Finite Group Databasebeta

← #27302300

Subgroups — #27302300

Order 13568.

420 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283420C211yesmaximal, characteristic
2#20276342C212yesmaximal, complemented
3C2×#16810404C212yesmaximal, complemented
4#20276320C211yesmaximal, characteristic, complemented
5C53×#2153C211yesmaximal, characteristic, complemented
6C2×#13133223C2211yescharacteristic
7C22×#11719912C2211yescharacteristic
8#16811135C411yescharacteristic
9C2×#13133157C2212yes—
10#16810702—22no—
11#16810696—22no—
12#16810650C411yescharacteristic
13#16810591C411yescharacteristic
14#16810586C411yescharacteristic
15#16810406—24no—
16#16810404—24no—
17#16810401—22no—
18C2×M32×C53C2212yes—
19C53×#405C2211yescharacteristic
20C53×#364—22no—
21C2×#11719945C2×C411yescharacteristic
22C22×C4×D53C2×C411yescharacteristic
23#13133223—22no—
24#13133223—22no—
25C2×#11719917C2×C411yescharacteristic
26C2×#11719912—22no—
27C2×#11719912—21no—
28C2×#11719912—21no—
29C2×#11719912—21no—
30C2×#11719912—22no—
31C22×#88322C2×C411yescharacteristic
32C2×C8×D53—21no—
33#13133201—21no—
34#13133197—21no—
35#13133192—21no—
36#13133187—21no—
37C4×D212—22no—
38#13133157—22no—
39#13133157D414yescomplemented
40C2×#11719908—22no—
41C4⋊#88322—22no—
42#13133129—22no—
43#13133129—22no—
44C2×M16×C53C2×C411yescharacteristic
45C22×C8×C53C2311yescharacteristic
46#117×C53C2×C411yescharacteristic
47M32×C53—22no—
48M32×C53—22no—
49C2×C16×C53—22no—
50#97×C53D412yes—
51C4×C8×C53D412yes—
52C23×D53M1611yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53—21no—
55C22×Dic53M1611yescharacteristic
56#11719945—22no—
57#11719945—22no—
58C2×D212—22no—
59C2×D212M1611yescharacteristic
60C2×C4×D53—21no—
61C2×C4×D53—21no—
62C2×C4×D53—22no—
63C2×C4×D53—21no—
64C2×C4×D53—22no—
65C2×Dic106M1611yescharacteristic
66M16×C53—22no—
67C2×C8×C53C22⋊C412yes—
68C2×C8×C53C2×D411yescharacteristic
69C2×C8×C53C2×D412yes—
70C2×C8×C53C2×D411yescharacteristic
71C2×C8×C53#4112yes—
72C4⋊C4×C53—21no—
73C22⋊C4×C53—21no—
74C42×C53C22⋊C412yes—
75#11719921—22no—
76#11719921—22no—
77C4⋊Dic53—22no—
78#11719917—22no—
79C2×#88322—21no—
80C2×#88322—21no—
81C2×#88322—22no—
82#11719912—42no—
83#11719912—41no—
84#11719912—41no—
85#11719912—41no—
86#11719912—41no—
87C8×D53—42no—
88C16×C53—42no—
89#11719908—24no—
90C23×C53C2×M1611yescharacteristic
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—22no—
94C22×D53—81no—
95C2×C4×C53—22no—
96C2×C4×C53C2×C22⋊C411yescharacteristic
97C2×C4×C53C4×D412yes—
98C2×C4×C53C2×M1611yescharacteristic
99C2×C4×C53#13112yes—
100C22⋊D53—42no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104D212—22no—
105D212—82no—
106C4×D53—41no—
107C4×D53—41no—
108C4×D53—41no—
109C4×D53—41no—
110C4×D53—42no—
111Dic106—22no—
112C8×C53—22no—
113C8×C53#12112yes—
114C8×C53#12411yescharacteristic
115C8×C53#12711yescharacteristic
116#88322—41no—
117#88322—41no—
118#13201—531nomaximal, Sylow
119C22×C53—41no—
120C22×C53#40711yescharacteristic
121C22×C53#43512yes—
122C2×D53—41no—
123C2×D53—42no—
124C2×D53—41no—
125C2×D53—41no—
126C2×D53—41no—
127C2×D53—82no—
128C4×C53—42no—
129C4×C53#39011yescharacteristic
130C4×C53#40811yescharacteristic
131C4×C53#43512yes—
132Dic53—41no—
133Dic53—41no—
134#2930—531no—
135#2153D5311yescharacteristic, complemented
