Finite Group Databasebeta

← #27150026

Subgroups — #27150026

Order 12416.

1102 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C24×Dic97C211yesmaximal, characteristic, complemented
2C2×#15690791C212yesmaximal, complemented
3#20260615C214yesmaximal, complemented
4#20260607C214yesmaximal, complemented
5#20260596C212yesmaximal, complemented
6#20260591C211yesmaximal, characteristic, complemented
7C2×#120×C97C211yesmaximal, characteristic
8C23×Dic97—28no—
9C23×Dic97—24no—
10C23×Dic97C2214yes—
11C23×Dic97C2212yes—
12#15690791—28no—
13C2×#13095490C2214yescomplemented
14C2×C22⋊Dic97C2214yes—
15C2×C22⋊Dic97C2212yes—
16C2×C22⋊Dic97C2214yescomplemented
17#15690717C2218yescomplemented
18C25×C97C2211yescharacteristic
19C22×D4×C97C2212yes—
20C22×D4×C97C2211yescharacteristic
21#120×C97—24no—
22C2×C22⋊C4×C97C2211yescharacteristic
23C2×C22⋊C4×C97C2212yes—
24C22×Dic97—232no—
25C22×Dic97—24no—
26C22×Dic97—28no—
27C22×Dic97—24no—
28C22×Dic97—216no—
29C22×Dic97D418yescomplemented
30C22×Dic97C2314yes—
31C22×Dic97C2314yes—
32#13095490—28no—
33C22⋊Dic97—24no—
34C22⋊Dic97—28no—
35C22⋊Dic97—28no—
36C24×C97—24no—
37C24×C97—28no—
38C24×C97D414yes—
39C24×C97C2311yescharacteristic
40C24×C97C2311yescharacteristic
41C24×C97C2312yes—
42C2×D4×C97—24no—
43C2×D4×C97—22no—
44C2×D4×C97—24no—
45C2×D4×C97—28no—
46C22×C4×C97C2311yescharacteristic
47C22×C4×C97C2312yes—
48C22⋊C4×C97—22no—
49C22⋊C4×C97—24no—
50C23×C97—232no—
51C23×C97—28no—
52C23×C97—216no—
53C23×C97—24no—
54C23×C97—24no—
55C23×C97—24no—
56C23×C97—21no—
57C23×C97—22no—
58C23×C97—24no—
59C23×C97C2×D414yes—
60C23×C97C2×D412yes—
61C23×C97C2×D414yes—
62C23×C97#4118yes—
63C23×C97C2411yescharacteristic
64D4×C97—44no—
65D4×C97—48no—
66C2×C4×C97—21no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—24no—
70C2×Dic97—232no—
71C2×Dic97—216no—
72C2×Dic97—24no—
73C2×Dic97—24no—
74C2×Dic97—24no—
75C2×Dic97—28no—
76C2×Dic97C2×D418yes—
77C22×C97—232no—
78C22×C97—24no—
79C22×C97—24no—
80C22×C97—24no—
81C22×C97—28no—
82C22×C97—216no—
83C22×C97—41no—
84C22×C97—42no—
85C22×C97—44no—
86C22×C97#12014yes—
87C22×C97#12318yescomplemented
88C22×C97C22×D412yes—
89C22×C97C22×D411yescharacteristic
90C22×C97C2×#4114yes—
91C4×C97—41no—
92C4×C97—42no—
93Dic97—28no—
94Dic97—44no—
95C2×C97—24no—
96C2×C97—28no—
97C2×C97—81no—
98C2×C97C2×#12011yescharacteristic
99C2×C97C2×#12312yes—
100C2×C97#52514yes—
101#2849—971nomaximal, Sylow
102C97#284911yescharacteristic, complemented, Sylow
103C24×C4—971no—
104C2×#123—972no—
105C2×#120D9711yescharacteristic
106#386—974no—
107#385—974no—
108#381—972no—
109#379—971no—
110C25C2×D9711yescharacteristic
111C22×D4C2×D9711yescharacteristic
112C22×D4C2×D9712yes—
113C23×C4—972no—
114C23×C4—974no—
115C23×C4—1948no—
116C23×C4—1944no—
117#123—1948no—
118#120—24no—
119C2×C4⋊C4—974no—
120C2×C22⋊C4—972no—
121C2×C22⋊C4—974no—
122C2×C22⋊C4—974no—
123C2×C22⋊C4C2×D9711yescharacteristic
124C2×C22⋊C4C2×D9712yes—
125#95—978no—
126C24—24no—
127C24—28no—
128C24C22⋊D9714yes—
129C24C22×D9711yescharacteristic
130C24C22×D9711yescharacteristic
131C24C22×D9712yes—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—24no—
135C2×D4—28no—
136C22×C4—978no—
137C22×C4—974no—
138C22×C4—974no—
139C22×C4—19416no—
140C22×C4—19432no—
141C22×C4—1944no—
142C22×C4—1948no—
143C22×C4—1944no—
144C22×C4C22×D9712yes—
145C22×C4C22×D9711yescharacteristic
146C4⋊C4—1948no—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—1948no—
150C22⋊C4—1944no—
151C22⋊C4—1948no—
152C23—232no—
153C23—28no—
154C23—216no—
155C23—24no—
156C23—24no—
157C23—24no—
158C23—22no—
159C23—21no—
160C23—24no—
161C23D4×D9714yes—
162C23C2×C22⋊D9712yes—
163C23C2×C22⋊D9714yes—
164C23#1309551018yes—
165C23C23×D9711yescharacteristic
166D4—44no—
167D4—48no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—978no—
173C2×C4—1944no—
174C2×C4—1948no—
175C2×C4—19432no—
176C2×C4—19416no—
177C2×C4—1944no—
178C2×C4—1944no—
179C22—232no—
180C22—24no—
181C22—24no—
182C22—24no—
183C22—28no—
184C22—216no—
185C22—41no—
186C22—42no—
187C22—44no—
188C22#1569077814yes—
189C22#1569079118yescomplemented
190C22C2×D4×D9712yes—
191C22C22×C22⋊D9711yescharacteristic
192C22C2×#1309551014yes—
193C4—41no—
194C4—42no—
195C4—3884no—
196C4—1948no—
197C2—24no—
198C2—28no—
199C2—81no—
200C2C2×#1569077811yescharacteristic
201C2C2×#1569079112yes—
202C2#2026124114yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting, prime radicalC2×#120×C97C2
hypercenterC2×#120D97
hypocenterC97#2849
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C24×Dic97C2semidirect
2. C2×#15690791C2semidirect
3. #20260615C2semidirect
4. #20260607C2semidirect
5. #20260596C2semidirect
6. #20260591C2semidirect
13. C2×#13095490C22semidirect
16. C2×C22⋊Dic97C22semidirect
17. #15690717C22semidirect
29. C22×Dic97D4semidirect
87. C22×C97#123semidirect
102. C97#2849semidirect
189. C22#15690791semidirect

Maximal subgroups

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

subgroupindex
C24×Dic972
C2×#156907912
#202606152
#202606072
#202605962
#202605912
C2×#120×C972
#284997