Finite Group Databasebeta

← C2×#20272886 #27304437

Subgroups — C2×#20272886 #27304437

Order 13568.

390 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16810416C212yesmaximal
2#20272886C214yesmaximal, direct product factor
3C2×C53×#490C211yesmaximal, characteristic
4C2×#13133132—22no—
5#16810416—22no—
6#16810416C2214yes—
7C2×#127×C53C2211yescharacteristic
8C2×#125×C53C411yescharacteristic, complemented
9C53×#490—22no—
10C53×#490C412yescomplemented
11C53×#490C2212yes—
12C2×#102×C53—22no—
13C2×#97×C53C411yescharacteristic, complemented
14C2×C22⋊Dic53—42no—
15#13133132—24no—
16#13133132—42no—
17C22×D4×C53—41no—
18C22×D4×C53D412yes—
19C22×C8×C53—21no—
20#127×C53—41no—
21#127×C53C2×C411yescharacteristic
22#127×C53D412yes—
23#127×C53C2311yescharacteristic
24#125×C53—21no—
25#125×C53—41no—
26#125×C53C2×C412yescomplemented
27C2×C4⋊C4×C53—21no—
28C2×C4⋊C4×C53—41no—
29C2×C42×C53C2×C411yescharacteristic
30#102×C53—24no—
31#102×C53—24no—
32#97×C53—21no—
33#97×C53C2×C412yescomplemented
34C24×C53—22no—
35C2×D4×C53—22no—
36C2×D4×C53—42no—
37C2×D4×C53—42no—
38C2×D4×C53—41no—
39C2×D4×C53—42no—
40C2×D4×C53—41no—
41C2×D4×C53—84no—
42C2×D4×C53C22⋊C412yes—
43C2×D4×C53C2×D412yes—
44C22×C4×C53—21no—
45C22×C4×C53—41no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×Dic53—82no—
48C2×C8×C53—22no—
49C2×C8×C53—41no—
50C2×C8×C53—41no—
51C4⋊C4×C53—22no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—41no—
55C42×C53—21no—
56C42×C53C22⋊C411yescharacteristic
57C42×C53C22×C411yescharacteristic
58C22⋊Dic53—44no—
59C22⋊Dic53—82no—
60C23×C53—22no—
61C23×C53—22no—
62C23×C53—42no—
63C23×C53—42no—
64C23×C53—84no—
65C23×C53#9911yescharacteristic
66D4×C53—42no—
67D4×C53—42no—
68D4×C53—82no—
69D4×C53—82no—
70D4×C53—84no—
71D4×C53—82no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—82no—
79C2×C4×C53#9911yescharacteristic
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×Dic53—84no—
82C2×Dic53—164no—
83C8×C53—42no—
84C2×#1412—531nomaximal, Sylow
85C22×C53—22no—
86C22×C53—42no—
87C22×C53—42no—
88C22×C53—82no—
89C22×C53—82no—
90C22×C53—84no—
91C22×C53—162no—
92C22×C53#35411yescharacteristic
93C22×C53#35411yescharacteristic
94C22×C53C2×#9911yescharacteristic
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—41no—
98C4×C53—41no—
99C4×C53—82no—
100Dic53—164no—
101C2×#490D5311yescharacteristic
102C2×#353—532no—
103#1412—534no—
104C2×C53—21no—
105C2×C53—21no—
106C2×C53—82no—
107C2×C53—82no—
108C2×C53#141212yescomplemented
109C2×C53C2×#35411yescharacteristic
110C2×#127C2×D5311yescharacteristic
111C2×#125Dic5311yescharacteristic, complemented
112#490—22no—
113#490Dic5312yescomplemented
114#490C2×D5312yes—
115C2×#102—22no—
116C2×#99—1062no—
117C2×#97Dic5311yescharacteristic, complemented
118#353—534no—
119#353—1062no—
120C53C2×#141211yescharacteristic, complemented, Sylow
121C22×D4—41no—
122C22×D4C22⋊D5312yes—
123C22×C8—21no—
124#127—41no—
125#127C2×Dic5311yescharacteristic
126#127C22⋊D5312yes—
127#127C22×D5311yescharacteristic
128#125—21no—
129#125—41no—
130#125C2×Dic5312yescomplemented
131C2×C4⋊C4—21no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—2122no—
134C2×C42C2×Dic5311yescharacteristic
135#102—24no—
136#102—24no—
137#99—1064no—
138#99—2122no—
139#97—21no—
140#97C2×Dic5312yescomplemented
141C24—22no—
142C2×D4—22no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—41no—
146C2×D4—42no—
147C2×D4—41no—
148C2×D4—84no—
149C2×D4C22⋊Dic5312yes—
150C2×D4C2×C22⋊D5312yes—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—4242no—
154C22×C4C22⋊Dic5311yescharacteristic
155C2×C8—22no—
156C2×C8—41no—
157C2×C8—41no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C22⋊C4—4242no—
163C22⋊C4—2124no—
164C42—21no—
165C42C22⋊Dic5311yescharacteristic
166C42C22×Dic5311yescharacteristic
167C23—22no—
168C23—22no—
169C23—42no—
170C23—42no—
171C23—84no—
172C23#1313313211yescharacteristic
173D4—42no—
174D4—42no—
175D4—82no—
176D4—82no—
177D4—84no—
178D4—82no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—8484no—
187C2×C4—4244no—
188C2×C4#1313313211yescharacteristic
189C2×C4C2×C22⋊Dic5311yescharacteristic
190C8—42no—
191C22—22no—
192C22—42no—
193C22—42no—
194C22—82no—
195C22—82no—
196C22—84no—
197C22—162no—
198C22#1681041811yescharacteristic
199C22#1681041811yescharacteristic
200C22C2×#1313313211yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—82no—
206C4—8484no—
207C2—21no—
208C2—21no—
209C2—82no—
210C2—82no—
211C2#2027288612yesdirect product factor
212C2C2×#1681041811yescharacteristic
subgroupG/H
centerC22#16810418
commutatorC42×C53C22×C4
Frattini#127C22×D53
FittingC2×C53×#490C2
socleC22×C53#354
prime radicalC2×#127×C53C22
cosocle#127×C53C23
hypercenterC2×#490D53
hypocenterC53C2×#1412
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
2. #20272886C2direct (N × H)
8. C2×#125×C53C4semidirect
10. C53×#490C4semidirect
13. C2×#97×C53C4semidirect
26. #125×C53C2×C4semidirect
33. #97×C53C2×C4semidirect
108. C2×C53#1412semidirect
111. C2×#125Dic53semidirect
113. #490Dic53semidirect
117. C2×#97Dic53semidirect
120. C53C2×#1412semidirect
130. #125C2×Dic53semidirect
140. #97C2×Dic53semidirect
211. C2#20272886direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168104162
#202728862
C2×C53×#4902
C2×#141253