Finite Group Databasebeta

← C2×#20289657 #28307962

Subgroups — C2×#20289657 #28307962

Order 13568.

598 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 128 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#20289657C214yesmaximal, direct product factor
3C2×#16810493C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811468—22no—
6#16811468C2214yes—
7C22×C4⋊Dic53—22no—
8C22×C4⋊Dic53C411yescharacteristic
9C22×C4⋊Dic53C2211yescharacteristic
10#16810493—22no—
11#16810493C412yes—
12#16810493C2212yes—
13C22×C42×C53C411yescharacteristic
14C22×#88324—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53D412yes—
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—24no—
21C2×C4⋊Dic53—24no—
22C2×C4⋊Dic53—41no—
23C2×C4⋊Dic53—41no—
24C2×C4⋊Dic53—48no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53C2×C412yes—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53C2311yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C2×C42×C53—21no—
34C2×C42×C53—22no—
35C2×C42×C53C814yescomplemented
36C2×C42×C53C2×C412yes—
37C24×C53C22⋊C411yescharacteristic
38C22×C4×C53—24no—
39C22×C4×C53—24no—
40C22×C4×C53—24no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C2×C812yes—
47C22×C4×C53M1612yes—
48C22×C4×C53C22×C411yescharacteristic
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—44no—
53C22×Dic53—44no—
54C22×Dic53—41no—
55C22×Dic53—41no—
56C22×Dic53C22⋊C412yes—
57C22×Dic53C2×D412yes—
58C2×#88324—44no—
59C2×#88324—84no—
60C42×C53—24no—
61C42×C53—22no—
62C42×C53C2×C814yescomplemented
63C4⋊Dic53—41no—
64C4⋊Dic53—42no—
65C4⋊Dic53—41no—
66C4⋊Dic53—48no—
67C4⋊Dic53—48no—
68C4⋊Dic53—42no—
69C4⋊Dic53—42no—
70C23×C53—22no—
71C23×C53—22no—
72C23×C53C22⋊C812yes—
73C23×C53#9911yescharacteristic
74C23×C53#9911yescharacteristic
75C23×C53#10112yes—
76C23×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53—216no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—24no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53C22⋊C812yes—
90C2×C4×C53C22×C811yescharacteristic
91C2×C4×C53C2×M1611yescharacteristic
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—44no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—44no—
98C2×Dic53—42no—
99C2×Dic53—82no—
100C2×Dic53—82no—
101C2×Dic53—82no—
102#88324—84no—
103C2×#1325—531nomaximal, Sylow
104C22×C53—24no—
105C22×C53—24no—
106C22×C53—21no—
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—21no—
110C22×C53#32412yes—
111C22×C53#32412yes—
112C22×C53#35312yes—
113C22×C53#35612yes—
114C22×C53C2×C22⋊C811yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10111yescharacteristic
117C4×C53—28no—
118C4×C53—22no—
119C4×C53—21no—
120C4×C53—21no—
121Dic53—42no—
122Dic53—42no—
123Dic53—82no—
124C2×#384—531no—
125C2×#324—532no—
126#1325—534no—
127C2×C53—21no—
128C2×C53—22no—
129C2×C53—21no—
130C2×C53#132514yescomplemented
131C2×C53C2×#32411yescharacteristic
132C2×C53C2×#35311yescharacteristic
133C2×C53C2×#35611yescharacteristic
134C22×C4⋊C4—531no—
135C22×C4⋊C4—531no—
136C22×C4⋊C4—1062no—
137C22×C42#615711yescharacteristic
138C2×C22⋊C8—1062no—
139#384—532no—
140#384—532no—
141#384—1062no—
142#324—534no—
143#324—1062no—
144C53C2×#132511yescharacteristic, complemented, Sylow
145C23×C4—21no—
146C23×C4—532no—
147C23×C4—1061no—
148C23×C4C2×#615711yescharacteristic
149C22×C8—2122no—
150C2×C4⋊C4—532no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—1064no—
155C2×C4⋊C4—1064no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—2121no—
158C2×C4⋊C4—2121no—
159C2×C4⋊C4—2121no—
160C2×C4⋊C4—2121no—
161C2×C4⋊C4—2128no—
162C2×C42—22no—
163C2×C42—21no—
164C2×C42#8832414yescomplemented
165C2×C42C2×#615712yes—
166C22⋊C8—1064no—
167C22⋊C8—2122no—
168C24C22⋊#615711yescharacteristic
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—532no—
177C22×C4—532no—
178C22×C4—1062no—
179C22×C4—1062no—
180C22×C4—1062no—
181C22×C4—2124no—
182C22×C4—2121no—
183C22×C4—2121no—
184C22×C4—2124no—
185C22×C4C2×#8832412yes—
186C22×C4#1171994012yes—
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22×#615711yescharacteristic
189C2×C8—4244no—
190C2×C8—2124no—
191C4⋊C4—2122no—
192C4⋊C4—2121no—
193C4⋊C4—2121no—
194C4⋊C4—2122no—
195C4⋊C4—2128no—
196C4⋊C4—2128no—
197C4⋊C4—2122no—
198C42—24no—
199C42—22no—
200C42C2×#8832414yescomplemented
201C23—22no—
202C23—22no—
203C23C22⋊#8832412yes—
204C23#1313327611yescharacteristic
205C23#1313327611yescharacteristic
206C23#1313327812yes—
207C23C2×C22⋊#615711yescharacteristic
208C2×C4—24no—
209C2×C4—216no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—1062no—
221C2×C4—1062no—
222C2×C4—4242no—
223C2×C4—4242no—
224C2×C4—4242no—
225C2×C4—2124no—
226C2×C4—2122no—
227C2×C4—2124no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4C22⋊#8832412yes—
231C2×C4C22×#8832411yescharacteristic
232C2×C4C2×#1171994011yescharacteristic
233C8—4244no—
234C22—24no—
235C22—24no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22#1681146812yes—
241C22#1681146812yes—
242C22#1681152812yes—
243C22#1681153412yes—
244C22C2×C22⋊#8832411yescharacteristic
245C22C2×#1313327611yescharacteristic
246C22C2×#1313327811yescharacteristic
247C4—28no—
248C4—22no—
249C4—21no—
250C4—21no—
251C4—4242no—
252C4—2122no—
253C4—2122no—
254C2—21no—
255C2—21no—
256C2—22no—
257C2#2028965714yesdirect product factor
258C2C2×#1681146811yescharacteristic
259C2C2×#1681152811yescharacteristic
260C2C2×#1681153411yescharacteristic
subgroupG/H
centerC23#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C42×C53C4
socleC23×C53#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC22×C42#6157
hypocenterC53C2×#1325
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #20289657C2direct (N × H)
35. C2×C42×C53C8semidirect
62. C42×C53C2×C8semidirect
130. C2×C53#1325semidirect
144. C53C2×#1325semidirect
164. C2×C42#88324semidirect
200. C42C2×#88324semidirect
257. C2#20289657direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168114682
#202896572
C2×#168104932
C2×#132553