Finite Group Databasebeta

← #28299643

Subgroups — #28299643

Order 13568.

462 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290643C211yesmaximal, characteristic
2#20290500C211yesmaximal, characteristic
3#20289534C214yesmaximal
4#20277282C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133272—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133182—21no—
13#16810528—21no—
14#16810519—21no—
15#16810519C2212yes—
16C4⋊C4×Dic53—21no—
17C4⋊C4×Dic53C412yes—
18C42×Dic53C2211yescharacteristic
19C53×#384C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—24no—
23#13133272—42no—
24C22×Dic106—21no—
25#13133217—24no—
26#13133182—42no—
27#13133182—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C412yes—
30C2×#11719919—22no—
31C2×C4×Dic53—22no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53C2311yescharacteristic
38#13133124—22no—
39#13133124—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C42×C53C2×C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—22no—
49C22×Dic53C22⋊C411yescharacteristic
50C22×Dic53C22⋊C411yescharacteristic
51C22×Dic53C2×D411yescharacteristic
52C22×Dic53C2×D411yescharacteristic
53C2×Dic106—24no—
54C2×Dic106—42no—
55C2×Dic106—42no—
56C2×#88324—24no—
57C2×#88324—42no—
58C2×#88324—42no—
59C2×#88324—42no—
60C2×#88324—42no—
61C2×#88324—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C42×C53—22no—
66C4⋊Dic53—24no—
67C4⋊Dic53—22no—
68C4⋊Dic53M1614yes—
69#11719919—42no—
70#11719919—42no—
71C4×Dic53—22no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—41no—
75C4×Dic53—41no—
76C4×Dic53—44no—
77C4×Dic53D812yes—
78C4×Dic53SD1612yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53#10212yes—
90C2×C4×C53#10412yes—
91C2×C4×C53C2×M1612yes—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—24no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53#10112yes—
104C2×Dic53#10212yes—
105C2×Dic53C2×D811yescharacteristic
106C2×Dic53C2×SD1611yescharacteristic
107C2×Dic53#13712yes—
108Dic106—48no—
109#88324—44no—
110#88324—82no—
111#7625—531nomaximal, Sylow
112C22×C53#40711yescharacteristic
113C22×C53#41011yescharacteristic
114C22×C53C2×#10111yescharacteristic
115C22×C53C2×#10211yescharacteristic
116C22×C53#41911yescharacteristic
117C22×C53C2×#10411yescharacteristic
118C22×C53#42111yescharacteristic
119C4×C53—22no—
120C4×C53—22no—
121C4×C53—22no—
122C4×C53—82no—
123Dic53—22no—
124Dic53—21no—
125Dic53—21no—
126Dic53—22no—
127Dic53—22no—
128Dic53—44no—
129C4×C4⋊Q8—531no—
130#1839—531no—
131#1767—531no—
132#1271—534no—
133C2×C53#146711yescharacteristic
134C2×C53#148611yescharacteristic
135C2×C53#149111yescharacteristic
136C2×C53#150211yescharacteristic
137C2×C53#151511yescharacteristic
138C2×C53#153111yescharacteristic
139C2×C53#154611yescharacteristic
140C2×C4⋊Q8—531no—
141C2×C4×Q8—1061no—
142C2×C4⋊C8—1061no—
143C2×#97—1061no—
144#391—1061no—
145#389—532no—
146#389—1061no—
147#384#615711yescharacteristic
148C4×C4⋊C4—532no—
149C4×C4⋊C4—1061no—
150C43—531no—
151#336—532no—
152#336—532no—
153#336—1061no—
154#336—1061no—
155C53#762511yescharacteristic, complemented, Sylow
156C22×Q8—1061no—
157C22×C8—1062no—
158C22×C8—1062no—
159C4⋊Q8—1064no—
160C4×Q8—2122no—
161C4×Q8—2122no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—532no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4—1062no—
167C2×C4⋊C4C2×#615712yes—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42—531no—
174C2×C42—1062no—
175C2×C42C2×#615711yescharacteristic
176C4⋊C8—1064no—
177#97—2122no—
178#95—1062no—
179#95—1062no—
180C2×Q8—1064no—
181C2×Q8—2122no—
182C2×Q8—2122no—
183C22×C4—22no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—531no—
187C22×C4—531no—
188C22×C4—1062no—
189C22×C4C22⋊#615711yescharacteristic
190C22×C4C22⋊#615711yescharacteristic
191C22×C4C22×#615711yescharacteristic
192C2×C8—1064no—
193C2×C8—2122no—
194C2×C8—2122no—
195C2×C8—2122no—
196C2×C8—2122no—
197C2×C8—2122no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—534no—
202C4⋊C4—1064no—
203C4⋊C4—1062no—
204C4⋊C4—2122no—
205C4⋊C4—2122no—
206C42—22no—
207C42—532no—
208C42—532no—
209C42—1062no—
210C42—1062no—
211C42—1062no—
212C42—2121no—
213C42—2121no—
214C42—2124no—
215C23C2×C22⋊#615711yescharacteristic
216Q8—2128no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—532no—
227C2×C4—532no—
228C2×C4—531no—
229C2×C4—531no—
230C2×C4—532no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—1064no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4#1313327912yes—
243C2×C4#1313328112yes—
244C2×C4D53⋊M1612yes—
245C8—4242no—
246C8—2124no—
247C22#1681159511yescharacteristic
248C22#1681160311yescharacteristic
249C22D53⋊#10111yescharacteristic
250C22C2×#1313327911yescharacteristic
251C22#1681161611yescharacteristic
252C22C2×#1313328111yescharacteristic
253C22#1681162211yescharacteristic
254C4—22no—
255C4—22no—
256C4—22no—
257C4—82no—
258C4—1061no—
259C4—1061no—
260C4—1062no—
261C4—1062no—
262C4—1062no—
263C4—2124no—
264C2#2028997911yescharacteristic
265C2#2029004211yescharacteristic
266C2#2029005911yescharacteristic
267C2#2029009511yescharacteristic
268C2#2029013911yescharacteristic
269C2#2029018811yescharacteristic
270C2#2029022511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7625
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
155. C53#7625semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202905002
#202895342
#202772822
#762553