Finite Group Databasebeta

← #120×#10587739 #31797252

Subgroups — #120×#10587739 #31797252

Order 17536.

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×#10587739C211yesmaximal, characteristic, complemented
2C2×D4×#10587739C213yesmaximal, complemented
3#22578351C213yesmaximal, complemented
4#22578350C213yesmaximal, complemented
5#22578344C211yesmaximal, characteristic, complemented
6C22⋊C4×#10587739C213yesmaximal, complemented
7#120×D137C211yesmaximal, characteristic
8C23×#10587739—212no—
9C23×#10587739C2213yes—
10C23×#10587739C2213yes—
11C23×#10587739C2211yescharacteristic, complemented
12D4×#10587739—212no—
13C2×C4⋊#10587739C2213yescomplemented
14C2×C22⋊#10587739C2213yes—
15C2×C22⋊#10587739C2213yes—
16C2×C22⋊#10587739C2213yescomplemented
17C2×C4×#10587739C2213yescomplemented
18(C2×#10587739)⋊C4C2216yescomplemented
19C24×D137C2211yescharacteristic
20C2×D4×D137C2213yes—
21#17984058C413yescomplemented
22#17984057C411yescharacteristic, complemented
23#17984056C413yescomplemented
24C22⋊C4×D137C2213yes—
25#120×C137C411yescharacteristic, complemented
26C22×#10587739—232no—
27C22×#10587739—212no—
28C22×#10587739—212no—
29C22×#10587739—26no—
30C22×#10587739—26no—
31C22×#10587739D4112yescomplemented
32C22×#10587739C2313yes—
33C22×#10587739C2311yescharacteristic
34C22×#10587739C2313yes—
35C22×#10587739C2311yescharacteristic
36C4⋊#10587739—26no—
37C22⋊#10587739—26no—
38C22⋊#10587739—26no—
39C22⋊#10587739—26no—
40C4×#10587739—26no—
41C23×D137—26no—
42C23×D137—26no—
43C23×D137C2×C413yescomplemented
44C23×D137C2313yes—
45C23×D137C2311yescharacteristic
46C2×D548C2×C413yescomplemented
47C2×C22⋊D137C2×C413yescomplemented
48C2×C22⋊D137C2×C413yescomplemented
49D4×D137—26no—
50D4×D137—26no—
51C2×C4×D137C2313yes—
52C22⋊Dic137C2×C413yescomplemented
53#14555487C2×C416yescomplemented
54C24×C137C2×C411yescharacteristic, complemented
55C2×D4×C137C2×C413yescomplemented
56C22⋊C4×C137C2×C413yescomplemented
57C2×#10587739—224no—
58C2×#10587739—26no—
59C2×#10587739—224no—
60C2×#10587739—23no—
61C2×#10587739—23no—
62C2×#10587739—26no—
63C2×#10587739—44no—
64C2×#10587739C2×D416yes—
65C2×#10587739C2×D416yes—
66C22×D137—224no—
67C22×D137—28no—
68C22×D137—26no—
69C22×D137—212no—
70C22×D137—26no—
71C22×D137—26no—
72C22×D137—23no—
73C22×D137—23no—
74C22×D137C22×C413yes—
75C22×D137C22×C411yescharacteristic, complemented
76C22×D137C2×D416yes—
77C22×D137#4116yes—
78C22×D137C2411yescharacteristic
79D548—26no—
80C22⋊D137—26no—
81C22⋊D137—26no—
82C2×Dic137C22×C413yescomplemented
83C4×D137—23no—
84C4×D137—23no—
85C23×C137—26no—
86C23×C137C22×C413yes—
87C23×C137C22×C411yescharacteristic, complemented
88D4×C137—26no—
89C2×C4×C137C22×C413yescomplemented
90C22×C137—28no—
91C22×C137—26no—
92C22×C137—23no—
93C22×C137C4×D416yescomplemented
94C22×C137C23×C411yescharacteristic
95C2×D137—224no—
96C2×D137—26no—
97C2×D137—26no—
98C2×D137—26no—
99C2×D137—212no—
100C2×D137—23no—
101C2×D137—41no—
102C2×D137—43no—
103C2×D137C4×D416yescomplemented
104C2×D137C22×D413yes—
105C2×D137C2×#4113yes—
106#10587739—26no—
107#10587739—41no—
108#10587739—43no—
109#10587739#12014yesdirect product factor
110C4×C137—23no—
111Dic137—23no—
112C2×C137—26no—
113C2×C137—41no—
114C2×C137C2×C4×D413yes—
115D137—26no—
116D137—41no—
117D137C2×#12011yescharacteristic
118D137#52513yes—
119C137C4×#12011yescharacteristic, complemented, Sylow
120C4×#120—1371nomaximal, Sylow
121C24×C4—1371no—
122C2×#120—1371no—
123C2×C4×D4—1373no—
124#386—1373no—
125#385—1373no—
126#379—1371no—
127C4×C22⋊C4—1373no—
128C25—1371no—
129C22×D4—1373no—
130C23×C4—27412no—
131C23×C4—1371no—
