Finite Group Databasebeta

← #28309138

Subgroups — #28309138

Order 13568.

414 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#13133278C211yesmaximal, characteristic
2#20289977C211yesmaximal, characteristic
3#20289700C214yesmaximal
4#20279703C211yesmaximal, characteristic
5#16811644—21no—
6C2×#13133278—21no—
7C4×#11719940—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×#13133217—21no—
12C2×#13133217C411yescharacteristic
13C2×C4×Dic106C2211yescharacteristic
14#16810555C412yes—
15#16810555C2212yes—
16#16810529—21no—
17#16810522—21no—
18#16810522—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21C4⋊#88324—41no—
22#13133278—24no—
23C22⋊#88324—22no—
24#13133272—41no—
25#13133272—41no—
26C4×#88324—41no—
27C22×Dic106—21no—
28C22×Dic106D411yescharacteristic
29#13133217—44no—
30#13133217—41no—
31#13133217—41no—
32C4×Dic106—41no—
33C4×Dic106—41no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53C2×C412yes—
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719919C2×C411yescharacteristic
40C2×#11719919C2311yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43#13133124—22no—
44#13133124—22no—
45C2×C4⋊C4×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48#95×C53C2×C412yes—
49C22×C4×C53—22no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C2×Dic106—22no—
59C2×Dic106—22no—
60C2×Dic106—41no—
61C2×Dic106—42no—
62C2×Dic106—42no—
63C2×Dic106—42no—
64#11719940—24no—
65#11719940—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C4⋊C4×C53—24no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—42no—
72C4⋊Dic53—41no—
73C4⋊Dic53—44no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—44no—
77C4⋊Dic53—42no—
78#11719919—41no—
79#11719919—41no—
80C4×Dic53—21no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C23×C53C2×C22⋊C411yescharacteristic
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—24no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53#9911yescharacteristic
94C2×C4×C53#9911yescharacteristic
95C2×C4×C53#10412yes—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53AGL(1,ℤ/8)12yes—
109Dic106—44no—
110Dic106—84no—
111#88324—44no—
112#88324—82no—
113#11782—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53C2×#9911yescharacteristic
117C22×C53#41811yescharacteristic
118C22×C53C2×#10411yescharacteristic
119C4×C53—21no—
120C4×C53—22no—
121C4×C53—44no—
122C4×C53—41no—
123C4×C53#35612yes—
124Dic53—22no—
125Dic53—41no—
126Dic53—41no—
127Dic53—41no—
128Dic53—82no—
129#2546—531no—
130C4×#101—531no—
131#1467—531no—
132#1346—534no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53#150711yescharacteristic
136C2×C53#212311yescharacteristic
137C2×C53C2×#35611yescharacteristic
138C2×C4⋊Q8—531no—
139C2×C4⋊Q8—1061no—
140C2×C4×Q8—531no—
141#432—1061no—
142C2×#101—1061no—
143C4×M16—1061no—
144#398—532no—
145#398—532no—
146#391—1061no—
147#389—1061no—
148#389—1061no—
149C4×C4⋊C4#615711yescharacteristic
150#343—532no—
151#324—532no—
152#324—1061no—
153C53#1178211yescharacteristic, complemented, Sylow
154C22×Q8—531no—
155C22×Q8—1061no—
156C2×M16—1062no—
157C4⋊Q8—2121no—
158C4⋊Q8—2121no—
159C4⋊Q8—2124no—
160C4×Q8—2121no—
161C4×Q8—2121no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—531no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—1062no—
168C2×C4⋊C4—1062no—
169C2×C4⋊C4—1061no—
170C2×C42—21no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42C2×#615711yescharacteristic
174C4⋊C8—2121no—
175#101—1064no—
176C22⋊C8—1062no—
177#97—2121no—
178#97—2121no—
179C4×C8—2121no—
180#95—1062no—
181#95—1062no—
182#95C2×#615712yes—
183C2×Q8—1062no—
184C2×Q8—1062no—
185C2×Q8—2121no—
186C2×Q8—2122no—
187C2×Q8—2122no—
188C2×Q8—2122no—
189C22×C4—22no—
190C22×C4—531no—
191C22×C4—531no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—1062no—
195C22×C4C22⋊#615711yescharacteristic
196C22×C4C22⋊#615711yescharacteristic
197C22×C4C22×#615711yescharacteristic
198M16—1064no—
199M16—2122no—
200C2×C8—2122no—
201C2×C8—2122no—
202C4⋊C4—24no—
203C4⋊C4—2121no—
204C4⋊C4—2121no—
205C4⋊C4—2121no—
206C4⋊C4—2122no—
207C4⋊C4—2124no—
208C4⋊C4—2124no—
209C4⋊C4—2121no—
210C4⋊C4—2121no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—1061no—
215C42—1061no—
216C42—1061no—
217C42—1061no—
218C23C2×C22⋊#615711yescharacteristic
219Q8—4244no—
220Q8—2124no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—532no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4—1061no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4—2121no—
242C2×C4#1313327611yescharacteristic
243C2×C4#1313327611yescharacteristic
244C2×C4#1313328212yes—
245C8—4242no—
246C8—2124no—
247C22—21no—
248C22—21no—
249C22C2×#1313327611yescharacteristic
250C22#1681161511yescharacteristic
251C22C2×#1313328211yescharacteristic
252C4—22no—
253C4—21no—
254C4—44no—
255C4—41no—
256C4—1062no—
257C4—4242no—
258C4—2121no—
259C4—2121no—
260C4—2121no—
261C4#1681153412yes—
262C2—21no—
263C2—21no—
264C2#2029011211yescharacteristic
265C2#2029097911yescharacteristic
266C2C2×#1681153411yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×#11719919C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#11782
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
153. C53#11782semidirect

Maximal subgroups

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

subgroupindex
C4×#131332782
#202899772
#202897002
#202797032
#1178253