Finite Group Databasebeta

← #27150016

Subgroups — #27150016

Order 12416.

1224 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15690782C213yesmaximal, complemented
3C2×#15690779C211yesmaximal, characteristic, complemented
4#20260615C213yesmaximal, complemented
5#20260612C213yesmaximal, complemented
6#20260593C213yesmaximal, complemented
7C2×#120×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C2213yes—
9C22×C22⋊D97C2211yescharacteristic, complemented
10C23×Dic97—212no—
11C23×Dic97C2213yes—
12C23×Dic97C2213yes—
13#15690782—212no—
14#15690779—24no—
15C2×#13095490C2213yescomplemented
16C2×C22⋊Dic97C2213yes—
17C2×C22⋊Dic97C2213yescomplemented
18C2×#13095488C2213yescomplemented
19#15690717C2216yescomplemented
20C25×C97C2211yescharacteristic
21C22×D4×C97C2213yes—
22#120×C97—24no—
23C2×C22⋊C4×C97C2213yes—
24C23×D97C2311yescharacteristic
25C2×C22⋊D97—26no—
26C2×C22⋊D97—26no—
27C2×C22⋊D97—26no—
28C2×C22⋊D97—26no—
29C22×Dic97—232no—
30C22×Dic97—26no—
31C22×Dic97—212no—
32C22×Dic97—212no—
33C22×Dic97D4112yescomplemented
34C22×Dic97C2313yes—
35C22×Dic97C2313yes—
36C22×Dic97C2311yescharacteristic
37#13095490—26no—
38C22⋊Dic97—26no—
39C22⋊Dic97—26no—
40#13095488—26no—
41C24×C97—26no—
42C24×C97—26no—
43C24×C97D414yescomplemented
44C24×C97C2311yescharacteristic
45C24×C97C2313yes—
46C2×D4×C97—26no—
47C2×D4×C97—26no—
48C2×D4×C97—26no—
49C22×C4×C97C2313yes—
50C22⋊C4×C97—26no—
51C23×C97—224no—
52C23×C97—28no—
53C23×C97—224no—
54C23×C97—26no—
55C23×C97—26no—
56C23×C97—21no—
57C23×C97—23no—
58C23×C97—23no—
59C23×C97C2×D416yes—
60C23×C97C2×D416yes—
61C23×C97#4116yes—
62C23×C97C2411yescharacteristic
63C22×D97—23no—
64C22×D97—21no—
65C22×D97—23no—
66D4×C97—412no—
67C2×C4×C97—23no—
68C2×C4×C97—23no—
69C2×C4×C97—23no—
70C22⋊D97—412no—
71C22⋊D97—44no—
72C2×Dic97—224no—
73C2×Dic97—26no—
74C2×Dic97—224no—
75C2×Dic97—23no—
76C2×Dic97—23no—
77C2×Dic97—23no—
78C2×Dic97C2×D416yes—
79C2×Dic97C2×D416yes—
80C22×C97—232no—
81C22×C97—26no—
82C22×C97—212no—
83C22×C97—26no—
84C22×C97—212no—
85C22×C97—41no—
86C22×C97—43no—
87C22×C97—43no—
88C22×C97#121112yescomplemented
89C22×C97C22×D413yes—
90C22×C97C22×D411yescharacteristic
91C22×C97C2×#4113yes—
92C2×D97—41no—
93C2×D97—43no—
94C2×D97—43no—
95C4×C97—43no—
96Dic97—26no—
97Dic97—43no—
98Dic97#12014yescomplemented
99C2×C97—212no—
100C2×C97—81no—
101C2×C97C2×#12011yescharacteristic
102C2×C97C2×#12113yes—
103C2×C97#52513yes—
104D97—81no—
105#2847—971nomaximal, Sylow
106C97#284711yescharacteristic, complemented, Sylow
107C24×C4—971no—
108C2×#121—973no—
109C2×#120—971no—
110C2×#120D9711yescharacteristic, complemented
