Finite Group Databasebeta

← #27294744

Subgroups — #27294744

Order 13568.

394 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13133136C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272745C214yesmaximal
4C53×#2567C211yesmaximal, characteristic
5#16810680—21no—
6C2×#13133136—21no—
7C4×#11719917—21no—
8#16810392C2212yes—
9#16810349—21no—
10#16810349C2212yes—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C4×Q8×C53C2211yescharacteristic
13C53×#395C412yes—
14C53×#395C2212yes—
15C53×#391—21no—
16C53×#389—21no—
17C53×#389—21no—
18C53×#384—21no—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—22no—
21C4⋊#88322—41no—
22#13133136—24no—
23C22⋊#88322—22no—
24#13133128—41no—
25#13133128—41no—
26C4×#88322—41no—
27C22×Q8×C53C2×C412yes—
28C22×Q8×C53D411yescharacteristic
29C4⋊Q8×C53—41no—
30C4⋊Q8×C53—41no—
31C4×Q8×C53—41no—
32C4×Q8×C53—41no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53—21no—
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45#95×C53—22no—
46#95×C53—22no—
47#95×C53C2×C412yes—
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—41no—
51C2×Q8×C53—42no—
52C2×Q8×C53—82no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C4⋊C4×C53—24no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—82no—
70C4⋊C4×C53—82no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—42no—
78#11719917—24no—
79#11719917—42no—
80C2×#88322—42no—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83Q8×C53—44no—
84Q8×C53—84no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—24no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#10412yes—
108C2×C4×C53#13712yes—
109#88322—44no—
110#88322—82no—
111#11795—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53C2×#9911yescharacteristic
115C22×C53#41911yescharacteristic
116C22×C53C2×#10411yescharacteristic
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—44no—
121C4×C53—41no—
122C4×C53—41no—
123C4×C53—41no—
124C4×C53—41no—
125C4×C53—82no—
126C4×C53#35612yes—
127#2567D5311yescharacteristic
128C4×#101—531no—
129#1467—531no—
130#1354—534no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#150811yescharacteristic
134C2×C53#212211yescharacteristic
135C2×C53C2×#35611yescharacteristic
136C2×C4⋊Q8Dic5311yescharacteristic
137C2×C4×Q8C2×D5311yescharacteristic
138#432—1061no—
139C2×#101—1061no—
140C4×M16—1061no—
141#395Dic5312yes—
142#395C2×D5312yes—
143#391—21no—
144#389—21no—
145#389—21no—
146#384—21no—
147C4×C4⋊C4Dic5311yescharacteristic
148#343—532no—
149#324—532no—
150#324—1061no—
151C53#1179511yescharacteristic, complemented, Sylow
152C22×Q8C2×Dic5312yes—
153C22×Q8C22⋊D5311yescharacteristic
154C2×M16—1062no—
155C4⋊Q8—41no—
156C4⋊Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic5311yescharacteristic
165C2×C4⋊C4C22⋊D5311yescharacteristic
166C2×C4⋊C4C22×D5311yescharacteristic
167C2×C42—21no—
168C2×C42C2×Dic5311yescharacteristic
169C2×C42C22⋊D5311yescharacteristic
170C2×C42C22⋊D5311yescharacteristic
171C4⋊C8—2121no—
172#101—1064no—
173C22⋊C8—1062no—
174#97—2121no—
175#97—2121no—
176C4×C8—2121no—
177#95—22no—
178#95—22no—
179#95C2×Dic5312yes—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—82no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4C22⋊Dic5311yescharacteristic
188C22×C4C22⋊Dic5311yescharacteristic
189C22×C4C22⋊Dic5311yescharacteristic
190C22×C4C22⋊Dic5311yescharacteristic
191C22×C4C22×Dic5311yescharacteristic
192C22×C4C2×C22⋊D5311yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194M16—1064no—
195M16—2122no—
196C2×C8—2122no—
197C2×C8—2122no—
198C4⋊C4—24no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C23C2×C22⋊Dic5311yescharacteristic
215Q8—44no—
216Q8—84no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4#1313313211yescharacteristic
238C2×C4#1313313211yescharacteristic
239C2×C4#1313314412yes—
240C2×C4#1313324812yes—
241C8—4242no—
242C8—2124no—
243C22—21no—
244C22—21no—
245C22C2×#1313313211yescharacteristic
246C22#1681063211yescharacteristic
247C22C2×#1313314411yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—44no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—82no—
257C4#1681042212yes—
258C2—21no—
259C2—21no—
260C2#2027323511yescharacteristic
261C2#2027636511yescharacteristic
262C2C2×#1681042211yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2567C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2567D53
hypocenterC53#11795
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
151. C53#11795semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730592
#202727452
C53×#25672
#1179553