Finite Group Databasebeta

← #120×#3866 #18011059

Subgroups — #120×#3866 #18011059

Order 4736.

1224 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 210 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×#3866C211yesmaximal, characteristic, complemented
2C2×D4×#3866C213yesmaximal, complemented
3#14558481C213yesmaximal, complemented
4#14558480C213yesmaximal, complemented
5#14558474C211yesmaximal, characteristic, complemented
6C22⋊C4×#3866C213yesmaximal, complemented
7#120×D37C211yesmaximal, characteristic
8C23×#3866—212no—
9C23×#3866C2213yes—
10C23×#3866C2213yes—
11C23×#3866C2211yescharacteristic, complemented
12D4×#3866—212no—
13C2×C4⋊#3866C2213yescomplemented
14C2×C22⋊#3866C2213yes—
15C2×C22⋊#3866C2213yes—
16C2×C22⋊#3866C2213yescomplemented
17C2×C4×#3866C2213yescomplemented
18(C2×#3866)⋊C4C2216yescomplemented
19C24×D37C2211yescharacteristic
20C2×D4×D37C2213yes—
21#11924506C413yescomplemented
22#11924505C411yescharacteristic, complemented
23#11924504C413yescomplemented
24C22⋊C4×D37C2213yes—
25#120×C37C411yescharacteristic, complemented
26C24×C37C2×C411yescharacteristic, complemented
27C23×D37—26no—
28C23×D37—26no—
29C23×D37C2×C413yescomplemented
30C23×D37C2311yescharacteristic
31C23×D37C2313yes—
32C22×#3866—232no—
33C22×#3866—26no—
34C22×#3866—212no—
35C22×#3866—212no—
36C22×#3866—26no—
37C22×#3866D4112yescomplemented
38C22×#3866C2313yes—
39C22×#3866C2311yescharacteristic
40C22×#3866C2313yes—
41C22×#3866C2311yescharacteristic
42C2×D4×C37C2×C413yescomplemented
43C2×C22⋊D37C2×C413yescomplemented
44C2×C22⋊D37C2×C413yescomplemented
45D4×D37—26no—
46D4×D37—26no—
47C2×D148C2×C413yescomplemented
48C2×C4×D37C2313yes—
49C22⋊#3866—26no—
50C22⋊#3866—26no—
51C22⋊#3866—26no—
52C4⋊#3866—26no—
53C4×#3866—26no—
54C22⋊C4×C37C2×C413yescomplemented
55C22⋊Dic37C2×C413yescomplemented
56#10596938C2×C416yescomplemented
57C23×C37—26no—
58C23×C37C22×C413yes—
59C23×C37C22×C411yescharacteristic, complemented
60C22×D37—28no—
61C22×D37—224no—
62C22×D37—26no—
63C22×D37—212no—
64C22×D37—26no—
65C22×D37—26no—
66C22×D37—23no—
67C22×D37—23no—
68C22×D37C22×C413yes—
69C22×D37C22×C411yescharacteristic, complemented
70C22×D37C2×D416yes—
71C22×D37#4116yes—
72C22×D37C2411yescharacteristic
73C2×#3866—224no—
74C2×#3866—26no—
75C2×#3866—224no—
76C2×#3866—23no—
77C2×#3866—23no—
78C2×#3866—26no—
79C2×#3866—44no—
80C2×#3866C2×D416yes—
81C2×#3866C2×D416yes—
82D4×C37—26no—
83C2×C4×C37C22×C413yescomplemented
84C22⋊D37—26no—
85C22⋊D37—26no—
86C2×Dic37C22×C413yescomplemented
87D148—26no—
88C4×D37—23no—
89C4×D37—23no—
90C22×C37—28no—
91C22×C37—26no—
92C22×C37—23no—
93C22×C37C4×D416yescomplemented
94C22×C37C23×C411yescharacteristic
95C2×D37—224no—
96C2×D37—26no—
97C2×D37—26no—
98C2×D37—26no—
99C2×D37—212no—
100C2×D37—23no—
101C2×D37—41no—
102C2×D37—43no—
103C2×D37C4×D416yescomplemented
104C2×D37C22×D413yes—
105C2×D37C2×#4113yes—
106#3866—26no—
107#3866—41no—
108#3866—43no—
109#3866#12014yesdirect product factor
110C4×C37—23no—
111Dic37—23no—
112C4×#120—371nomaximal, Sylow
113C2×C37—26no—
114C2×C37—41no—
115C2×C37C2×C4×D413yes—
116D37—26no—
117D37—41no—
118D37C2×#12011yescharacteristic
119D37#52513yes—
120C24×C4—371no—
121C2×#120—371no—
122C2×C4×D4—373no—
123#386—373no—
124#385—373no—
125#379—371no—
126C4×C22⋊C4—373no—
127C37C4×#12011yescharacteristic, complemented, Sylow
128C25—371no—
129C22×D4—373no—
130C23×C4—373no—
131C23×C4—371no—
