Finite Group Databasebeta

← #27286434

Subgroups — #27286434

Order 13568.

712 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#16810913C212yesmaximal, complemented
2#20275516C211yesmaximal, characteristic, complemented
3C2×D4⋊#88322C212yesmaximal, complemented
4#20272598C211yesmaximal, characteristic, complemented
5C53×#2537C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810913—24no—
8#16810913—24no—
9C2×#13133148—22no—
10C2×C4⋊#88322C2212yes—
11C2×#13133140—22no—
12C2×#13133138—22no—
13C2×C4×#88322C2211yescharacteristic
14D4⋊#88322—24no—
15C2×#121×C53—22no—
16C2×C4×D4×C53C2212yes—
17C53×#394—22no—
18C53×#384C2211yescharacteristic
19C53×#380—22no—
20C22×Dic106—21no—
21C22×#88322—21no—
22C22×#88322—22no—
23#13133217—21no—
24#13133217—22no—
25#13133217D412yescomplemented
26C2×C4⋊Dic53—22no—
27#13133148—28no—
28C4⋊#88322—22no—
29C4⋊#88322D414yescomplemented
30#13133140—44no—
31#13133138—28no—
32C4×#88322—22no—
33C22×D4×C53—22no—
34C22×D4×C53D412yes—
35C23×C4×C53—22no—
36#121×C53—44no—
37#121×C53—44no—
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41C4×D4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53D412yes—
44C2×C4⋊C4×C53D412yes—
45C2×C22⋊C4×C53—24no—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53C2311yescharacteristic
48#95×C53—22no—
49C24×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—22no—
54C2×D4×C53—22no—
55C2×D4×C53—48no—
56C2×D4×C53—82no—
57C2×D4×C53SD1614yes—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×Dic53—41no—
69C2×Dic106—22no—
70C2×Dic106—42no—
71C2×Dic106—42no—
72C2×Dic106—41no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—22no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C22⋊C4×C53—48no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—42no—
81C42×C53C2×D411yescharacteristic
82C42×C53C2×D412yes—
83C42×C53#4111yescharacteristic
84C4⋊Dic53—24no—
85C4⋊Dic53—42no—
86C2×#88322—22no—
87C2×#88322—24no—
88C2×#88322—41no—
89C2×#88322—41no—
90C2×#88322—42no—
91C2×#88322—42no—
92C23×C53—24no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53—42no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53#12011yescharacteristic
100D4×C53—24no—
101D4×C53—24no—
102D4×C53—42no—
103D4×C53—42no—
104D4×C53—88no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—44no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—82no—
113C2×C4×C53—82no—
114C2×C4×C53—82no—
115C2×C4×C53—82no—
116C2×C4×C53#12012yes—
117C2×C4×C53#12111yescharacteristic
118C2×C4×C53#12112yes—
119C2×C4×C53#12711yescharacteristic
120C2×C4×C53C2×SD1612yes—
121C2×C4×C53#13512yes—
122C2×C4×C53AGL(1,ℤ/8)12yes—
123C2×C4×C53#13712yes—
124C2×Dic53—42no—
125C2×Dic53—81no—
126C2×Dic53—81no—
127Dic106—44no—
128Dic106—82no—
129#88322—42no—
130#88322—44no—
131#10235—531nomaximal, Sylow
132C22×C53—44no—
133C22×C53—44no—
134C22×C53—44no—
135C22×C53—82no—
136C22×C53—82no—
137C22×C53—82no—
138C22×C53—82no—
139C22×C53#39211yescharacteristic
140C22×C53#45012yes—
141C22×C53#45212yes—
142C22×C53#45311yescharacteristic
143C22×C53#45411yescharacteristic
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—82no—
147C4×C53—82no—
148C4×C53#44812yes—
149C4×C53#44912yes—
150C4×C53#46512yes—
151C4×C53#46812yes—
152Dic53—82no—
153C2×#476—532no—
154#2537D5311yescharacteristic
155#1956—531no—
156C2×D4⋊C8—532no—
157#1296—531no—
158C2×C53—84no—
159C2×C53#162911yescharacteristic
160C2×C53#163411yescharacteristic
161C2×C53#163512yes—
162C2×C53#200012yes—
163C2×C53#200611yescharacteristic
164C2×C4⋊Q8—531no—
165C2×#121—22no—
166C2×C4×D4C2×D5312yes—
167#476—1064no—
168#476—1064no—
169C2×#106—1062no—
170C2×C4⋊C8—532no—
171C2×#103—1062no—
172C2×#102—1062no—
173C2×C4×C8—531no—
174#394—22no—
175#384C2×D5311yescharacteristic
176#380—22no—
177D4⋊C8—1064no—
178C53#1023511yescharacteristic, complemented, Sylow
179C22×Q8—1061no—
180C22×D4—22no—
181C22×D4C22⋊D5312yes—
182C23×C4—22no—
183C22×C8—1061no—
184C22×C8—1062no—
185C4⋊Q8—532no—
186C4⋊Q8—1062no—
187C4⋊Q8—1061no—
188#121—44no—
189#121—44no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4C22⋊D5312yes—
197C2×C4⋊C4C22⋊D5312yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C42C22×D5311yescharacteristic
201#106—1068no—
202C4⋊C8—534no—
203C4⋊C8—1062no—
204#103—2124no—
205#102—1068no—
206C4×C8—1062no—
207#95—22no—
208C24—22no—
209C2×Q8—1062no—
210C2×Q8—2122no—
211C2×Q8—2122no—
212C2×Q8—2121no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—48no—
219C2×D4—82no—
220C2×D4#1171992314yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—2121no—
230C22×C4C2×C22⋊D5311yescharacteristic
231C22×C4C2×C22⋊D5312yes—
232C2×C8—1062no—
233C2×C8—1064no—
234C2×C8—2121no—
235C2×C8—2121no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—1064no—
244C4⋊C4—2122no—
245C22⋊C4—48no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42D4×D5311yescharacteristic
249C42D4×D5312yes—
250C42#1171994711yescharacteristic
251C23—24no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23#1313318611yescharacteristic
259Q8—4242no—
260Q8—2124no—
261D4—24no—
262D4—24no—
263D4—42no—
264D4—42no—
265D4—88no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—4241no—
278C2×C4—4241no—
279C2×C4—2122no—
280C2×C4#1313318512yes—
281C2×C4#1313318811yescharacteristic
282C2×C4#1313318912yes—
283C2×C4#1313321311yescharacteristic
284C2×C4C2×#1171992312yes—
285C2×C4#1313323712yes—
286C2×C4#1313324312yes—
287C2×C4#1313324712yes—
288C8—2122no—
289C8—2124no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22#1681053111yescharacteristic
298C22#1681077412yes—
299C22#1681078512yes—
300C22#1681079111yescharacteristic
301C22#1681079611yescharacteristic
302C4—21no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—4242no—
307C4#1681076112yes—
308C4#1681076612yes—
309C4#1681085012yes—
310C4#1681086712yes—
311C2—84no—
312C2#2027383111yescharacteristic
313C2#2027386011yescharacteristic
314C2#2027386612yes—
315C2#2027571812yes—
316C2#2027575711yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2537C2
socleC23×C53#120
hypercenter#2537D53
hypocenterC53#10235
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16810913C2semidirect
2. #20275516C2semidirect
3. C2×D4⋊#88322C2semidirect
4. #20272598C2semidirect
25. #13133217D4semidirect
29. C4⋊#88322D4semidirect
178. C53#10235semidirect

Maximal subgroups

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

subgroupindex
C2×#168109132
#202755162
C2×D4⋊#883222
#202725982
C53×#25372
#1023553