Finite Group Databasebeta

← #27294671

Subgroups — #27294671

Order 13568.

446 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 91 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#20273059C211yesmaximal, characteristic, complemented
3#20272721C212yesmaximal, complemented
4#20272718C212yesmaximal
5C53×#2536C211yesmaximal, characteristic
6#16810680—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C22⋊Q8×C53—21no—
13C2×C4×Q8×C53C2211yescharacteristic
14C2×C4×D4×C53C411yescharacteristic
15C53×#398C411yescharacteristic
16C53×#398C2211yescharacteristic
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#388—21no—
20C53×#388—21no—
21C53×#382C411yescharacteristic
22C53×#380—21no—
23C2×#11719917—22no—
24C4⋊#88322—41no—
25#13133136—24no—
26C22⋊#88322—22no—
27#13133128—41no—
28#13133128—41no—
29C4×#88322—41no—
30C22×Q8×C53D411yescharacteristic
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53—21no—
33C22⋊Q8×C53—44no—
34C4×Q8×C53—41no—
35C4×Q8×C53—41no—
36C4×D4×C53—41no—
37C4×D4×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53C2311yescharacteristic
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53—21no—
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—42no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
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—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—21no—
96#11719917—24no—
97#11719917—42no—
98C2×#88322—42no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53C2×C22⋊C411yescharacteristic
108Q8×C53—44no—
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
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—21no—
120C2×C4×C53—21no—
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—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—84no—
135C2×C4×C53#9911yescharacteristic
136C2×C4×C53#9911yescharacteristic
137C2×C4×C53#10412yes—
138C2×C4×C53AGL(1,ℤ/8)12yes—
139#88322—44no—
140#88322—82no—
141#11778—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53C2×#9911yescharacteristic
152C22×C53#41811yescharacteristic
153C22×C53C2×#10411yescharacteristic
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—81no—
163C4×C53—81no—
164C4×C53—81no—
165#2536D5311yescharacteristic
166C4×#101—531no—
167#1467—531no—
168#1346—532no—
169#1345—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—82no—
173C2×C53#150711yescharacteristic
174C2×C53#212011yescharacteristic
175C2×C53#213311yescharacteristic
176C2×C22⋊Q8—21no—
177C2×C4×Q8C2×D5311yescharacteristic
178C2×C4×D4Dic5311yescharacteristic
179#432—1061no—
180C2×#101—1061no—
181C4×M16—1061no—
182#398Dic5311yescharacteristic
183#398C2×D5311yescharacteristic
184#394Dic5311yescharacteristic
185#394C2×D5311yescharacteristic
186#388—21no—
187#388—21no—
188#382Dic5311yescharacteristic
189#380—21no—
190#343—532no—
191#324—532no—
192#324—1061no—
193C53#1177811yescharacteristic, complemented, Sylow
194C22×Q8C22⋊D5311yescharacteristic
195C22×D4C2×Dic5311yescharacteristic
196C23×C4—21no—
197C2×M16—1062no—
198C22⋊Q8—44no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×Dic5311yescharacteristic
207C2×C4⋊C4C2×Dic5311yescharacteristic
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C4⋊C4C22×D5311yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42C2×Dic5311yescharacteristic
216C2×C42C22⋊D5311yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—2121no—
219#101—1064no—
220C22⋊C8—1062no—
221#97—2121no—
222#97—2121no—
223C4×C8—2121no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C2×Dic5311yescharacteristic
228#95C2×Dic5311yescharacteristic
229C24—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×D4—41no—
234C2×D4—42no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C22⋊Dic5311yescharacteristic
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22×Dic5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—1064no—
249M16—2122no—
250C2×C8—2122no—
251C2×C8—2122no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊Dic5311yescharacteristic
279Q8—44no—
280D4—42no—
281D4—42no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
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—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—84no—
306C2×C4#1313313211yescharacteristic
307C2×C4#1313313211yescharacteristic
308C2×C4#1313314412yes—
309C2×C4#1313324312yes—
310C8—4242no—
311C8—2124no—
312C22—21no—
313C22—21no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22C2×#1313313211yescharacteristic
322C22#1681062811yescharacteristic
323C22C2×#1313314411yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—82no—
338C2#2027323111yescharacteristic
339C2#2027635611yescharacteristic
340C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2536C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2536D53
hypocenterC53#11778
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. C4×#13133136C2semidirect
2. #20273059C2semidirect
3. #20272721C2semidirect
193. C53#11778semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730592
#202727212
#202727182
C53×#25362
#1177853