Finite Group Databasebeta

← #28302322

Subgroups — #28302322

Order 13568.

814 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 120 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#20290441C212yesmaximal
2C2×#16811480C214yesmaximal, complemented
3C2×#16811176C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272C2214yes—
6#16811489—22no—
7#16811480—28no—
8#16811176—28no—
9C2×#13133216C412yes—
10C2×#13133208C2212yes—
11C2×#13133194—22no—
12C2×C4×Dic106—22no—
13C2×#13133176C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15C2×#117×C53C411yescharacteristic
16C22×#88324—24no—
17C22⋊#88324—28no—
18#13133272—28no—
19C22×Dic106—22no—
20C23×Dic53D412yes—
21#13133216—24no—
22#13133216—24no—
23#13133208—24no—
24#13133208D418yes—
25#13133194—216no—
26C4×Dic106—216no—
27#13133176—24no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53C2×C412yes—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—24no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53D412yes—
37C2×C4×Dic53C2311yescharacteristic
38C23×C4×C53C2×C411yescharacteristic
39#117×C53—24no—
40C2×C4⋊C4×C53C2×C412yes—
41C2×C22⋊C4×C53—21no—
42C2×C42×C53—21no—
43C24×C53C22⋊C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53—22no—
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—24no—
52C22×Dic53—24no—
53C22×Dic53—42no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×Dic106—28no—
57C2×Dic106—44no—
58C2×Dic106—44no—
59C2×#88324—28no—
60C2×#88324—44no—
61C2×#88324—44no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53C22⋊C414yes—
64C22⋊C4×C53—24no—
65C42×C53—24no—
66C22⋊Dic53—24no—
67C4⋊Dic53—22no—
68C4⋊Dic53—28no—
69C4⋊Dic53C22⋊C414yes—
70#11719919—28no—
71#11719919—28no—
72C4×Dic53—24no—
73C4×Dic53—22no—
74C4×Dic53—24no—
75C4×Dic53C2×D414yes—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53#10112yes—
81C23×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53—28no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—24no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53#10112yes—
91C2×C4×C53C2×C22⋊C412yes—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—24no—
97C2×Dic53—48no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53#10414yes—
101C2×Dic53#12014yes—
102Dic106—416no—
103#88324—48no—
104#8338—531nomaximal, Sylow
105C22×C53—22no—
106C22×C53—24no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—42no—
110C22×C53#37911yescharacteristic
111C22×C53C2×#10111yescharacteristic
112C22×C53C2×#10111yescharacteristic
113C22×C53C2×#10412yes—
114C22×C53#42112yes—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—44no—
118Dic53—24no—
119Dic53—24no—
120Dic53—44no—
121C2×#533—531no—
122#1749—532no—
123C2×#330—534no—
124C2×C53—42no—
125C2×C53#152611yescharacteristic
126C2×C53#153112yes—
127C2×C53#153711yescharacteristic
128C2×C53#178411yescharacteristic
129C2×C53#178912yes—
130#533—1068no—
131C2×C4⋊Q8—532no—
132C2×#125—532no—
133C2×C22⋊Q8—1062no—
134C2×C4×Q8—1062no—
135C2×#117—531no—
136C2×#117#615711yescharacteristic
137C22×C42—531no—
138C2×C22⋊C8—1062no—
139C2×#97—534no—
140#336—1062no—
141#330—1068no—
142C53#833811yescharacteristic, complemented, Sylow
143C22×Q8—1062no—
144C23×C4—532no—
145C23×C4C2×#615711yescharacteristic
146C22×C8—1064no—
147C4⋊Q8—1064no—
148C4⋊Q8—1064no—
149#125—538no—
150#125—1064no—
151C22⋊Q8—10616no—
152C4×Q8—10616no—
153#117—24no—
154#117—1064no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4C2×#615712yes—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—1061no—
162C2×C42—21no—
163C2×C42—531no—
164C2×C42—532no—
165C2×C42—531no—
166C2×C42—1064no—
167C2×C42—1061no—
168C22⋊C8—1068no—
169#97—1068no—
170C24C22⋊#615711yescharacteristic
171C2×Q8—1068no—
172C2×Q8—2124no—
173C2×Q8—2124no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—532no—
179C22×C4—532no—
180C22×C4—1064no—
181C22×C4—1064no—
182C22×C4—1062no—
183C22×C4—2122no—
184C22×C4C22⋊#615711yescharacteristic
185C22×C4C22×#615711yescharacteristic
186C2×C8—1068no—
187C2×C8—2124no—
188C2×C8—2124no—
189C4⋊C4—22no—
190C4⋊C4—534no—
191C4⋊C4—1068no—
192C4⋊C4—1068no—
193C4⋊C4—1068no—
194C4⋊C4—1062no—
195C4⋊C4C22⋊#615714yes—
196C22⋊C4—24no—
197C22⋊C4—1064no—
198C42—24no—
199C42—534no—
200C42—1064no—
201C42—1064no—
202C42—1062no—
203C23—21no—
204C23—21no—
205C23—22no—
206C23—22no—
207C23#1313327812yes—
208C23C2×C22⋊#615711yescharacteristic
209Q8—21216no—
210C2×C4—28no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—534no—
219C2×C4—534no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1064no—
225C2×C4—2128no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4#1313327812yes—
229C2×C4C2×C22⋊#615712yes—
230C8—2128no—
231C22—24no—
232C22—22no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22#1681156411yescharacteristic
237C22C2×#1313327811yescharacteristic
238C22C2×#1313327811yescharacteristic
239C22D53⋊#10412yes—
240C22#1681162012yes—
241C4—22no—
242C4—22no—
243C4—44no—
244C4—1064no—
245C4—1064no—
246C4—2124no—
247C2—42no—
248C2#2029017211yescharacteristic
249C2#2029018712yes—
250C2#2029020611yescharacteristic
251C2#2029054911yescharacteristic
252C2#2029055712yes—
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#117#6157
hypocenterC53#8338
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#16811480C2semidirect
142. C53#8338semidirect

Maximal subgroups

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

subgroupindex
#202904412
C2×#168114802
C2×#168111762
#833853