Finite Group Databasebeta

← C2×#20289646 #28307914

Subgroups — C2×#20289646 #28307914

Order 13568.

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

normal subgroups only

This group has 618 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#20289646C214yesmaximal, direct product factor
3C2×#16810529C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811468—22no—
6#16811468C2214yes—
7C23×Dic106C411yescharacteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810529—22no—
10#16810529C412yes—
11#16810529C2212yes—
12C2×#13133124—22no—
13C22×C42×C53C411yescharacteristic
14C22×#88324—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C22×Dic106—21no—
18C22×Dic106—22no—
19C22×Dic106—41no—
20C22×Dic106—41no—
21C22×Dic106—42no—
22C22×Dic106C814yescomplemented
23C22×Dic106C2×C412yes—
24C23×Dic53—21no—
25C23×Dic53D412yes—
26C2×C4⋊Dic53—41no—
27C2×C4⋊Dic53—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53D412yes—
30C2×C4⋊Dic53C2311yescharacteristic
31#13133124—24no—
32#13133124—24no—
33C23×C4×C53—21no—
34C23×C4×C53C2×C411yescharacteristic
35C2×C42×C53—21no—
36C2×C42×C53—42no—
37C2×C42×C53C2×C412yes—
38C24×C53C22⋊C411yescharacteristic
39C22×C4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—22no—
42C22×C4×C53—24no—
43C22×C4×C53—22no—
44C22×C4×C53—42no—
45C22×C4×C53—42no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C2×C812yes—
48C22×C4×C53M1612yes—
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—24no—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—41no—
55C22×Dic53—41no—
56C22×Dic53—42no—
57C22×Dic53—44no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×Dic106—24no—
61C2×Dic106—22no—
62C2×Dic106—44no—
63C2×Dic106—44no—
64C2×Dic106—42no—
65C2×Dic106—42no—
66C2×Dic106—44no—
67C2×Dic106—44no—
68C2×Dic106—42no—
69C2×Dic106—42no—
70C2×Dic106C2×C814yescomplemented
71C2×#88324—44no—
72C2×#88324—84no—
73C42×C53—42no—
74C42×C53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—42no—
77C4⋊Dic53—41no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53C22⋊C812yes—
81C23×C53#9911yescharacteristic
82C23×C53#9911yescharacteristic
83C23×C53#10012yes—
84C23×C53C2×C22⋊C411yescharacteristic
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—24no—
94C2×C4×C53—48no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53C22⋊C812yes—
98C2×C4×C53C22×C811yescharacteristic
99C2×C4×C53C2×M1611yescharacteristic
100C2×Dic53—24no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—44no—
104C2×Dic53—44no—
105C2×Dic53—44no—
106C2×Dic53—42no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53—82no—
111Dic106—48no—
112Dic106—44no—
113Dic106—44no—
114#88324—84no—
115C2×#1319—531nomaximal, Sylow
116C22×C53—24no—
117C22×C53—24no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#32312yes—
123C22×C53#32312yes—
124C22×C53#35412yes—
125C22×C53#35612yes—
126C22×C53C2×C22⋊C811yescharacteristic
127C22×C53C2×#9911yescharacteristic
128C22×C53C2×#10011yescharacteristic
129C4×C53—22no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—44no—
133Dic53—42no—
134Dic53—42no—
135Dic53—44no—
136C2×#391—531no—
137C2×#324—532no—
138#1319—534no—
139C2×C53—21no—
140C2×C53—22no—
141C2×C53—21no—
142C2×C53#131914yescomplemented
143C2×C53C2×#32311yescharacteristic
144C2×C53C2×#35411yescharacteristic
145C2×C53C2×#35611yescharacteristic
146C23×Q8—531no—
147C22×C4⋊C4—531no—
148C22×C42#615711yescharacteristic
149C2×C22⋊C8—1062no—
150#391—532no—
151#391—532no—
152#391—1062no—
153C2×#95—1062no—
154#324—534no—
155#324—1062no—
156C53C2×#131911yescharacteristic, complemented, Sylow
157C22×Q8—532no—
158C22×Q8—534no—
159C22×Q8—1062no—
160C22×Q8—1061no—
161C22×Q8—2121no—
162C22×Q8—2121no—
163C22×Q8—2122no—
164C23×C4—21no—
165C23×C4—532no—
166C23×C4—1061no—
167C23×C4C2×#615711yescharacteristic
168C22×C8—2122no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—2121no—
173C2×C4⋊C4—2121no—
174C2×C42—21no—
175C2×C42—42no—
176C2×C42C2×#615712yes—
177C22⋊C8—1064no—
178C22⋊C8—2122no—
179#95—1064no—
180#95—1064no—
181C24C22⋊#615711yescharacteristic
182C2×Q8—534no—
183C2×Q8—1064no—
184C2×Q8—1062no—
185C2×Q8—2122no—
186C2×Q8—2122no—
187C2×Q8—2124no—
188C2×Q8—2124no—
189C2×Q8—2122no—
190C2×Q8—2122no—
191C2×Q8—2124no—
192C2×Q8—2124no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—532no—
201C22×C4—532no—
202C22×C4—1062no—
203C22×C4—1064no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4—2124no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4—2122no—
210C22×C4D53⋊C812yes—
211C22×C4#1171993612yes—
212C22×C4C22⋊#615711yescharacteristic
213C22×C4C22×#615711yescharacteristic
214C2×C8—4244no—
215C2×C8—2124no—
216C4⋊C4—2122no—
217C4⋊C4—2121no—
218C4⋊C4—2121no—
219C42—42no—
220C42—42no—
221C23—22no—
222C23—22no—
223C23#1313327412yes—
224C23#1313327611yescharacteristic
225C23#1313327611yescharacteristic
226C23#1313327712yes—
227C23C2×C22⋊#615711yescharacteristic
228Q8—2124no—
229Q8—2128no—
230Q8—2124no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—48no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—1064no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—4242no—
247C2×C4—2124no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2124no—
251C2×C4—2124no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4#1313327412yes—
255C2×C4C2×D53⋊C811yescharacteristic
256C2×C4C2×#1171993611yescharacteristic
257C8—4244no—
258C22—24no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22#1681146712yes—
265C22#1681146712yes—
266C22#1681153012yes—
267C22#1681153412yes—
268C22C2×#1313327411yescharacteristic
269C22C2×#1313327611yescharacteristic
270C22C2×#1313327711yescharacteristic
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—2124no—
276C4—2122no—
277C4—2122no—
278C2—21no—
279C2—21no—
280C2—22no—
281C2#2028964614yesdirect product factor
282C2C2×#1681146711yescharacteristic
283C2C2×#1681153011yescharacteristic
284C2C2×#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×#1319
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. #20289646C2direct (N × H)
22. C22×Dic106C8semidirect
70. C2×Dic106C2×C8semidirect
142. C2×C53#1319semidirect
156. C53C2×#1319semidirect
281. C2#20289646direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168114682
#202896462
C2×#168105292
C2×#131953