Finite Group Databasebeta

← C23×#10587550 #17979512

Subgroups — C23×#10587550 #17979512

Order 4352.

5758 conjugacy classes of subgroups; only the 1208 normal subgroups are shown.

normal subgroups only

This group has 5758 conjugacy classes of subgroups — too many to draw. Only its 1208 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
1C23×#63243C211yesmaximal, characteristic, complemented
2C23×C4×#595C211yesmaximal, characteristic, complemented
3C22×#10587550C2128yesmaximal, direct product factor
4C23×#63256C211yesmaximal, characteristic
5C24×#595—21no—
6C22×#63243—27no—
7C22×#63243C22114yescomplemented
8C23×#3624—21no—
9C22×C4×#595—27no—
10C22×C4×#595C22114yescomplemented
11C2×#10587550C221112yesdirect product factor
12C22×#63256—27no—
13C22×#63256—27no—
14C22×#63256C4114yescomplemented
15C22×#63256C22114yes—
16C23×D68—21no—
17C23×D68C411yescharacteristic, complemented
18C23×C4×D17—21no—
19C23×C4×D17C2211yescharacteristic
20C23×Q8×C17C411yescharacteristic, complemented
21C24×D17—21no—
22C24×D17D411yescharacteristic
23C23×#595—216no—
24C23×#595—214no—
25C22×Q8×C17—27no—
26C22×Q8×C17C2×C4114yescomplemented
27C23×C4×C17—21no—
28C23×C4×C17C2×C411yescharacteristic
29C23×Dic17D411yescharacteristic
30C2×#63256—284no—
31C2×#63256—242no—
32C2×#63256—242no—
33C2×#63256C2×C4184yescomplemented
34C2×#63256C23128yes—
35C22×D68—214no—
36C22×D68—214no—
37C22×D68—27no—
38C22×D68C2×C4114yescomplemented
39C22×C4×D17—214no—
40C22×C4×D17—214no—
41C22×C4×D17C2×C417yes—
42C22×C4×D17D417yes—
43C22×C4×D17D417yes—
44C22×C4×D17C2317yes—
45C22×#3624—214no—
46C2×C4×#595—242no—
47C2×C4×#595C23128yescomplemented
48C2×#63243—242no—
49C2×#63243C23128yescomplemented
50#10587550C23164yesdirect product factor
51C24×C17C22⋊C411yescharacteristic
52C23×D17—28no—
53C23×D17—214no—
54C23×D17—48no—
55C23×D17C22⋊C417yes—
56C23×D17C2×D417yes—
57C22×#595—2112no—
58C22×#595—228no—
59C2×Q8×C17—242no—
60C2×Q8×C17C22×C4128yescomplemented
61C22×C4×C17—214no—
62C22×C4×C17C22⋊C417yes—
63C22×C4×C17C22×C417yes—
64C22×Dic17C22⋊C417yes—
65C22×Dic17C2×D417yes—
66#63256—2168no—
67#63256—228no—
68#63256—228no—
69#63256C22×C4156yescomplemented
70#63256C2418yes—
71C2×D68—284no—
72C2×D68—228no—
73C2×D68—242no—
74C2×D68C22×C4128yescomplemented
75C2×C4×D17—284no—
76C2×C4×D17—228no—
77C2×C4×D17C22⋊C4142yes—
78C2×C4×D17C22×C4121yes—
79C2×C4×D17C2×D4121yes—
80C2×C4×D17C2×D4121yes—
81C2×C4×D17C2417yes—
82C2×#3624—228no—
83C4×#595—228no—
84C4×#595C2418yescomplemented
85#63243—228no—
86#63243C2418yescomplemented
87C23×#104—171nomaximal, Sylow
88C23×C17#10418yescomplemented
89C23×C17C2×C22⋊C417yes—
90C22×D17—228no—
91C22×D17—228no—
92C22×D17—228no—
93C22×D17—456no—
94C22×D17C2×C22⋊C4121yes—
95C22×D17C22×D417yes—
96C2×#595—2112no—
97C2×#595—28no—
98Q8×C17—228no—
99Q8×C17C23×C418yescomplemented
100C2×C4×C17—228no—
101C2×C4×C17C2×C22⋊C4121yes—
102C2×C4×C17C23×C417yes—
103C2×Dic17C2×C22⋊C4121yes—
104C2×Dic17C22×D417yes—
105D68—256no—
106D68—28no—
107D68—228no—
108D68C23×C418yescomplemented
109C4×D17—256no—
110C4×D17—28no—
111C4×D17C2×C22⋊C4142yes—
112C4×D17C23×C417yes—
113C4×D17C22×D417yes—
114C4×D17C22×D417yes—
115C4×D17C2511yescharacteristic
116#3624—28no—
117C23×#41—171no—
118C23×M16—171no—
119C23×C42—171no—
120C22×#104—1728no—
121C22×C17C2×#104128yescomplemented
122C22×C17C22×C22⋊C417yes—
123C2×D17—214no—
124C2×D17—242no—
125C2×D17—28no—
126C2×D17—456no—
127C2×D17C22×C22⋊C417yes—
128C2×D17C23×D411yescharacteristic
129#595—216no—
130C4×C17—28no—
131C4×C17C22×C22⋊C417yes—
132C4×C17C24×C411yescharacteristic
133Dic17C22×C22⋊C417yes—
134Dic17C23×D411yescharacteristic
135C22×#41—1714no—
136C22×#41—1714no—
137C22×#41—347no—
138C22×#41—347no—
139C23×Q8#59511yescharacteristic, complemented
140C23×D4—171no—
