Finite Group Databasebeta

← C2×#20289860 #28312491

Subgroups — C2×#20289860 #28312491

Order 13568.

340 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 64 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×#16811532C211yesmaximal, characteristic
2C2×#16811530C211yesmaximal, characteristic
3#20289860C214yesmaximal, direct product factor
4C2×#16810980C211yesmaximal, characteristic
5C2×#13133278—21no—
6C2×#13133276—21no—
7#16811532—21no—
8#16811532C2212yes—
9#16811530—21no—
10#16811530C2212yes—
11C2×#13133205C2211yescharacteristic
12#16810980—21no—
13#16810980—21no—
14#16810980C412yescomplemented
15#16810980C2212yes—
16C2×#13133140—21no—
17C2×#13133137—21no—
18C2×C4×#88322C411yescharacteristic, complemented
19C2×#125×C53C411yescharacteristic, complemented
20C2×#11719940—41no—
21C2×C22⋊#6157—41no—
22#13133278—22no—
23#13133278—41no—
24#13133276—22no—
25#13133276—41no—
26C22×Dic106D411yescharacteristic
27C22×D212D411yescharacteristic
28C22×#88322—21no—
29#13133205—41no—
30#13133205C2×C411yescharacteristic
31#13133205D411yescharacteristic
32#13133205D411yescharacteristic
33#13133205C2311yescharacteristic
34C2×#11719921—41no—
35#13133140—22no—
36#13133140—22no—
37#13133137—22no—
38#13133137—22no—
39C4×#88322—21no—
40C4×#88322C2×C412yescomplemented
41#125×C53—21no—
42#125×C53—41no—
43#125×C53C2×C412yescomplemented
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—41no—
46C2×C42×C53C2×C411yescharacteristic
47C23×D53—21no—
48C22×#6157—81no—
49C22×C4×C53—21no—
50C22×C4×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×Dic53—21no—
53C2×D212—21no—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212C22⋊C411yescharacteristic
57C2×D212C2×D411yescharacteristic
58C2×Dic106—21no—
59C2×Dic106—41no—
60C2×Dic106—41no—
61C2×Dic106C22⋊C411yescharacteristic
62C2×Dic106C2×D411yescharacteristic
63C22⋊#6157—42no—
64C22⋊#6157—81no—
65#11719940—42no—
66#11719940—81no—
67C2×#88324—81no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C42×C53—21no—
73C42×C53C22⋊C411yescharacteristic
74C42×C53C22×C411yescharacteristic
75#11719921—41no—
76#11719921—41no—
77#11719921—41no—
78#11719921—41no—
79C2×#88322—22no—
80C2×#88322—41no—
81C2×#88322—41no—
82C23×C53#9911yescharacteristic
83C22×D53—21no—
84C22×D53—21no—
85C22×D53—41no—
86C22×D53—41no—
87C22×D53—82no—
88C2×#6157—82no—
89C2×#6157—162no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—82no—
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53C2×C22⋊C411yescharacteristic
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103D212—41no—
104D212—41no—
105D212—81no—
106Dic106—41no—
107Dic106—41no—
108Dic106—81no—
109#88324—82no—
110#88322—42no—
111C2×#1414—531nomaximal, Sylow
112C22×C53—22no—
113C22×C53#35411yescharacteristic
114C22×C53#35411yescharacteristic
115C22×C53C2×#9911yescharacteristic
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—82no—
119C2×D53—81no—
120C2×D53—81no—
121C2×D53—161no—
122#6157—162no—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—41no—
126C4×C53—41no—
127C4×C53—82no—
128Dic53—41no—
129Dic53—41no—
130C2×#488—531no—
131C2×#355—531no—
132C2×#354—531no—
133#1414—534no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53#141412yescomplemented
137C2×C53C2×#35411yescharacteristic
138D53—81no—
139D53—81no—
140C2×#125#615711yescharacteristic, complemented
141C2×#124—531no—
142#488—532no—
143#488—532no—
144#488—1061no—
145#488—1061no—
146C2×#103—1061no—
147C2×#102—1061no—
148C2×#101—1061no—
149C2×#99—1061no—
150C2×C4×C8—531no—
151#355—532no—
152#355—1061no—
153#354—532no—
154#354—1061no—
155C53C2×#141411yescharacteristic, complemented, Sylow
156C22×Q8—531no—
157C22×D4—531no—
158C2×M16—2121no—
159C22×C8—1061no—
160#125—21no—
161#125—41no—
162#125C2×#615712yescomplemented
163#124—531no—
164#124—531no—
165#124—531no—
166#124—531no—
167#124—2121no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—2121no—
171C2×C22⋊C4—2121no—
172C2×C42C2×#615711yescharacteristic
173#103—1062no—
174#103—1062no—
175#102—1062no—
176#102—1062no—
177#101—1062no—
178#101—2121no—
179#99—1062no—
180#99—2121no—
181C4×C8—532no—
182C4×C8—1061no—
183C24—1061no—
184C2×Q8—531no—
185C2×Q8—531no—
186C2×Q8—1061no—
187C2×Q8—2121no—
188C2×Q8—2121no—
189C2×D4—531no—
190C2×D4—531no—
191C2×D4—1061no—
192C2×D4—2121no—
193C2×D4—2121no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—1061no—
197C22×C4—4241no—
198C22×C4C22⋊#615711yescharacteristic
199M16—4241no—
200M16—2122no—
201C2×C8—1062no—
202C2×C8—4241no—
203C2×C8—2121no—
204C2×C8—2121no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C22⋊C4—4241no—
210C22⋊C4—2121no—
211C22⋊C4—2121no—
212C22⋊C4—2121no—
213C22⋊C4—2121no—
214C22⋊C4—2122no—
215C42—21no—
216C42C22⋊#615711yescharacteristic
217C42C22×#615711yescharacteristic
218C23—1061no—
219C23—1061no—
220C23—4242no—
221C23—2121no—
222C23—2121no—
223C23#1313327511yescharacteristic
224Q8—4241no—
225Q8—2121no—
226Q8—2121no—
227D4—4241no—
228D4—2121no—
229D4—2121no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—8482no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—4242no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4#1313327511yescharacteristic
244C2×C4C2×C22⋊#615711yescharacteristic
245C8—4242no—
246C8—2122no—
247C22—22no—
248C22—8481no—
249C22—4241no—
250C22—4241no—
251C22—4242no—
252C22—2121no—
253C22—2121no—
254C22#1681152911yescharacteristic
255C22#1681152911yescharacteristic
256C22C2×#1313327511yescharacteristic
257C4—21no—
258C4—21no—
259C4—41no—
260C4—41no—
261C4—82no—
262C4—8482no—
263C4—2121no—
264C4—2121no—
265C2—21no—
266C2—21no—
267C2—4241no—
268C2—4241no—
269C2#2028986012yesdirect product factor
270C2C2×#1681152911yescharacteristic
subgroupG/H
centerC22#16811529
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×#125×C53C4
socleC22×C53#354
prime radicalC22×D212D4
cosocle#13133205C23
hypercenterC2×#125#6157
hypocenterC53C2×#1414
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
3. #20289860C2direct (N × H)
14. #16810980C4semidirect
18. C2×C4×#88322C4semidirect
19. C2×#125×C53C4semidirect
40. C4×#88322C2×C4semidirect
43. #125×C53C2×C4semidirect
136. C2×C53#1414semidirect
140. C2×#125#6157semidirect
155. C53C2×#1414semidirect
162. #125C2×#6157semidirect
269. C2#20289860direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115322
C2×#168115302
#202898602
C2×#168109802
C2×#141453