Finite Group Databasebeta

← #28299704

Subgroups — #28299704

Order 13568.

646 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 124 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#20290571C212yesmaximal
2#20289536C214yesmaximal, complemented
3C2×#16810528C211yesmaximal, characteristic
4C2×#13133284—22no—
5C2×C22⋊#88324—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—
15#13133284—44no—
16C22⋊#88324—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—44no—
41C22×C4×C53—42no—
42C22×C4×C53C22⋊C411yescharacteristic
43C22×C4×C53C22×C411yescharacteristic
44C22×Dic53—24no—
45C22×Dic53—22no—
46C22×Dic53—42no—
47C22×Dic53—42no—
48C22×Dic53—42no—
49C22×Dic53—42no—
50C22×Dic53C22⋊C412yes—
51C22×Dic53SD1614yes—
52C22×Dic53Q1614yes—
53C22×Dic53C2×D412yes—
54C2×Dic106—24no—
55C2×Dic106—22no—
56C2×Dic106—42no—
57C2×Dic106—416no—
58C2×Dic106—42no—
59C2×Dic106—42no—
60C2×Dic106—42no—
61C2×Dic106M1614yes—
62C2×#88324—44no—
63C2×#88324—44no—
64C2×#88324—44no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—42no—
67C4×Dic53—24no—
68C4×Dic53—41no—
69C4×Dic53—41no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—22no—
73C23×C53#10012yes—
74C23×C53#10314yes—
75C23×C53C2×C22⋊C411yescharacteristic
76C2×C4×C53—28no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—42no—
82C2×C4×C53—44no—
83C2×C4×C53—84no—
84C2×C4×C53#9912yes—
85C2×C4×C53C2×M1612yes—
86C2×Dic53—22no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—24no—
91C2×Dic53—44no—
92C2×Dic53—44no—
93C2×Dic53—48no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53#10314yes—
97C2×Dic53C2×SD1612yes—
98C2×Dic53C2×Q1612yes—
99Dic106—48no—
100Dic106—44no—
101Dic106—44no—
102#88324—84no—
103#7643—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—24no—
106C22×C53—24no—
107C22×C53—22no—
108C22×C53#33314yes—
109C22×C53#40711yescharacteristic
110C22×C53C2×#9911yescharacteristic
111C22×C53C2×#10011yescharacteristic
112C22×C53C2×#10312yes—
113C22×C53#42112yes—
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—82no—
117Dic53—22no—
118Dic53—24no—
119Dic53—22no—
120Dic53—44no—
121C2×#391—531no—
122#1797—532no—
123#1272—534no—
124C2×C53—24no—
125C2×C53#146211yescharacteristic
126C2×C53#148712yes—
127C2×C53#152012yes—
128C2×C53#153311yescharacteristic
129C2×C53C2×#33311yescharacteristic
130C23×Q8—531no—
131C22×C4⋊C4#615711yescharacteristic
132C22×C42—531no—
133C2×C4⋊C8—1062no—
134C2×C22⋊C8—1062no—
135#391—532no—
136#391—532no—
137#391—1062no—
138C2×#95—1062no—
139#336—534no—
140C53#764311yescharacteristic, complemented, Sylow
141C22×Q8—532no—
142C22×Q8—1064no—
143C22×Q8—1061no—
144C22×Q8—2122no—
145C22×Q8—2122no—
146C23×C4—21no—
147C23×C4—532no—
148C23×C4—1061no—
149C23×C4C2×#615711yescharacteristic
150C22×C8—1064no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4C2×#615712yes—
154C2×C42—532no—
155C2×C42—531no—
156C2×C42—531no—
157C2×C42—2122no—
158C4⋊C8—2124no—
159C22⋊C8—2124no—
160#95—1064no—
161#95—1064no—
162C24C22⋊#615711yescharacteristic
163C2×Q8—534no—
164C2×Q8—1064no—
165C2×Q8—1062no—
166C2×Q8—21216no—
167C2×Q8—2122no—
168C2×Q8—2122no—
169C2×Q8—2122no—
170C2×Q8—2122no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—44no—
176C22×C4—42no—
177C22×C4—532no—
178C22×C4—532no—
179C22×C4—534no—
180C22×C4—534no—
181C22×C4—1064no—
182C22×C4—1062no—
183C22×C4—2122no—
184C22×C4—2122no—
185C22×C4—2122no—
186C22×C4—2122no—
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22×#615711yescharacteristic
189C2×C8—2124no—
190C2×C8—2124no—
191C2×C8—2124no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C42—1064no—
195C42—2121no—
196C42—2121no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23#1313327712yes—
201C23#1313328014yes—
202C23C2×C22⋊#615711yescharacteristic
203Q8—2124no—
204Q8—2128no—
205Q8—2124no—
206C2×C4—28no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—84no—
214C2×C4—534no—
215C2×C4—532no—
216C2×C4—532no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1064no—
222C2×C4—2124no—
223C2×C4—2124no—
224C2×C4—2128no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4#1313327612yes—
228C2×C4D53⋊M1612yes—
229C8—4244no—
230C22—24no—
231C22—22no—
232C22—22no—
233C22—24no—
234C22#1681148314yes—
235C22#1681159511yescharacteristic
236C22C2×#1313327611yescharacteristic
237C22C2×#1313327711yescharacteristic
238C22C2×#1313328012yes—
239C22#1681162112yes—
240C4—22no—
241C4—22no—
242C4—82no—
243C4—1062no—
244C4—1064no—
245C4—1062no—
246C4—2124no—
247C2—24no—
248C2#2028996211yescharacteristic
249C2#2029004512yes—
250C2#2029015412yes—
251C2#2029019411yescharacteristic
252C2C2×#1681148311yescharacteristic
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#7643
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
140. C53#7643semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202895362
C2×#168105282
#764353