132C23×C4—1373no—
133C23×C4—1373no—
134#120—1371no—
135#120—1373no—
136#120—1373no—
137#120#1058773911yescharacteristic, direct product factor
138C4×D4—27412no—
139C2×C4⋊C4—1373no—
140C2×C22⋊C4—1373no—
141C2×C22⋊C4—1373no—
142C2×C22⋊C4—1373no—
143C2×C22⋊C4—1373no—
144C2×C42—1373no—
145#95—1376no—
146C24—2746no—
147C24—2746no—
148C24—1373no—
149C24—1371no—
150C24—1373no—
151C24C2×#1058773911yescharacteristic, complemented
152C2×D4—2746no—
153C2×D4—2746no—
154C2×D4—1373no—
155C2×D4—1373no—
156C2×D4—1373no—
157C2×D4C2×#1058773913yescomplemented
158C22×C4—27432no—
159C22×C4—2746no—
160C22×C4—27412no—
161C22×C4—27412no—
162C22×C4—2746no—
163C22×C4—1373no—
164C22×C4—1371no—
165C22×C4—1373no—
166C22×C4—13712no—
167C22×C4—1373no—
168C22×C4—1371no—
169C4⋊C4—2746no—
170C22⋊C4—2746no—
171C22⋊C4—2746no—
172C22⋊C4—2746no—
173C22⋊C4—1373no—
174C22⋊C4—1376no—
175C22⋊C4C2×#1058773913yescomplemented
176C42—2746no—
177C23—26no—
178C23—2748no—
179C23—27424no—
180C23—2746no—
181C23—27412no—
182C23—2746no—
183C23—2743no—
184C23—2743no—
185C23—2746no—
186C23—1371no—
187C23—1376no—
188C23—1376no—
189C23—1373no—
190C23—1371no—
191C23C22×#1058773913yes—
192C23C22×#1058773911yescharacteristic, complemented
193D4—26no—
194D4—2746no—
195D4—2746no—
196D4—2746no—
197C2×C4—2743no—
198C2×C4—2743no—
199C2×C4—27424no—
200C2×C4—2746no—
201C2×C4—27424no—
202C2×C4—2743no—
203C2×C4—2743no—
204C2×C4—2746no—
205C2×C4—5484no—
206C2×C4—1376no—
207C2×C4—1373no—
208C2×C4—1376no—
209C2×C4C22×#1058773913yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—27424no—
214C22—2746no—
215C22—27412no—
216C22—2746no—
217C22—2746no—
218C22—2743no—
219C22—5481no—
220C22—5483no—
221C22—1376no—
222C22—1373no—
223C22—1373no—
224C22D4×#1058773916yescomplemented
225C22C23×#1058773911yescharacteristic
226C4—23no—
227C4—2746no—
228C4—2743no—
229C4—5481no—
230C4—5483no—
231C4—1374no—
232C2—26no—
233C2—41no—
234C2—2746no—
235C2—5481no—
236C2—1371no—
237C2—1373no—
238C2C2×D4×#1058773913yes—
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
Fitting#120×C137C4
prime radical#120×D137C2
cosocleC22×D137C24
hypercenter#120#10587739
hypocenterC137C4×#120
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#10587739C2semidirect
2. C2×D4×#10587739C2semidirect
3. #22578351C2semidirect
4. #22578350C2semidirect
5. #22578344C2semidirect
6. C22⋊C4×#10587739C2semidirect
11. C23×#10587739C22semidirect
13. C2×C4⋊#10587739C22semidirect
16. C2×C22⋊#10587739C22semidirect
17. C2×C4×#10587739C22semidirect
18. (C2×#10587739)⋊C4C22semidirect
21. #17984058C4semidirect
22. #17984057C4semidirect
23. #17984056C4semidirect
25. #120×C137C4semidirect
31. C22×#10587739D4semidirect
43. C23×D137C2×C4semidirect
46. C2×D548C2×C4semidirect
47. C2×C22⋊D137C2×C4semidirect
48. C2×C22⋊D137C2×C4semidirect
52. C22⋊Dic137C2×C4semidirect
53. #14555487C2×C4semidirect
54. C24×C137C2×C4semidirect
55. C2×D4×C137C2×C4semidirect
56. C22⋊C4×C137C2×C4semidirect
75. C22×D137C22×C4semidirect
82. C2×Dic137C22×C4semidirect
87. C23×C137C22×C4semidirect
89. C2×C4×C137C22×C4semidirect
93. C22×C137C4×D4semidirect
103. C2×D137C4×D4semidirect
109. #10587739#120direct (N × H)
119. C137C4×#120semidirect
137. #120#10587739direct (N × H)
151. C24C2×#10587739semidirect
157. C2×D4C2×#10587739semidirect
175. C22⋊C4C2×#10587739semidirect
192. C23C22×#10587739semidirect
209. C2×C4C22×#10587739semidirect
224. C22D4×#10587739semidirect

Maximal subgroups

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

subgroupindex
C24×#105877392
C2×D4×#105877392
#225783512
#225783502
#225783442
C22⋊C4×#105877392
#120×D1372
C4×#120137