Finite Group Databasebeta

← #120×#10597010 #34061154

Subgroups — #120×#10597010 #34061154

Order 19072.

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×#10597010C211yesmaximal, characteristic, complemented
2C2×D4×#10597010C213yesmaximal, complemented
3#23750573C213yesmaximal, complemented
4#23750572C213yesmaximal, complemented
5#23750566C211yesmaximal, characteristic, complemented
6C22⋊C4×#10597010C213yesmaximal, complemented
7#120×D149C211yesmaximal, characteristic
8C23×#10597010—212no—
9C23×#10597010C2213yes—
10C23×#10597010C2213yes—
11C23×#10597010C2211yescharacteristic, complemented
12D4×#10597010—212no—
13C2×C4⋊#10597010C2213yescomplemented
14C2×C22⋊#10597010C2213yes—
15C2×C22⋊#10597010C2213yes—
16C2×C22⋊#10597010C2213yescomplemented
17C2×C4×#10597010C2213yescomplemented
18(C2×#10597010)⋊C4C2216yescomplemented
19C24×D149C2211yescharacteristic
20C2×D4×D149C2213yes—
21#18011881C413yescomplemented
22#18011880C411yescharacteristic, complemented
23#18011879C413yescomplemented
24C22⋊C4×D149C2213yes—
25#120×C149C411yescharacteristic, complemented
26C22×#10597010—232no—
27C22×#10597010—212no—
28C22×#10597010—212no—
29C22×#10597010—26no—
30C22×#10597010—26no—
31C22×#10597010D4112yescomplemented
32C22×#10597010C2313yes—
33C22×#10597010C2311yescharacteristic
34C22×#10597010C2313yes—
35C22×#10597010C2311yescharacteristic
36C4⋊#10597010—26no—
37C22⋊#10597010—26no—
38C22⋊#10597010—26no—
39C22⋊#10597010—26no—
40C4×#10597010—26no—
41C23×D149—26no—
42C23×D149—26no—
43C23×D149C2×C413yescomplemented
44C23×D149C2313yes—
45C23×D149C2311yescharacteristic
46C2×D596C2×C413yescomplemented
47C2×C22⋊D149C2×C413yescomplemented
48C2×C22⋊D149C2×C413yescomplemented
49D4×D149—26no—
50D4×D149—26no—
51C2×C4×D149C2313yes—
52C22⋊Dic149C2×C413yescomplemented
53#14558713C2×C416yescomplemented
54C24×C149C2×C411yescharacteristic, complemented
55C2×D4×C149C2×C413yescomplemented
56C22⋊C4×C149C2×C413yescomplemented
57C2×#10597010—224no—
58C2×#10597010—26no—
59C2×#10597010—224no—
60C2×#10597010—23no—
61C2×#10597010—23no—
62C2×#10597010—26no—
63C2×#10597010—44no—
64C2×#10597010C2×D416yes—
65C2×#10597010C2×D416yes—
66C22×D149—224no—
67C22×D149—28no—
68C22×D149—26no—
69C22×D149—212no—
70C22×D149—26no—
71C22×D149—26no—
72C22×D149—23no—
73C22×D149—23no—
74C22×D149C22×C413yes—
75C22×D149C22×C411yescharacteristic, complemented
76C22×D149C2×D416yes—
77C22×D149#4116yes—
78C22×D149C2411yescharacteristic
79D596—26no—
80C22⋊D149—26no—
81C22⋊D149—26no—
82C2×Dic149C22×C413yescomplemented
83C4×D149—23no—
84C4×D149—23no—
85C23×C149—26no—
86C23×C149C22×C413yes—
87C23×C149C22×C411yescharacteristic, complemented
88D4×C149—26no—
89C2×C4×C149C22×C413yescomplemented
90C22×C149—28no—
91C22×C149—26no—
92C22×C149—23no—
93C22×C149C4×D416yescomplemented
94C22×C149C23×C411yescharacteristic
95C2×D149—224no—
96C2×D149—26no—
97C2×D149—26no—
98C2×D149—26no—
99C2×D149—212no—
100C2×D149—23no—
101C2×D149—41no—
102C2×D149—43no—
103C2×D149C4×D416yescomplemented
104C2×D149C22×D413yes—
105C2×D149C2×#4113yes—
106#10597010—26no—
107#10597010—41no—
108#10597010—43no—
109#10597010#12014yesdirect product factor
110C4×C149—23no—
111Dic149—23no—
112C2×C149—26no—
113C2×C149—41no—
114C2×C149C2×C4×D413yes—
115D149—26no—
116D149—41no—
117D149C2×#12011yescharacteristic
118D149#52513yes—
119C149C4×#12011yescharacteristic, complemented, Sylow
120C4×#120—1491nomaximal, Sylow
121C24×C4—1491no—
122C2×#120—1491no—
123C2×C4×D4—1493no—
124#386—1493no—
125#385—1493no—
126#379—1491no—
127C4×C22⋊C4—1493no—
128C25—1491no—
129C22×D4—1493no—
130C23×C4—29812no—
131C23×C4—1491no—
132C23×C4—1493no—
