Finite Group Databasebeta

← #27293403

Subgroups — #27293403

Order 13568.

364 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 83 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#20283430C211yesmaximal, characteristic
2#20276334C212yesmaximal, complemented
3#20272665C212yesmaximal, complemented
4#20272653C211yesmaximal, characteristic, complemented
5C53×#1321C211yesmaximal, characteristic, complemented
6C2×#13133225C2211yescharacteristic
7C2×#13133224C411yescharacteristic
8#16811169C411yescharacteristic
9M32×D53—22no—
10#16810697—22no—
11#16810691—22no—
12#16810646C2211yescharacteristic
13#16810589C411yescharacteristic
14C2×#13133127C411yescharacteristic
15#16810405C2212yes—
16#16810404—22no—
17C53×#408C2211yescharacteristic
18C53×#349C2212yes—
19C2×#11719945C2×C411yescharacteristic
20#13133225—42no—
21#13133224—42no—
22C2×#11719917—21no—
23C2×#11719917C2×C411yescharacteristic
24C2×#11719912D412yes—
25C22×#88322—21no—
26C22×#88322C2×C411yescharacteristic
27C2×C8×D53D412yes—
28#13133215C811yescharacteristic
29#13133213C811yescharacteristic
30#13133206C811yescharacteristic
31#13133204C811yescharacteristic
32#13133189—22no—
33C4×C22⋊D53—22no—
34C2×C4×Dic53C2×C411yescharacteristic
35#13133157—22no—
36#13133156—42no—
37C16×D53—42no—
38#13133146—22no—
39#13133129—22no—
40#13133127—21no—
41#13133127—21no—
42C2×#41×C53C2×C411yescharacteristic
43C2×M16×C53C2311yescharacteristic
44C22×C8×C53C2×C411yescharacteristic
45M32×C53—22no—
46C22⋊C8×C53—22no—
47#41×C53—41no—
48#41×C53—41no—
49C2×Q8×C53C2×C811yescharacteristic
50C2×D4×C53—21no—
51C2×D4×C53C2×C811yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53C22×C411yescharacteristic
54C2×C22⋊D53—22no—
55C2×C22⋊D53M1612yes—
56C22×Dic53—21no—
57#11719945—41no—
58#11719945—41no—
59C2×D212C2×C811yescharacteristic
60C2×C4×D53C22⋊C412yes—
61C2×Dic106C2×C811yescharacteristic
62M16×C53—41no—
63C2×C8×C53—21no—
64C2×C8×C53—41no—
65C2×C8×C53C2×D412yes—
66C22⋊Dic53—22no—
67#11719921—22no—
68#11719919—22no—
69C4×Dic53—21no—
70C4×Dic53C2×C811yescharacteristic
71C4×Dic53C2×C811yescharacteristic
72#11719917—42no—
73#11719917—41no—
74C2×#88322—22no—
75C2×#88322—21no—
76C2×#88322—42no—
77C2×#88322C22⋊C411yescharacteristic
78C2×#88322C22⋊C411yescharacteristic
79#11719912—42no—
80C8×D53—22no—
81C8×D53—22no—
82C16×C53—42no—
83#11719908—42no—
84C23×C53—21no—
85C23×C53C2×M1611yescharacteristic
86C22×D53C22⋊C812yes—
87Q8×C53—41no—
88D4×C53—41no—
89D4×C53—42no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53C2×C22⋊C411yescharacteristic
96C2×C4×C53C22×C811yescharacteristic
97C22⋊D53—42no—
98C22⋊D53—82no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—81no—
102C2×Dic53C22⋊C812yes—
103D212—41no—
104C4×D53—22no—
105C4×D53—22no—
106Dic106—41no—
107C8×C53—22no—
108C8×C53—41no—
109#88322—21no—
110#88322—41no—
111#88322#9912yes—
112#11486—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—41no—
115C22×C53—42no—
116C22×C53—41no—
117C22×C53C2×C22⋊C811yescharacteristic
118C2×D53—22no—
119C2×D53—42no—
120C4×C53—21no—
121C4×C53—41no—
122C4×C53C2×#9911yescharacteristic
