Finite Group Databasebeta

← C2×#20289642 #28307894

Subgroups — C2×#20289642 #28307894

Order 13568.

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

normal subgroups only

This group has 726 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×#16811466C212yesmaximal
2#20289642C214yesmaximal, direct product factor
3C2×#16810484C211yesmaximal, characteristic
4C2×C22⋊#88324—22no—
5#16811466—22no—
6#16811466C2214yes—
7C24×Dic53C411yescharacteristic
8C22×C22⋊Dic53C2211yescharacteristic
9#16810484—22no—
10#16810484C412yes—
11#16810484C2212yes—
12C2×#13133124—22no—
13C22×C22⋊C4×C53C411yescharacteristic
14C22×#88324—42no—
15C22⋊#88324—24no—
16C22⋊#88324—42no—
17C23×Dic53—21no—
18C23×Dic53—21no—
19C23×Dic53—22no—
20C23×Dic53—41no—
21C23×Dic53—41no—
22C23×Dic53—42no—
23C23×Dic53C814yescomplemented
24C23×Dic53C2×C412yes—
25C23×Dic53D412yes—
26C2×C22⋊Dic53—41no—
27C2×C22⋊Dic53—41no—
28C2×C22⋊Dic53C2×C411yescharacteristic
29C2×C22⋊Dic53D412yes—
30C2×C22⋊Dic53C2311yescharacteristic
31#13133124—24no—
32#13133124—24no—
33C25×C53C2×C411yescharacteristic
34C23×C4×C53—21no—
35C2×C22⋊C4×C53—21no—
36C2×C22⋊C4×C53—42no—
37C2×C22⋊C4×C53C2×C412yes—
38C24×C53—21no—
39C24×C53—21no—
40C24×C53—22no—
41C24×C53—44no—
42C24×C53C22⋊C411yescharacteristic
43C24×C53C22⋊C411yescharacteristic
44C24×C53C2×C812yes—
45C24×C53M1612yes—
46C24×C53C22×C411yescharacteristic
47C22×C4×C53—22no—
48C22×C4×C53—44no—
49C22×C4×C53—42no—
50C22×Dic53—24no—
51C22×Dic53—24no—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—22no—
55C22×Dic53—22no—
56C22×Dic53—44no—
57C22×Dic53—44no—
58C22×Dic53—42no—
59C22×Dic53—42no—
60C22×Dic53—41no—
61C22×Dic53—41no—
62C22×Dic53—42no—
63C22×Dic53—44no—
64C22×Dic53—44no—
65C22×Dic53—42no—
66C22×Dic53—42no—
67C22×Dic53—44no—
68C22×Dic53C22⋊C412yes—
69C22×Dic53C2×C814yescomplemented
70C22×Dic53C2×D412yes—
71C2×#88324—44no—
72C2×#88324—84no—
73C22⋊C4×C53—42no—
74C22⋊C4×C53—42no—
75C22⋊Dic53—41no—
76C22⋊Dic53—42no—
77C22⋊Dic53—41no—
78C23×C53—24no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—42no—
88C23×C53—44no—
89C23×C53—44no—
90C23×C53—42no—
91C23×C53—48no—
92C23×C53—44no—
93C23×C53—44no—
94C23×C53C22⋊C812yes—
95C23×C53C22⋊C812yes—
96C23×C53#9911yescharacteristic
97C23×C53#9911yescharacteristic
98C23×C53#10012yes—
99C23×C53C2×C22⋊C411yescharacteristic
100C23×C53C22×C811yescharacteristic
101C23×C53C2×M1611yescharacteristic
102C2×C4×C53—44no—
103C2×C4×C53—42no—
104C2×C4×C53—84no—
105C2×Dic53—24no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—48no—
109C2×Dic53—44no—
110C2×Dic53—44no—
111C2×Dic53—44no—
112C2×Dic53—44no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115C2×Dic53—44no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53—82no—
119#88324—84no—
120C2×#1317—531nomaximal, Sylow
121C22×C53—24no—
122C22×C53—24no—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—22no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—42no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—44no—
134C22×C53—44no—
135C22×C53—44no—
136C22×C53—44no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53#32312yes—
140C22×C53#32312yes—
141C22×C53#35214yes—
142C22×C53C2×C22⋊C811yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53C2×#10011yescharacteristic
145C4×C53—82no—
146Dic53—42no—
147Dic53—42no—
148Dic53—44no—
149C2×#381—531no—
150C2×#323—532no—
