Finite Group Databasebeta

← #27139853

Subgroups — #27139853

Order 12416.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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×#15690727C211yesmaximal, characteristic
2#20260615C211yesmaximal, characteristic
3C2×#99×C97C211yesmaximal, characteristic
4C23×Dic97C411yescharacteristic, complemented
5C2×#13095495—21no—
6C2×C22⋊Dic97—21no—
7C2×C22⋊Dic97C411yescharacteristic, complemented
8#15690727—21no—
9#15690727C412yescomplemented
10#15690717—21no—
11C22×D4×C97C2211yescharacteristic
12C2×C22⋊C4×C97—21no—
13#99×C97—21no—
14#99×C97C412yescomplemented
15C22×Dic97—21no—
16C22×Dic97—21no—
17C22×Dic97—42no—
18C22×Dic97—41no—
19#13095495—22no—
20#13095495—41no—
21C2×#11715895—41no—
22C22⋊Dic97—42no—
23C22⋊Dic97—41no—
24C24×C97C2×C411yescharacteristic
25C24×C97D411yescharacteristic
26C2×D4×C97—41no—
27C2×D4×C97—41no—
28C2×D4×C97C2×C411yescharacteristic
29C2×D4×C97C2×C411yescharacteristic
30C2×D4×C97D411yescharacteristic
31C2×D4×C97Q811yescharacteristic
32C22×C4×C97—41no—
33C22×C4×C97D411yescharacteristic
34C22⋊C4×C97—22no—
35C22⋊C4×C97—41no—
36C23×C97—21no—
37C23×C97—21no—
38C23×C97—21no—
39C23×C97—21no—
40C23×C97—41no—
41C23×C97—41no—
42C23×C97—41no—
43C23×C97—41no—
44C23×C97C4211yescharacteristic
45C23×C97C22⋊C411yescharacteristic
46C23×C97C22⋊C411yescharacteristic
47C23×C97C4⋊C411yescharacteristic
48C23×C97C4⋊C411yescharacteristic
49D4×C97—41no—
50D4×C97—41no—
51D4×C97—41no—
52D4×C97—41no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—42no—
56C2×C4×C97—81no—
57C2×C4×C97—81no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C4⋊C411yescharacteristic
60C2×Dic97—22no—
61C2×Dic97—41no—
62C2×Dic97—42no—
63C2×Dic97—41no—
64C2×Dic97—42no—
65C2×Dic97—41no—
66C2×Dic97—41no—
67C2×Dic97—41no—
68#11715895—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97—42no—
80C22×C97—41no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97—81no—
85C22×C97#9511yescharacteristic
86C22×C97#9911yescharacteristic
87C22×C97#9911yescharacteristic
88C4×C97—21no—
89C4×C97—21no—
90C4×C97—82no—
91Dic97—42no—
92Dic97—81no—
93C2×C97—21no—
94C2×C97—21no—
95C2×C97—41no—
96C2×C97—41no—
97C2×C97—41no—
98C2×C97—41no—
99C2×C97#34211yescharacteristic
100C2×C97AΣL(1,𝔽16)11yescharacteristic
101C2×C97#35211yescharacteristic
102#1390—971nomaximal, Sylow
103C97#139011yescharacteristic, complemented, Sylow
104C2×#100—971no—
105C2×#99D9711yescharacteristic
106#386—971no—
107C22×D4C2×D9711yescharacteristic
108C23×C4—971no—
109C2×M16—1941no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—971no—
112C2×C22⋊C4—1941no—
113#100—972no—
114#100—1941no—
115#99—21no—
116#99Dic9712yescomplemented
117#95—1941no—
118C24C4×D9711yescharacteristic
119C24C22⋊D9711yescharacteristic
120C2×D4—41no—
121C2×D4—41no—
122C2×D4Dic19411yescharacteristic
123C2×D4C4×D9711yescharacteristic
124C2×D4C2×Dic9711yescharacteristic
125C2×D4C22⋊D9711yescharacteristic
126C22×C4—41no—
127C22×C4—3881no—
128C22×C4—3882no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4D38811yescharacteristic
132M16—3881no—
133M16—1942no—
134C2×C8—3881no—
135C22⋊C4—22no—
136C22⋊C4—41no—
137C22⋊C4—3882no—
138C22⋊C4—3881no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23C4×Dic9711yescharacteristic
148C23#1309548811yescharacteristic
149C23C22⋊Dic9711yescharacteristic
150C23#1309549011yescharacteristic
151C23#1309549011yescharacteristic
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—3881no—
162C2×C4—3881no—
163C2×C4—3881no—
164C2×C4—3881no—
165C2×C4—3882no—
166C2×C4—3881no—
167C2×C4—3882no—
168C2×C4—1942no—
169C2×C4#1309548811yescharacteristic
170C2×C4C4⋊Dic9711yescharacteristic
171C8—3882no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22#1569071711yescharacteristic
189C22#1569072411yescharacteristic
190C22#1569072411yescharacteristic
191C4—21no—
192C4—21no—
193C4—82no—
194C4—7761no—
195C4—3882no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—41no—
201C2—41no—
202C2#2026050111yescharacteristic
203C2#2026052111yescharacteristic
204C2#2026052411yescharacteristic
subgroupG/H
centerC22#15690724
commutatorC23×C97C42
FrattiniC22×D4C2×D97
FittingC2×#99×C97C2
socleC22×C97#99
prime radical, cosocleC22×D4×C97C22
hypercenterC2×#99D97
hypocenterC97#1390
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
4. C23×Dic97C4semidirect
7. C2×C22⋊Dic97C4semidirect
9. #15690727C4semidirect
14. #99×C97C4semidirect
103. C97#1390semidirect
116. #99Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907272
#202606152
C2×#99×C972
#139097