Finite Group Databasebeta

← #120×#1181 #16835955

Subgroups — #120×#1181 #16835955

Order 3712.

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×#1181C211yesmaximal, characteristic, complemented
2C2×D4×#1181C213yesmaximal, complemented
3#14272111C213yesmaximal, complemented
4#14272110C213yesmaximal, complemented
5#14272104C211yesmaximal, characteristic, complemented
6C22⋊C4×#1181C213yesmaximal, complemented
7#120×D29C211yesmaximal, characteristic
8C24×D29C2211yescharacteristic
9C23×#1181—212no—
10C23×#1181C2213yes—
11C23×#1181C2213yes—
12C23×#1181C2211yescharacteristic, complemented
13C2×D4×D29C2213yes—
14C2×C22⋊#1181C2213yes—
15C2×C22⋊#1181C2213yes—
16C2×C22⋊#1181C2213yescomplemented
17D4×#1181—212no—
18C2×C4⋊#1181C2213yescomplemented
19C2×C4×#1181C2213yescomplemented
20C29×#120C411yescharacteristic, complemented
21#11745315C411yescharacteristic, complemented
22#11745299C413yescomplemented
23#11745244C413yescomplemented
24C22⋊C4×D29C2213yes—
25(C2×#1181)⋊C4C2216yescomplemented
26C24×C29C2×C411yescharacteristic, complemented
27C23×D29—26no—
28C23×D29—26no—
29C23×D29C2×C413yescomplemented
30C23×D29C2311yescharacteristic
31C23×D29C2313yes—
32C22×#1181—232no—
33C22×#1181—26no—
34C22×#1181—212no—
35C22×#1181—212no—
36C22×#1181—26no—
37C22×#1181D4112yescomplemented
38C22×#1181C2313yes—
39C22×#1181C2311yescharacteristic
40C22×#1181C2313yes—
41C22×#1181C2311yescharacteristic
42C2×D4×C29C2×C413yescomplemented
43C2×C22⋊D29C2×C413yescomplemented
44C2×C22⋊D29C2×C413yescomplemented
45D4×D29—26no—
46D4×D29—26no—
47C2×D116C2×C413yescomplemented
48C2×C4×D29C2313yes—
49C22⋊#1181—26no—
50C22⋊#1181—26no—
51C22⋊#1181—26no—
52C4⋊#1181—26no—
53C4×#1181—26no—
54C22⋊C4×C29C2×C413yescomplemented
55C22⋊Dic29C2×C413yescomplemented
56#90781C2×C416yescomplemented
57C23×C29—26no—
58C23×C29C22×C413yes—
59C23×C29C22×C411yescharacteristic, complemented
60C22×D29—28no—
61C22×D29—224no—
62C22×D29—26no—
63C22×D29—212no—
64C22×D29—26no—
65C22×D29—26no—
66C22×D29—23no—
67C22×D29—23no—
68C22×D29C22×C413yes—
69C22×D29C22×C411yescharacteristic, complemented
70C22×D29C2×D416yes—
71C22×D29#4116yes—
72C22×D29C2411yescharacteristic
73C2×#1181—224no—
74C2×#1181—26no—
75C2×#1181—224no—
76C2×#1181—23no—
77C2×#1181—23no—
78C2×#1181—26no—
79C2×#1181—44no—
80C2×#1181C2×D416yes—
81C2×#1181C2×D416yes—
82D4×C29—26no—
83C2×C4×C29C22×C413yescomplemented
84C22⋊D29—26no—
85C22⋊D29—26no—
86C2×Dic29C22×C413yescomplemented
87D116—26no—
88C4×D29—23no—
89C4×D29—23no—
90C4×#120—291nomaximal, Sylow
91C22×C29—28no—
92C22×C29—26no—
93C22×C29—23no—
94C22×C29C4×D416yescomplemented
95C22×C29C23×C411yescharacteristic
96C2×D29—224no—
97C2×D29—26no—
98C2×D29—26no—
99C2×D29—26no—
100C2×D29—212no—
101C2×D29—23no—
102C2×D29—41no—
103C2×D29—43no—
104C2×D29C4×D416yescomplemented
105C2×D29C22×D413yes—
106C2×D29C2×#4113yes—
107#1181—26no—
108#1181—41no—
109#1181—43no—
110#1181#12014yesdirect product factor
111C4×C29—23no—
112Dic29—23no—
113C24×C4—291no—
114C2×#120—291no—
115C2×C4×D4—293no—
116#386—293no—
117#385—293no—
118#379—291no—
119C4×C22⋊C4—293no—
120C2×C29—26no—
121C2×C29—41no—
122C2×C29C2×C4×D413yes—
123D29—26no—
124D29—41no—
125D29C2×#12011yescharacteristic
126D29#52513yes—
127C25—291no—
128C22×D4—293no—
129C23×C4—293no—
130C23×C4—291no—
131C23×C4—293no—
132C23×C4—5812no—
133#120—291no—
134#120—293no—
135#120—293no—
