Finite Group Databasebeta

← #28308176

Subgroups — #28308176

Order 13568.

598 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 84 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#20289963C212yesmaximal
2#20289646C214yesmaximal, complemented
3C2×#16810529C211yesmaximal, characteristic
4#16811593—22no—
5#16811468—22no—
6#16811468C2214yes—
7C23×Dic106C411yescharacteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810529—22no—
10#16810529C412yes—
11#16810529C2212yes—
12C2×#13133124—22no—
13C22×C42×C53C411yescharacteristic
14C2×#11719940—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C22×Dic106—21no—
18C22×Dic106—24no—
19C22×Dic106—41no—
20C22×Dic106—41no—
21C22×Dic106—42no—
22C22×Dic106C2×C412yes—
23C23×Dic53—21no—
24C23×Dic53D412yes—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53D412yes—
29C2×C4⋊Dic53C2311yescharacteristic
30#13133124—24no—
31#13133124—24no—
32C23×C4×C53—21no—
33C23×C4×C53C2×C411yescharacteristic
34C2×C42×C53—21no—
35C2×C42×C53—42no—
36C2×C42×C53C2×C412yes—
37C24×C53C22⋊C411yescharacteristic
38C22×C4×C53—21no—
39C22×C4×C53—21no—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—24no—
43C22×C4×C53—22no—
44C22×C4×C53—42no—
45C22×C4×C53—42no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—24no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—42no—
53C22×Dic53—41no—
54C22×Dic53—41no—
55C22×Dic53—44no—
56C22×Dic53C22⋊C412yes—
57C22×Dic53C2×D412yes—
58C2×Dic106—24no—
59C2×Dic106—22no—
60C2×Dic106—42no—
61C2×Dic106—48no—
62C2×Dic106—48no—
63C2×Dic106—42no—
64C2×Dic106—42no—
65C2×Dic106—42no—
66C2×Dic106M1614yescomplemented
67#11719940—84no—
68C2×#88324—44no—
69C42×C53—42no—
70C42×C53—42no—
71C4⋊Dic53—41no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53#9911yescharacteristic
79C23×C53#9911yescharacteristic
80C23×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53—24no—
82C2×C4×C53—24no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—24no—
88C2×C4×C53—48no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53#10012yes—
92C2×C4×C53C2×M1612yes—
93C2×Dic53—24no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—44no—
97C2×Dic53—44no—
98C2×Dic53—44no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—82no—
104Dic106—48no—
105Dic106—44no—
106Dic106—44no—
107#88324—84no—
108#11506—531nomaximal, Sylow
109C22×C53—24no—
110C22×C53—24no—
111C22×C53—21no—
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—21no—
117C22×C53#35412yes—
118C22×C53#35612yes—
119C22×C53#40711yescharacteristic
120C22×C53C2×#9911yescharacteristic
121C22×C53C2×#10011yescharacteristic
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—44no—
125Dic53—42no—
126Dic53—42no—
127Dic53—44no—
128C2×#391—531no—
129#1463—532no—
130#1319—534no—
131C2×C53—21no—
132C2×C53—22no—
133C2×C53—22no—
134C2×C53—21no—
135C2×C53#146211yescharacteristic
136C2×C53C2×#35411yescharacteristic
137C2×C53C2×#35611yescharacteristic
138C23×Q8—531no—
139C22×C4⋊C4—531no—
140C22×C42#615711yescharacteristic
141#407—1062no—
142#391—532no—
143#391—532no—
144#391—1062no—
145C2×#95—1062no—
146#324—534no—
147#324—1062no—
148C53#1150611yescharacteristic, complemented, Sylow
149C22×Q8—532no—
150C22×Q8—1064no—
151C22×Q8—1061no—
152C22×Q8—2122no—
153C22×Q8—2121no—
154C22×Q8—2121no—
155C23×C4—21no—
156C23×C4—532no—
157C23×C4—1061no—
158C23×C4C2×#615711yescharacteristic
159C2×M16—2122no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—2121no—
164C2×C4⋊C4—2121no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C2×#615712yes—
168C22⋊C8—1064no—
169C22⋊C8—2122no—
170#95—1064no—
171#95—1064no—
172C24C22⋊#615711yescharacteristic
173C2×Q8—534no—
174C2×Q8—1064no—
175C2×Q8—1062no—
176C2×Q8—2128no—
177C2×Q8—2128no—
178C2×Q8—2122no—
179C2×Q8—2122no—
180C2×Q8—2122no—
181C2×Q8—2122no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—532no—
191C22×C4—532no—
192C22×C4—1062no—
193C22×C4—1064no—
194C22×C4—1062no—
195C22×C4—1062no—
196C22×C4—2124no—
197C22×C4—2122no—
198C22×C4—2121no—
199C22×C4—2121no—
200C22×C4C22⋊#615711yescharacteristic
201C22×C4C22×#615711yescharacteristic
202M16—4244no—
203C2×C8—2124no—
204C4⋊C4—2122no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C42—42no—
208C42—42no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23#1313327611yescharacteristic
214C23#1313327611yescharacteristic
215C23C2×C22⋊#615711yescharacteristic
216Q8—2124no—
217Q8—2128no—
218Q8—2124no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—48no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—1064no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—4242no—
233C2×C4—2124no—
234C2×C4—2124no—
235C2×C4—2124no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2122no—
240C2×C4#1313327712yes—
241C2×C4D53⋊M1612yes—
242C8—4244no—
243C22—24no—
244C22—24no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22#1681153012yes—
252C22#1681153412yes—
253C22#1681159511yescharacteristic
254C22C2×#1313327611yescharacteristic
255C22C2×#1313327711yescharacteristic
256C4—22no—
257C4—22no—
258C4—44no—
259C4—2122no—
260C4—2122no—
261C4—2124no—
262C2—21no—
263C2—22no—
264C2—22no—
265C2—21no—
266C2#2028996211yescharacteristic
267C2C2×#1681153011yescharacteristic
268C2C2×#1681153411yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53#11506
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289646C2semidirect
66. C2×Dic106M16semidirect
148. C53#11506semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202896462
C2×#168105292
#1150653