Finite Group Databasebeta

← C2×#20289849 #28312428

Subgroups — C2×#20289849 #28312428

Order 13568.

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

normal subgroups only

This group has 474 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×#16811528C212yesmaximal
2#20289849C214yesmaximal, direct product factor
3C2×#16810996C211yesmaximal, characteristic
4C2×#13133276—22no—
5#16811528—22no—
6#16811528C2214yes—
7C22×#11719913—22no—
8C2×#13133217C411yescharacteristic, complemented
9C2×#13133214C2211yescharacteristic
10#16810996—22no—
11#16810996C412yescomplemented
12#16810996C2212yes—
13C2×C4×C8×C53C411yescharacteristic, complemented
14C2×C22⋊#6157—42no—
15#13133276—24no—
16#13133276—42no—
17C22×Dic106—21no—
18C22×D212—41no—
19C22×D212D412yes—
20C2×#11719913—24no—
21C2×#11719913—24no—
22C2×#11719913—44no—
23C2×#11719913—44no—
24#13133217—21no—
25#13133217—41no—
26#13133217C2×C412yescomplemented
27#13133214—41no—
28#13133214C2×C411yescharacteristic
29#13133214D412yes—
30#13133214C2311yescharacteristic
31C2×C4⋊Dic53—41no—
32C22×C8×C53—21no—
33C2×C42×C53C2×C411yescharacteristic
34C4×C8×C53—21no—
35C4×C8×C53C2×C412yescomplemented
36C23×D53—22no—
37C22×#6157—82no—
38C22×C4×C53—21no—
39C22×C4×C53C22⋊C411yescharacteristic
40C22×Dic53—41no—
41C2×D212—22no—
42C2×D212—42no—
43C2×D212—42no—
44C2×D212—41no—
45C2×D212—42no—
46C2×D212—41no—
47C2×D212—84no—
48C2×D212C22⋊C412yes—
49C2×D212C2×D412yes—
50C2×Dic106—22no—
51C2×Dic106—42no—
52C2×Dic106—42no—
53C2×Dic106—41no—
54C22⋊#6157—44no—
55C22⋊#6157—82no—
56C2×C8×C53—22no—
57C2×C8×C53—22no—
58C2×C8×C53—22no—
59C42×C53—21no—
60C42×C53C22⋊C411yescharacteristic
61C42×C53C22×C411yescharacteristic
62C4⋊Dic53—42no—
63C4⋊Dic53—42no—
64#11719913—48no—
65#11719913—48no—
66C23×C53#9911yescharacteristic
67C22×D53—22no—
68C22×D53—22no—
69C22×D53—42no—
70C22×D53—42no—
71C22×D53—84no—
72C2×#6157—84no—
73C2×#6157—164no—
74C2×C4×C53—24no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53#9911yescharacteristic
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×Dic53—42no—
82C2×Dic53—81no—
83C2×Dic53—81no—
84D212—42no—
85D212—42no—
86D212—82no—
87D212—82no—
88D212—84no—
89D212—82no—
90Dic106—44no—
91Dic106—82no—
92C8×C53—24no—
93C2×#1409—531nomaximal, Sylow
94C22×C53—22no—
95C22×C53#35311yescharacteristic
96C22×C53#35311yescharacteristic
97C22×C53C2×#9911yescharacteristic
98C2×D53—42no—
99C2×D53—42no—
100C2×D53—84no—
101C2×D53—82no—
102C2×D53—82no—
103C2×D53—162no—
104#6157—164no—
105C4×C53—22no—
106C4×C53—22no—
107C4×C53—21no—
108C4×C53—21no—
109Dic53—82no—
110C2×#492—531no—
111C2×#353—532no—
112#1409—534no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#140912yescomplemented
116C2×C53C2×#35311yescharacteristic
117D53—82no—
118D53—82no—
119C22×SD16—1062no—
120C2×C4⋊Q8—531no—
121C2×#127—531no—
122#492—532no—
123#492—532no—
124#492—1062no—
125C2×#99—1062no—
126C2×C4×C8#615711yescharacteristic, complemented
127#353—534no—
128#353—1062no—
129C53C2×#140911yescharacteristic, complemented, Sylow
130C22×Q8—1061no—
131C22×D4—532no—
132C22×D4—2121no—
133C2×SD16—1064no—
134C2×SD16—1064no—
135C2×SD16—2124no—
136C2×SD16—2124no—
137C22×C8—21no—
138C4⋊Q8—532no—
139C4⋊Q8—1061no—
140C4⋊Q8—2121no—
141#127—532no—
142#127—531no—
143#127—531no—
144#127—2121no—
145C2×C4⋊C4—2121no—
146C2×C22⋊C4—2122no—
147C2×C42C2×#615711yescharacteristic
148#99—1064no—
149#99—2122no—
150C4×C8—21no—
151C4×C8C2×#615712yescomplemented
152C24—1062no—
153C2×Q8—1062no—
154C2×Q8—2122no—
155C2×Q8—2122no—
156C2×Q8—2121no—
157C2×D4—532no—
158C2×D4—532no—
159C2×D4—1062no—
160C2×D4—4244no—
161C2×D4—2122no—
162C2×D4—2121no—
163C2×D4—2121no—
164C2×D4—2122no—
165C2×D4—2122no—
166C22×C4—21no—
167C22×C4—4242no—
168C22×C4—2121no—
169C22×C4C22⋊#615711yescharacteristic
170SD16—2128no—
171SD16—2128no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C4⋊C4—2122no—
176C4⋊C4—2122no—
177C22⋊C4—4242no—
178C22⋊C4—2124no—
179C42—21no—
180C42C22⋊#615711yescharacteristic
181C42C22×#615711yescharacteristic
182C23—1062no—
183C23—1062no—
184C23—4244no—
185C23—2122no—
186C23—2122no—
187C23#1313327611yescharacteristic
188Q8—4242no—
189Q8—2124no—
190D4—4244no—
191D4—4242no—
192D4—4242no—
193D4—4242no—
194D4—2122no—
195D4—2122no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—8484no—
202C2×C4—4244no—
203C2×C4—4241no—
204C2×C4—4241no—
205C2×C4—2122no—
206C2×C4#1313327611yescharacteristic
207C2×C4C2×C22⋊#615711yescharacteristic
208C8—24no—
209C22—22no—
210C22—8482no—
211C22—4242no—
212C22—4242no—
213C22—4244no—
214C22—2122no—
215C22—2122no—
216C22#1681152811yescharacteristic
217C22#1681152811yescharacteristic
218C22C2×#1313327611yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—8484no—
224C4—4242no—
225C2—21no—
226C2—21no—
227C2—4242no—
228C2—4242no—
229C2#2028984912yesdirect product factor
230C2C2×#1681152811yescharacteristic
subgroupG/H
centerC22#16811528
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×C4×C8×C53C4
socleC22×C53#353
prime radicalC2×#13133214C22
cosocle#13133214C23
hypercenterC2×C4×C8#6157
hypocenterC53C2×#1409
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8

Complements

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

NH ≅ G/Ntype
2. #20289849C2direct (N × H)
8. C2×#13133217C4semidirect
11. #16810996C4semidirect
13. C2×C4×C8×C53C4semidirect
26. #13133217C2×C4semidirect
35. C4×C8×C53C2×C4semidirect
115. C2×C53#1409semidirect
126. C2×C4×C8#6157semidirect
129. C53C2×#1409semidirect
151. C4×C8C2×#6157semidirect
229. C2#20289849direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115282
#202898492
C2×#168109962
C2×#140953