Finite Group Databasebeta

← #27143345

Subgroups — #27143345

Order 12416.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C22×C8×D97C211yesmaximal, characteristic
2C2×#15690749C212yesmaximal, complemented
3C22⋊#13095486C211yesmaximal, characteristic, complemented
4#20260514C212yesmaximal, complemented
5C22⋊C16×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C411yescharacteristic
7C22×#11715895C411yescharacteristic
8C2×C8×D97—24no—
9C2×C8×D97C411yescharacteristic
10C2×C8×D97C411yescharacteristic
11C2×C8×D97C2211yescharacteristic
12C2×C8×D97C2211yescharacteristic
13#15690749—24no—
14C2×#13095486C2212yescomplemented
15C22×C8×C97C2211yescharacteristic
16C2×C16×C97C2212yescomplemented
17C23×D97C811yescharacteristic
18C22×Dic97C811yescharacteristic
19C2×C4×D97—22no—
20C2×C4×D97—22no—
21C2×C4×D97C811yescharacteristic
22C2×C4×D97C811yescharacteristic
23C2×C4×D97C2×C411yescharacteristic
24C2×C4×D97C2×C411yescharacteristic
25C2×#11715895—22no—
26C2×#11715895C2×C411yescharacteristic
27C2×#11715895C2×C411yescharacteristic
28C8×D97—22no—
29C8×D97—22no—
30C8×D97—22no—
31C8×D97D414yes—
32#13095486—22no—
33C22×C4×C97C2×C411yescharacteristic
34C2×C8×C97—22no—
35C2×C8×C97C2×C411yescharacteristic
36C2×C8×C97C2311yescharacteristic
37C16×C97—22no—
38C23×C97C2×C811yescharacteristic
39C22×D97—22no—
40C22×D97—22no—
41C22×D97—22no—
42C22×D97C2×C811yescharacteristic
43C22×D97C2×C811yescharacteristic
44C2×C4×C97—21no—
45C2×C4×C97—21no—
46C2×C4×C97C2×C811yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—22no—
49C2×Dic97C2×C811yescharacteristic
50C2×Dic97C2×C811yescharacteristic
51C4×D97—22no—
52C4×D97—21no—
53C4×D97—21no—
54C4×D97—22no—
55C4×D97C22⋊C412yes—
56C4×D97M1612yes—
57C8×C97—21no—
58C8×C97C2×D412yes—
59#11715895—21no—
60#11715895C22⋊C412yes—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97M3212yescomplemented
64C22×C97C22×C811yescharacteristic
65C2×D97—28no—
66C2×D97—21no—
67C2×D97—21no—
68C2×D97—21no—
69C2×D97—21no—
70C2×D97C22⋊C812yes—
71C2×D97M3212yescomplemented
72C4×C97—21no—
73C4×C97C2×C22⋊C411yescharacteristic
74C4×C97C2×M1611yescharacteristic
75Dic97—21no—
76Dic97C22⋊C812yes—
77C2×C97—22no—
78C2×C97C2×C22⋊C811yescharacteristic
79C2×C97C2×M3211yescharacteristic
80C2×C97C2×M3211yescharacteristic
81D97—22no—
82D97—22no—
83#2112—971nomaximal, Sylow
84C97#211211yescharacteristic, complemented, Sylow
85C23×C8—971no—
86C2×M32—972no—
87C22⋊C16—971no—
88C22⋊C16—972no—
89C22⋊C16D9711yescharacteristic, complemented
90C23×C4—971no—
91C22×C8—971no—
92C22×C8—971no—
93C22×C8—971no—
94C22×C8—971no—
95C22×C8—971no—
96C22×C8—1944no—
97C22×C8C2×D9711yescharacteristic
98M32—1944no—
99C2×C16—972no—
100C2×C16C2×D9712yescomplemented
101C24—971no—
102C22×C4—971no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4—1942no—
108C22×C4—1942no—
109C22×C4C4×D9711yescharacteristic
110C2×C8—22no—
111C2×C8—971no—
112C2×C8—974no—
113C2×C8—971no—
114C2×C8—1942no—
115C2×C8—1942no—
116C2×C8—1942no—
117C2×C8—1942no—
118C2×C8C4×D9711yescharacteristic
119C2×C8C22×D9711yescharacteristic
120C16—22no—
121C16—1942no—
122C23—971no—
123C23—971no—
124C23—1942no—
125C23—1942no—
126C23—1942no—
127C23C8×D9711yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—971no—
131C2×C4—972no—
132C2×C4—972no—
133C2×C4—971no—
134C2×C4—1942no—
135C2×C4—1942no—
136C2×C4—1942no—
137C2×C4—1941no—
138C2×C4—1941no—
139C2×C4C8×D9711yescharacteristic
140C2×C4C2×C4×D9711yescharacteristic
141C8—21no—
142C8—972no—
143C8—1941no—
144C8D4×D9712yes—
145C22—21no—
146C22—21no—
147C22—972no—
148C22—972no—
149C22—1948no—
150C22—1941no—
151C22—1941no—
152C22—1941no—
153C22—1941no—
154C22#1569074912yescomplemented
155C22C2×C8×D9711yescharacteristic
156C4—21no—
157C4—972no—
158C4—1941no—
159C4C22⋊C4×D9711yescharacteristic
160C4M16×D9711yescharacteristic
161C2—22no—
162C2—1942no—
163C2—1942no—
164C2C22⋊C8×D9711yescharacteristic
165C2M32×D9711yescharacteristic
166C2C2×#1569074911yescharacteristic
subgroupG/H
center, FrattiniC2×C8C22×D97
commutator, socleC22×C97C22×C8
FittingC22⋊C16×C97C2
prime radicalC23×D97C8
cosocleC2×C8×C97C23
hypercenterC22⋊C16D97
hypocenterC97#2112
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#15690749C2semidirect
3. C22⋊#13095486C2semidirect
4. #20260514C2semidirect
5. C22⋊C16×C97C2semidirect
14. C2×#13095486C22semidirect
16. C2×C16×C97C22semidirect
63. C22×C97M32semidirect
71. C2×D97M32semidirect
84. C97#2112semidirect
89. C22⋊C16D97semidirect
100. C2×C16C2×D97semidirect
154. C22#15690749semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D972
C2×#156907492
C22⋊#130954862
#202605142
C22⋊C16×C972
#211297