Finite Group Databasebeta

← #120×Dic97 #27144239

Subgroups — #120×Dic97 #27144239

Order 12416.

1224 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 289 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×D4×Dic97C213yesmaximal, complemented
3#20260615C213yesmaximal, complemented
4#20260610C213yesmaximal, complemented
5#20260591C211yesmaximal, characteristic, complemented
6C22⋊C4×Dic97C213yesmaximal, complemented
7C2×#120×C97C211yesmaximal, characteristic
8C23×Dic97—212no—
9C23×Dic97C2213yes—
10C23×Dic97C2213yes—
11C23×Dic97C2211yescharacteristic, complemented
12D4×Dic97—212no—
13C2×C4⋊Dic97C2213yescomplemented
14C2×C22⋊Dic97C2213yes—
15C2×C22⋊Dic97C2213yes—
16C2×C22⋊Dic97C2213yescomplemented
17C2×C4×Dic97C2213yescomplemented
18#15690717C2216yescomplemented
19C25×C97C2211yescharacteristic
20C22×D4×C97C2213yes—
21#120×C97C418yescomplemented
22C2×C22⋊C4×C97C2213yes—
23C22×Dic97—232no—
24C22×Dic97—26no—
25C22×Dic97—212no—
26C22×Dic97—212no—
27C22×Dic97—26no—
28C22×Dic97D4112yescomplemented
29C22×Dic97C2313yes—
30C22×Dic97C2311yescharacteristic
31C22×Dic97C2313yes—
32C22×Dic97C2311yescharacteristic
33C4⋊Dic97—26no—
34C22⋊Dic97—26no—
35C22⋊Dic97—26no—
36C22⋊Dic97—26no—
37C4×Dic97—26no—
38C24×C97—26no—
39C24×C97—26no—
40C24×C97C2×C414yescomplemented
41C24×C97C2311yescharacteristic
42C24×C97C2313yes—
43C2×D4×C97—26no—
44C2×D4×C97—26no—
45C2×D4×C97C2×C4112yescomplemented
46C22×C4×C97C2313yes—
47C22⋊C4×C97C2×C4112yescomplemented
48C23×C97—224no—
49C23×C97—28no—
50C23×C97—224no—
51C23×C97—26no—
52C23×C97—26no—
53C23×C97—23no—
54C23×C97—23no—
55C23×C97C22×C416yes—
56C23×C97C22×C412yescomplemented
57C23×C97C2×D416yes—
58C23×C97#4116yes—
59C23×C97C2411yescharacteristic
60D4×C97—224no—
61C2×C4×C97—23no—
62C2×C4×C97—23no—
63C2×C4×C97C22×C416yescomplemented
64C2×Dic97—224no—
65C2×Dic97—26no—
66C2×Dic97—224no—
67C2×Dic97—23no—
68C2×Dic97—23no—
69C2×Dic97—26no—
70C2×Dic97—44no—
71C2×Dic97C2×D416yes—
72C2×Dic97C2×D416yes—
73C22×C97—232no—
74C22×C97—212no—
75C22×C97—26no—
76C22×C97—26no—
77C22×C97—212no—
78C22×C97—26no—
79C22×C97—41no—
80C22×C97—43no—
81C22×C97C4×D4112yescomplemented
82C22×C97C23×C411yescharacteristic
83C22×C97C22×D413yes—
84C22×C97C2×#4113yes—
85C4×C97—26no—
86Dic97—26no—
87Dic97—41no—
88Dic97—43no—
89Dic97#12014yesdirect product factor
90C2×C97—212no—
91C2×C97—42no—
92C2×C97C2×C4×D413yes—
93C2×C97C2×#12011yescharacteristic
94C2×C97#52513yes—
95C4×#120—971nomaximal, Sylow
96C97C4×#12011yescharacteristic, complemented, Sylow
97C24×C4—971no—
98C2×#120D9711yescharacteristic
99C2×C4×D4—973no—
100#386—973no—
101#385—973no—
102#379—971no—
103C4×C22⋊C4—973no—
104C25C2×D9711yescharacteristic
105C22×D4C2×D9713yes—
106C23×C4—973no—
107C23×C4—971no—
