Finite Group Databasebeta

← #27293523

Subgroups — #27293523

Order 13568.

330 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 69 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#20283347C211yesmaximal, characteristic
2#20276524C212yesmaximal, complemented
3#20272666C211yesmaximal, characteristic, complemented
4#20272650C212yesmaximal
5C53×#1327C211yesmaximal, characteristic, complemented
6C2×C42×D53C411yescharacteristic
7#16810679—22no—
8#16810675—22no—
9#16810647C411yescharacteristic
10#16810586C411yescharacteristic
11#16810586C2211yescharacteristic
12M16×Dic53C411yescharacteristic
13M16×Dic53C2211yescharacteristic
14#16810443—22no—
15#16810405C2212yes—
16#16810400—22no—
17C53×#423C2211yescharacteristic
18C53×#349C2212yes—
19C22×C4×D53—21no—
20C22×C4×D53C2×C411yescharacteristic
21C2×#11719917C2×C411yescharacteristic
22C2×#11719917C2×C411yescharacteristic
23C2×C4×Dic53—21no—
24C2×C4×Dic53C2×C411yescharacteristic
25C42×D53—21no—
26C42×D53—21no—
27C42×D53—21no—
28C42×D53—21no—
29#13133157—22no—
30C4⋊#88322—22no—
31#13133129—22no—
32#13133129—22no—
33#13133127—21no—
34#13133127D412yes—
35C8×Dic53—21no—
36C8×Dic53D412yes—
37C2×M16×C53C2×C411yescharacteristic
38C2×M16×C53C2311yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40M32×C53—22no—
41C4⋊C8×C53—22no—
42C23×D53M1611yescharacteristic
43C22×C4×C53—21no—
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53—21no—
46C22×Dic53M1611yescharacteristic
47C2×C4×D53—21no—
48C2×C4×D53—21no—
49C2×C4×D53—21no—
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—44no—
54C2×C4×D53—41no—
55C2×C4×D53—41no—
56M16×C53—41no—
57M16×C53—41no—
58C2×C8×C53—21no—
59C2×C8×C53C2×D412yes—
60C42×C53—21no—
61C42×C53M1611yescharacteristic
62C42×C53M1611yescharacteristic
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—21no—
66C4×Dic53—21no—
67C4×Dic53—21no—
68C4×Dic53C22⋊C411yescharacteristic
69C4×Dic53C22⋊C411yescharacteristic
70#11719917—41no—
71#11719917—41no—
72C2×#88322—21no—
73C2×#88322C22⋊C412yes—
74C16×C53—42no—
75#11719908—42no—
76C23×C53C2×M1611yescharacteristic
77C22×D53—21no—
78C22×D53—42no—
79C22×D53—41no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—42no—
85C2×C4×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53C2×M1611yescharacteristic
87C2×Dic53—21no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—42no—
91C2×Dic53—41no—
92C4×D53—24no—
93C4×D53—42no—
94C4×D53—42no—
95C4×D53—42no—
96C4×D53—42no—
97C4×D53—41no—
98C4×D53—41no—
99C8×C53—22no—
100C8×C53—41no—
101#88322—22no—
102#88322—41no—
103#11521—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—41no—
106C22×C53#40711yescharacteristic
107C2×D53—22no—
108C2×D53—41no—
109C2×D53—42no—
110C2×D53—41no—
111C2×D53—42no—
112C4×C53—22no—
113C4×C53—21no—
114C4×C53—41no—
115C4×C53#41011yescharacteristic
116C4×C53#41311yescharacteristic
117Dic53—22no—
118Dic53—41no—
119Dic53—41no—
120#2920—531no—
121#2165—532no—
122#1327—531no—
123#1327D5311yescharacteristic, complemented
124#1320—532no—
125C2×C53—21no—
126C2×C53—41no—
127C2×C53#146411yescharacteristic
128D53—42no—
129C22×C42—531no—
130#432—1062no—
131#431—1062no—
132#423—531no—
133#423C2×D5311yescharacteristic
134#407—531no—
135#407—531no—
136C4×M16—531no—
137C4×M16—531no—
138#364—1062no—
139#349—532no—
140#349C2×D5312yes—
141#347—1062no—
142C53#1152111yescharacteristic, complemented, Sylow
143C23×C4—531no—
144C23×C4—1061no—
145C2×M16—531no—
146C2×M16—531no—
147C2×M16C4×D5311yescharacteristic
148C2×M16C22×D5311yescharacteristic
149C2×C42—531no—
150C2×C42—1061no—
151C2×C42—1061no—
152C2×C42—1061no—
153C2×C42—1061no—
154C2×C42—1061no—
155C2×C42C4×D5311yescharacteristic
156M32—22no—
157M32—1062no—
158C4⋊C8—22no—
159C4⋊C8—1062no—
160C22⋊C8—1062no—
161C22⋊C8—1062no—
162#97—532no—
163#97—1061no—
164C4×C8—532no—
165C4×C8—1061no—
166C24—531no—
167C22×C4—21no—
168C22×C4—531no—
169C22×C4—1061no—
170C22×C4—1061no—
171C22×C4—1061no—
172C22×C4—1061no—
173C22×C4—1061no—
174C22×C4—1061no—
175C22×C4—1061no—
176C22×C4—2121no—
177C22×C4—2121no—
178C22×C4—2124no—
179C22×C4C2×C4×D5311yescharacteristic
180M16—41no—
181M16—41no—
182M16—2121no—
183M16—2121no—
184C2×C8—21no—
185C2×C8—532no—
186C2×C8—1061no—
187C2×C8D4×D5312yes—
188C42—21no—
189C42—531no—
190C42—531no—
191C42—1061no—
192C42—1061no—
193C42—1061no—
194C42—1061no—
195C42—1061no—
196C42#1171991211yescharacteristic
197C42#1171991211yescharacteristic
198C16—42no—
199C16—2122no—
200C23—1061no—
201C23—2121no—
202C23—2122no—
203C23M16×D5311yescharacteristic
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—1064no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4—2121no—
219C2×C4—2121no—
220C2×C4—2121no—
221C2×C4C22⋊C4×D5311yescharacteristic
222C2×C4C2×#1171991211yescharacteristic
223C8—22no—
224C8—41no—
225C8—1062no—
226C8—2121no—
227C22—21no—
228C22—41no—
229C22—1062no—
230C22—2121no—
231C22—2122no—
232C22—2122no—
233C22—2121no—
234C22#1681058511yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—1062no—
239C4—2121no—
240C4—2121no—
241C4#1681059511yescharacteristic
242C4#1681060611yescharacteristic
243C2—21no—
244C2—41no—
245C2—2122no—
246C2#2027304811yescharacteristic
subgroupG/H
centerC4#16810595
commutatorC22×C4×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1327C2
socleC2×C53#1464
prime radicalC23×D53M16
cosocleC2×M16×C53C23
hypercenter#1327D53
hypocenterC53#11521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #20276524C2semidirect
3. #20272666C2semidirect
5. C53×#1327C2semidirect
123. #1327D53semidirect
142. C53#11521semidirect

Maximal subgroups

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

subgroupindex
#202833472
#202765242
#202726662
#202726502
C53×#13272
#1152153