Finite Group Databasebeta

← #27300908

Subgroups — #27300908

Order 13568.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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#20283782C211yesmaximal, characteristic
2#20275099C211yesmaximal, characteristic
3#20275003C212yesmaximal, complemented
4#20272835C212yesmaximal, complemented
5C53×#2126C211yesmaximal, characteristic
6#16811390C2211yescharacteristic
7#16810701—22no—
8D4×#88322—22no—
9#16810676—21no—
10#16810650—22no—
11#16810640—21no—
12#16810597C2211yescharacteristic
13#16810590—21no—
14#16810590C2212yes—
15#16810393—22no—
16#16810383—21no—
17C53×#532C2211yescharacteristic
18C53×#410C2212yes—
19C53×#354C412yescomplemented
20C53×#353C412yescomplemented
21#13133224—42no—
22C2×#11719917—21no—
23C2×#11719917—22no—
24C2×#11719917—21no—
25C2×#11719917D412yes—
26C22×#88322—21no—
27C22×#88322D412yes—
28#13133176—21no—
29C4⋊#88322—21no—
30C4⋊#88322—21no—
31C4⋊#88322—44no—
32#13133144—42no—
33#13133132—22no—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36C22⋊#88322—22no—
37C4×#88322—22no—
38C2×#41×C53D411yescharacteristic
39C2×#41×C53C2311yescharacteristic
40C4⋊Q8×C53Q811yescharacteristic
41#127×C53C2×C411yescharacteristic
42#124×C53C2×C411yescharacteristic
43#124×C53Q811yescharacteristic
44#121×C53—22no—
45C4×D4×C53—22no—
46#117×C53—22no—
47C2×C42×C53D411yescharacteristic
48#99×C53—22no—
49#99×C53C2×C414yes—
50#41×C53—41no—
51#41×C53—41no—
52#41×C53—41no—
53#41×C53—41no—
54C2×Q8×C53C4⋊C411yescharacteristic
55C2×Q8×C53C2×Q811yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53C4⋊C411yescharacteristic
59C2×D4×C53C22×C411yescharacteristic
60C2×D4×C53#4112yes—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—42no—
67C22⋊C4×C53—22no—
68C22⋊C4×C53—22no—
69C22⋊C4×C53—24no—
70C42×C53—21no—
71C42×C53—42no—
72C42×C53C4⋊C411yescharacteristic
73C42×C53C4⋊C411yescharacteristic
74C22⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4×Dic53—41no—
77#11719917—22no—
78#11719917—42no—
79#11719917—41no—
80#11719917—42no—
81#11719917—81no—
82C2×#88322—24no—
83C2×#88322—22no—
84C2×#88322—22no—
85C2×#88322—41no—
86C2×#88322—42no—
87C2×#88322—81no—
88C23×C53—21no—
89C23×C53C4×D412yes—
90C23×C53#12311yescharacteristic
91Q8×C53—41no—
92Q8×C53—41no—
93D4×C53—41no—
94D4×C53—42no—
95D4×C53—41no—
96D4×C53—44no—
97D4×C53—81no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—44no—
109C2×C4×C53C2×C4⋊C411yescharacteristic
110C2×C4×C53#12011yescharacteristic
111C2×C4×C53C22⋊Q812yes—
112C2×Dic53—81no—
113#88322—42no—
114#88322—42no—
115#88322—42no—
116#12968—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—22no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53#38511yescharacteristic
124C4×C53—22no—
125C4×C53—21no—
126C4×C53—22no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—84no—
130C4×C53B(4,𝔽2)11yescharacteristic
131C4×C53#45811yescharacteristic
132Dic53—161no—
133#2979—531no—
134#2126D5311yescharacteristic
135#1884—531no—
136#1868—532no—
137#1391—532no—
138C2×C53—21no—
139C2×C53—41no—
140C2×C53—42no—
141C2×C53—81no—
142C2×C53#186711yescharacteristic
143#568—531no—
144#532C2×D5311yescharacteristic
145#436—1062no—
146C8×D4—1062no—
147#431—1061no—
148#424—1062no—
149#421—1061no—
150#410—531no—
151#410C2×D5312yes—
152#408—532no—
153#408—1061no—
154#354Dic5312yescomplemented
155#353Dic5312yescomplemented
156#344—1062no—
157#339—1061no—
158C53#1296811yescharacteristic, complemented, Sylow
159C2×#41D21211yescharacteristic
160C2×#41C22×D5311yescharacteristic
161#131—2122no—
162C2×M16—532no—
163C2×M16—1061no—
164C2×M16—1062no—
165C2×M16—1061no—
166C22×C8—532no—
167C22×C8—1061no—
168C4⋊Q8Dic10611yescharacteristic
169#127C2×Dic5311yescharacteristic
170#124Dic10611yescharacteristic
171#124C2×Dic5311yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—22no—
175#117—1061no—
176C2×C42D21211yescharacteristic
177C4⋊C8—1061no—
178C4⋊C8—1061no—
179C4⋊C8—2124no—
180#104—2122no—
181#99—22no—
182#99—1062no—
183#99C2×Dic5314yes—
184C22⋊C8—1062no—
185C22⋊C8—1062no—
186C22⋊C8—1062no—
187C4×C8—1062no—
188#41—41no—
189#41—41no—
190#41—41no—
191#41—41no—
192C2×Q8C4⋊Dic5311yescharacteristic
193C2×Q8C2×Dic10611yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4C4⋊Dic5311yescharacteristic
197C2×D4#1171994712yes—
198C2×D4C22×Dic5311yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×D21211yescharacteristic
202C22×C4D4×D5312yes—
203M16—1062no—
204M16—4241no—
205M16—2122no—
206M16—2121no—
207M16—2122no—
208C2×C8—1064no—
209C2×C8—1062no—
210C2×C8—1062no—
211C2×C8—4241no—
212C2×C8—2121no—
213C2×C8—2122no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—2121no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—24no—
220C22⋊C4—2121no—
221C42—21no—
222C42—42no—
223C42—2121no—
224C42C4⋊Dic5311yescharacteristic
225C42C4⋊Dic5311yescharacteristic
226C23—21no—
227C23D4×Dic5312yes—
228C23#1313320111yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—42no—
232D4—41no—
233D4—41no—
234D4—44no—
235D4—81no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—4241no—
248C2×C4C2×C4⋊Dic5311yescharacteristic
249C2×C4#1313318711yescharacteristic
250C2×C4#1313319412yes—
251C8—2122no—
252C8—2122no—
253C8—2122no—
254C22—22no—
255C22—21no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1681049811yescharacteristic
261C4—22no—
262C4—21no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—84no—
267C4—8481no—
268C4#1681081111yescharacteristic
269C4#1681081411yescharacteristic
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2#2027499811yescharacteristic
subgroupG/H
centerC4#16810811
commutatorC2×D4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#2126C2
socleC2×C53#1867
prime radicalC53×#532C22
cosocleC2×#41×C53C23
hypercenter#2126D53
hypocenterC53#12968
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2126

Complements

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

NH ≅ G/Ntype
3. #20275003C2semidirect
4. #20272835C2semidirect
19. C53×#354C4semidirect
20. C53×#353C4semidirect
154. #354Dic53semidirect
155. #353Dic53semidirect
158. C53#12968semidirect

Maximal subgroups

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

subgroupindex
#202837822
#202750992
#202750032
#202728352
C53×#21262
#1296853