108C23×C4—973no—
109C23×C4—19412no—
110#120Dic9718yesdirect product factor
111C4×D4—19412no—
112C2×C4⋊C4—973no—
113C2×C22⋊C4—973no—
114C2×C22⋊C4—973no—
115C2×C22⋊C4—973no—
116C2×C22⋊C4C2×D9713yes—
117C2×C42—973no—
118#95—976no—
119C24—26no—
120C24—26no—
121C24C2×Dic9714yescomplemented
122C24C22×D9711yescharacteristic
123C24C22×D9713yes—
124C2×D4—26no—
125C2×D4—26no—
126C2×D4C2×Dic97112yescomplemented
127C22×C4—971no—
128C22×C4—9712no—
129C22×C4—973no—
130C22×C4—973no—
131C22×C4—971no—
132C22×C4—1946no—
133C22×C4—19412no—
134C22×C4—19412no—
135C22×C4—19432no—
136C22×C4—1946no—
137C22×C4C22×D9713yes—
138C4⋊C4—1946no—
139C22⋊C4—1946no—
140C22⋊C4—1946no—
141C22⋊C4—1946no—
142C22⋊C4C2×Dic97112yescomplemented
143C42—1946no—
144C23—28no—
145C23—224no—
146C23—224no—
147C23—26no—
148C23—26no—
149C23—23no—
150C23—23no—
151C23C22×Dic9716yes—
152C23C22×Dic9712yescomplemented
153C23D4×D9716yes—
154C23#1309551016yes—
155C23C23×D9711yescharacteristic
156D4—224no—
157C2×C4—23no—
158C2×C4—23no—
159C2×C4—976no—
160C2×C4—976no—
161C2×C4—3884no—
162C2×C4—19424no—
163C2×C4—1943no—
164C2×C4—1943no—
165C2×C4—1946no—
166C2×C4—19424no—
167C2×C4—1946no—
168C2×C4C22×Dic9716yescomplemented
169C22—232no—
170C22—212no—
171C22—26no—
172C22—26no—
173C22—212no—
174C22—26no—
175C22—41no—
176C22—43no—
177C22D4×Dic97112yescomplemented
178C22C23×Dic9711yescharacteristic
179C22C2×D4×D9713yes—
180C22C2×#1309551013yes—
181C4—26no—
182C4—974no—
183C4—3881no—
184C4—3883no—
185C4—1946no—
186C2—212no—
187C2—42no—
188C2C2×D4×Dic9713yes—
189C2#120×D9711yescharacteristic
190C2#2026124113yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting, prime radicalC2×#120×C97C2
socle, cosocleC23×C97C24
hypercenterC2×#120D97
hypocenterC97C4×#120
LCSG ▷ C22×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×D4×Dic97C2semidirect
3. #20260615C2semidirect
4. #20260610C2semidirect
5. #20260591C2semidirect
6. C22⋊C4×Dic97C2semidirect
11. C23×Dic97C22semidirect
13. C2×C4⋊Dic97C22semidirect
16. C2×C22⋊Dic97C22semidirect
17. C2×C4×Dic97C22semidirect
18. #15690717C22semidirect
21. #120×C97C4semidirect
28. C22×Dic97D4semidirect
40. C24×C97C2×C4semidirect
45. C2×D4×C97C2×C4semidirect
47. C22⋊C4×C97C2×C4semidirect
56. C23×C97C22×C4semidirect
63. C2×C4×C97C22×C4semidirect
81. C22×C97C4×D4semidirect
89. Dic97#120direct (N × H)
96. C97C4×#120semidirect
110. #120Dic97direct (N × H)
121. C24C2×Dic97semidirect
126. C2×D4C2×Dic97semidirect
142. C22⋊C4C2×Dic97semidirect
152. C23C22×Dic97semidirect
168. C2×C4C22×Dic97semidirect
177. C22D4×Dic97semidirect

Maximal subgroups

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

subgroupindex
C24×Dic972
C2×D4×Dic972
#202606152
#202606102
#202605912
C22⋊C4×Dic972
C2×#120×C972
C4×#12097