Finite Group Databasebeta

← #16949833

Subgroups — #16949833

Order 4352.

1018 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#11762608C214yesmaximal, complemented
2#14528708C212yesmaximal
3C22×C17×C4⋊Q8C211yesmaximal, characteristic
4C2×#10587507—22no—
5C2×#10587507C2214yes—
6#11762773—22no—
7#11762608—28no—
8C23×Q8×C17C411yescharacteristic
9C23×Q8×C17C2211yescharacteristic
10C2×C17×C4⋊Q8—22no—
11C2×C17×C4⋊Q8—28no—
12C2×C17×C4⋊Q8C412yes—
13C2×C17×C4⋊Q8C2212yes—
14C22×C4⋊C4×C17—22no—
15C22×C42×C17C411yescharacteristic
16C22×Q8×C17—24no—
17C22×Q8×C17—24no—
18C22×Q8×C17—21no—
19C22×Q8×C17—42no—
20C22×Q8×C17—42no—
21C22×Q8×C17—42no—
22C22×Q8×C17—42no—
23C22×Q8×C17C2×C412yes—
24C22×Q8×C17C2×C411yescharacteristic
25C22×Q8×C17D412yes—
26C22×Q8×C17C2311yescharacteristic
27C23×C4×C17—21no—
28C23×C4×C17—21no—
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17D412yes—
31C17×C4⋊Q8—216no—
32C17×C4⋊Q8—24no—
33C17×C4⋊Q8—24no—
34C17×C4⋊Q8—24no—
35C17×C4⋊Q8D418yes—
36C2×C4⋊C4×C17—216no—
37C2×C4⋊C4×C17—24no—
38C2×C4⋊C4×C17—24no—
39C2×C42×C17—21no—
40C2×C42×C17—24no—
41C2×C42×C17C2×C412yes—
42C2×#63223—24no—
43#10587507—28no—
44#10587507—44no—
45C22⋊#3622—44no—
46C24×C17C22⋊C411yescharacteristic
47C2×Q8×C17—24no—
48C2×Q8×C17—24no—
49C2×Q8×C17—22no—
50C2×Q8×C17—22no—
51C2×Q8×C17—24no—
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17—42no—
55C2×Q8×C17—41no—
56C2×Q8×C17—41no—
57C2×Q8×C17—416no—
58C2×Q8×C17—416no—
59C2×Q8×C17—42no—
60C2×Q8×C17—42no—
61C2×Q8×C17—42no—
62C2×Q8×C17—42no—
63C2×Q8×C17C22⋊C414yes—
64C2×Q8×C17C2×D414yes—
65C22×C4×C17—28no—
66C22×C4×C17—24no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—24no—
72C22×C4×C17—24no—
73C22×C4×C17—22no—
74C22×C4×C17—24no—
75C22×C4×C17—42no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17C22⋊C412yes—
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D412yes—
82C4⋊C4×C17—216no—
83C4⋊C4×C17—216no—
84C42×C17—24no—
85C42×C17—22no—
86C42×C17C22⋊C414yes—
87#63223—44no—
88#63223—44no—
89C2×#3622—44no—
90#12700—171nomaximal, Sylow
91C23×C17—24no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17#9912yes—
95C23×C17C2×C22⋊C411yescharacteristic
96Q8×C17—44no—
97Q8×C17—48no—
98Q8×C17—44no—
99Q8×C17—48no—
100Q8×C17—44no—
101Q8×C17—44no—
102C2×C4×C17—216no—
103C2×C4×C17—28no—
104C2×C4×C17—28no—
105C2×C4×C17—21no—
106C2×C4×C17—24no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—22no—
112C2×C4×C17—24no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—48no—
117C2×C4×C17—48no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17#9912yes—
121C2×C4×C17C2×C22⋊C412yes—
122C2×C4×C17#12014yes—
123#3622—84no—
124C22×C4⋊Q8D1711yescharacteristic
125C2×#356—174no—
126#1784—172no—
127C22×C17—24no—
128C22×C17—24no—
129C22×C17—22no—
130C22×C17—22no—
131C22×C17—22no—
132C22×C17#35614yescomplemented
133C22×C17#37911yescharacteristic
134C22×C17C2×#9911yescharacteristic
135C22×C17C2×#9911yescharacteristic
136C4×C17—28no—
137C4×C17—24no—
138C4×C17—22no—
139C4×C17—22no—
140C4×C17—42no—
141C4×C17—44no—
142C23×Q8Dic1711yescharacteristic
143C23×Q8C2×D1711yescharacteristic
144C2×C4⋊Q8—22no—
145C2×C4⋊Q8—28no—
146C2×C4⋊Q8Dic1712yes—
147C2×C4⋊Q8C2×D1712yes—
148C22×C4⋊C4—22no—
149C22×C42Dic1711yescharacteristic
150C2×#101—174no—
151C2×#101—342no—
152#407—342no—
153#356—348no—
154C2×C17—21no—
155C2×C17—24no—
156C2×C17—21no—
157C2×C17#178111yescharacteristic
158C2×C17C2×#35612yes—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—21no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8C2×Dic1712yes—
167C22×Q8C2×Dic1711yescharacteristic
168C22×Q8C22⋊D1712yes—
169C22×Q8C22×D1711yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4C2×Dic1711yescharacteristic
173C23×C4C22⋊D1712yes—
174C2×M16—344no—
175C4⋊Q8—216no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179C4⋊Q8C22⋊D1718yes—
180C2×C4⋊C4—216no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C42—21no—
184C2×C42—24no—
185C2×C42C2×Dic1712yes—
186#101—348no—
187#101—684no—
188C22⋊C8—684no—
189C17#1270011yescharacteristic, complemented, Sylow
190C24C22⋊Dic1711yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—24no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—416no—
202C2×Q8—416no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22⋊Dic1714yes—
208C2×Q8C2×C22⋊D1714yes—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C22⋊Dic1712yes—
223C22×C4C22⋊Dic1711yescharacteristic
224C22×C4C22×Dic1711yescharacteristic
225C22×C4C2×C22⋊D1712yes—
226M16—684no—
227M16—684no—
228C2×C8—684no—
229C4⋊C4—216no—
230C4⋊C4—216no—
231C42—24no—
232C42—22no—
233C42C22⋊Dic1714yes—
234C23—21no—
235C23—24no—
236C23—21no—
237C23#1058750512yes—
238C23C2×C22⋊Dic1711yescharacteristic
239Q8—44no—
240Q8—48no—
241Q8—44no—
242Q8—48no—
243Q8—44no—
244Q8—44no—
245C2×C4—28no—
246C2×C4—216no—
247C2×C4—28no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4#1058750512yes—
264C2×C4C2×C22⋊Dic1712yes—
265C2×C4#1058763914yes—
266C8—1364no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22#1176260814yescomplemented
273C22#1176266511yescharacteristic
274C22C2×#1058750511yescharacteristic
275C22C2×#1058750511yescharacteristic
276C4—28no—
277C4—24no—
278C4—22no—
279C4—22no—
280C4—42no—
281C4—44no—
282C2—21no—
283C2—21no—
284C2—24no—
285C2#1452869911yescharacteristic
286C2C2×#1176260812yes—
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC22×C17×C4⋊Q8C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC22×Q8×C17C23
hypercenterC22×C4⋊Q8D17
hypocenterC17#12700
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11762608C2semidirect
132. C22×C17#356semidirect
189. C17#12700semidirect
272. C22#11762608semidirect

Maximal subgroups

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

subgroupindex
C2×#117626082
#145287082
C22×C17×C4⋊Q82
#1270017