Finite Group Databasebeta

← #120×#11924078 #49175248

Subgroups — #120×#11924078 #49175248

Order 37504.

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×#11924078C211yesmaximal, characteristic, complemented
2C2×D4×#11924078C213yesmaximal, complemented
3#34037077C213yesmaximal, complemented
4#34037076C213yesmaximal, complemented
5#34037070C211yesmaximal, characteristic, complemented
6C22⋊C4×#11924078C213yesmaximal, complemented
7#120×D293C211yesmaximal, characteristic
8C23×#11924078—212no—
9C23×#11924078C2213yes—
10C23×#11924078C2213yes—
11C23×#11924078C2211yescharacteristic, complemented
12D4×#11924078—212no—
13C2×C4⋊#11924078C2213yescomplemented
14C2×C22⋊#11924078C2213yes—
15C2×C22⋊#11924078C2213yes—
16C2×C22⋊#11924078C2213yescomplemented
17C2×C4×#11924078C2213yescomplemented
18(C2×#11924078)⋊C4C2216yescomplemented
19C24×D293C2211yescharacteristic
20C2×D4×D293C2213yes—
21#22627594C413yescomplemented
22#22627593C411yescharacteristic, complemented
23#22627592C413yescomplemented
24C22⋊C4×D293C2213yes—
25#120×C293C411yescharacteristic, complemented
26C22×#11924078—232no—
27C22×#11924078—212no—
28C22×#11924078—212no—
29C22×#11924078—26no—
30C22×#11924078—26no—
31C22×#11924078D4112yescomplemented
32C22×#11924078C2313yes—
33C22×#11924078C2311yescharacteristic
34C22×#11924078C2313yes—
35C22×#11924078C2311yescharacteristic
36C4⋊#11924078—26no—
37C22⋊#11924078—26no—
38C22⋊#11924078—26no—
39C22⋊#11924078—26no—
40C4×#11924078—26no—
41C23×D293—26no—
42C23×D293—26no—
43C23×D293C2×C413yescomplemented
44C23×D293C2313yes—
45C23×D293C2311yescharacteristic
46C2×D1172C2×C413yescomplemented
47C2×C22⋊D293C2×C413yescomplemented
48C2×C22⋊D293C2×C413yescomplemented
49D4×D293—26no—
50D4×D293—26no—
51C2×C4×D293C2313yes—
52C22⋊Dic293C2×C413yescomplemented
53#17989779C2×C416yescomplemented
54C24×C293C2×C411yescharacteristic, complemented
55C2×D4×C293C2×C413yescomplemented
56C22⋊C4×C293C2×C413yescomplemented
57C2×#11924078—224no—
58C2×#11924078—26no—
59C2×#11924078—224no—
60C2×#11924078—23no—
61C2×#11924078—23no—
62C2×#11924078—26no—
63C2×#11924078—44no—
64C2×#11924078C2×D416yes—
65C2×#11924078C2×D416yes—
66C22×D293—224no—
67C22×D293—28no—
68C22×D293—26no—
69C22×D293—212no—
70C22×D293—26no—
71C22×D293—26no—
72C22×D293—23no—
73C22×D293—23no—
74C22×D293C22×C413yes—
75C22×D293C22×C411yescharacteristic, complemented
76C22×D293C2×D416yes—
77C22×D293#4116yes—
78C22×D293C2411yescharacteristic
79D1172—26no—
80C22⋊D293—26no—
81C22⋊D293—26no—
82C2×Dic293C22×C413yescomplemented
83C4×D293—23no—
84C4×D293—23no—
85C23×C293—26no—
86C23×C293C22×C413yes—
87C23×C293C22×C411yescharacteristic, complemented
88D4×C293—26no—
89C2×C4×C293C22×C413yescomplemented
90C22×C293—28no—
91C22×C293—26no—
92C22×C293—23no—
93C22×C293C4×D416yescomplemented
94C22×C293C23×C411yescharacteristic
95C2×D293—224no—
96C2×D293—26no—
97C2×D293—26no—
98C2×D293—26no—
99C2×D293—212no—
100C2×D293—23no—
101C2×D293—41no—
102C2×D293—43no—
103C2×D293C4×D416yescomplemented
104C2×D293C22×D413yes—
105C2×D293C2×#4113yes—
106#11924078—26no—
107#11924078—41no—
108#11924078—43no—
109#11924078#12014yesdirect product factor
110C4×C293—23no—
111Dic293—23no—
112C2×C293—26no—
113C2×C293—41no—
114C2×C293C2×C4×D413yes—
115D293—26no—
116D293—41no—
117D293C2×#12011yescharacteristic
118D293#52513yes—
119C293C4×#12011yescharacteristic, complemented, Sylow
120C4×#120—2931nomaximal, Sylow
121C24×C4—2931no—
122C2×#120—2931no—
123C2×C4×D4—2933no—
124#386—2933no—
125#385—2933no—
126#379—2931no—
127C4×C22⋊C4—2933no—
128C25—2931no—
129C22×D4—2933no—
130C23×C4—2933no—
131C23×C4—2931no—
132C23×C4—2933no—
