Finite Group Databasebeta

← #120×#595 #14545247

Subgroups — #120×#595 #14545247

Order 2176.

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×#595C211yesmaximal, characteristic, complemented
2C2×D4×#595C213yesmaximal, complemented
3#11763757C213yesmaximal, complemented
4#11763756C213yesmaximal, complemented
5#11763750C211yesmaximal, characteristic, complemented
6C22⋊C4×#595C213yesmaximal, complemented
7#120×D17C211yesmaximal, characteristic
8C24×D17C2211yescharacteristic
9C23×#595—212no—
10C23×#595C2213yes—
11C23×#595C2213yes—
12C23×#595C2211yescharacteristic, complemented
13C2×D4×D17C2213yes—
14C2×C22⋊#595C2213yes—
15C2×C22⋊#595C2213yes—
16C2×C22⋊#595C2213yescomplemented
17D4×#595—212no—
18C2×C4⋊#595C2213yescomplemented
19C2×C4×#595C2213yescomplemented
20C17×#120C411yescharacteristic, complemented
21#10587639C411yescharacteristic, complemented
22#10587623C413yescomplemented
23#10587568C413yescomplemented
24C22⋊C4×D17C2213yes—
25(C2×#595)⋊C4C2216yescomplemented
26C24×C17C2×C411yescharacteristic, complemented
27C23×D17—26no—
28C23×D17—26no—
29C23×D17C2×C413yescomplemented
30C23×D17C2311yescharacteristic
31C23×D17C2313yes—
32C22×#595—232no—
33C22×#595—26no—
34C22×#595—212no—
35C22×#595—212no—
36C22×#595—26no—
37C22×#595D4112yescomplemented
38C22×#595C2313yes—
39C22×#595C2311yescharacteristic
40C22×#595C2313yes—
41C22×#595C2311yescharacteristic
42C2×D4×C17C2×C413yescomplemented
43C2×C22⋊D17C2×C413yescomplemented
44C2×C22⋊D17C2×C413yescomplemented
45D4×D17—26no—
46D4×D17—26no—
47C2×D68C2×C413yescomplemented
48C2×C4×D17C2313yes—
49C22⋊#595—26no—
50C22⋊#595—26no—
51C22⋊#595—26no—
52C4⋊#595—26no—
53C4×#595—26no—
54C22⋊C4×C17C2×C413yescomplemented
55C22⋊Dic17C2×C413yescomplemented
56#63227C2×C416yescomplemented
57C23×C17—26no—
58C23×C17C22×C413yes—
59C23×C17C22×C411yescharacteristic, complemented
60C22×D17—28no—
61C22×D17—224no—
62C22×D17—26no—
63C22×D17—212no—
64C22×D17—26no—
65C22×D17—26no—
66C22×D17—23no—
67C22×D17—23no—
68C22×D17C22×C413yes—
69C22×D17C22×C411yescharacteristic, complemented
70C22×D17C2×D416yes—
71C22×D17#4116yes—
72C22×D17C2411yescharacteristic
73C2×#595—224no—
74C2×#595—26no—
75C2×#595—224no—
76C2×#595—23no—
77C2×#595—23no—
78C2×#595—26no—
79C2×#595—44no—
80C2×#595C2×D416yes—
81C2×#595C2×D416yes—
82D4×C17—26no—
83C2×C4×C17C22×C413yescomplemented
84C22⋊D17—26no—
85C22⋊D17—26no—
86C2×Dic17C22×C413yescomplemented
87D68—26no—
88C4×D17—23no—
89C4×D17—23no—
90C4×#120—171nomaximal, Sylow
91C22×C17—28no—
92C22×C17—26no—
93C22×C17—23no—
94C22×C17C4×D416yescomplemented
95C22×C17C23×C411yescharacteristic
96C2×D17—224no—
97C2×D17—26no—
98C2×D17—26no—
99C2×D17—26no—
100C2×D17—212no—
101C2×D17—23no—
102C2×D17—41no—
103C2×D17—43no—
104C2×D17C4×D416yescomplemented
105C2×D17C22×D413yes—
106C2×D17C2×#4113yes—
107#595—26no—
108#595—41no—
109#595—43no—
110#595#12014yesdirect product factor
111C4×C17—23no—
112Dic17—23no—
113C24×C4—171no—
114C2×#120—171no—
115C2×C4×D4—173no—
116#386—173no—
117#385—173no—
118#379—171no—
119C4×C22⋊C4—173no—
120C2×C17—26no—
121C2×C17—41no—
122C2×C17C2×C4×D413yes—
123D17—26no—
124D17—41no—
125D17C2×#12011yescharacteristic
126D17#52513yes—
127C25—171no—
128C22×D4—173no—
129C23×C4—173no—
130C23×C4—171no—
131C23×C4—173no—
132C23×C4—3412no—
133#120—171no—
134#120—173no—
135#120—173no—