141C23×D4—341no—
142C24×C4—171no—
143C24×C4—341no—
144C24×C4—341no—
145C22×M16—1714no—
146C22×M16—347no—
147C23×C8—341no—
148C22×C42—1714no—
149C22×C42—347no—
150C2×#104—17112no—
151C2×C17C22×#104114yescomplemented
152C2×C17C23×C22⋊C411yescharacteristic
153D17—21no—
154D17—27no—
155D17—48no—
156C25—171no—
157C25—341no—
158C2×#41—1784no—
159C2×#41—1728no—
160C2×#41—3484no—
161C2×#41—3442no—
162C2×#41—3442no—
163C22×Q8—27no—
164C22×Q8C2×#595114yescomplemented
165C22×D4—1714no—
166C22×D4—3414no—
167C22×D4—3414no—
168C22×D4—347no—
169C23×C4—21no—
170C23×C4—171no—
171C23×C4—177no—
172C23×C4—177no—
173C23×C4—177no—
174C23×C4—177no—
175C23×C4—3414no—
176C23×C4—3414no—
177C23×C4—3414no—
178C23×C4—3416no—
179C23×C4C2×#59511yescharacteristic
180C2×M16—1728no—
181C2×M16—3442no—
182C22×C8—3414no—
183C2×C42—1728no—
184C2×C42—3442no—
185#104—1764no—
186C17C23×#10411yescharacteristic, complemented, Sylow
187C24—177no—
188C24—177no—
189C24—348no—
190C24—3414no—
191C24—688no—
192C24C22⋊#59511yescharacteristic
193#41—1756no—
194#41—178no—
195#41—3428no—
196#41—3428no—
197#41—34168no—
198C2×Q8—242no—
199C2×Q8C22×#595128yescomplemented
200C2×D4—1728no—
201C2×D4—3428no—
202C2×D4—3484no—
203C2×D4—3442no—
204C22×C4—214no—
205C22×C4—177no—
206C22×C4—177no—
207C22×C4—1742no—
208C22×C4—1721no—
209C22×C4—1721no—
210C22×C4—1721no—
211C22×C4—177no—
212C22×C4—3428no—
213C22×C4—3484no—
214C22×C4—3428no—
215C22×C4—34112no—
216C22×C4C22⋊#59517yes—
217C22×C4C22×#59517yes—
218M16—178no—
219M16—3428no—
220C2×C8—3428no—
221C42—178no—
222C42—3428no—
223C23—177no—
224C23—1721no—
225C23—3428no—
226C23—3428no—
227C23—3428no—
228C23—6856no—
229C23#1058755018yesdirect product factor
230C23C2×C22⋊#59517yes—
231Q8—228no—
232Q8C23×#59518yescomplemented
233D4—178no—
234D4—348no—
235D4—3456no—
236D4—3428no—
237C2×C4—228no—
238C2×C4—177no—
239C2×C4—1721no—
240C2×C4—177no—
241C2×C4—177no—
242C2×C4—1742no—
243C2×C4—171no—
244C2×C4—177no—
245C2×C4—348no—
246C2×C4—3456no—
247C2×C4—348no—
248C2×C4—34112no—
249C2×C4C2×C22⋊#595121yes—
250C2×C4C23×#59517yes—
251C8—348no—
252C22—171no—
253C22—177no—
254C22—3414no—
255C22—3442no—
256C22—348no—
257C22—6856no—
258C22C2×#10587550128yesdirect product factor
259C22C22×C22⋊#59517yes—
260C4—28no—
261C4—171no—
262C4—177no—
263C4—3416no—
264C4C22×C22⋊#59517yes—
265C4C24×#59511yescharacteristic
266C2—341no—
267C2—347no—
268C2—688no—
269C2C22×#10587550114yesdirect product factor
270C2C23×C22⋊#59511yescharacteristic
subgroupG/H
centerC24C22⋊#595
commutatorC4×C17C24×C4
FrattiniC4C24×#595
FittingC23×Q8×C17C4
socleC24×C17C22⋊C4
prime radicalC23×#63256C2
cosocleC4×D17C25
hypercenterC23×Q8#595
hypocenterC17C23×#104
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#63243C2semidirect
2. C23×C4×#595C2semidirect
3. C22×#10587550C2direct (N × H)
7. C22×#63243C22semidirect
10. C22×C4×#595C22semidirect
11. C2×#10587550C22direct (N × H)
14. C22×#63256C4semidirect
17. C23×D68C4semidirect
20. C23×Q8×C17C4semidirect
26. C22×Q8×C17C2×C4semidirect
33. C2×#63256C2×C4semidirect
38. C22×D68C2×C4semidirect
47. C2×C4×#595C23semidirect
49. C2×#63243C23semidirect
50. #10587550C23direct (N × H)
60. C2×Q8×C17C22×C4semidirect
69. #63256C22×C4semidirect
74. C2×D68C22×C4semidirect
84. C4×#595C24semidirect
86. #63243C24semidirect
88. C23×C17#104semidirect
99. Q8×C17C23×C4semidirect
108. D68C23×C4semidirect
121. C22×C17C2×#104semidirect
139. C23×Q8#595semidirect
151. C2×C17C22×#104semidirect
164. C22×Q8C2×#595semidirect
186. C17C23×#104semidirect
199. C2×Q8C22×#595semidirect
229. C23#10587550direct (N × H)
232. Q8C23×#595semidirect
258. C22C2×#10587550direct (N × H)
269. C2C22×#10587550direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#632432
C23×C4×#5952
C22×#105875502
C23×#632562
C23×#10417