111#386—973no—
112#386—973no—
113#380—973no—
114C25C2×D9711yescharacteristic
115C22×D4—973no—
116C22×D4—971no—
117C22×D4C2×D9713yes—
118C23×C4—973no—
119C23×C4—973no—
120C23×C4—19412no—
121#121—19412no—
122#120—24no—
123#120—1944no—
124C2×C4⋊C4—973no—
125C2×C22⋊C4—973no—
126C2×C22⋊C4—973no—
127C2×C22⋊C4—973no—
128C2×C22⋊C4C2×D9713yes—
129#95—976no—
130C24—26no—
131C24—26no—
132C24—971no—
133C24C22⋊D9714yescomplemented
134C24C22×D9711yescharacteristic
135C24C22×D9713yes—
136C2×D4—26no—
137C2×D4—26no—
138C2×D4—26no—
139C2×D4—1946no—
140C2×D4—1946no—
141C2×D4—1946no—
142C2×D4—1946no—
143C22×C4—971no—
144C22×C4—9712no—
145C22×C4—973no—
146C22×C4—973no—
147C22×C4—19412no—
148C22×C4—19412no—
149C22×C4—19432no—
150C22×C4—1946no—
151C22×C4C22×D9713yes—
152C4⋊C4—1946no—
153C22⋊C4—26no—
154C22⋊C4—1946no—
155C22⋊C4—1946no—
156C22⋊C4—1946no—
157C23—28no—
158C23—224no—
159C23—224no—
160C23—26no—
161C23—26no—
162C23—23no—
163C23—21no—
164C23—23no—
165C23—1943no—
166C23—1941no—
167C23—1943no—
168C23D4×D9716yes—
169C23C2×C22⋊D9716yes—
170C23#1309551016yes—
171C23C23×D9711yescharacteristic
172D4—412no—
173D4—3884no—
174D4—38812no—
175C2×C4—23no—
176C2×C4—23no—
177C2×C4—23no—
178C2×C4—976no—
179C2×C4—976no—
180C2×C4—19424no—
181C2×C4—1943no—
182C2×C4—1943no—
183C2×C4—1943no—
184C2×C4—19424no—
185C2×C4—1946no—
186C22—232no—
187C22—26no—
188C22—212no—
189C22—26no—
190C22—212no—
191C22—41no—
192C22—43no—
193C22—43no—
194C22—3881no—
195C22—3883no—
196C22—3883no—
197C22#15690782112yescomplemented
198C22C2×D4×D9713yes—
199C22C22×C22⋊D9711yescharacteristic
200C22C2×#1309551013yes—
201C4—43no—
202C4—974no—
203C4—3883no—
204C4—1946no—
205C2—212no—
206C2—81no—
207C2—7761no—
208C2#120×D9711yescharacteristic
209C2C2×#1569078213yes—
210C2#2026124113yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×#120×C97C2
hypercenterC2×#120D97
hypocenterC97#2847
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×#15690782C2semidirect
3. C2×#15690779C2semidirect
4. #20260615C2semidirect
5. #20260612C2semidirect
6. #20260593C2semidirect
7. C2×#120×C97C2semidirect
9. C22×C22⋊D97C22semidirect
15. C2×#13095490C22semidirect
17. C2×C22⋊Dic97C22semidirect
18. C2×#13095488C22semidirect
19. #15690717C22semidirect
33. C22×Dic97D4semidirect
43. C24×C97D4semidirect
88. C22×C97#121semidirect
98. Dic97#120semidirect
106. C97#2847semidirect
110. C2×#120D97semidirect
133. C24C22⋊D97semidirect
197. C22#15690782semidirect

Maximal subgroups

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

subgroupindex
C24×Dic972
C2×#156907822
C2×#156907792
#202606152
#202606122
#202605932
C2×#120×C972
#284797