Finite Group Databasebeta

← #28308190

Subgroups — #28308190

Order 13568.

510 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289966C212yesmaximal
2#20289646C214yesmaximal, complemented
3#20277299C211yesmaximal, characteristic
4#16811643—22no—
5#16811468C2214yes—
6#16811466—22no—
7C23×Dic106C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810529C412yes—
10#16810529C2212yes—
11#16810528—22no—
12#16810484—22no—
13C2×#117×C53C411yescharacteristic
14#13133284—42no—
15C22⋊#88324—24no—
16#13133272—42no—
17C22×Dic106—21no—
18C22×Dic106—24no—
19C22×Dic106—42no—
20C22×Dic106—42no—
21C22×Dic106C2×C412yes—
22C23×Dic53—21no—
23#13133176—42no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×C4⋊Dic53C2311yescharacteristic
26C2×C22⋊Dic53D412yes—
27C2×C4×Dic53D412yes—
28#13133124—24no—
29#13133124—24no—
30C23×C4×C53C2×C411yescharacteristic
31#117×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C22⋊C4×C53—21no—
34C2×C42×C53C2×C412yes—
35C24×C53C22⋊C411yescharacteristic
36C22×C4×C53—22no—
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C22×C411yescharacteristic
42C22×Dic53—24no—
43C22×Dic53—22no—
44C22×Dic53—42no—
45C22×Dic53—42no—
46C22×Dic53C22⋊C412yes—
47C22×Dic53C2×D412yes—
48C2×Dic106—24no—
49C2×Dic106—22no—
50C2×Dic106—42no—
51C2×Dic106—416no—
52C2×Dic106—42no—
53C2×Dic106—42no—
54C2×Dic106—42no—
55C2×Dic106M1614yes—
56C2×#88324—44no—
57C4⋊C4×C53—42no—
58C22⋊C4×C53—42no—
59C42×C53—42no—
60C22⋊Dic53—42no—
61C4⋊Dic53—41no—
62C4⋊Dic53—41no—
63C4×Dic53—22no—
64C4×Dic53—22no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53#10012yes—
70C23×C53C2×C22⋊C411yescharacteristic
71C2×C4×C53—28no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53#9912yes—
80C2×C4×C53C2×M1612yes—
81C2×Dic53—24no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—48no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89Dic106—48no—
90Dic106—44no—
91Dic106—44no—
92#88324—84no—
93#11510—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—22no—
97C22×C53—42no—
98C22×C53—42no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53#40711yescharacteristic
102C22×C53C2×#9911yescharacteristic
103C22×C53C2×#10011yescharacteristic
104C4×C53—22no—
105C4×C53—22no—
106C4×C53—82no—
107Dic53—42no—
108Dic53—42no—
109Dic53—44no—
110#2310—531no—
111#1464—532no—
112#1319—534no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—42no—
116C2×C53#146211yescharacteristic
117C2×C53#212911yescharacteristic
118C2×C53#213511yescharacteristic
119C23×Q8—531no—
120C2×#117—531no—
121C2×#117#615711yescharacteristic
122#432—1062no—
123#391—532no—
124#391—532no—
125#391—1062no—
126#381—1062no—
127#324—534no—
128#323—1062no—
129C53#1151011yescharacteristic, complemented, Sylow
130C22×Q8—532no—
131C22×Q8—1064no—
132C22×Q8—1061no—
133C22×Q8—2122no—
134C22×Q8—2122no—
135C23×C4—1061no—
136C23×C4C2×#615711yescharacteristic
137#117—42no—
138#117—2122no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—531no—
141C2×C4⋊C4—531no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—532no—
144C2×C42—532no—
145C2×C42C2×#615712yes—
146C4⋊C8—2122no—
147C22⋊C8—1064no—
148#97—2122no—
149#95—1064no—
150#95—1064no—
151C24C22⋊#615711yescharacteristic
152C2×Q8—534no—
153C2×Q8—1064no—
154C2×Q8—1062no—
155C2×Q8—21216no—
156C2×Q8—2122no—
157C2×Q8—2122no—
158C2×Q8—2122no—
159C2×Q8—2122no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—532no—
165C22×C4—532no—
166C22×C4—1064no—
167C22×C4—1062no—
168C22×C4—2122no—
169C22×C4—2122no—
170C22×C4C22⋊#615711yescharacteristic
171C22×C4C22×#615711yescharacteristic
172C2×C8—2124no—
173C4⋊C4—42no—
174C4⋊C4—2121no—
175C4⋊C4—2121no—
176C22⋊C4—42no—
177C22⋊C4—2122no—
178C42—42no—
179C42—1062no—
180C42—1062no—
181C23—21no—
182C23—21no—
183C23—41no—
184C23—41no—
185C23#1313327712yes—
186C23C2×C22⋊#615711yescharacteristic
187Q8—2124no—
188Q8—2128no—
189Q8—2124no—
190C2×C4—28no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—1064no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—2128no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4#1313327612yes—
207C2×C4D53⋊M1612yes—
208C8—4244no—
209C22—21no—
210C22—21no—
211C22—22no—
212C22—42no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22#1681159511yescharacteristic
217C22C2×#1313327611yescharacteristic
218C22C2×#1313327711yescharacteristic
219C4—22no—
220C4—22no—
221C4—82no—
222C4—2124no—
223C4—2122no—
224C4—2122no—
225C2—21no—
226C2—21no—
227C2—42no—
228C2#2028996211yescharacteristic
229C2#2029099911yescharacteristic
230C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11510
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289646C2semidirect
129. C53#11510semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896462
#202772992
#1151053