no—
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—816no—
178C2×D53—82no—
179C2×D53—82no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53AΣL(1,𝔽16)12yes—
187Dic53—82no—
188#2408—531no—
189#1800D5311yescharacteristic, complemented
190#1459—531no—
191#1345—534no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—42no—
195C2×C53—42no—
196C2×C53#151911yescharacteristic
197C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
198C2×C53#213411yescharacteristic
199D53—84no—
200C23×D4—531no—
201C2×#121—1061no—
202C2×#120—1061no—
203C2×C4×D4C2×D5311yescharacteristic
204#423—21no—
205C2×#100—21no—
206C2×C22⋊C8—1061no—
207#394—532no—
208#392—532no—
209#390—1061no—
210#380—531no—
211#379—1061no—
212#343C2×D5312yescomplemented
213#324—532no—
214#323—1061no—
215C53#1188011yescharacteristic, complemented, Sylow
216C25—1061no—
217C22×D4—532no—
218C22×D4—1062no—
219C22×D4—1061no—
220C22×D4—2121no—
221C22×D4—2121no—
222C22×D4—2124no—
223C22×D4C22⋊D5311yescharacteristic
224C23×C4C4×D5311yescharacteristic
225C23×C4D21211yescharacteristic
226C2×M16—22no—
227C22×C8—2121no—
228#121—2124no—
229#120—2122no—
230#120—2122no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4C22×D5311yescharacteristic
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1062no—
238C2×C22⋊C4—1062no—
239C2×C22⋊C4C4×D5311yescharacteristic
240C2×C22⋊C4D21211yescharacteristic
241C2×C42C22⋊D5311yescharacteristic
242C4⋊C8—41no—
243C4⋊C8—41no—
244#100—42no—
245C22⋊C8—1062no—
246C22⋊C8—2121no—
247#95—532no—
248C24—1062no—
249C24—4242no—
250C24—2122no—
251C24—2122no—
252C24—2124no—
253C24—2122no—
254C24#1171992111yescharacteristic
255C24#1171992111yescharacteristic
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—1064no—
259C2×D4—4242no—
260C2×D4—2122no—
261C2×D4—2122no—
262C2×D4—2128no—
263C2×D4—2124no—
264C2×D4—2124no—
265C2×D4—2124no—
266C2×D4—2124no—
267C2×D4—2124no—
268C2×D4—2121no—
269C2×D4—2121no—
270C2×D4#1171992212yes—
271C2×D4#1171992412yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—1062no—
279C22×C4#1171992111yescharacteristic
280C22×C4#1171992111yescharacteristic
281C22×C4C2×C4×D5311yescharacteristic
282C22×C4C2×D21211yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284M16—42no—
285M16—42no—
286C2×C8—42no—
287C2×C8—4242no—
288C2×C8—2122no—
289C4⋊C4—41no—
290C4⋊C4—2122no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—4242no—
294C22⋊C4—2124no—
295C22⋊C4—2122no—
296C22⋊C4—2122no—
297C22⋊C4—2122no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—1064no—
309C23—4244no—
310C23—4244no—
311C23—4248no—
312C23—4248no—
313C23—2122no—
314C23—2122no—
315C23—2122no—
316C23—2128no—
317C23—2122no—
318C23—2122no—
319C23—2122no—
320C23#1313313712yes—
321C23#1313313712yes—
322C23C2×#1171992111yescharacteristic
323D4—42no—
324D4—42no—
325D4—4248no—
326D4—2124no—
327D4—2124no—
328D4—2124no—
329D4—2124no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4#1313313111yescharacteristic
347C2×C4#1313313111yescharacteristic
348C2×C4C2×#1171992211yescharacteristic
349C2×C4C2×#1171992411yescharacteristic
350C8—82no—
351C8—4242no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—4242no—
365C22—4242no—
366C22—42416no—
367C22—4242no—
368C22—4242no—
369C22—2124no—
370C22—2124no—
371C22—2124no—
372C22C2×#1313313111yescharacteristic
373C22C2×#1313313711yescharacteristic
374C22#1681064111yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—4242no—
382C4#1681041112yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—4244no—
388C2#2027328211yescharacteristic
389C2C2×#1681041111yescharacteristic
390C2#2027640611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radical#20278071C2
cosocleC2×C4⋊C4×C53C23
hypercenter#1800D53
hypocenterC53#11880
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20273033C2semidirect
3. #20272717C2semidirect
4. C53×#1800C2semidirect
14. #16810349C22semidirect
19. C53×#343C22semidirect
189. #1800D53semidirect
212. #343C2×D53semidirect
215. C53#11880semidirect

Maximal subgroups

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

subgroupindex
#202780712
#202730332
#202727172
C53×#18002
#1188053