Finite Group Databasebeta

← #120×#89137 #28379495

Subgroups — #120×#89137 #28379495

Order 13952.

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×#89137C211yesmaximal, characteristic, complemented
2C2×D4×#89137C213yesmaximal, complemented
3#20295974C213yesmaximal, complemented
4#20295973C213yesmaximal, complemented
5#20295967C211yesmaximal, characteristic, complemented
6C22⋊C4×#89137C213yesmaximal, complemented
7#120×D109C211yesmaximal, characteristic
8C23×#89137—212no—
9C23×#89137C2213yes—
10C23×#89137C2213yes—
11C23×#89137C2211yescharacteristic, complemented
12D4×#89137—212no—
13C2×C4⋊#89137C2213yescomplemented
14C2×C22⋊#89137C2213yes—
15C2×C22⋊#89137C2213yes—
16C2×C22⋊#89137C2213yescomplemented
17C2×C4×#89137C2213yescomplemented
18(C2×#89137)⋊C4C2216yescomplemented
19C24×D109C2211yescharacteristic
20C2×D4×D109C2213yes—
21#16812669C413yescomplemented
22#16812668C411yescharacteristic, complemented
23#16812667C413yescomplemented
24C22⋊C4×D109C2213yes—
25#120×C109C411yescharacteristic, complemented
26C22×#89137—232no—
27C22×#89137—26no—
28C22×#89137—212no—
29C22×#89137—212no—
30C22×#89137—26no—
31C22×#89137D4112yescomplemented
32C22×#89137C2313yes—
33C22×#89137C2311yescharacteristic
34C22×#89137C2313yes—
35C22×#89137C2311yescharacteristic
36C4⋊#89137—26no—
37C22⋊#89137—26no—
38C22⋊#89137—26no—
39C22⋊#89137—26no—
40C4×#89137—26no—
41C23×D109—26no—
42C23×D109—26no—
43C23×D109C2×C413yescomplemented
44C23×D109C2311yescharacteristic
45C23×D109C2313yes—
46C2×D436C2×C413yescomplemented
47C2×C22⋊D109C2×C413yescomplemented
48C2×C22⋊D109C2×C413yescomplemented
49D4×D109—26no—
50D4×D109—26no—
51C2×C4×D109C2313yes—
52C22⋊Dic109C2×C413yescomplemented
53#13182006C2×C416yescomplemented
54C24×C109C2×C411yescharacteristic, complemented
55C2×D4×C109C2×C413yescomplemented
56C22⋊C4×C109C2×C413yescomplemented
57C23×C109—26no—
58C23×C109C22×C413yes—
59C23×C109C22×C411yescharacteristic, complemented
60C22×D109—224no—
61C22×D109—28no—
62C22×D109—26no—
63C22×D109—212no—
64C22×D109—26no—
65C22×D109—26no—
66C22×D109—23no—
67C22×D109—23no—
68C22×D109C22×C413yes—
69C22×D109C22×C411yescharacteristic, complemented
70C22×D109C2×D416yes—
71C22×D109#4116yes—
72C22×D109C2411yescharacteristic
73C2×#89137—224no—
74C2×#89137—26no—
75C2×#89137—224no—
76C2×#89137—23no—
77C2×#89137—23no—
78C2×#89137—26no—
79C2×#89137—44no—
80C2×#89137C2×D416yes—
81C2×#89137C2×D416yes—
82D4×C109—26no—
83C2×C4×C109C22×C413yescomplemented
84C22⋊D109—26no—
85C22⋊D109—26no—
86C2×Dic109C22×C413yescomplemented
87D436—26no—
88C4×D109—23no—
89C4×D109—23no—
90C22×C109—28no—
91C22×C109—26no—
92C22×C109—23no—
93C22×C109C4×D416yescomplemented
94C22×C109C23×C411yescharacteristic
95C2×D109—224no—
96C2×D109—26no—
97C2×D109—26no—
98C2×D109—26no—
99C2×D109—212no—
100C2×D109—23no—
101C2×D109—41no—
102C2×D109—43no—
103C2×D109C4×D416yescomplemented
104C2×D109C22×D413yes—
105C2×D109C2×#4113yes—
106#89137—26no—
107#89137—41no—
108#89137—43no—
109#89137#12014yesdirect product factor
110C4×C109—23no—
111Dic109—23no—
112C2×C109—26no—
113C2×C109—41no—
114C2×C109C2×C4×D413yes—
115D109—26no—
116D109—41no—
117D109C2×#12011yescharacteristic
118D109#52513yes—
119C4×#120—1091nomaximal, Sylow
120C109C4×#12011yescharacteristic, complemented, Sylow
121C24×C4—1091no—
122C2×#120—1091no—
123C2×C4×D4—1093no—
124#386—1093no—
125#385—1093no—
126#379—1091no—
127C4×C22⋊C4—1093no—
128C25—1091no—
129C22×D4—1093no—
130C23×C4—1093no—
131C23×C4—1091no—
132C23×C4—1093no—
133C23×C4—21812no—