136#120#118111yescharacteristic, direct product factor
137C4×D4—5812no—
138C2×C4⋊C4—293no—
139C2×C22⋊C4—293no—
140C2×C22⋊C4—293no—
141C2×C22⋊C4—293no—
142C2×C22⋊C4—293no—
143C2×C42—293no—
144#95—296no—
145C29C4×#12011yescharacteristic, complemented, Sylow
146C24—293no—
147C24—293no—
148C24—291no—
149C24—586no—
150C24—586no—
151C24C2×#118111yescharacteristic, complemented
152C2×D4—293no—
153C2×D4—293no—
154C2×D4—293no—
155C2×D4—586no—
156C2×D4—586no—
157C2×D4C2×#118113yescomplemented
158C22×C4—291no—
159C22×C4—293no—
160C22×C4—2912no—
161C22×C4—293no—
162C22×C4—293no—
163C22×C4—291no—
164C22×C4—5812no—
165C22×C4—5812no—
166C22×C4—586no—
167C22×C4—586no—
168C22×C4—5832no—
169C4⋊C4—586no—
170C22⋊C4—293no—
171C22⋊C4—296no—
172C22⋊C4—586no—
173C22⋊C4—586no—
174C22⋊C4—586no—
175C22⋊C4C2×#118113yescomplemented
176C42—586no—
177C23—26no—
178C23—291no—
179C23—296no—
180C23—296no—
181C23—293no—
182C23—291no—
183C23—586no—
184C23—586no—
185C23—5812no—
186C23—583no—
187C23—583no—
188C23—588no—
189C23—5824no—
190C23—586no—
191C23C22×#118113yes—
192C23C22×#118111yescharacteristic, complemented
193D4—26no—
194D4—586no—
195D4—586no—
196D4—586no—
197C2×C4—296no—
198C2×C4—296no—
199C2×C4—293no—
200C2×C4—583no—
201C2×C4—583no—
202C2×C4—5824no—
203C2×C4—586no—
204C2×C4—5824no—
205C2×C4—583no—
206C2×C4—583no—
207C2×C4—586no—
208C2×C4—1164no—
209C2×C4C22×#118113yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—293no—
214C22—293no—
215C22—296no—
216C22—586no—
217C22—586no—
218C22—5812no—
219C22—5824no—
220C22—586no—
221C22—583no—
222C22—1161no—
223C22—1163no—
224C22D4×#118116yescomplemented
225C22C23×#118111yescharacteristic
226C4—23no—
227C4—294no—
228C4—586no—
229C4—583no—
230C4—1161no—
231C4—1163no—
232C2—26no—
233C2—41no—
234C2—291no—
235C2—293no—
236C2—586no—
237C2—1161no—
238C2C2×D4×#118113yes—
subgroupG/H
center, FrattiniC22C23×#1181
commutator, socleC22×C29C23×C4
FittingC29×#120C4
prime radical#120×D29C2
cosocleC22×D29C24
hypercenter#120#1181
hypocenterC29C4×#120
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#1181C2semidirect
2. C2×D4×#1181C2semidirect
3. #14272111C2semidirect
4. #14272110C2semidirect
5. #14272104C2semidirect
6. C22⋊C4×#1181C2semidirect
12. C23×#1181C22semidirect
16. C2×C22⋊#1181C22semidirect
18. C2×C4⋊#1181C22semidirect
19. C2×C4×#1181C22semidirect
20. C29×#120C4semidirect
21. #11745315C4semidirect
22. #11745299C4semidirect
23. #11745244C4semidirect
25. (C2×#1181)⋊C4C22semidirect
26. C24×C29C2×C4semidirect
29. C23×D29C2×C4semidirect
37. C22×#1181D4semidirect
42. C2×D4×C29C2×C4semidirect
43. C2×C22⋊D29C2×C4semidirect
44. C2×C22⋊D29C2×C4semidirect
47. C2×D116C2×C4semidirect
54. C22⋊C4×C29C2×C4semidirect
55. C22⋊Dic29C2×C4semidirect
56. #90781C2×C4semidirect
59. C23×C29C22×C4semidirect
69. C22×D29C22×C4semidirect
83. C2×C4×C29C22×C4semidirect
86. C2×Dic29C22×C4semidirect
94. C22×C29C4×D4semidirect
104. C2×D29C4×D4semidirect
110. #1181#120direct (N × H)
136. #120#1181direct (N × H)
145. C29C4×#120semidirect
151. C24C2×#1181semidirect
157. C2×D4C2×#1181semidirect
175. C22⋊C4C2×#1181semidirect
192. C23C22×#1181semidirect
209. C2×C4C22×#1181semidirect
224. C22D4×#1181semidirect

Maximal subgroups

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

subgroupindex
C24×#11812
C2×D4×#11812
#142721112
#142721102
#142721042
C22⋊C4×#11812
#120×D292
C4×#12029