Finite Group Databasebeta

← C2×#20289656 #28307961

Subgroups — C2×#20289656 #28307961

Order 13568.

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

normal subgroups only

This group has 510 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
1C2×#16811468C212yesmaximal
2#20289656C214yesmaximal, direct product factor
3C2×#16810488C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811468—22no—
6#16811468C2214yes—
7C22×C4⋊Dic53C2211yescharacteristic
8C22×C4×Dic53C411yescharacteristic
9#16810488—22no—
10#16810488C412yes—
11#16810488C2212yes—
12C2×#13133124—22no—
13C22×C4⋊C4×C53C411yescharacteristic
14C22×#88324—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—41no—
20C2×C4⋊Dic53—41no—
21C2×C4⋊Dic53C2×C411yescharacteristic
22C2×C4⋊Dic53D412yes—
23C2×C4⋊Dic53C2311yescharacteristic
24C2×C4×Dic53—21no—
25C2×C4×Dic53—22no—
26C2×C4×Dic53C814yescomplemented
27C2×C4×Dic53C2×C412yes—
28#13133124—24no—
29#13133124—24no—
30C23×C4×C53—21no—
31C23×C4×C53C2×C411yescharacteristic
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—41no—
34C2×C4⋊C4×C53—41no—
35C2×C4⋊C4×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—21no—
38C22×C4×C53—21no—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—44no—
42C22×C4×C53—41no—
43C22×C4×C53—41no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C2×C812yes—
46C22×C4×C53M1612yes—
47C22×C4×C53C22×C411yescharacteristic
48C22×Dic53—24no—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—42no—
53C22×Dic53—42no—
54C22×Dic53—44no—
55C22×Dic53C22⋊C412yes—
56C22×Dic53C2×D412yes—
57C2×#88324—44no—
58C2×#88324—84no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊Dic53—41no—
62C4⋊Dic53—42no—
63C4⋊Dic53—41no—
64C4×Dic53—24no—
65C4×Dic53—22no—
66C4×Dic53C2×C814yescomplemented
67C23×C53—22no—
68C23×C53—22no—
69C23×C53C22⋊C812yes—
70C23×C53#9911yescharacteristic
71C23×C53#9911yescharacteristic
72C23×C53#10112yes—
73C23×C53C2×C22⋊C411yescharacteristic
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—44no—
83C2×C4×C53—42no—
84C2×C4×C53—82no—
85C2×C4×C53—82no—
86C2×C4×C53C22⋊C812yes—
87C2×C4×C53C22×C811yescharacteristic
88C2×C4×C53C2×M1611yescharacteristic
89C2×Dic53—24no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—48no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—44no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—82no—
99#88324—84no—
100C2×#1324—531nomaximal, Sylow
101C22×C53—24no—
102C22×C53—24no—
103C22×C53—21no—
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—21no—
107C22×C53#32412yes—
108C22×C53#32412yes—
109C22×C53#35412yes—
110C22×C53#35512yes—
111C22×C53C2×C22⋊C811yescharacteristic
112C22×C53C2×#9911yescharacteristic
113C22×C53C2×#10111yescharacteristic
114C4×C53—22no—
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—82no—
118Dic53—44no—
119Dic53—42no—
120Dic53—42no—
121C2×#382—531no—
122C2×#324—532no—
123#1324—534no—
124C2×C53—21no—
125C2×C53—22no—
126C2×C53—21no—
127C2×C53#132414yescomplemented
128C2×C53C2×#32411yescharacteristic
129C2×C53C2×#35411yescharacteristic
130C2×C53C2×#35511yescharacteristic
131C22×C4⋊C4—531no—
132C22×C4⋊C4#615711yescharacteristic
133C22×C42—531no—
134C2×C22⋊C8—1062no—
135#382—532no—
136#382—532no—
137#382—1062no—
138C2×#95—1062no—
139#324—534no—
140#324—1062no—
141C53C2×#132411yescharacteristic, complemented, Sylow
142C23×C4—21no—
143C23×C4—532no—
144C23×C4—1061no—
145C23×C4C2×#615711yescharacteristic
146C22×C8—2122no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—2121no—
154C2×C4⋊C4—2121no—
155C2×C4⋊C4C2×#615712yes—
156C2×C42—532no—
157C2×C42—534no—
158C2×C42—1062no—
159C2×C42—1061no—
160C22⋊C8—1064no—
161C22⋊C8—2122no—
162#95—1064no—
163#95—1064no—
164C24C22⋊#615711yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—44no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—532no—
173C22×C4—532no—
174C22×C4—1062no—
175C22×C4—1064no—
176C22×C4—1062no—
177C22×C4—1062no—
178C22×C4—2124no—
179C22×C4—2122no—
180C22×C4—2122no—
181C22×C4D53⋊C812yes—
182C22×C4#1171993612yes—
183C22×C4C22⋊#615711yescharacteristic
184C22×C4C22×#615711yescharacteristic
185C2×C8—4244no—
186C2×C8—2124no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—2122no—
190C4⋊C4—2121no—
191C4⋊C4—2121no—
192C42—534no—
193C42—1064no—
194C42—1062no—
195C23—22no—
196C23—22no—
197C23#1313327412yes—
198C23#1313327511yescharacteristic
199C23#1313327511yescharacteristic
200C23#1313327812yes—
201C23C2×C22⋊#615711yescharacteristic
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—1064no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—4242no—
218C2×C4—2128no—
219C2×C4—2122no—
220C2×C4—2122no—
221C2×C4—2124no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4#1313327412yes—
225C2×C4C2×D53⋊C811yescharacteristic
226C2×C4C2×#1171993611yescharacteristic
227C8—4244no—
228C22—24no—
229C22—24no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22#1681146912yes—
235C22#1681146912yes—
236C22#1681152912yes—
237C22#1681153112yes—
238C22C2×#1313327411yescharacteristic
239C22C2×#1313327511yescharacteristic
240C22C2×#1313327811yescharacteristic
241C4—22no—
242C4—21no—
243C4—21no—
244C4—82no—
245C4—2124no—
246C4—2122no—
247C4—2122no—
248C2—21no—
249C2—21no—
250C2—22no—
251C2#2028965614yesdirect product factor
252C2C2×#1681146911yescharacteristic
253C2C2×#1681152911yescharacteristic
254C2C2×#1681153111yescharacteristic
subgroupG/H
centerC23#13133275
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53C2×#1324
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289656C2direct (N × H)
26. C2×C4×Dic53C8semidirect
66. C4×Dic53C2×C8semidirect
127. C2×C53#1324semidirect
141. C53C2×#1324semidirect
251. C2#20289656direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168114682
#202896562
C2×#168104882
C2×#132453