Finite Group Databasebeta

← C2×#20289856 #28312466

Subgroups — C2×#20289856 #28312466

Order 13568.

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

normal subgroups only

This group has 390 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#20289856C214yesmaximal, direct product factor
3C2×#16810988C211yesmaximal, characteristic
4C2×#13133276—22no—
5#16811528—22no—
6#16811528C2214yes—
7C2×#13133214C2211yescharacteristic
8C2×#13133209C411yescharacteristic, complemented
9#16810988—22no—
10#16810988C412yescomplemented
11#16810988C2212yes—
12C2×#13133139—22no—
13C2×#97×C53C411yescharacteristic, complemented
14C2×C22⋊#6157—42no—
15#13133276—24no—
16#13133276—42no—
17C22×D212—41no—
18C22×D212D412yes—
19#13133214—41no—
20#13133214C2×C411yescharacteristic
21#13133214D412yes—
22#13133214C2311yescharacteristic
23#13133209—21no—
24#13133209—41no—
25#13133209C2×C412yescomplemented
26C2×C4⋊Dic53—21no—
27C2×#11719919—41no—
28#13133139—24no—
29#13133139—24no—
30C22×C8×C53—21no—
31C2×C42×C53C2×C411yescharacteristic
32#97×C53—21no—
33#97×C53C2×C412yescomplemented
34C23×D53—22no—
35C22×#6157—82no—
36C22×C4×C53—21no—
37C22×C4×C53C22⋊C411yescharacteristic
38C22×Dic53—41no—
39C2×D212—22no—
40C2×D212—42no—
41C2×D212—42no—
42C2×D212—41no—
43C2×D212—42no—
44C2×D212—41no—
45C2×D212—84no—
46C2×D212C22⋊C412yes—
47C2×D212C2×D412yes—
48C22⋊#6157—44no—
49C22⋊#6157—82no—
50C2×C8×C53—22no—
51C2×C8×C53—41no—
52C2×C8×C53—41no—
53C42×C53—21no—
54C42×C53C22⋊C411yescharacteristic
55C42×C53C22×C411yescharacteristic
56C4⋊Dic53—22no—
57C4⋊Dic53—41no—
58#11719919—42no—
59#11719919—42no—
60C23×C53#9911yescharacteristic
61C22×D53—22no—
62C22×D53—22no—
63C22×D53—42no—
64C22×D53—42no—
65C22×D53—84no—
66C2×#6157—84no—
67C2×#6157—164no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—42no—
73C2×C4×C53#9911yescharacteristic
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×Dic53—42no—
76C2×Dic53—82no—
77D212—42no—
78D212—42no—
79D212—82no—
80D212—82no—
81D212—84no—
82D212—82no—
83C8×C53—42no—
84C2×#1412—531nomaximal, Sylow
85C22×C53—22no—
86C22×C53#35411yescharacteristic
87C22×C53#35411yescharacteristic
88C22×C53C2×#9911yescharacteristic
89C2×D53—42no—
90C2×D53—42no—
91C2×D53—84no—
92C2×D53—82no—
93C2×D53—82no—
94C2×D53—162no—
95#6157—164no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—41no—
99C4×C53—41no—
100Dic53—82no—
101C2×#490—531no—
102C2×#353—532no—
103#1412—534no—
104C2×C53—21no—
105C2×C53—21no—
106C2×C53#141212yescomplemented
107C2×C53C2×#35411yescharacteristic
108D53—82no—
109D53—82no—
110C2×#127—531no—
111C2×#125—531no—
112#490—532no—
113#490—532no—
114#490—1062no—
115C2×#102—1062no—
116C2×#99—1062no—
117C2×#97#615711yescharacteristic, complemented
118#353—534no—
119#353—1062no—
120C53C2×#141211yescharacteristic, complemented, Sylow
121C22×D4—532no—
122C22×D4—2121no—
123C22×C8—21no—
124#127—532no—
125#127—531no—
126#127—531no—
127#127—2121no—
128#125—532no—
129#125—1061no—
130#125—2121no—
131C2×C4⋊C4—1061no—
132C2×C4⋊C4—2121no—
133C2×C22⋊C4—2122no—
134C2×C42C2×#615711yescharacteristic
135#102—1064no—
136#102—1064no—
137#99—1064no—
138#99—2122no—
139#97—21no—
140#97C2×#615712yescomplemented
141C24—1062no—
142C2×D4—532no—
143C2×D4—532no—
144C2×D4—1062no—
145C2×D4—4244no—
146C2×D4—2122no—
147C2×D4—2121no—
148C2×D4—2121no—
149C2×D4—2122no—
150C2×D4—2122no—
151C22×C4—21no—
152C22×C4—4242no—
153C22×C4—2121no—
154C22×C4C22⋊#615711yescharacteristic
155C2×C8—22no—
156C2×C8—41no—
157C2×C8—41no—
158C4⋊C4—1062no—
159C4⋊C4—2122no—
160C4⋊C4—2122no—
161C4⋊C4—2121no—
162C22⋊C4—4242no—
163C22⋊C4—2124no—
164C42—21no—
165C42C22⋊#615711yescharacteristic
166C42C22×#615711yescharacteristic
167C23—1062no—
168C23—1062no—
169C23—4244no—
170C23—2122no—
171C23—2122no—
172C23#1313327611yescharacteristic
173D4—4242no—
174D4—4244no—
175D4—4242no—
176D4—4242no—
177D4—2122no—
178D4—2122no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—8484no—
185C2×C4—4244no—
186C2×C4—4242no—
187C2×C4—2122no—
188C2×C4#1313327611yescharacteristic
189C2×C4C2×C22⋊#615711yescharacteristic
190C8—42no—
191C22—22no—
192C22—8482no—
193C22—4242no—
194C22—4242no—
195C22—4244no—
196C22—2122no—
197C22—2122no—
198C22#1681153011yescharacteristic
199C22#1681153011yescharacteristic
200C22C2×#1313327611yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—8484no—
206C4—4242no—
207C2—21no—
208C2—21no—
209C2—4242no—
210C2—4242no—
211C2#2028985612yesdirect product factor
212C2C2×#1681153011yescharacteristic
subgroupG/H
centerC22#16811530
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×#97×C53C4
socleC22×C53#354
prime radicalC2×#13133214C22
cosocle#13133214C23
hypercenterC2×#97#6157
hypocenterC53C2×#1412
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
2. #20289856C2direct (N × H)
8. C2×#13133209C4semidirect
10. #16810988C4semidirect
13. C2×#97×C53C4semidirect
25. #13133209C2×C4semidirect
33. #97×C53C2×C4semidirect
106. C2×C53#1412semidirect
117. C2×#97#6157semidirect
120. C53C2×#1412semidirect
140. #97C2×#6157semidirect
211. C2#20289856direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168115282
#202898562
C2×#168109882
C2×#141253