Finite Group Databasebeta

← #27294769

Subgroups — #27294769

Order 13568.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280481C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272714C214yesmaximal
4C53×#1820C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic, complemented
6C2×#13133196—21no—
7#16810680—21no—
8#16810553—21no—
9#16810553C412yescomplemented
10#16810548—21no—
11#16810548C2212yes—
12#16810529—21no—
13#16810489C411yescharacteristic, complemented
14#16810349—21no—
15#16810347C2212yes—
16C2×#117×C53C2211yescharacteristic
17C53×#381—21no—
18C53×#342—21no—
19C53×#342C2212yes—
20C22×Dic106—21no—
21#13133217—41no—
22#13133217—41no—
23#13133196—44no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—22no—
28C2×C22⋊Dic53—22no—
29C4⋊#88322—41no—
30C22⋊#88322—22no—
31#13133128—41no—
32#13133124—22no—
33#13133124—21no—
34#13133124—22no—
35#13133124C2×C412yes—
36C23×C4×C53D411yescharacteristic
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53D411yescharacteristic
43C2×C22⋊C4×C53C2311yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46#95×C53—41no—
47#95×C53—41no—
48C24×C53C2×D411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—41no—
53C22×C4×C53—42no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53—22no—
62C2×Dic106—42no—
63C2×Dic106—42no—
64C2×Dic106—41no—
65C2×Dic106—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C22⋊C4×C53—41no—
69C22⋊C4×C53—41no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C22⋊Dic53—44no—
77C4⋊Dic53—42no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80#11719919—44no—
81C2×#88322—42no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—41no—
85C23×C53—41no—
86C23×C53C2×C22⋊C411yescharacteristic
87C23×C53AGL(1,ℤ/8)12yes—
88C2×C4×C53—21no—
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—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—41no—
103C2×C4×C53—82no—
104C2×C4×C53—82no—
105C2×C4×C53—82no—
106C2×C4×C53#9911yescharacteristic
107C2×C4×C53#9911yescharacteristic
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114Dic106—82no—
115Dic106—82no—
116#88322—82no—
117#11801—531nomaximal, Sylow
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—22no—
121C22×C53—42no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—81no—
125C22×C53C2×#9911yescharacteristic
126C22×C53#41811yescharacteristic
127C22×C53#42111yescharacteristic
128C4×C53—21no—
129C4×C53—21no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—162no—
136Dic53—82no—
137Dic53—82no—
138#2621—531no—
139#1820D5311yescharacteristic
140#1464—531no—
141#1344—534no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53—42no—
145C2×C53#151011yescharacteristic
146C2×C53#212311yescharacteristic
147C2×C53#212911yescharacteristic
148C2×C4⋊Q8—531no—
149C2×C22⋊Q8—1061no—
150C2×#117C2×D5311yescharacteristic
151#432—1061no—
152#398—532no—
153#398—1061no—
154#397—532no—
155#397—1061no—
156#391—1061no—
157#382—531no—
158#381—21no—
159#342—21no—
160#342C2×D5312yes—
161#324—1061no—
162#323—532no—
163C53#1180111yescharacteristic, complemented, Sylow
164C22×Q8—1061no—
165C23×C4C22⋊D5311yescharacteristic
166C4⋊Q8—2121no—
167C4⋊Q8—2121no—
168C22⋊Q8—2124no—
169#117—41no—
170#117—41no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4C4×D5311yescharacteristic
176C2×C4⋊C4D21211yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—1062no—
179C2×C22⋊C4C22⋊D5311yescharacteristic
180C2×C22⋊C4C22×D5311yescharacteristic
181C2×C42C4×D5311yescharacteristic
182C2×C42D21211yescharacteristic
183C4⋊C8—2121no—
184C22⋊C8—1062no—
185#97—2121no—
186#95—41no—
187#95—41no—
188#95—532no—
189#95—1062no—
190#95—1061no—
191#95—1062no—
192C24C2×C22⋊D5311yescharacteristic
193C2×Q8—2122no—
194C2×Q8—2122no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—1062no—
203C22×C4—1062no—
204C22×C4#1171992111yescharacteristic
205C22×C4#1171992111yescharacteristic
206C22×C4#1171992111yescharacteristic
207C22×C4#1171992111yescharacteristic
208C22×C4C2×C4×D5311yescharacteristic
209C22×C4C2×D21211yescharacteristic
210C2×C8—2122no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—2122no—
214C4⋊C4—2124no—
215C4⋊C4—2122no—
216C4⋊C4—2122no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—2124no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23C2×#1171992111yescharacteristic
231C23#1313324212yes—
232Q8—4242no—
233Q8—4242no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4#1313313111yescharacteristic
259C2×C4#1313313111yescharacteristic
260C8—4242no—
261C22—21no—
262C22—21no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22C2×#1313313111yescharacteristic
269C22#1681062711yescharacteristic
270C22#1681064111yescharacteristic
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—162no—
279C4—4242no—
280C4—4242no—
281C2—21no—
282C2—21no—
283C2—42no—
284C2#2027324311yescharacteristic
285C2#2027636811yescharacteristic
286C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1820C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1820D53
hypocenterC53#11801
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

Complements

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

NH ≅ G/Ntype
5. C2×#13133217C4semidirect
9. #16810553C4semidirect
13. #16810489C4semidirect
163. C53#11801semidirect

Maximal subgroups

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

subgroupindex
#202804812
#202730492
#202727142
C53×#18202
#1180153