136#120#59511yescharacteristic, direct product factor
137C4×D4—3412no—
138C2×C4⋊C4—173no—
139C2×C22⋊C4—173no—
140C2×C22⋊C4—173no—
141C2×C22⋊C4—173no—
142C2×C22⋊C4—173no—
143C2×C42—173no—
144#95—176no—
145C17C4×#12011yescharacteristic, complemented, Sylow
146C24—173no—
147C24—173no—
148C24—171no—
149C24—346no—
150C24—346no—
151C24C2×#59511yescharacteristic, complemented
152C2×D4—173no—
153C2×D4—173no—
154C2×D4—173no—
155C2×D4—346no—
156C2×D4—346no—
157C2×D4C2×#59513yescomplemented
158C22×C4—171no—
159C22×C4—1712no—
160C22×C4—173no—
161C22×C4—173no—
162C22×C4—173no—
163C22×C4—171no—
164C22×C4—3412no—
165C22×C4—3412no—
166C22×C4—346no—
167C22×C4—346no—
168C22×C4—3432no—
169C4⋊C4—346no—
170C22⋊C4—173no—
171C22⋊C4—176no—
172C22⋊C4—346no—
173C22⋊C4—346no—
174C22⋊C4—346no—
175C22⋊C4C2×#59513yescomplemented
176C42—346no—
177C23—26no—
178C23—171no—
179C23—176no—
180C23—176no—
181C23—173no—
182C23—171no—
183C23—346no—
184C23—346no—
185C23—3412no—
186C23—343no—
187C23—343no—
188C23—348no—
189C23—3424no—
190C23—346no—
191C23C22×#59513yes—
192C23C22×#59511yescharacteristic, complemented
193D4—26no—
194D4—346no—
195D4—346no—
196D4—346no—
197C2×C4—176no—
198C2×C4—176no—
199C2×C4—173no—
200C2×C4—343no—
201C2×C4—343no—
202C2×C4—3424no—
203C2×C4—346no—
204C2×C4—3424no—
205C2×C4—343no—
206C2×C4—343no—
207C2×C4—346no—
208C2×C4—684no—
209C2×C4C22×#59513yescomplemented
210C22—28no—
211C22—26no—
212C22—23no—
213C22—173no—
214C22—173no—
215C22—176no—
216C22—346no—
217C22—346no—
218C22—3412no—
219C22—3424no—
220C22—346no—
221C22—343no—
222C22—681no—
223C22—683no—
224C22D4×#59516yescomplemented
225C22C23×#59511yescharacteristic
226C4—23no—
227C4—174no—
228C4—346no—
229C4—343no—
230C4—681no—
231C4—683no—
232C2—26no—
233C2—41no—
234C2—171no—
235C2—173no—
236C2—346no—
237C2—681no—
238C2C2×D4×#59513yes—
subgroupG/H
center, FrattiniC22C23×#595
commutator, socleC22×C17C23×C4
FittingC17×#120C4
prime radical#120×D17C2
cosocleC22×D17C24
hypercenter#120#595
hypocenterC17C4×#120
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×#595C2semidirect
2. C2×D4×#595C2semidirect
3. #11763757C2semidirect
4. #11763756C2semidirect
5. #11763750C2semidirect
6. C22⋊C4×#595C2semidirect
12. C23×#595C22semidirect
16. C2×C22⋊#595C22semidirect
18. C2×C4⋊#595C22semidirect
19. C2×C4×#595C22semidirect
20. C17×#120C4semidirect
21. #10587639C4semidirect
22. #10587623C4semidirect
23. #10587568C4semidirect
25. (C2×#595)⋊C4C22semidirect
26. C24×C17C2×C4semidirect
29. C23×D17C2×C4semidirect
37. C22×#595D4semidirect
42. C2×D4×C17C2×C4semidirect
43. C2×C22⋊D17C2×C4semidirect
44. C2×C22⋊D17C2×C4semidirect
47. C2×D68C2×C4semidirect
54. C22⋊C4×C17C2×C4semidirect
55. C22⋊Dic17C2×C4semidirect
56. #63227C2×C4semidirect
59. C23×C17C22×C4semidirect
69. C22×D17C22×C4semidirect
83. C2×C4×C17C22×C4semidirect
86. C2×Dic17C22×C4semidirect
94. C22×C17C4×D4semidirect
104. C2×D17C4×D4semidirect
110. #595#120direct (N × H)
136. #120#595direct (N × H)
145. C17C4×#120semidirect
151. C24C2×#595semidirect
157. C2×D4C2×#595semidirect
175. C22⋊C4C2×#595semidirect
192. C23C22×#595semidirect
209. C2×C4C22×#595semidirect
224. C22D4×#595semidirect

Maximal subgroups

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

subgroupindex
C24×#5952
C2×D4×#5952
#117637572
#117637562
#117637502
C22⋊C4×#5952
#120×D172
C4×#12017