Finite Group Databasebeta

← #27281515

Subgroups — #27281515

Order 13568.

706 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 115 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#20283756C211yesmaximal, characteristic, complemented
2#20279661C211yesmaximal, characteristic
3#20273384C212yesmaximal, complemented
4#20272616C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7C2×#13133213—21no—
8C2×#13133191—22no—
9#16810701—24no—
10#16810676C2212yes—
11#16810587—22no—
12#16810532C414yes—
13#16810524—21no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810366—24no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#423C2212yes—
18C2×#100×C53—21no—
19C53×#343—21no—
20C22×D212—21no—
21C22×D212—21no—
22C22×D212C2×C412yes—
23C22×C22⋊D53—24no—
24C2×#11719917—24no—
25#13133214—22no—
26#13133214—22no—
27#13133213—44no—
28#13133191—48no—
29C2×C4⋊Dic53—21no—
30C2×#11719921—24no—
31C2×C4×Dic53—21no—
32C4⋊#88322D414yescomplemented
33C22⋊#88322—24no—
34C22⋊#88322—22no—
35C4×#88322—22no—
36#13133124C2×C412yes—
37C22×D4×C53D411yescharacteristic
38C23×C4×C53D412yes—
39C2×M16×C53—22no—
40C4×D4×C53—22no—
41C4×D4×C53—22no—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2×C412yes—
44C2×C42×C53C2311yescharacteristic
45C4⋊C8×C53—22no—
46C4⋊C8×C53—22no—
47#100×C53—42no—
48C24×C53C22⋊C412yes—
49C23×D53—22no—
50C2×D4×C53—22no—
51C2×D4×C53—21no—
52C2×D4×C53—22no—
53C2×D4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—44no—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C2×C22⋊D53—48no—
62C2×C22⋊D53—48no—
63C2×C22⋊D53—84no—
64C22×Dic53—22no—
65C2×D212—24no—
66C2×D212—44no—
67C2×D212—41no—
68C2×D212—42no—
69C2×D212—41no—
70C2×D212—42no—
71C2×D212—41no—
72C2×D212—41no—
73C2×D212—82no—
74M16×C53—42no—
75M16×C53—42no—
76C2×C8×C53—42no—
77C4⋊C4×C53—22no—
78C22⋊C4×C53—24no—
79C42×C53C2×D412yes—
80C42×C53#4112yes—
81#11719921—48no—
82C4⋊Dic53—24no—
83C4×Dic53—42no—
84#11719917—84no—
85C2×#88322—24no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53C2×C22⋊C411yescharacteristic
92C22×D53—44no—
93C22×D53—44no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—42no—
97D4×C53—42no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—44no—
103C2×C4×C53—42no—
104C2×C4×C53—82no—
105C2×C4×C53C4×D412yes—
106C2×C4×C53#12011yescharacteristic
107C2×C4×C53#12112yes—
108C2×C4×C53#12311yescharacteristic
109C2×C4×C53AGL(1,ℤ/8)12yes—
110C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C53#13711yescharacteristic
112C22⋊D53—816no—
113C2×Dic53—24no—
114C2×Dic53—42no—
115C2×Dic53—42no—
116D212—84no—
117D212—82no—
118D212—82no—
119D212—82no—
120D212—82no—
121C8×C53—82no—
122#88322—44no—
123#9405—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53#38611yescharacteristic
132C22×C53#41711yescharacteristic
133C22×C53#41811yescharacteristic
134C2×D53—84no—
135C2×D53—84no—
136C2×D53—82no—
137C2×D53—82no—
138C2×D53—82no—
139C4×C53—21no—
140C4×C53—22no—
141C4×C53—42no—
142C4×C53#45312yes—
143C4×C53#46812yes—
144C4×C53#46912yes—
145Dic53—44no—
146#2975—531no—
147#2541—531no—
148#1800D5311yescharacteristic, complemented
149#1545—532no—
150#1303—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—82no—
154C2×C53#188711yescharacteristic
