Finite Group Databasebeta

← #27294461

Subgroups — #27294461

Order 13568.

434 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#13133136C211yesmaximal, characteristic, complemented
2#20273039C211yesmaximal, characteristic, complemented
3#20272742C212yesmaximal, complemented
4#20272718C212yesmaximal
5C53×#2398C211yesmaximal, characteristic
6#16810676—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C4×Q8×C53C2211yescharacteristic
13C2×C4×D4×C53C411yescharacteristic
14C53×#400C411yescharacteristic
15C53×#400C2211yescharacteristic
16C53×#397—21no—
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#393—21no—
20C4×C22⋊C4×C53—21no—
21C4×C22⋊C4×C53—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25#13133136—24no—
26C22⋊#88322—22no—
27#13133128—41no—
28C4×#88322—41no—
29C4×#88322—41no—
30C22×Q8×C53D411yescharacteristic
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53—21no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C4×D4×C53—41no—
36C4×D4×C53—41no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—42no—
93#11719917—24no—
94#11719917—42no—
95C2×#88322—42no—
96C2×#88322—42no—
97C23×C53—22no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—44no—
106D4×C53—42no—
107D4×C53—42no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—82no—
132C2×C4×C53—82no—
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53#9911yescharacteristic
135C2×C4×C53#10412yes—
136C2×C4×C53#13512yes—
137#88322—44no—
138#88322—82no—
139#11734—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53C2×#9911yescharacteristic
150C22×C53#41611yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C4×C53—21no—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—81no—
160C4×C53—81no—
161C4×C53—81no—
162C4×C53#35512yes—
163#2398D5311yescharacteristic
164C4×#101—531no—
165#1461—531no—
166#1353—532no—
167#1345—532no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—82no—
171C2×C53#150111yescharacteristic
172C2×C53#212011yescharacteristic
173C2×C53C2×#35511yescharacteristic
174C2×C4×Q8C2×D5311yescharacteristic
175C2×C4×D4Dic5311yescharacteristic
176#431—1061no—
177C2×#101—1061no—
178C4×M16—1061no—
179#400Dic5311yescharacteristic
180#400C2×D5311yescharacteristic
181#397—21no—
182#394Dic5311yescharacteristic
183#394C2×D5311yescharacteristic
184#393—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187#376Dic5311yescharacteristic
188#343—532no—
189#324—532no—
190#324—1061no—
191C53#1173411yescharacteristic, complemented, Sylow
192C22×Q8C22⋊D5311yescharacteristic
193C22×D4C2×Dic5311yescharacteristic
194C23×C4—21no—
195C2×M16—1062no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4C2×Dic5311yescharacteristic
201C2×C4⋊C4C2×Dic5311yescharacteristic
202C2×C4⋊C4C22⋊D5311yescharacteristic
203C2×C4⋊C4C22×D5311yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C2×Dic5311yescharacteristic
211C2×C42C22⋊D5311yescharacteristic
212C2×C42C22⋊D5311yescharacteristic
213C4⋊C8—2121no—
214#101—1064no—
215C22⋊C8—1062no—
216#97—2121no—
217C4×C8—2121no—
218C4×C8—2121no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C2×Dic5311yescharacteristic
225#95C2×Dic5311yescharacteristic
226C24—21no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—41no—
231C2×D4—42no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4C22⋊Dic5311yescharacteristic
240C22×C4C22⋊Dic5311yescharacteristic
241C22×C4C22⋊Dic5311yescharacteristic
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22×Dic5311yescharacteristic
244C22×C4C2×C22⋊D5311yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246M16—1064no—
247M16—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—42no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C2×C22⋊Dic5311yescharacteristic
274Q8—44no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#1313313211yescharacteristic
303C2×C4#1313313211yescharacteristic
304C2×C4#1313314412yes—
305C2×C4#1313323712yes—
306C8—4242no—
307C8—2124no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22C2×#1313313211yescharacteristic
318C22#1681061911yescharacteristic
319C22C2×#1313314411yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4#1681042012yes—
331C2—21no—
332C2—21no—
333C2—82no—
334C2#2027320611yescharacteristic
335C2#2027635611yescharacteristic
336C2C2×#1681042011yescharacteristic
subgroupG/H
centerC2×C4#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2398C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2398D53
hypocenterC53#11734
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. C4×#13133136C2semidirect
2. #20273039C2semidirect
3. #20272742C2semidirect
191. C53#11734semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730392
#202727422
#202727182
C53×#23982
#1173453