Finite Group Databasebeta

← C4×#15690868 #27157480

Subgroups — C4×#15690868 #27157480

Order 12416.

412 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 96 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×#15690868C212yesmaximal
2C4×C22⋊#87604C212yesmaximal, complemented
3#20261613C212yesmaximal, complemented
4C2×C4×C22⋊D97C211yesmaximal, characteristic
5C2×C22⋊#87604C2214yes—
6C2×C4×#87604—22no—
7#15690868C418yesdirect product factor
8(C2×#87604)⋊C4—22no—
9C22×C22⋊D97C2211yescharacteristic
10C22×C4×D97C2211yescharacteristic
11C4×C22⋊D97—22no—
12C4×C22⋊D97—22no—
13C2×#13095490C411yescharacteristic, complemented
14C2×C22⋊Dic97C411yescharacteristic, complemented
15C2×#13095488C2211yescharacteristic
16C2×C4×Dic97C411yescharacteristic, complemented
17C23×C4×C97C411yescharacteristic, complemented
18C22×#87604—24no—
19C22⋊#87604—24no—
20C22⋊#87604C2×C418yescomplemented
21C4×#87604—44no—
22C23×D97C2311yescharacteristic
23C2×C22⋊D97—22no—
24C2×C22⋊D97—22no—
25C2×C22⋊D97C2×C414yes—
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C411yescharacteristic
29C2×C4×D97—42no—
30C2×C4×D97D412yes—
31C2×C4×D97D412yes—
32#13095490—22no—
33C22⋊Dic97—22no—
34#13095488—22no—
35C4×Dic97—21no—
36C4×Dic97—21no—
37C24×C97C2×C411yescharacteristic
38C22×C4×C97—21no—
39C22×C4×C97—21no—
40C22×C4×C97—42no—
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97C2×C411yescharacteristic
43C23×C97—21no—
44C23×C97—21no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97C4212yescomplemented
48C23×C97C22×C411yescharacteristic
49C22×D97—41no—
50C22×D97—41no—
51C22×D97C22×C412yes—
52C22×D97C2×D412yes—
53C22×D97#4112yes—
54C2×#87604—28no—
55C2×#87604—44no—
56C2×#87604—44no—
57C2×C4×C97—22no—
58C2×C4×C97—21no—
59C2×C4×C97—42no—
60C2×C4×C97—42no—
61C2×C4×C97—41no—
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C22⋊D97—28no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97—21no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70C2×Dic97C4212yescomplemented
71C2×Dic97C22⋊C412yes—
72C4×D97—24no—
73C4×D97—42no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—22no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97C2×C4211yescharacteristic
82C22×C97C2×C22⋊C411yescharacteristic
83C22×C97#11711yescharacteristic
84C2×D97—22no—
85C2×D97—22no—
86C2×D97—42no—
87C2×D97—42no—
88C2×D97C4×D414yes—
89#87604—48no—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97#9912yescomplemented
93Dic97—22no—
94Dic97—22no—
95Dic97—41no—
96C2×C97—21no—
97C2×C97—21no—
98C2×C97—42no—
99C2×C97C4×C22⋊C411yescharacteristic
100C2×C97C2×#9911yescharacteristic
101C2×C97#41011yescharacteristic
102D97—24no—
103C4×#99—971nomaximal, Sylow
104C97C4×#9911yescharacteristic, complemented, Sylow
105C2×C4×D4—971no—
106C2×#99—972no—
107C4×C22⋊C4—972no—
108#342—972no—
109C22×D4—971no—
110C23×C4—971no—
111C23×C4#8760411yescharacteristic, complemented
112C4×D4—1942no—
113C4×D4—1942no—
114C2×C4⋊C4—971no—
115C2×C22⋊C4—974no—
116C2×C22⋊C4—971no—
117C2×C22⋊C4—971no—
118C2×C42—971no—
119C2×C42—1942no—
120#99—978no—
121#95—1942no—
122C24—971no—
123C24C2×#8760411yescharacteristic
124C2×D4—974no—
125C2×D4—1942no—
126C2×D4—1942no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—972no—
133C22×C4—972no—
134C22×C4—971no—
135C22×C4—3882no—
136C22×C4—1944no—
137C22×C4C2×#8760411yescharacteristic
138C22×C4C2×#8760411yescharacteristic
139C4⋊C4—1942no—
140C22⋊C4—978no—
141C22⋊C4—1942no—
142C22⋊C4—1942no—
143C22⋊C4—1944no—
144C42—3884no—
145C42—1941no—
146C42—1941no—
147C23—21no—
148C23—21no—
149C23—41no—
150C23—41no—
151C23—972no—
152C23—972no—
153C23—972no—
154C23—3881no—
155C23—3881no—
156C23C4×#8760412yescomplemented
157C23C22×#8760411yescharacteristic
158D4—1948no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—972no—
165C2×C4—972no—
166C2×C4—3882no—
167C2×C4—3884no—
168C2×C4—3884no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4—1948no—
172C2×C4—1941no—
173C2×C4—1941no—
174C2×C4—1941no—
175C2×C4—1944no—
176C2×C4C22⋊#8760411yescharacteristic
177C2×C4C22⋊#8760411yescharacteristic
178C22—21no—
179C22—21no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—974no—
186C22—3882no—
187C22—3882no—
188C22—1942no—
189C22—1942no—
190C22C2×C4×#8760411yescharacteristic
191C22C2×C22⋊#8760411yescharacteristic
192C22#1569088911yescharacteristic
193C4—21no—
194C4—41no—
195C4—3888no—
196C4—3881no—
197C4—1942no—
198C4—1942no—
199C4#1569086812yesdirect product factor
200C2—21no—
201C2—21no—
202C2—42no—
203C2—1944no—
204C2C4×C22⋊#8760411yescharacteristic
205C2C2×#1569086811yescharacteristic
206C2#2026171411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC22×C97C2×C42
FrattiniC23C22×#87604
FittingC23×C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C22
cosocleC23×D97C23
hypercenterC23×C4#87604
hypocenterC97C4×#99
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊#87604C2semidirect
3. #20261613C2semidirect
7. #15690868C4direct (N × H)
13. C2×#13095490C4semidirect
14. C2×C22⋊Dic97C4semidirect
16. C2×C4×Dic97C4semidirect
17. C23×C4×C97C4semidirect
20. C22⋊#87604C2×C4semidirect
47. C23×C97C42semidirect
70. C2×Dic97C42semidirect
92. C4×C97#99semidirect
104. C97C4×#99semidirect
111. C23×C4#87604semidirect
156. C23C4×#87604semidirect
199. C4#15690868direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908682
C4×C22⋊#876042
#202616132
C2×C4×C22⋊D972
C4×#9997