133C23×C4—1493no—
134#120—1491no—
135#120—1493no—
136#120—1493no—
137#120#1059701011yescharacteristic, direct product factor
138C4×D4—29812no—
139C2×C4⋊C4—1493no—
140C2×C22⋊C4—1493no—
141C2×C22⋊C4—1493no—
142C2×C22⋊C4—1493no—
143C2×C22⋊C4—1493no—
144C2×C42—1493no—
145#95—1496no—
146C24—2986no—
147C24—2986no—
148C24—1493no—
149C24—1491no—
150C24—1493no—
151C24C2×#1059701011yescharacteristic, complemented
152C2×D4—2986no—
153C2×D4—2986no—
154C2×D4—1493no—
155C2×D4—1493no—
156C2×D4—1493no—
157C2×D4C2×#1059701013yescomplemented
158C22×C4—29832no—
159C22×C4—2986no—
160C22×C4—29812no—
161C22×C4—29812no—
162C22×C4—2986no—
163C22×C4—1493no—
164C22×C4—1491no—
165C22×C4—14912no—
166C22×C4—1493no—
167C22×C4—1493no—
168C22×C4—1491no—
169C4⋊C4—2986no—
170C22⋊C4—2986no—
171C22⋊C4—2986no—
172C22⋊C4—2986no—
173C22⋊C4—1493no—
174C22⋊C4—1496no—
175C22⋊C4C2×#1059701013yescomplemented
176C42—2986no—
177C23—26no—
178C23—2988no—
179C23—29824no—
180C23—2986no—
181C23—29812no—
182C23—2986no—
183C23—2983no—
184C23—2983no—
185C23—2986no—
186C23—1491no—
187C23—1496no—
188C23—1496no—
189C23—1493no—
190C23—1491no—
191C23C22×#1059701013yes—
192C23C22×#1059701011yescharacteristic, complemented
193D4—26no—
194D4—2986no—
195D4—2986no—
196D4—2986no—
197C2×C4—2983no—
198C2×C4—2983no—
199C2×C4—29824no—
200C2×C4—2986no—
201C2×C4—29824no—
202C2×C4—2983no—
203C2×C4—2983no—
204C2×C4—2986no—
205C2×C4—5964no—
206C2×C4—1496no—
207C2×C4—1493no—
208C2×C4—1496no—
209C2×C4C22×#1059701013yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—29824no—
214C22—2986no—
215C22—29812no—
216C22—2986no—
217C22—2986no—
218C22—2983no—
219C22—5961no—
220C22—5963no—
221C22—1496no—
222C22—1493no—
223C22—1493no—
224C22D4×#1059701016yescomplemented
225C22C23×#1059701011yescharacteristic
226C4—23no—
227C4—2986no—
228C4—2983no—
229C4—5961no—
230C4—5963no—
231C4—1494no—
232C2—26no—
233C2—41no—
234C2—2986no—
235C2—5961no—
236C2—1491no—
237C2—1493no—
238C2C2×D4×#1059701013yes—
subgroupG/H
center, FrattiniC22C23×#10597010
commutator, socleC22×C149C23×C4
Fitting#120×C149C4
prime radical#120×D149C2
cosocleC22×D149C24
hypercenter#120#10597010
hypocenterC149C4×#120
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#10597010C2semidirect
2. C2×D4×#10597010C2semidirect
3. #23750573C2semidirect
4. #23750572C2semidirect
5. #23750566C2semidirect
6. C22⋊C4×#10597010C2semidirect
11. C23×#10597010C22semidirect
13. C2×C4⋊#10597010C22semidirect
16. C2×C22⋊#10597010C22semidirect
17. C2×C4×#10597010C22semidirect
18. (C2×#10597010)⋊C4C22semidirect
21. #18011881C4semidirect
22. #18011880C4semidirect
23. #18011879C4semidirect
25. #120×C149C4semidirect
31. C22×#10597010D4semidirect
43. C23×D149C2×C4semidirect
46. C2×D596C2×C4semidirect
47. C2×C22⋊D149C2×C4semidirect
48. C2×C22⋊D149C2×C4semidirect
52. C22⋊Dic149C2×C4semidirect
53. #14558713C2×C4semidirect
54. C24×C149C2×C4semidirect
55. C2×D4×C149C2×C4semidirect
56. C22⋊C4×C149C2×C4semidirect
75. C22×D149C22×C4semidirect
82. C2×Dic149C22×C4semidirect
87. C23×C149C22×C4semidirect
89. C2×C4×C149C22×C4semidirect
93. C22×C149C4×D4semidirect
103. C2×D149C4×D4semidirect
109. #10597010#120direct (N × H)
119. C149C4×#120semidirect
137. #120#10597010direct (N × H)
151. C24C2×#10597010semidirect
157. C2×D4C2×#10597010semidirect
175. C22⋊C4C2×#10597010semidirect
192. C23C22×#10597010semidirect
209. C2×C4C22×#10597010semidirect
224. C22D4×#10597010semidirect

Maximal subgroups

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

subgroupindex
C24×#105970102
C2×D4×#105970102
#237505732
#237505722
#237505662
C22⋊C4×#105970102
#120×D1492
C4×#120149