136#2117—532no—
137C2×#349—532no—
138#2110—531no—
139C2×C53—21no—
140C2×C53—22no—
141C2×C53#195511yescharacteristic
142D53—81no—
143D53—81no—
144C2×#131—531no—
145C22×M16—531no—
146#525—531no—
147C2×M32—532no—
148C2×M32C2×D5312yes—
149#436—1062no—
150#435—1062no—
151#424—531no—
152#408—531no—
153#407—531no—
154#405C2×D5311yescharacteristic
155#364—22no—
156#350—1064no—
157#349—1064no—
158#347—1062no—
159C53#1320111yescharacteristic, complemented, Sylow
160C2×#41—531no—
161C23×C4—531no—
162#131—1062no—
163#131—1062no—
164C2×M16—531no—
165C2×M16—1061no—
166C2×M16—1062no—
167C2×M16—1062no—
168C2×M16—1061no—
169C2×M16—1061no—
170C2×M16C4×D5311yescharacteristic
171C22×C8—531no—
172C22×C8—1061no—
173C22×C8C22×D5311yescharacteristic
174#123—1061no—
175C22⋊Q8—1061no—
176#121—1061no—
177#120—1061no—
178C4×D4—1062no—
179#117C4×D5311yescharacteristic
180M32—22no—
181M32—22no—
182M32—534no—
183M32—1062no—
184C2×C16—22no—
185C2×C16—1062no—
186C4⋊C8—1062no—
187C22⋊C8—1062no—
188C22⋊C8—1062no—
189#97D21212yes—
190C4×C8C22⋊D5312yes—
191C24—531no—
192#41—1062no—
193#41—1062no—
194C2×Q8—531no—
195C2×D4—531no—
196C2×D4—1062no—
197C2×D4—1061no—
198C22×C4—531no—
199C22×C4—1061no—
200C22×C4—1061no—
201C22×C4—1062no—
202C22×C4—1061no—
203C22×C4—1062no—
204C22×C4C2×C4×D5311yescharacteristic
205M16—22no—
206M16—1062no—
207M16—2122no—
208M16—2121no—
209M16—2121no—
210M16—2121no—
211M16—2121no—
212C2×C8—1061no—
213C2×C8—1061no—
214C2×C8—1062no—
215C2×C8—2122no—
216C2×C8#1171992112yes—
217C2×C8C2×D21211yescharacteristic
218C2×C8D4×D5312yes—
219C2×C8#1171994912yes—
220C2×C8C2×C22⋊D5311yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—1062no—
223C22⋊C4—21no—
224C22⋊C4—1062no—
225C22⋊C4—1062no—
226C42#1171992112yes—
227C16—42no—
228C16—1064no—
229C23—1062no—
230C23—1061no—
231C23—1061no—
232C23—4241no—
233C23M16×D5311yescharacteristic
234Q8—1062no—
235D4—1062no—
236D4—4242no—
237D4—2122no—
238C2×C4—22no—
239C2×C4—1061no—
240C2×C4—1061no—
241C2×C4—1062no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4—2122no—
247C2×C4C2×#1171992111yescharacteristic
248C2×C4#1313317912yes—
249C2×C4M16×D5311yescharacteristic
250C2×C4#1313322312yes—
251C8—22no—
252C8—2121no—
253C8—2121no—
254C8#1313319212yes—
255C8#1313320611yescharacteristic
256C8#1313321311yescharacteristic
257C22—41no—
258C22—4242no—
259C22—2121no—
260C22—2122no—
261C22—2121no—
262C22—2121no—
263C22—2121no—
264C22#1681058611yescharacteristic
265C22#1681069312yes—
266C4—42no—
267C4—2121no—
268C4—2121no—
269C4#1681052411yescharacteristic
270C4#1681058911yescharacteristic
271C4#1681069312yes—
272C2—21no—
273C2—22no—
274C2—4241no—
275C2—4241no—
276C2#2027551211yescharacteristic
subgroupG/H
centerC4#16810524
commutatorC22×C4×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2153C2
socleC2×C53#1955
prime radical#16811135C4
cosocleC22×C8×C53C23
hypercenter#2153D53
hypocenterC53#13201
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #20276342C2semidirect
3. C2×#16810404C2semidirect
4. #20276320C2semidirect
5. C53×#2153C2semidirect
39. #13133157D4semidirect
135. #2153D53semidirect
159. C53#13201semidirect

Maximal subgroups

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

subgroupindex
#202834202
#202763422
C2×#168104042
#202763202
C53×#21532
#1320153