Finite Group Databasebeta

← #28308423

Subgroups — #28308423

Order 13568.

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

normal subgroups only

This group has 358 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
1#20289977C211yesmaximal, characteristic
2#20289957C211yesmaximal, characteristic
3#20289656C214yesmaximal
4#20279931C211yesmaximal, characteristic
5#16811644—21no—
6#16811640—21no—
7#16811469—21no—
8#16811469—21no—
9#16811468C2212yes—
10#16811468C2212yes—
11C2×C4×Dic106C2211yescharacteristic
12#16810561—21no—
13#16810561—21no—
14#16810530—21no—
15#16810530—21no—
16#16810528C412yes—
17#16810488C2212yes—
18C4×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20#13133284—41no—
21C4⋊#88324—41no—
22C22⋊#88324—22no—
23C22⋊#88324—22no—
24#13133272—41no—
25C4×#88324—41no—
26C22×Dic106C2×C411yescharacteristic
27C4×Dic106—41no—
28C4×Dic106—41no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2311yescharacteristic
31C2×#11719919D411yescharacteristic
32C2×#11719919D411yescharacteristic
33C2×C4×Dic53C2×C412yes—
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36#13133124—21no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124—22no—
41C22×Q8×C53—21no—
42C4⋊Q8×C53—41no—
43C4⋊Q8×C53—41no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C42×C53C2×C411yescharacteristic
47C2×Q8×C53—44no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C22×C4×C53—22no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53—22no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C2×Dic106—22no—
60C2×Dic106—22no—
61C2×Dic106—41no—
62C2×#88324—42no—
63C2×#88324—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C42×C53—21no—
67C42×C53—21no—
68C4⋊Dic53—41no—
69C4⋊Dic53—42no—
70#11719919—41no—
71#11719919—41no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—24no—
77C23×C53C2×C22⋊C411yescharacteristic
78Q8×C53—82no—
79Q8×C53—82no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—41no—
85C2×C4×C53—44no—
86C2×C4×C53—42no—
87C2×C4×C53#9911yescharacteristic
88C2×C4×C53#9911yescharacteristic
89C2×C4×C53#13112yes—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53#10112yes—
103Dic106—44no—
104#88324—82no—
105#88324—82no—
106#11573—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53#40811yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53C2×#10111yescharacteristic
112C4×C53—21no—
113C4×C53—22no—
114C4×C53—41no—
115C4×C53—82no—
116C4×C53#35512yes—
117Dic53—22no—
118Dic53—41no—
119Dic53—41no—
120Dic53—41no—
121Dic53—82no—
122#2568—531no—
123#1467—531no—
124#1461—531no—
125#1324—534no—
126C2×C53—21no—
127C2×C53—21no—
128C2×C53#147211yescharacteristic
129C2×C53#212911yescharacteristic
130C2×C53C2×#35511yescharacteristic
131C2×C4⋊Q8#615711yescharacteristic
132C2×C4×Q8—531no—
133#432—1061no—
134#431—1061no—
135#400—1061no—
136#400—1061no—
137#391—532no—
138#391—1061no—
139#391—1061no—
140#382—532no—
141C4×C4⋊C4—531no—
142#324—532no—
143#324—532no—
144#324—1061no—
145#324—1061no—
146C53#1157311yescharacteristic, complemented, Sylow
147C22×Q8—21no—
148C22×Q8—531no—
149C4⋊Q8—41no—
150C4⋊Q8—41no—
151C4×Q8—2121no—
152C4×Q8—2121no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4C2×#615712yes—
159C2×C42—531no—
160C2×C42—531no—
161C2×C42—532no—
162C2×C42C2×#615711yescharacteristic
163C4⋊C8—2121no—
164C4⋊C8—2121no—
165C22⋊C8—1062no—
166C22⋊C8—1062no—
167#97—2121no—
168C4×C8—2121no—
169#95—1062no—
170#95—1062no—
171#95—1062no—
172#95—1062no—
173#95—1061no—
174C2×Q8—44no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—1062no—
178C2×Q8—1062no—
179C2×Q8—2121no—
180C22×C4—22no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—1062no—
186C22×C4C22⋊#615711yescharacteristic
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22×#615711yescharacteristic
189C2×C8—2122no—
190C2×C8—2122no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—2121no—
194C4⋊C4—2121no—
195C4⋊C4—2121no—
196C4⋊C4—2122no—
197C42—21no—
198C42—21no—
199C42—1061no—
200C42—1061no—
201C42—1061no—
202C42—1061no—
203C42—1064no—
204C23C2×C22⋊#615711yescharacteristic
205Q8—82no—
206Q8—82no—
207Q8—2124no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—532no—
216C2×C4—1061no—
217C2×C4—1061no—
218C2×C4—1061no—
219C2×C4—1061no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1061no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4—2121no—
227C2×C4—2121no—
228C2×C4#1313327511yescharacteristic
229C2×C4#1313327511yescharacteristic
230C2×C4#1313330512yes—
231C8—4242no—
232C8—4242no—
233C22—21no—
234C22—21no—
235C22#1681159811yescharacteristic
236C22C2×#1313327511yescharacteristic
237C22D53⋊#10111yescharacteristic
238C4—21no—
239C4—22no—
240C4—41no—
241C4—82no—
242C4—1062no—
243C4—4242no—
244C4—2121no—
245C4—2121no—
246C4—2121no—
247C4#1681153112yes—
248C2—21no—
249C2—21no—
250C2#2028999511yescharacteristic
251C2#2029099711yescharacteristic
252C2C2×#1681153111yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11573
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
146. C53#11573semidirect

Maximal subgroups

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

subgroupindex
#202899772
#202899572
#202896562
#202799312
#1157353