Finite Group Databasebeta

← #27279618

Subgroups — #27279618

Order 13568.

602 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 127 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#20283756C211yesmaximal, characteristic, complemented
2#20273106C212yesmaximal, complemented
3#20273036C211yesmaximal, characteristic, complemented
4#20272544C212yesmaximal, complemented
5C53×#2328C211yesmaximal, characteristic
6#16810701—24no—
7#16810676C2212yes—
8#16810676C2212yes—
9#16810587—22no—
10D4⋊#88322—24no—
11#16810347—22no—
12C2×#127×C53C411yescharacteristic
13C2×C4×D4×C53C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C53×#390C411yescharacteristic
16C53×#390C411yescharacteristic
17C53×#386—22no—
18C53×#386—22no—
19C53×#382C411yescharacteristic
20C53×#376C2211yescharacteristic
21C2×#11719917—24no—
22C4⋊#88322—22no—
23C4⋊#88322—22no—
24C22⋊#88322—24no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27C4×#88322—22no—
28C4×#88322D414yescomplemented
29C22×D4×C53—22no—
30C22×D4×C53C2×C411yescharacteristic
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53—21no—
33C23×C4×C53D412yes—
34#127×C53—22no—
35#127×C53—22no—
36C4×D4×C53—22no—
37C4×D4×C53—22no—
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53D412yes—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53C2311yescharacteristic
49#95×C53—22no—
50#95×C53—21no—
51#95×C53—21no—
52C24×C53—21no—
53C24×C53C22⋊C412yes—
54C2×D4×C53—28no—
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—82no—
63C2×D4×C53—82no—
64C2×D4×C53M1612yescomplemented
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—42no—
71C22×C4×C53—82no—
72C22×C4×C53—82no—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C4⋊C4×C53—22no—
78C4⋊C4×C53—22no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—82no—
83C22⋊C4×C53—82no—
84C42×C53—21no—
85C42×C53—22no—
86C42×C53M1612yescomplemented
87C42×C53C2×D412yes—
88C42×C53#4112yes—
89#11719917—84no—
90C2×#88322—24no—
91C2×#88322—42no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53C2×C22⋊C411yescharacteristic
103D4×C53—24no—
104D4×C53—42no—
105D4×C53—42no—
106D4×C53—42no—
107D4×C53—88no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—44no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—82no—
122C2×C4×C53—84no—
123C2×C4×C53C4×D412yes—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53#12112yes—
126C2×C4×C53#12311yescharacteristic
127C2×C4×C53C2×M1611yescharacteristic
128C2×C4×C53C2×M1611yescharacteristic
129C2×C4×C53#13112yes—
130#88322—44no—
131#88322—82no—
132#9105—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—44no—
136C22×C53—44no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—82no—
142C22×C53—82no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53#38611yescharacteristic
146C22×C53#40711yescharacteristic
147C22×C53#40811yescharacteristic
148C4×C53—22no—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53—81no—
154C4×C53—81no—
155C4×C53#43512yes—
156C4×C53#43612yes—
157#2975—531no—
158#2328D5311yescharacteristic
159#1478—532no—
160#1460—531no—
161#1274—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53—82no—
165C2×C53—82no—
166C2×C53#187011yescharacteristic
167C2×C53#191011yescharacteristic
168C2×C53#191111yescharacteristic
169C2×#127Dic5311yescharacteristic
170C2×C4×D4C2×D5311yescharacteristic
171C2×C4×D4C2×D5311yescharacteristic
172#436—1064no—
173#431—532no—
174#431—532no—
175#407—1062no—
176#390Dic5311yescharacteristic
177#390Dic5311yescharacteristic
178#386—22no—
179#386—22no—
180#382Dic5311yescharacteristic
181#376C2×D5311yescharacteristic
182D4⋊C8—1064no—
183#323—1062no—
184C53#910511yescharacteristic, complemented, Sylow
185C22×D4—22no—
186C22×D4C2×Dic5311yescharacteristic
187C22×D4C2×Dic5311yescharacteristic
188C23×C4—21no—
189C23×C4C22⋊D5312yes—
190C2×M16—1064no—
191#127—22no—
192#127—22no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C2×C4⋊C4C2×Dic5311yescharacteristic
198C2×C4⋊C4C2×Dic5311yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊D5312yes—
203C2×C42C2×Dic5311yescharacteristic
204C2×C42C2×Dic5311yescharacteristic
205C2×C42C22×D5311yescharacteristic
206C4⋊C8—1062no—
207C4⋊C8—1062no—
208C22⋊C8—1064no—
209C22⋊C8—1062no—
210C22⋊C8—1062no—
211C4×C8—534no—
212C4×C8—1062no—
213#95—21no—
214#95—21no—
215#95—22no—
216C24—21no—
217C24C22⋊Dic5312yes—
218C2×D4—28no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—82no—
227C2×D4—82no—
228C2×D4#1171991712yescomplemented
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C22⋊Dic5312yes—
238C22×C4C22×Dic5311yescharacteristic
239C22×C4C2×C22⋊D5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241M16—4244no—
242C2×C8—1064no—
243C2×C8—2122no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C22⋊C4—24no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—21no—
252C42—22no—
253C42#1171991712yescomplemented
254C42D4×D5312yes—
255C42#1171994712yes—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic5311yescharacteristic
267D4—24no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4D4×Dic5312yes—
288C2×C4#1313318511yescharacteristic
289C2×C4#1313318912yes—
290C2×C4#1313319811yescharacteristic
291C2×C4C2×#1171991711yescharacteristic
292C2×C4C2×#1171991711yescharacteristic
293C2×C4#1313322412yes—
294C8—4242no—
295C8—2124no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22#1681050311yescharacteristic
309C22#1681058711yescharacteristic
310C22#1681059011yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—81no—
318C4#1681069512yes—
319C4#1681070112yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#2027501311yescharacteristic
325C2#2027524311yescharacteristic
326C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2328C2
socleC22×C53#386
prime radicalC2×#127×C53C4
cosocleC2×C42×C53C23
hypercenter#2328D53
hypocenterC53#9105
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #20283756C2semidirect
2. #20273106C2semidirect
3. #20273036C2semidirect
4. #20272544C2semidirect
28. C4×#88322D4semidirect
64. C2×D4×C53M16semidirect
86. C42×C53M16semidirect
184. C53#9105semidirect
228. C2×D4#11719917semidirect
253. C42#11719917semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202731062
#202730362
#202725442
C53×#23282
#910553