Finite Group Databasebeta

← #28301888

Subgroups — #28301888

Order 13568.

516 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 134 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#20290500C212yesmaximal
2C2×#16811483C212yesmaximal
3#20289593C212yesmaximal
4C4×#13133215C211yesmaximal, characteristic
5C2×#13133284C2214yes—
6C2×#13133272—22no—
7#16811489—22no—
8#16811483C418yescomplemented
9C2×#13133215C2211yescharacteristic
10C2×C4×Dic106C411yescharacteristic
11#16810530C411yescharacteristic
12#16810528C411yescharacteristic
13#16810518—22no—
14#16810492C2211yescharacteristic
15C4×#11719919—22no—
16C42×Dic53C2211yescharacteristic
17C2×C4×Q8×C53C411yescharacteristic
18C22×#88324—24no—
19#13133284—24no—
20#13133284C2×C418yes—
21#13133272—44no—
22C22×Dic106C2×C411yescharacteristic
23#13133215—22no—
24#13133215C2×C414yes—
25C4×Dic106—42no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—22no—
29C2×#11719919—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33C2×C4×Dic53D412yes—
34C2×C4×Dic53C2311yescharacteristic
35#13133124—22no—
36#13133124—22no—
37C22×Q8×C53C2×C411yescharacteristic
38C4×Q8×C53—42no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53—21no—
42C2×C42×C53C2×C411yescharacteristic
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53C4212yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×Dic106—21no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106C4212yes—
60C2×#88324—28no—
61C2×#88324—44no—
62C2×#88324—44no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—41no—
65C42×C53—22no—
66C42×C53—42no—
67C4⋊Dic53—41no—
68#11719919—28no—
69#11719919—42no—
70#11719919—42no—
71#11719919—42no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—24no—
75C4×Dic53—41no—
76C4×Dic53—42no—
77C4×Dic53C22×C412yes—
78C4×Dic53#4112yes—
79C23×C53C2×C22⋊C411yescharacteristic
80Q8×C53—42no—
81Q8×C53—42no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53#10012yes—
93C2×C4×C53C2×C4211yescharacteristic
94C2×C4×C53#11711yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53C4×D414yes—
104C2×Dic53AGL(1,ℤ/8)12yes—
105C2×Dic53#13714yes—
106C2×Dic53#13712yes—
107Dic106—42no—
108Dic106—42no—
109#88324—48no—
110#8232—531nomaximal, Sylow
111C22×C53C4×C22⋊C411yescharacteristic
112C22×C53C2×#10011yescharacteristic
113C22×C53#41311yescharacteristic
114C22×C53#41712yes—
115C22×C53#41912yes—
116C4×C53—22no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—41no—
120C4×C53—42no—
121C4×C53—81no—
122Dic53—42no—
123Dic53—42no—
124Dic53—81no—
125Dic53#44014yes—
126Dic53#44114yes—
127C4×C4⋊Q8—531no—
128#1767—532no—
129C2×#333—532no—
130#1296—532no—
131C2×C53#155811yescharacteristic
132C2×C53#155912yes—
133C2×C53#156011yescharacteristic
134C2×C53C4×#10011yescharacteristic
135C2×C53#176312yes—
136C2×C4⋊Q8—531no—
137C2×C4×Q8—531no—
138C2×C4×Q8#615711yescharacteristic
139C2×C4⋊C8—534no—
140C2×#97—1062no—
141#391—531no—
142#391—531no—
143#389—1062no—
144#384—531no—
145C4×C4⋊C4—1062no—
146C43—531no—
147#336—1062no—
148#333—538no—
149C53#823211yescharacteristic, complemented, Sylow
150C22×Q8—531no—
151C22×Q8C2×#615711yescharacteristic
152C22×C8—1064no—
153C4⋊Q8—534no—
154C4⋊Q8—1062no—
155C4×Q8—42no—
156C4×Q8—2122no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4C2×#615711yescharacteristic
163C2×C42—21no—
164C2×C42—531no—
165C2×C42—532no—
166C2×C42—532no—
167C2×C42—531no—
168C2×C42—1061no—
169C2×C42C2×#615711yescharacteristic
170C4⋊C8—538no—
171C4⋊C8—1064no—
172#97—2124no—
173#95—1062no—
174#95—1062no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—532no—
179C2×Q8—1061no—
180C2×Q8—2121no—
181C2×Q8—2121no—
182C2×Q8C4×#615712yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—532no—
186C22×C4—532no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4C22⋊#615711yescharacteristic
190C22×C4C22⋊#615711yescharacteristic
191C22×C4C22×#615711yescharacteristic
192C2×C8—1068no—
193C2×C8—2124no—
194C2×C8—2124no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C4⋊C4—1068no—
198C4⋊C4—2121no—
199C4⋊C4—2122no—
200C4⋊C4—2122no—
201C4⋊C4—2122no—
202C42—22no—
203C42—42no—
204C42—532no—
205C42—532no—
206C42—1062no—
207C42—1064no—
208C42—1062no—
209C42—2121no—
210C42—2122no—
211C23C2×C22⋊#615711yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—2122no—
215Q8—2122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—534no—
227C2×C4—532no—
228C2×C4—532no—
229C2×C4—534no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—1062no—
234C2×C4—2121no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4#1313327712yes—
239C2×C4C2×C4×#615711yescharacteristic
240C2×C4#1313329611yescharacteristic
241C8—2128no—
242C22C4×C22⋊#615711yescharacteristic
243C22C2×#1313327711yescharacteristic
244C22#1681160911yescharacteristic
245C22#1681161412yes—
246C22#1681161612yes—
247C4—21no—
248C4—21no—
249C4—22no—
250C4—42no—
251C4—41no—
252C4—81no—
253C4—534no—
254C4—534no—
255C4—4241no—
256C4—2122no—
257C4—2122no—
258C2#2029024811yescharacteristic
259C2#2029025012yes—
260C2#2029025211yescharacteristic
261C2C4×#1313327711yescharacteristic
262C2#2029048712yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#8232
LCSG ▷ C2×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
8. #16811483C4semidirect
149. C53#8232semidirect

Maximal subgroups

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

subgroupindex
#202905002
C2×#168114832
#202895932
C4×#131332152
#823253