Finite Group Databasebeta

← #28299656

Subgroups — #28299656

Order 13568.

658 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 112 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
2#20289536C214yesmaximal, complemented
3C2×#16810528C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5C2×#13133272—22no—
6#16811489C2214yes—
7C23×Dic106C411yescharacteristic
8C22×C4×Dic53C2211yescharacteristic
9#16810528—22no—
10#16810528C412yes—
11#16810528C2212yes—
12C2×#13133124—22no—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—28no—
16#13133272—44no—
17C22×Dic106—21no—
18C22×Dic106—24no—
19C22×Dic106—42no—
20C22×Dic106—42no—
21C22×Dic106C2×C412yes—
22C23×Dic53—21no—
23C23×Dic53D412yes—
24C2×C4×Dic53—42no—
25C2×C4×Dic53C2×C411yescharacteristic
26C2×C4×Dic53D412yes—
27C2×C4×Dic53C2311yescharacteristic
28#13133124—24no—
29#13133124—24no—
30C23×C4×C53—21no—
31C23×C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—42no—
34C2×C4⋊C4×C53C2×C412yes—
35C24×C53C22⋊C411yescharacteristic
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53—42no—
41C22×C4×C53—44no—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C22×C411yescharacteristic
44C22×Dic53—24no—
45C22×Dic53—24no—
46C22×Dic53—22no—
47C22×Dic53—24no—
48C22×Dic53—42no—
49C22×Dic53—42no—
50C22×Dic53—42no—
51C22×Dic53C22⋊C412yes—
52C22×Dic53C2×D412yes—
53C2×Dic106—24no—
54C2×Dic106—22no—
55C2×Dic106—42no—
56C2×Dic106—416no—
57C2×Dic106—42no—
58C2×Dic106—42no—
59C2×Dic106—42no—
60C2×Dic106M1614yes—
61C2×#88324—28no—
62C2×#88324—44no—
63C2×#88324—44no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C4×Dic53—24no—
67C4×Dic53—41no—
68C4×Dic53—41no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53#10012yes—
73C23×C53#10414yes—
74C23×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53—28no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—42no—
81C2×C4×C53—44no—
82C2×C4×C53—84no—
83C2×C4×C53#9912yes—
84C2×C4×C53C2×M1612yes—
85C2×Dic53—22no—
86C2×Dic53—22no—
87C2×Dic53—24no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—22no—
91C2×Dic53—48no—
92C2×Dic53—48no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53#13714yes—
96Dic106—48no—
97Dic106—44no—
98Dic106—44no—
99#88324—48no—
100#7629—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53—22no—
103C22×C53—24no—
104C22×C53—24no—
105C22×C53#32914yes—
106C22×C53#40711yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#10011yescharacteristic
109C22×C53#41912yes—
110C22×C53C2×#10412yes—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—82no—
114Dic53—24no—
115Dic53—44no—
116Dic53—42no—
117C2×#391—531no—
118#1749—532no—
119#1272—534no—
120C2×C53—24no—
121C2×C53#146211yescharacteristic
122C2×C53#149112yes—
123C2×C53#150812yes—
124C2×C53C2×#32911yescharacteristic
125C2×C53#155811yescharacteristic
126C23×Q8—531no—
127C22×C4⋊C4#615711yescharacteristic
128C22×C42—531no—
129C2×C22⋊C8—1062no—
130C2×#97—1062no—
131#391—532no—
132#391—532no—
133#391—1062no—
134C2×#95—1062no—
135#336—534no—
136C53#762911yescharacteristic, complemented, Sylow
137C22×Q8—532no—
138C22×Q8—1064no—
139C22×Q8—1061no—
140C22×Q8—2122no—
141C22×Q8—2122no—
142C23×C4—21no—
143C23×C4—532no—
144C23×C4—1061no—
145C23×C4C2×#615711yescharacteristic
146C22×C8—1064no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C2×#615712yes—
150C2×C42—532no—
151C2×C42—531no—
152C2×C42—531no—
153C2×C42—2122no—
154C22⋊C8—1068no—
155#97—2124no—
156#95—1064no—
157#95—1064no—
158C24C22⋊#615711yescharacteristic
159C2×Q8—534no—
160C2×Q8—1064no—
161C2×Q8—1062no—
162C2×Q8—21216no—
163C2×Q8—2122no—
164C2×Q8—2122no—
165C2×Q8—2122no—
166C2×Q8—2122no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—532no—
174C22×C4—532no—
175C22×C4—1064no—
176C22×C4—1064no—
177C22×C4—1064no—
178C22×C4—1062no—
179C22×C4—2122no—
180C22×C4—2122no—
181C22×C4—2122no—
182C22×C4C22⋊#615711yescharacteristic
183C22×C4C22×#615711yescharacteristic
184C2×C8—1068no—
185C2×C8—2124no—
186C2×C8—2124no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C42—1064no—
190C42—2121no—
191C42—2121no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23#1313327712yes—
196C23#1313328114yes—
197C23C2×C22⋊#615711yescharacteristic
198Q8—2124no—
199Q8—2128no—
200Q8—2124no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—84no—
209C2×C4—534no—
210C2×C4—1062no—
211C2×C4—1062no—
212C2×C4—1064no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—2128no—
217C2×C4—2128no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4#1313327612yes—
221C2×C4D53⋊M1612yes—
222C8—2128no—
223C22—24no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22#1681147814yes—
228C22#1681159511yescharacteristic
229C22C2×#1313327611yescharacteristic
230C22C2×#1313327711yescharacteristic
231C22#1681161612yes—
232C22C2×#1313328112yes—
233C4—22no—
234C4—22no—
235C4—82no—
236C4—1064no—
237C4—2122no—
238C4—2124no—
239C2—24no—
240C2#2028996211yescharacteristic
241C2#2029005912yes—
242C2#2029011412yes—
243C2C2×#1681147811yescharacteristic
244C2#2029024811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#7629
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289536C2semidirect
136. C53#7629semidirect

Maximal subgroups

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

subgroupindex
#202904412
#202895362
C2×#168105282
#762953