Finite Group Databasebeta

← #27294745

Subgroups — #27294745

Order 13568.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279920C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic
3#20272746C214yesmaximal
4C53×#1467C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133136—21no—
7C4×#11719917—21no—
8#16810543C412yes—
9#16810543C2212yes—
10#16810530—21no—
11#16810521—21no—
12#16810521—21no—
13#16810492—21no—
14C4×C4⋊Dic53C411yescharacteristic
15#16810392C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#432—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106C2×C412yes—
21C2×#11719917—22no—
22#13133217—41no—
23#13133217—41no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×C4×Dic53—21no—
30#13133136—24no—
31#13133128—41no—
32C4×#88322—41no—
33#13133124—22no—
34#13133124—22no—
35#13133124C2×C412yes—
36C22×Q8×C53D411yescharacteristic
37C4×Q8×C53—41no—
38C4×Q8×C53—41no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C22⋊C8×C53—22no—
47#97×C53—41no—
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×Dic106—42no—
61C2×Dic106—82no—
62C2×C8×C53—42no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C42×C53—21no—
67C42×C53—21no—
68C42×C53—21no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—21no—
72C4⋊Dic53—24no—
73C4⋊Dic53—42no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76#11719919—82no—
77#11719919—82no—
78C4×Dic53—42no—
79#11719917—24no—
80#11719917—42no—
81C2×#88322—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83Q8×C53—44no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53#9911yescharacteristic
99C2×C4×C53#9911yescharacteristic
100C2×C4×C53#10412yes—
101C2×C4×C53#13712yes—
102C2×Dic53—24no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108Dic106—84no—
109C8×C53—82no—
110#88322—44no—
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—41no—
121C4×C53—41no—
122C4×C53—41no—
123C4×C53—41no—
124C4×C53#35612yes—
125Dic53—44no—
126Dic53—82no—
127#2567—531no—
128C4×#101—531no—
129#1467D5311yescharacteristic
130#1354—534no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#150811yescharacteristic
134C2×C53#212211yescharacteristic
135C2×C53C2×#35611yescharacteristic
136C2×C4⋊Q8—531no—
137C2×C4×Q8C2×D5311yescharacteristic
138#432—21no—
139C2×#101—1061no—
140C4×M16—1061no—
141#395—532no—
142#395—532no—
143#391—1061no—
144#389—1061no—
145#389—1061no—
146#384—1061no—
147C4×C4⋊C4—531no—
148#343—532no—
149#324—21no—
150#324C2×D5312yes—
151C53#1179511yescharacteristic, complemented, Sylow
152C22×Q8—532no—
153C22×Q8D21211yescharacteristic
154C2×M16—1062no—
155C4⋊Q8—2121no—
156C4⋊Q8—2121no—
157C4×Q8—41no—
158C4×Q8—41no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4C4×D5311yescharacteristic
165C2×C4⋊C4C22⋊D5311yescharacteristic
166C2×C4⋊C4C22×D5311yescharacteristic
167C2×C42—1061no—
168C2×C42C4×D5311yescharacteristic
169C2×C42D21211yescharacteristic
170C2×C42C22⋊D5311yescharacteristic
171C4⋊C8—41no—
172#101—1064no—
173C22⋊C8—22no—
174#97—41no—
175#97—2121no—
176C4×C8—2121no—
177#95—532no—
178#95—1062no—
179#95—1062no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—4242no—
184C2×Q8—2122no—
185C22×C4—1062no—
186C22×C4—1062no—
187C22×C4#1171992111yescharacteristic
188C22×C4#1171992111yescharacteristic
189C22×C4#1171992111yescharacteristic
190C22×C4#1171992111yescharacteristic
191C22×C4C2×C4×D5311yescharacteristic
192C22×C4C2×D21211yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194M16—1064no—
195M16—2122no—
196C2×C8—42no—
197C2×C8—2122no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—1064no—
202C4⋊C4—4242no—
203C4⋊C4—4242no—
204C4⋊C4—2122no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—2122no—
214C23C2×#1171992111yescharacteristic
215Q8—44no—
216Q8—4244no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—1064no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4#1313313111yescharacteristic
238C2×C4#1313313111yescharacteristic
239C2×C4#1313314312yes—
240C2×C4#1313325012yes—
241C8—82no—
242C8—2124no—
243C22—21no—
244C22—21no—
245C22C2×#1313313111yescharacteristic
246C22#1681063311yescharacteristic
247C22C2×#1313314311yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—4242no—
256C4—2124no—
257C4#1681042112yes—
258C2—21no—
259C2—21no—
260C2#2027323611yescharacteristic
261C2#2027636411yescharacteristic
262C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11795
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

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
#202799202
C4×#131331362
#202727462
C53×#14672
#1179553