132C23×C4—373no—
133C23×C4—7412no—
134#120—371no—
135#120—373no—
136#120—373no—
137#120#386611yescharacteristic, direct product factor
138C4×D4—7412no—
139C2×C4⋊C4—373no—
140C2×C22⋊C4—373no—
141C2×C22⋊C4—373no—
142C2×C22⋊C4—373no—
143C2×C22⋊C4—373no—
144C2×C42—373no—
145#95—376no—
146C24—373no—
147C24—373no—
148C24—371no—
149C24—746no—
150C24—746no—
151C24C2×#386611yescharacteristic, complemented
152C2×D4—373no—
153C2×D4—373no—
154C2×D4—373no—
155C2×D4—746no—
156C2×D4—746no—
157C2×D4C2×#386613yescomplemented
158C22×C4—371no—
159C22×C4—373no—
160C22×C4—3712no—
161C22×C4—373no—
162C22×C4—373no—
163C22×C4—371no—
164C22×C4—7412no—
165C22×C4—7412no—
166C22×C4—746no—
167C22×C4—746no—
168C22×C4—7432no—
169C4⋊C4—746no—
170C22⋊C4—373no—
171C22⋊C4—376no—
172C22⋊C4—746no—
173C22⋊C4—746no—
174C22⋊C4—746no—
175C22⋊C4C2×#386613yescomplemented
176C42—746no—
177C23—26no—
178C23—371no—
179C23—376no—
180C23—376no—
181C23—373no—
182C23—371no—
183C23—746no—
184C23—746no—
185C23—7412no—
186C23—743no—
187C23—743no—
188C23—748no—
189C23—7424no—
190C23—746no—
191C23C22×#386613yes—
192C23C22×#386611yescharacteristic, complemented
193D4—26no—
194D4—746no—
195D4—746no—
196D4—746no—
197C2×C4—376no—
198C2×C4—376no—
199C2×C4—373no—
200C2×C4—743no—
201C2×C4—743no—
202C2×C4—7424no—
203C2×C4—746no—
204C2×C4—7424no—
205C2×C4—743no—
206C2×C4—743no—
207C2×C4—746no—
208C2×C4—1484no—
209C2×C4C22×#386613yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—373no—
214C22—373no—
215C22—376no—
216C22—746no—
217C22—746no—
218C22—7412no—
219C22—7424no—
220C22—746no—
221C22—743no—
222C22—1481no—
223C22—1483no—
224C22D4×#386616yescomplemented
225C22C23×#386611yescharacteristic
226C4—23no—
227C4—374no—
228C4—746no—
229C4—743no—
230C4—1481no—
231C4—1483no—
232C2—26no—
233C2—41no—
234C2—371no—
235C2—373no—
236C2—746no—
237C2—1481no—
238C2C2×D4×#386613yes—
subgroupG/H
center, FrattiniC22C23×#3866
commutator, socleC22×C37C23×C4
Fitting#120×C37C4
prime radical#120×D37C2
cosocleC22×D37C24
hypercenter#120#3866
hypocenterC37C4×#120
LCSG ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#3866C2semidirect
2. C2×D4×#3866C2semidirect
3. #14558481C2semidirect
4. #14558480C2semidirect
5. #14558474C2semidirect
6. C22⋊C4×#3866C2semidirect
11. C23×#3866C22semidirect
13. C2×C4⋊#3866C22semidirect
16. C2×C22⋊#3866C22semidirect
17. C2×C4×#3866C22semidirect
18. (C2×#3866)⋊C4C22semidirect
21. #11924506C4semidirect
22. #11924505C4semidirect
23. #11924504C4semidirect
25. #120×C37C4semidirect
26. C24×C37C2×C4semidirect
29. C23×D37C2×C4semidirect
37. C22×#3866D4semidirect
42. C2×D4×C37C2×C4semidirect
43. C2×C22⋊D37C2×C4semidirect
44. C2×C22⋊D37C2×C4semidirect
47. C2×D148C2×C4semidirect
54. C22⋊C4×C37C2×C4semidirect
55. C22⋊Dic37C2×C4semidirect
56. #10596938C2×C4semidirect
59. C23×C37C22×C4semidirect
69. C22×D37C22×C4semidirect
83. C2×C4×C37C22×C4semidirect
86. C2×Dic37C22×C4semidirect
93. C22×C37C4×D4semidirect
103. C2×D37C4×D4semidirect
109. #3866#120direct (N × H)
127. C37C4×#120semidirect
137. #120#3866direct (N × H)
151. C24C2×#3866semidirect
157. C2×D4C2×#3866semidirect
175. C22⋊C4C2×#3866semidirect
192. C23C22×#3866semidirect
209. C2×C4C22×#3866semidirect
224. C22D4×#3866semidirect

Maximal subgroups

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

subgroupindex
C24×#38662
C2×D4×#38662
#145584812
#145584802
#145584742
C22⋊C4×#38662
#120×D372
C4×#12037