123C4×C53#41311yescharacteristic
124Dic53—22no—
125Dic53—41no—
126Dic53—41no—
127#2932—531no—
128#2115—532no—
129#1327—532no—
130#1321—531no—
131#1321D5311yescharacteristic, complemented
132C2×C53—21no—
133C2×C53—41no—
134C2×C53—81no—
135C2×C53#145911yescharacteristic
136D53—42no—
137C2×#131—531no—
138C2×#131—531no—
139#532—531no—
140C2×M32—1062no—
141#435—1062no—
142#435—1062no—
143#423—531no—
144#408—531no—
145#408C2×D5311yescharacteristic
146C2×#97—531no—
147#349—532no—
148#349—1062no—
149#349C2×D5312yes—
150C53#1148611yescharacteristic, complemented, Sylow
151C2×#41—531no—
152C2×#41C4×D5311yescharacteristic
153#131—2122no—
154#131—2122no—
155C2×M16—532no—
156C2×M16—531no—
157C2×M16—1061no—
158C2×M16C22×D5311yescharacteristic
159C22×C8—531no—
160C22×C8—532no—
161C22×C8—1061no—
162C22×C8C4×D5311yescharacteristic
163C4⋊Q8—531no—
164#127—531no—
165#124—531no—
166#124—531no—
167#121—1062no—
168C4×D4—1062no—
169C2×C42—531no—
170M32—22no—
171M32—1062no—
172M32—2122no—
173C2×C16—2122no—
174C4⋊C8—1062no—
175C22⋊C8—22no—
176C22⋊C8—1062no—
177#97—1061no—
178#97—1061no—
179#41—41no—
180#41—41no—
181#41—2121no—
182#41—2121no—
183C2×Q8—531no—
184C2×Q8C8×D5311yescharacteristic
185C2×D4—21no—
186C2×D4—532no—
187C2×D4—531no—
188C2×D4—1062no—
189C2×D4C8×D5311yescharacteristic
190C22×C4—21no—
191C22×C4—532no—
192C22×C4—1061no—
193C22×C4C2×C4×D5311yescharacteristic
194M16—41no—
195M16—2122no—
196M16—2122no—
197M16—2121no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—531no—
201C2×C8—531no—
202C2×C8—1062no—
203C2×C8—1062no—
204C2×C8—1062no—
205C2×C8—1061no—
206C2×C8—2122no—
207C2×C8D4×D5312yes—
208C4⋊C4—1062no—
209C22⋊C4—1062no—
210C22⋊C4—1062no—
211C42—531no—
212C42—531no—
213C42—1061no—
214C16—42no—
215C16—2122no—
216C23—21no—
217C23—532no—
218C23M16×D5311yescharacteristic
219Q8—41no—
220Q8—2121no—
221D4—41no—
222D4—42no—
223D4—4242no—
224D4—2121no—
225D4—2122no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—532no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4—4241no—
237C2×C4C22⋊C4×D5311yescharacteristic
238C2×C4C2×C8×D5311yescharacteristic
239C8—22no—
240C8—41no—
241C8—532no—
242C8—1061no—
243C8—2121no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—41no—
248C22—1062no—
249C22—2122no—
250C22C22⋊C8×D5311yescharacteristic
251C4—21no—
252C4—41no—
253C4—1062no—
254C4—2121no—
255C4—2121no—
256C4#99×D5311yescharacteristic
257C4#1681060411yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—2122no—
262C2#2027303011yescharacteristic
subgroupG/H
centerC4#99×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×M16C22×D53
FittingC53×#1321C2
socleC2×C53#1459
prime radical#16811169C4
cosocleC2×M16×C53C23
hypercenter#1321D53
hypocenterC53#11486
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #20276334C2semidirect
3. #20272665C2semidirect
4. #20272653C2semidirect
5. C53×#1321C2semidirect
131. #1321D53semidirect
150. C53#11486semidirect

Maximal subgroups

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

subgroupindex
#202834302
#202763342
#202726652
#202726532
C53×#13212
#1148653