Finite Group Databasebeta

← #27300918

Subgroups — #27300918

Order 13568.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277889C211yesmaximal, characteristic
2#20275119C211yesmaximal, characteristic
3#20274932C212yesmaximal, complemented
4#20272841C212yesmaximal, complemented
5C2×C53×#355C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810665—22no—
8#16810628—21no—
9C2×#13133136C2211yescharacteristic
10#16810530C2212yes—
11#16810516—22no—
12#16810503—21no—
13#16810498—22no—
14#16810489—21no—
15#16810384—22no—
16#16810383—21no—
17C2×#124×C53C2211yescharacteristic
18C2×#101×C53C2212yes—
19C53×#355C414yescomplemented
20C23×Dic53—21no—
21C2×#11719917—21no—
22Q8×Dic53—41no—
23Q8×Dic53—41no—
24#13133176—21no—
25#13133176—22no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53D412yes—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—21no—
31C2×C4×Dic53D412yes—
32#13133151—42no—
33#13133149—42no—
34#13133138—41no—
35#13133138—41no—
36#13133136—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40C22×Q8×C53C2311yescharacteristic
41C22×D4×C53D411yescharacteristic
42C2×M16×C53—22no—
43#124×C53—22no—
44#124×C53C2×C412yes—
45#124×C53Q812yes—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53D411yescharacteristic
48#101×C53—22no—
49#101×C53C2×C414yescomplemented
50C24×C53—21no—
51C2×Q8×C53—41no—
52C2×Q8×C53—41no—
53C2×Q8×C53C22×C411yescharacteristic
54C2×Q8×C53C2×Q811yescharacteristic
55C2×Q8×C53#4112yes—
56C2×D4×C53—21no—
57C2×D4×C53—41no—
58C2×D4×C53—41no—
59C2×D4×C53C4⋊C412yes—
60C22×C4×C53—21no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D412yes—
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—82no—
66C22×Dic53—81no—
67M16×C53—24no—
68M16×C53—42no—
69C2×C8×C53—42no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—24no—
72C42×C53—21no—
73C42×C53C4⋊C412yes—
74C22⋊Dic53—41no—
75C22⋊Dic53—82no—
76C22⋊Dic53—81no—
77C4⋊Dic53—21no—
78C4⋊Dic53—21no—
79C4⋊Dic53—22no—
80C4⋊Dic53—22no—
81C4⋊Dic53—42no—
82C4⋊Dic53—81no—
83C4⋊Dic53—81no—
84C4×Dic53—41no—
85C4×Dic53—42no—
86C4×Dic53—42no—
87#11719917—41no—
88#11719917—41no—
89C2×#88322—41no—
90C23×C53—22no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#12011yescharacteristic
96Q8×C53—41no—
97Q8×C53—41no—
98Q8×C53—42no—
99D4×C53—42no—
100D4×C53—81no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—22no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53C2×C4⋊C411yescharacteristic
109C2×C4×C53C4×D412yes—
110C2×C4×C53C22⋊Q812yes—
111C2×C4×C53#12311yescharacteristic
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—42no—
120C2×Dic53—84no—
121C8×C53—44no—
122#88322—81no—
123#12970—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53#38511yescharacteristic
132C22×C53B(4,𝔽2)11yescharacteristic
133C22×C53#45811yescharacteristic
134C4×C53—21no—
135C4×C53—22no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—42no—
139Dic53—82no—
140Dic53—81no—
141Dic53—82no—
142Dic53—81no—
143#2379—531no—
144C2×#355D5311yescharacteristic
145#1888—531no—
146#1857—532no—
147#1394—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—82no—
151C2×C53#186711yescharacteristic
152C2×C53#220211yescharacteristic
153C2×C53#220311yescharacteristic
154C2×#124C2×D5311yescharacteristic
155#519—531no—
156#428—1062no—
157#418—1061no—
158C2×#101—531no—
159C2×#101C2×D5312yes—
160#391—532no—
161#388—1062no—
162#386—1061no—
163#385—1062no—
164#382—1061no—
165#355Dic5314yescomplemented
166#339—1062no—
167#339—1061no—
168C53#1297011yescharacteristic, complemented, Sylow
169C22×Q8C22×D5311yescharacteristic
170C22×D4D21211yescharacteristic
171C23×C4—1061no—
172C2×M16—22no—
173C2×M16—1061no—
174#124—22no—
175#124Dic10612yes—
176#124C2×Dic5312yes—
177C4×Q8—2121no—
178C4×Q8—2121no—
179#117—1061no—
180#117—1062no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—532no—
188C2×C42D21211yescharacteristic
189#107—2122no—
190#106—2122no—
191#102—2121no—
192#102—2121no—
193#101—22no—
194#101—1062no—
195#101C2×Dic5314yescomplemented
196#95—1062no—
197#95—1062no—
198#95—1062no—
199C24—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8C2×Dic10611yescharacteristic
203C2×Q8#1171994712yes—
204C2×Q8C22×Dic5311yescharacteristic
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊Dic5312yes—
209C22×C4—21no—
210C22×C4—1062no—
211C22×C4—1062no—
212C22×C4—4242no—
213C22×C4—4241no—
214C22×C4C2×D21211yescharacteristic
215C22×C4D4×D5312yes—
216M16—24no—
217M16—42no—
218M16—2121no—
219M16—2121no—
220C2×C8—42no—
221C2×C8—2121no—
222C4⋊C4—1061no—
223C4⋊C4—1061no—
224C4⋊C4—1062no—
225C4⋊C4—1062no—
226C4⋊C4—4241no—
227C4⋊C4—4241no—
228C4⋊C4—2122no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—4242no—
232C22⋊C4—4241no—
233C22⋊C4—2121no—
234C42—21no—
235C42—2122no—
236C42—2122no—
237C42—2121no—
238C42C4⋊Dic5312yes—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23#1313318711yescharacteristic
245Q8—41no—
246Q8—42no—
247Q8—41no—
248D4—42no—
249D4—81no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—4244no—
258C2×C4—2122no—
259C2×C4—2122no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2122no—
265C2×C4—2122no—
266C2×C4C2×C4⋊Dic5311yescharacteristic
267C2×C4D4×Dic5312yes—
268C2×C4#1313319412yes—
269C2×C4#1313320111yescharacteristic
270C8—44no—
271C8—4241no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#1681049811yescharacteristic
280C22#1681081111yescharacteristic
281C22#1681081411yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—4242no—
288C4—4241no—
289C4—4242no—
290C4—4241no—
291C2—21no—
292C2—21no—
293C2—82no—
294C2#2027499811yescharacteristic
295C2#2027673311yescharacteristic
296C2#2027673811yescharacteristic
subgroupG/H
centerC22#16810811
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#355C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
cosocleC22×Q8×C53C23
hypercenterC2×#355D53
hypocenterC53#12970
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #20274932C2semidirect
4. #20272841C2semidirect
19. C53×#355C4semidirect
49. #101×C53C2×C4semidirect
165. #355Dic53semidirect
168. C53#12970semidirect
195. #101C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202778892
#202751192
#202749322
#202728412
C2×C53×#3552
#1297053