155C2×C53#189511yescharacteristic
156C2×C53#191411yescharacteristic
157D53—162no—
158C2×#127—531no—
159C2×#127—1061no—
160C2×#121—1062no—
161C2×C4×D4C2×D5311yescharacteristic
162#436—1064no—
163#431—532no—
164#423C2×D5312yes—
165C2×#100—21no—
166#407—1062no—
167#392—534no—
168#390—1061no—
169C4×C4⋊C4—531no—
170#343—21no—
171#331—1064no—
172C53#940511yescharacteristic, complemented, Sylow
173C22×D4—532no—
174C22×D4—1064no—
175C22×D4—1061no—
176C22×D4—1061no—
177C22×D4C22⋊D5311yescharacteristic
178C23×C4D21212yes—
179C2×M16—22no—
180C2×M16—1064no—
181#127—1062no—
182#127—1062no—
183#127—2124no—
184#121—2128no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4C22⋊D5311yescharacteristic
189C2×C22⋊C4—1064no—
190C2×C22⋊C4C4×D5312yes—
191C2×C42—1061no—
192C2×C42C22×D5311yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—534no—
196#100—42no—
197C22⋊C8—1064no—
198C22⋊C8—1062no—
199C4×C8—1062no—
200#95—532no—
201C24—1062no—
202C24#1171992112yes—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—1064no—
208C2×D4—4242no—
209C2×D4—4244no—
210C2×D4—2124no—
211C2×D4—2128no—
212C2×D4—2128no—
213C2×D4—2121no—
214C2×D4—2122no—
215C2×D4—2121no—
216C2×D4—2122no—
217C2×D4—2121no—
218C2×D4—2121no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—44no—
222C22×C4—1062no—
223C22×C4#1171992112yes—
224C22×C4C2×C4×D5311yescharacteristic
225C22×C4C2×D21211yescharacteristic
226C22×C4C2×C22⋊D5311yescharacteristic
227M16—42no—
228M16—42no—
229M16—4244no—
230C2×C8—42no—
231C2×C8—1064no—
232C4⋊C4—22no—
233C4⋊C4—1064no—
234C22⋊C4—24no—
235C22⋊C4—2128no—
236C42—2122no—
237C42D4×D5312yes—
238C42#1171994712yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—2124no—
245C23—2124no—
246C23—2122no—
247C23—2122no—
248C23—2122no—
249C23C2×#1171992111yescharacteristic
250D4—42no—
251D4—42no—
252D4—4244no—
253D4—4242no—
254D4—42416no—
255D4—4242no—
256D4—4242no—
257D4—4242no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—1064no—
265C2×C4—2122no—
266C2×C4—2122no—
267C2×C4#1313317712yes—
268C2×C4#1313318711yescharacteristic
269C2×C4#1313318912yes—
270C2×C4#1313320111yescharacteristic
271C2×C4#1313324212yes—
272C2×C4#1313324311yescharacteristic
273C2×C4#1313324811yescharacteristic
274C8—82no—
275C8—2124no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—4244no—
284C22—4244no—
285C22—4242no—
286C22—4242no—
287C22—4242no—
288C22#1681050011yescharacteristic
289C22#1681062211yescharacteristic
290C22#1681062711yescharacteristic
291C4—22no—
292C4—21no—
293C4—42no—
294C4—2124no—
295C4#1681079312yes—
296C4#1681087112yes—
297C4#1681087412yes—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—8482no—
302C2#2027511111yescharacteristic
303C2#2027515111yescharacteristic
304C2#2027526011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1800C2
socleC22×C53#386
prime radical#20279661C2
cosocleC2×C42×C53C23
hypercenter#1800D53
hypocenterC53#9405
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #20283756C2semidirect
3. #20273384C2semidirect
4. #20272616C2semidirect
5. C53×#1800C2semidirect
32. C4⋊#88322D4semidirect
148. #1800D53semidirect
172. C53#9405semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202796612
#202733842
#202726162
C53×#18002
#940553