133C23×C4—58612no—
134#120—2931no—
135#120—2933no—
136#120—2933no—
137#120#1192407811yescharacteristic, direct product factor
138C4×D4—58612no—
139C2×C4⋊C4—2933no—
140C2×C22⋊C4—2933no—
141C2×C22⋊C4—2933no—
142C2×C22⋊C4—2933no—
143C2×C22⋊C4—2933no—
144C2×C42—2933no—
145#95—2936no—
146C24—2933no—
147C24—2933no—
148C24—2931no—
149C24—5866no—
150C24—5866no—
151C24C2×#1192407811yescharacteristic, complemented
152C2×D4—2933no—
153C2×D4—2933no—
154C2×D4—2933no—
155C2×D4—5866no—
156C2×D4—5866no—
157C2×D4C2×#1192407813yescomplemented
158C22×C4—2931no—
159C22×C4—2933no—
160C22×C4—29312no—
161C22×C4—2933no—
162C22×C4—2933no—
163C22×C4—2931no—
164C22×C4—58612no—
165C22×C4—58612no—
166C22×C4—5866no—
167C22×C4—5866no—
168C22×C4—58632no—
169C4⋊C4—5866no—
170C22⋊C4—2933no—
171C22⋊C4—2936no—
172C22⋊C4—5866no—
173C22⋊C4—5866no—
174C22⋊C4—5866no—
175C22⋊C4C2×#1192407813yescomplemented
176C42—5866no—
177C23—26no—
178C23—2931no—
179C23—2936no—
180C23—2936no—
181C23—2933no—
182C23—2931no—
183C23—5866no—
184C23—5866no—
185C23—58612no—
186C23—5863no—
187C23—5863no—
188C23—5868no—
189C23—58624no—
190C23—5866no—
191C23C22×#1192407813yes—
192C23C22×#1192407811yescharacteristic, complemented
193D4—26no—
194D4—5866no—
195D4—5866no—
196D4—5866no—
197C2×C4—2936no—
198C2×C4—2936no—
199C2×C4—2933no—
200C2×C4—5863no—
201C2×C4—5863no—
202C2×C4—58624no—
203C2×C4—5866no—
204C2×C4—58624no—
205C2×C4—5863no—
206C2×C4—5863no—
207C2×C4—5866no—
208C2×C4—11724no—
209C2×C4C22×#1192407813yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—2933no—
214C22—2933no—
215C22—2936no—
216C22—5866no—
217C22—5866no—
218C22—58612no—
219C22—58624no—
220C22—5866no—
221C22—5863no—
222C22—11721no—
223C22—11723no—
224C22D4×#1192407816yescomplemented
225C22C23×#1192407811yescharacteristic
226C4—23no—
227C4—2934no—
228C4—5866no—
229C4—5863no—
230C4—11721no—
231C4—11723no—
232C2—26no—
233C2—41no—
234C2—2931no—
235C2—2933no—
236C2—5866no—
237C2—11721no—
238C2C2×D4×#1192407813yes—
subgroupG/H
center, FrattiniC22C23×#11924078
commutator, socleC22×C293C23×C4
Fitting#120×C293C4
prime radical#120×D293C2
cosocleC22×D293C24
hypercenter#120#11924078
hypocenterC293C4×#120
LCSG ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#11924078C2semidirect
2. C2×D4×#11924078C2semidirect
3. #34037077C2semidirect
4. #34037076C2semidirect
5. #34037070C2semidirect
6. C22⋊C4×#11924078C2semidirect
11. C23×#11924078C22semidirect
13. C2×C4⋊#11924078C22semidirect
16. C2×C22⋊#11924078C22semidirect
17. C2×C4×#11924078C22semidirect
18. (C2×#11924078)⋊C4C22semidirect
21. #22627594C4semidirect
22. #22627593C4semidirect
23. #22627592C4semidirect
25. #120×C293C4semidirect
31. C22×#11924078D4semidirect
43. C23×D293C2×C4semidirect
46. C2×D1172C2×C4semidirect
47. C2×C22⋊D293C2×C4semidirect
48. C2×C22⋊D293C2×C4semidirect
52. C22⋊Dic293C2×C4semidirect
53. #17989779C2×C4semidirect
54. C24×C293C2×C4semidirect
55. C2×D4×C293C2×C4semidirect
56. C22⋊C4×C293C2×C4semidirect
75. C22×D293C22×C4semidirect
82. C2×Dic293C22×C4semidirect
87. C23×C293C22×C4semidirect
89. C2×C4×C293C22×C4semidirect
93. C22×C293C4×D4semidirect
103. C2×D293C4×D4semidirect
109. #11924078#120direct (N × H)
119. C293C4×#120semidirect
137. #120#11924078direct (N × H)
151. C24C2×#11924078semidirect
157. C2×D4C2×#11924078semidirect
175. C22⋊C4C2×#11924078semidirect
192. C23C22×#11924078semidirect
209. C2×C4C22×#11924078semidirect
224. C22D4×#11924078semidirect

Maximal subgroups

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

subgroupindex
C24×#119240782
C2×D4×#119240782
#340370772
#340370762
#340370702
C22⋊C4×#119240782
#120×D2932
C4×#120293