151#1317—534no—
152C2×C53—21no—
153C2×C53—22no—
154C2×C53—21no—
155C2×C53—42no—
156C2×C53—41no—
157C2×C53—41no—
158C2×C53#131714yescomplemented
159C2×C53C2×#32311yescharacteristic
160C2×C53C2×#35212yes—
161C24×C4—531no—
162C22×C22⋊C4—531no—
163C22×C22⋊C4#615711yescharacteristic
164C2×C22⋊C8—1062no—
165#381—532no—
166#381—532no—
167#381—1062no—
168C2×#95—1062no—
169#323—534no—
170#323—1062no—
171C53C2×#131711yescharacteristic, complemented, Sylow
172C25C2×#615711yescharacteristic
173C23×C4—21no—
174C23×C4—532no—
175C23×C4—532no—
176C23×C4—534no—
177C23×C4—1062no—
178C23×C4—1061no—
179C23×C4—1061no—
180C23×C4—2121no—
181C23×C4—2121no—
182C23×C4—2122no—
183C22×C8—2122no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4—531no—
188C2×C22⋊C4—531no—
189C2×C22⋊C4—2121no—
190C2×C22⋊C4—2121no—
191C2×C22⋊C4C2×#615712yes—
192C22⋊C8—1064no—
193C22⋊C8—2122no—
194#95—1064no—
195#95—1064no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—44no—
200C24D53⋊C812yes—
201C24#1171993612yes—
202C24C22⋊#615711yescharacteristic
203C24C22⋊#615711yescharacteristic
204C24C22×#615711yescharacteristic
205C22×C4—22no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—532no—
209C22×C4—532no—
210C22×C4—534no—
211C22×C4—1062no—
212C22×C4—1064no—
213C22×C4—1064no—
214C22×C4—1062no—
215C22×C4—1062no—
216C22×C4—1062no—
217C22×C4—2124no—
218C22×C4—2122no—
219C22×C4—2122no—
220C22×C4—2124no—
221C22×C4—2124no—
222C22×C4—2121no—
223C22×C4—2121no—
224C22×C4—2122no—
225C22×C4—2122no—
226C22×C4—2122no—
227C22×C4—2124no—
228C22×C4—2124no—
229C2×C8—4244no—
230C2×C8—2124no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—2121no—
234C22⋊C4—2122no—
235C22⋊C4—2121no—
236C23—24no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—44no—
246C23—42no—
247C23—42no—
248C23—44no—
249C23—48no—
250C23—44no—
251C23—44no—
252C23#1313327412yes—
253C23#1313327412yes—
254C23#1313327611yescharacteristic
255C23#1313327611yescharacteristic
256C23#1313327712yes—
257C23C2×C22⋊#615711yescharacteristic
258C23C2×D53⋊C811yescharacteristic
259C23C2×#1171993611yescharacteristic
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—1064no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—4242no—
267C2×C4—2128no—
268C2×C4—2124no—
269C2×C4—2124no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2124no—
273C2×C4—2124no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2124no—
277C8—4244no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—21no—
284C22—21no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22#1681146712yes—
297C22#1681146712yes—
298C22#1681152614yes—
299C22C2×#1313327411yescharacteristic
300C22C2×#1313327611yescharacteristic
301C22C2×#1313327711yescharacteristic
302C4—82no—
303C4—2122no—
304C4—2122no—
305C4—2124no—
306C2—21no—
307C2—21no—
308C2—22no—
309C2—42no—
310C2—41no—
311C2—41no—
312C2#2028964214yesdirect product factor
313C2C2×#1681146711yescharacteristic
314C2C2×#1681152612yes—
subgroupG/H
centerC23#13133276
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53C2×#1317
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289642C2direct (N × H)
23. C23×Dic53C8semidirect
69. C22×Dic53C2×C8semidirect
158. C2×C53#1317semidirect
171. C53C2×#1317semidirect
312. C2#20289642direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168114662
#202896422
C2×#168104842
C2×#131753