Finite Group Databasebeta

← #28299737

Subgroups — #28299737

Order 13568.

456 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 122 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#20290632C212yesmaximal
2#20289536C212yesmaximal
3#20289535C212yesmaximal
4C4×#13133215C211yesmaximal, characteristic
5C2×C4⋊#88324—22no—
6#16811489—22no—
7#16811489C2214yes—
8C2×#13133215C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810528C2211yescharacteristic
12#16810518—22no—
13#16810492C411yescharacteristic
14C4×#11719919—22no—
15C42×Dic53C2211yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C22×#88324—24no—
18C4⋊#88324—44no—
19C22×Dic106C2×C411yescharacteristic
20#13133215—22no—
21#13133215—22no—
22C4×Dic106—42no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×#11719919—21no—
25C2×#11719919—22no—
26C2×#11719919—22no—
27C2×C4×Dic53—21no—
28C2×C4×Dic53C2×C411yescharacteristic
29C2×C4×Dic53D412yes—
30C2×C4×Dic53D412yes—
31C2×C4×Dic53C2311yescharacteristic
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53C2×C411yescharacteristic
35C4×Q8×C53—42no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C42×C53—21no—
39C2×C42×C53C2×C411yescharacteristic
40C2×Q8×C53—21no—
41C2×Q8×C53—41no—
42C2×Q8×C53—41no—
43C2×Q8×C53M1612yes—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22×C411yescharacteristic
49C22×Dic53—21no—
50C22×Dic53—21no—
51C22×Dic53C22⋊C412yes—
52C22×Dic53C2×D412yes—
53C2×Dic106—21no—
54C2×Dic106—41no—
55C2×Dic106—41no—
56C2×Dic106M1612yes—
57C2×#88324—44no—
58C2×#88324—44no—
59C2×#88324—44no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—41no—
62C42×C53—22no—
63C42×C53—42no—
64C4⋊Dic53—41no—
65#11719919—28no—
66#11719919—44no—
67#11719919—42no—
68C4×Dic53—21no—
69C4×Dic53—21no—
70C4×Dic53—42no—
71C4×Dic53—41no—
72C4×Dic53—42no—
73C4×Dic53SD1614yes—
74C4×Dic53Q1614yes—
75C23×C53C2×C22⋊C411yescharacteristic
76Q8×C53—42no—
77Q8×C53—42no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53#10012yes—
88C2×C4×C53#10314yes—
89C2×C4×C53C2×M1611yescharacteristic
90C2×C4×C53C2×M1611yescharacteristic
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100C2×Dic53#10314yes—
101C2×Dic53C2×SD1612yes—
102C2×Dic53C2×Q1612yes—
103Dic106—42no—
104Dic106—42no—
105#88324—84no—
106#7653—531nomaximal, Sylow
107C22×C53#40711yescharacteristic
108C22×C53#41011yescharacteristic
109C22×C53C2×#10011yescharacteristic
110C22×C53C2×#10312yes—
111C22×C53#42112yes—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—41no—
115C4×C53—42no—
116C4×C53—81no—
117C4×C53#33314yes—
118Dic53—24no—
119Dic53—22no—
120Dic53—22no—
121Dic53—42no—
122Dic53—81no—
123C4×C4⋊Q8—531no—
124#1831—532no—
125#1272—532no—
126#1272—532no—
127C2×C53#146611yescharacteristic
128C2×C53#148712yes—
129C2×C53#151812yes—
130C2×C53#153611yescharacteristic
131C2×C53C2×#33311yescharacteristic
132C2×C4⋊Q8—531no—
133C2×C4×Q8—531no—
134C2×C4×Q8#615711yescharacteristic
135C2×C4⋊C8—1062no—
136#391—531no—
137#391—531no—
138#389—1062no—
139#384—531no—
140C4×C4⋊C4—1062no—
141C43—531no—
142#336—534no—
143#336—1062no—
144C53#765311yescharacteristic, complemented, Sylow
145C22×Q8—531no—
146C22×Q8C2×#615711yescharacteristic
147C22×C8—1064no—
148C4⋊Q8—1062no—
149C4⋊Q8—1062no—
150C4×Q8—42no—
151C4×Q8—2122no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—1062no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4C2×#615711yescharacteristic
158C2×C42—21no—
159C2×C42—532no—
160C2×C42—531no—
161C2×C42—532no—
162C2×C42—531no—
163C2×C42—1061no—
164C2×C42C2×#615711yescharacteristic
165C4⋊C8—2124no—
166#95—1062no—
167#95—1062no—
168C2×Q8—21no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—532no—
172C2×Q8—1061no—
173C2×Q8—2121no—
174C2×Q8—2121no—
175C2×Q8#1171993612yes—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—532no—
179C22×C4—532no—
180C22×C4—1061no—
181C22×C4—1061no—
182C22×C4C22⋊#615711yescharacteristic
183C22×C4C22⋊#615711yescharacteristic
184C22×C4C22×#615711yescharacteristic
185C2×C8—2124no—
186C2×C8—2124no—
187C2×C8—2124no—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—1068no—
191C4⋊C4—2121no—
192C4⋊C4—2122no—
193C4⋊C4—2124no—
194C42—22no—
195C42—42no—
196C42—534no—
197C42—534no—
198C42—1061no—
199C42—1061no—
200C42—2121no—
201C42—2122no—
202C42—2122no—
203C23C2×C22⋊#615711yescharacteristic
204Q8—42no—
205Q8—42no—
206Q8—2122no—
207Q8—2122no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—534no—
218C2×C4—532no—
219C2×C4—532no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—2121no—
226C2×C4—2122no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4#1313327712yes—
230C2×C4#1313328014yes—
231C2×C4D53⋊M1611yescharacteristic
232C2×C4C2×#1171993611yescharacteristic
233C8—4244no—
234C22#1681159411yescharacteristic
235C22#1681160211yescharacteristic
236C22C2×#1313327711yescharacteristic
237C22C2×#1313328012yes—
238C22#1681162012yes—
239C4—21no—
240C4—21no—
241C4—41no—
242C4—42no—
243C4—81no—
244C4—1062no—
245C4—1062no—
246C4—1064no—
247C4—4241no—
248C4—2122no—
249C4#1681148314yes—
250C2#2028997411yescharacteristic
251C2#2029004412yes—
252C2#2029014812yes—
253C2#2029020311yescharacteristic
254C2C2×#1681148311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7653
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
144. C53#7653semidirect

Maximal subgroups

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

subgroupindex
#202906322
#202895362
#202895352
C4×#131332152
#765353