Finite Group Databasebeta

← #27294783

Subgroups — #27294783

Order 13568.

472 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274798C211yesmaximal, characteristic, complemented
2#20273049C211yesmaximal, characteristic, complemented
3#20272721C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2392C211yesmaximal, characteristic
6#16810680—21no—
7#16810587—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×C4×D4×C53—21no—
13C2×C4×D4×C53C411yescharacteristic
14C2×#117×C53C2211yescharacteristic
15C53×#398C2211yescharacteristic
16C53×#397C411yescharacteristic
17C53×#396—21no—
18C53×#394—21no—
19C53×#394C2211yescharacteristic
20C53×#392C411yescharacteristic
21C4×C4⋊C4×C53C411yescharacteristic
22C4×C22⋊C4×C53—21no—
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28#13133128—41no—
29C22×D4×C53—21no—
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53D411yescharacteristic
33C4×D4×C53—44no—
34C4×D4×C53—41no—
35C4×D4×C53—41no—
36#117×C53—41no—
37#117×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C24×C53C22⋊C411yescharacteristic
59C2×D4×C53—42no—
60C2×D4×C53—41no—
61C2×D4×C53—42no—
62C2×D4×C53—42no—
63C2×D4×C53—42no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—41no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—42no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—42no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—42no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—42no—
98#11719917—42no—
99#11719917—42no—
100C2×#88322—42no—
101C2×#88322—42no—
102C23×C53—22no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53C2×C22⋊C411yescharacteristic
115D4×C53—42no—
116D4×C53—48no—
117D4×C53—42no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C2×C4×C53#9911yescharacteristic
144C2×C4×C53#9911yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)12yes—
146#88322—82no—
147#88322—82no—
148#11804—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53C2×#9911yescharacteristic
164C22×C53#41811yescharacteristic
165C22×C53#42111yescharacteristic
166C4×C53—22no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—81no—
175C4×C53—81no—
176#2392D5311yescharacteristic
177#1822—531no—
178#1464—531no—
179#1346—532no—
180#1345—532no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—81no—
184C2×C53—82no—
185C2×C53#151011yescharacteristic
186C2×C53#212011yescharacteristic
187C2×C53#213211yescharacteristic
188C2×C4×D4—21no—
189C2×C4×D4Dic5311yescharacteristic
190C2×#117C2×D5311yescharacteristic
191#432—1061no—
192#407—1061no—
193#398C2×D5311yescharacteristic
194#397Dic5311yescharacteristic
195#396—21no—
196#394—21no—
197#394C2×D5311yescharacteristic
198#392Dic5311yescharacteristic
199C4×C4⋊C4Dic5311yescharacteristic
200C4×C22⋊C4—21no—
201#343—532no—
202#343—1061no—
203#324—532no—
204#323—1061no—
205C53#1180411yescharacteristic, complemented, Sylow
206C22×D4—21no—
207C22×D4C2×Dic5311yescharacteristic
208C23×C4—21no—
209C23×C4C22⋊D5311yescharacteristic
210C2×M16—1062no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—44no—
214#117—41no—
215#117—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×Dic5311yescharacteristic
219C2×C4⋊C4C22⋊D5311yescharacteristic
220C2×C4⋊C4C22×D5311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4C2×Dic5311yescharacteristic
227C2×C22⋊C4C22⋊D5311yescharacteristic
228C2×C42—21no—
229C2×C42C2×Dic5311yescharacteristic
230C2×C42C22⋊D5311yescharacteristic
231C4⋊C8—2121no—
232C22⋊C8—1062no—
233C22⋊C8—2121no—
234C22⋊C8—2121no—
235#97—2121no—
236#95—21no—
237#95—21no—
238#95C2×Dic5311yescharacteristic
239#95C2×Dic5311yescharacteristic
240C24—21no—
241C24C22⋊Dic5311yescharacteristic
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C22⋊Dic5311yescharacteristic
259C22×C4C22⋊Dic5311yescharacteristic
260C22×C4C22⋊Dic5311yescharacteristic
261C22×C4C22×Dic5311yescharacteristic
262C22×C4C2×C22⋊D5311yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264M16—2122no—
265M16—2122no—
266C2×C8—2122no—
267C2×C8—2122no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—42no—
285C23—22no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Dic5311yescharacteristic
298D4—42no—
299D4—48no—
300D4—42no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#1313313211yescharacteristic
327C2×C4#1313313211yescharacteristic
328C2×C4#1313324312yes—
329C8—4242no—
330C8—4242no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#1313313211yescharacteristic
346C22#1681062811yescharacteristic
347C22#1681064011yescharacteristic
348C4—22no—
349C4—21no—
350C4—21no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—81no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—82no—
362C2#2027324411yescharacteristic
363C2#2027635611yescharacteristic
364C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2392C2
socleC22×C53C2×#99
prime radicalC53×#392C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2392D53
hypocenterC53#11804
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2392

Complements

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

NH ≅ G/Ntype
1. #20274798C2semidirect
2. #20273049C2semidirect
3. #20272721C2semidirect
4. #20272718C2semidirect
205. C53#11804semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730492
#202727212
#202727182
C53×#23922
#1180453