134#120—1091no—
135#120—1093no—
136#120—1093no—
137#120#8913711yescharacteristic, direct product factor
138C4×D4—21812no—
139C2×C4⋊C4—1093no—
140C2×C22⋊C4—1093no—
141C2×C22⋊C4—1093no—
142C2×C22⋊C4—1093no—
143C2×C22⋊C4—1093no—
144C2×C42—1093no—
145#95—1096no—
146C24—1093no—
147C24—1093no—
148C24—1091no—
149C24—2186no—
150C24—2186no—
151C24C2×#8913711yescharacteristic, complemented
152C2×D4—1093no—
153C2×D4—1093no—
154C2×D4—1093no—
155C2×D4—2186no—
156C2×D4—2186no—
157C2×D4C2×#8913713yescomplemented
158C22×C4—1091no—
159C22×C4—10912no—
160C22×C4—1093no—
161C22×C4—1093no—
162C22×C4—1093no—
163C22×C4—1091no—
164C22×C4—2186no—
165C22×C4—21812no—
166C22×C4—21812no—
167C22×C4—2186no—
168C22×C4—21832no—
169C4⋊C4—2186no—
170C22⋊C4—1093no—
171C22⋊C4—1096no—
172C22⋊C4—2186no—
173C22⋊C4—2186no—
174C22⋊C4—2186no—
175C22⋊C4C2×#8913713yescomplemented
176C42—2186no—
177C23—26no—
178C23—1091no—
179C23—1096no—
180C23—1096no—
181C23—1093no—
182C23—1091no—
183C23—2183no—
184C23—2183no—
185C23—2186no—
186C23—2186no—
187C23—21812no—
188C23—2188no—
189C23—21824no—
190C23—2186no—
191C23C22×#8913713yes—
192C23C22×#8913711yescharacteristic, complemented
193D4—26no—
194D4—2186no—
195D4—2186no—
196D4—2186no—
197C2×C4—1096no—
198C2×C4—1096no—
199C2×C4—1093no—
200C2×C4—4364no—
201C2×C4—2183no—
202C2×C4—2183no—
203C2×C4—21824no—
204C2×C4—2183no—
205C2×C4—2183no—
206C2×C4—2186no—
207C2×C4—2186no—
208C2×C4—21824no—
209C2×C4C22×#8913713yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—1093no—
214C22—1093no—
215C22—1096no—
216C22—4361no—
217C22—4363no—
218C22—2186no—
219C22—2186no—
220C22—21812no—
221C22—21824no—
222C22—2183no—
223C22—2186no—
224C22D4×#8913716yescomplemented
225C22C23×#8913711yescharacteristic
226C4—23no—
227C4—1094no—
228C4—4361no—
229C4—4363no—
230C4—2186no—
231C4—2183no—
232C2—26no—
233C2—41no—
234C2—1091no—
235C2—1093no—
236C2—4361no—
237C2—2186no—
238C2C2×D4×#8913713yes—
subgroupG/H
center, FrattiniC22C23×#89137
commutator, socleC22×C109C23×C4
Fitting#120×C109C4
prime radical#120×D109C2
cosocleC22×D109C24
hypercenter#120#89137
hypocenterC109C4×#120
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#89137C2semidirect
2. C2×D4×#89137C2semidirect
3. #20295974C2semidirect
4. #20295973C2semidirect
5. #20295967C2semidirect
6. C22⋊C4×#89137C2semidirect
11. C23×#89137C22semidirect
13. C2×C4⋊#89137C22semidirect
16. C2×C22⋊#89137C22semidirect
17. C2×C4×#89137C22semidirect
18. (C2×#89137)⋊C4C22semidirect
21. #16812669C4semidirect
22. #16812668C4semidirect
23. #16812667C4semidirect
25. #120×C109C4semidirect
31. C22×#89137D4semidirect
43. C23×D109C2×C4semidirect
46. C2×D436C2×C4semidirect
47. C2×C22⋊D109C2×C4semidirect
48. C2×C22⋊D109C2×C4semidirect
52. C22⋊Dic109C2×C4semidirect
53. #13182006C2×C4semidirect
54. C24×C109C2×C4semidirect
55. C2×D4×C109C2×C4semidirect
56. C22⋊C4×C109C2×C4semidirect
59. C23×C109C22×C4semidirect
69. C22×D109C22×C4semidirect
83. C2×C4×C109C22×C4semidirect
86. C2×Dic109C22×C4semidirect
93. C22×C109C4×D4semidirect
103. C2×D109C4×D4semidirect
109. #89137#120direct (N × H)
120. C109C4×#120semidirect
137. #120#89137direct (N × H)
151. C24C2×#89137semidirect
157. C2×D4C2×#89137semidirect
175. C22⋊C4C2×#89137semidirect
192. C23C22×#89137semidirect
209. C2×C4C22×#89137semidirect
224. C22D4×#89137semidirect

Maximal subgroups

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

subgroupindex
C24×#891372
C2×D4×#891372
#202959742
#202959732
#202959672
C22⋊C4×#891372
#120×D1092
C4×#120109