Finite Group Databasebeta

← #27273040

Subgroups — #27273040

Order 13568.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#20282706C211yesmaximal, characteristic
2#20276049C211yesmaximal, characteristic
3#20276044C211yesmaximal, characteristic
4#20275961C211yesmaximal, characteristic
5#20272539C211yesmaximal, characteristic
6#20272537C211yesmaximal, characteristic
7C53×#1271C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×#13133151—21no—
10C2×#13133149—21no—
11C2×#13133140—21no—
12C2×#13133140—21no—
13#16810563—21no—
14#16810562—21no—
15#16810561—21no—
16#16810521C2211yescharacteristic
17#16810521C2211yescharacteristic
18#16810510—21no—
19#16810509—21no—
20#16810504—21no—
21#16810386—21no—
22#16810386—21no—
23#16810378C2211yescharacteristic
24C53×#389C2211yescharacteristic
25C53×#336C2211yescharacteristic
26C53×#336C2211yescharacteristic
27C22×Dic106—21no—
28C22×#88322—21no—
29#13133217—42no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53D411yescharacteristic
33C2×C4⋊Dic53D411yescharacteristic
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×C4×Dic53—21no—
39#13133151—42no—
40#13133149—42no—
41#13133140—42no—
42#13133140—42no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C42×C53C2311yescharacteristic
58#95×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—41no—
69C2×Dic106—22no—
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106—41no—
75C2×C8×C53—22no—
76C2×C8×C53—22no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C4⋊C4×C53—21no—
82C4⋊C4×C53—21no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C42×C53—41no—
87C4⋊Dic53—22no—
88C4⋊Dic53—22no—
89C4⋊Dic53—21no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92C4⋊Dic53—21no—
93C4⋊Dic53—41no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97#11719919—41no—
98#11719919—41no—
99#11719919—41no—
100#11719919—41no—
101C4×Dic53—81no—
102C2×#88322—41no—
103C2×#88322—41no—
104C2×#88322—41no—
105C23×C53#12011yescharacteristic
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53#13511yescharacteristic
119C2×C4×C53#13511yescharacteristic
120C2×C4×C53#13511yescharacteristic
121C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C53#13711yescharacteristic
123C2×C4×C53#13711yescharacteristic
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—42no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—81no—
138C2×Dic53—81no—
139Dic106—42no—
140Dic106—42no—
141Dic106—81no—
142Dic106—81no—
143C8×C53—42no—
144C8×C53—42no—
145#88322—81no—
146#7775—531nomaximal, Sylow
147C22×C53#44911yescharacteristic
148C22×C53#45211yescharacteristic
149C22×C53#45211yescharacteristic
150C22×C53#45411yescharacteristic
151C22×C53#45611yescharacteristic
152C22×C53#45611yescharacteristic
153C22×C53B(4,𝔽2)11yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—81no—
161C4×C53—81no—
162Dic53—82no—
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166Dic53—81no—
167#2810—531no—
168#2058—531no—
169#2057—531no—
170#2043—531no—
171#1271—531no—
172#1271—531no—
173#1271D5311yescharacteristic
174C2×C53#159711yescharacteristic
175C2×C53#160311yescharacteristic
176C2×C53#160311yescharacteristic
177C2×C53#163811yescharacteristic
178C2×C53#165011yescharacteristic
179C2×C53#165211yescharacteristic
180C2×C53#166311yescharacteristic
181C2×C4⋊Q8—531no—
182C2×#107—1061no—
183C2×#106—1061no—
184C2×#103—1061no—
185C2×#103—1061no—
186#400—1061no—
187#400—1061no—
188#400—1061no—
189#389—531no—
190#389—531no—
191#389C2×D5311yescharacteristic
192#387—1061no—
193#387—1061no—
194#387—1061no—
195#340—1061no—
196#340—1061no—
197#336—531no—
198#336C2×D5311yescharacteristic
199#336C2×D5311yescharacteristic
200C53#777511yescharacteristic, complemented, Sylow
201C22×Q8—1061no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—1061no—
205C4⋊Q8—2122no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4D21211yescharacteristic
216C2×C4⋊C4D21211yescharacteristic
217C2×C42—1061no—
218C2×C42C22×D5311yescharacteristic
219#107—2122no—
220#106—2122no—
221#103—2122no—
222#103—2122no—
223#95—21no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C2×Q8—1062no—
234C2×Q8—2121no—
235C2×Q8—2121no—
236C2×Q8—2121no—
237C2×Q8—2121no—
238C2×Q8—2121no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1061no—
245C22×C4—2121no—
246C22×C4C2×D21211yescharacteristic
247C22×C4D4×D5311yescharacteristic
248C22×C4D4×D5311yescharacteristic
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—2121no—
256C2×C8—2121no—
257C2×C8—2121no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—1062no—
264C4⋊C4—1062no—
265C4⋊C4—1061no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C4⋊C4—2121no—
276C4⋊C4—2121no—
277C42—41no—
278C42—4241no—
279C23#1313318711yescharacteristic
280Q8—4241no—
281Q8—4241no—
282Q8—2122no—
283Q8—2122no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—4241no—
297C2×C4—4241no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2122no—
311C2×C4#1313323611yescharacteristic
312C2×C4#1313323811yescharacteristic
313C2×C4#1313323911yescharacteristic
314C2×C4#1313324511yescharacteristic
315C2×C4#1313324611yescharacteristic
316C2×C4#1313324911yescharacteristic
317C8—42no—
318C8—42no—
319C8—4241no—
320C22#1681077011yescharacteristic
321C22#1681078611yescharacteristic
322C22#1681078811yescharacteristic
323C22#1681079811yescharacteristic
324C22#1681080711yescharacteristic
325C22#1681080711yescharacteristic
326C22#1681081111yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—81no—
334C4—81no—
335C4—4241no—
336C4—4241no—
337C4—4241no—
338C4—4242no—
339C4—4241no—
340C2#2027364611yescharacteristic
341C2#2027367911yescharacteristic
342C2#2027368111yescharacteristic
343C2#2027388811yescharacteristic
344C2#2027397311yescharacteristic
345C2#2027398711yescharacteristic
346C2#2027404411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1271C2
socle, prime radicalC23×C53#120
hypercenter#1271D53
hypocenterC53#7775
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
200. C53#7775semidirect

Maximal subgroups

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

subgroupindex
#202827062
#202760492
#202760442
#202759612
#202725392
#202725372
C53×#12712
#777553