Finite Group Databasebeta

← #17952568

Subgroups — #17952568

Order 4352.

802 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 132 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
1#14544640C212yesmaximal
2C2×#11763668C214yesmaximal, complemented
3C2×#11763362C211yesmaximal, characteristic
4C2×#10587544C2214yes—
5C2×C22⋊#3624—22no—
6#11763675—22no—
7#11763668—28no—
8#11763362—28no—
9C2×#10587574C2212yes—
10C2×#10587576C412yes—
11C2×#10587564—22no—
12C2×C4×Dic34—22no—
13C2×#10587612C411yescharacteristic
14C22×C4×Dic17C2211yescharacteristic
15C2×C17×#117C411yescharacteristic
16C23×C4×C17C2×C411yescharacteristic
17C23×Dic17D412yes—
18C22×Dic34—22no—
19C22×#3624—24no—
20C17×#117—24no—
21C2×C4⋊C4×C17C2×C412yes—
22C2×C22⋊C4×C17—21no—
23C2×C42×C17—21no—
24C2×C22⋊Dic17—21no—
25#10587612—24no—
26C2×C4⋊Dic17—22no—
27C2×C4⋊Dic17C2×C412yes—
28C2×#63225—22no—
29C2×#63225—22no—
30C2×C4×Dic17—24no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17D412yes—
34C2×C4×Dic17C2311yescharacteristic
35#10587576—24no—
36#10587576—24no—
37#10587574—24no—
38#10587574D418yes—
39#10587564—216no—
40C4×Dic34—216no—
41C22⋊#3624—44no—
42#10587544—28no—
43C24×C17C22⋊C411yescharacteristic
44C22×C4×C17—22no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17C22⋊C411yescharacteristic
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—22no—
51C22×Dic17—24no—
52C22×Dic17—24no—
53C22×Dic17—22no—
54C22×Dic17—22no—
55C22×Dic17C22⋊C412yes—
56C22×Dic17C2×D412yes—
57C2×Dic34—28no—
58C2×Dic34—44no—
59C2×Dic34—42no—
60C2×Dic34—42no—
61C2×#3624—44no—
62C2×#3624—44no—
63C2×#3624—44no—
64C4⋊C4×C17—22no—
65C4⋊C4×C17C22⋊C414yes—
66C22⋊C4×C17—24no—
67C42×C17—24no—
68C22⋊Dic17—24no—
69C4⋊Dic17—28no—
70C4⋊Dic17—22no—
71C4⋊Dic17C22⋊C414yes—
72#63225—28no—
73#63225—28no—
74C4×Dic17—24no—
75C4×Dic17—22no—
76C4×Dic17—24no—
77C4×Dic17C2×D414yes—
78#8345—171nomaximal, Sylow
79C23×C17—21no—
80C23×C17—21no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17#10112yes—
84C23×C17C2×C22⋊C411yescharacteristic
85C2×C4×C17—28no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—24no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17#10112yes—
94C2×C4×C17C2×C22⋊C412yes—
95C2×Dic17—24no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—42no—
100C2×Dic17—42no—
101C2×Dic17—44no—
102C2×Dic17—44no—
103C2×Dic17#12014yes—
104C2×Dic17#13514yes—
105C2×Dic17AGL(1,ℤ/8)12yes—
106C2×Dic17#13712yes—
107Dic34—48no—
108Dic34—48no—
109#3624—84no—
110C2×#533—171no—
111#1797—172no—
112C2×#332—174no—
113C22×C17—24no—
114C22×C17—22no—
115C22×C17—41no—
116C22×C17—41no—
117C22×C17—42no—
118C22×C17#37911yescharacteristic
119C22×C17C2×#10111yescharacteristic
120C22×C17C2×#10111yescharacteristic
121C22×C17#41612yes—
122C22×C17#41712yes—
123C4×C17—22no—
124C4×C17—22no—
125C4×C17—44no—
126Dic17—44no—
127Dic17—42no—
128Dic17#44914yes—
129Dic17#45214yes—
130#533—348no—
131C2×C4⋊Q8—172no—
132C2×#125—172no—
133C2×C22⋊Q8—342no—
134C2×C4×Q8—342no—
135C2×#117—171no—
136C2×#117#59511yescharacteristic
137C22×C42—171no—
138C2×C4⋊C8—174no—
139C2×C22⋊C8—342no—
140#336—342no—
141#332—348no—
142C2×C17—42no—
143C2×C17#154411yescharacteristic
144C2×C17#155212yes—
145C2×C17#156111yescharacteristic
146C2×C17#178411yescharacteristic
147C2×C17#178812yes—
148C22×Q8—342no—
149C23×C4—172no—
150C23×C4C2×#59511yescharacteristic
151C22×C8—344no—
152C4⋊Q8—178no—
153C4⋊Q8—344no—
154#125—344no—
155#125—344no—
156C22⋊Q8—3416no—
157C4×Q8—3416no—
158#117—24no—
159#117—344no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4—342no—
162C2×C4⋊C4—342no—
163C2×C4⋊C4—342no—
164C2×C4⋊C4C2×#59512yes—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—341no—
167C2×C42—21no—
168C2×C42—171no—
169C2×C42—172no—
170C2×C42—171no—
171C2×C42—344no—
172C2×C42—341no—
173C4⋊C8—348no—
174C22⋊C8—684no—
175C17#834511yescharacteristic, complemented, Sylow
176C24C22⋊#59511yescharacteristic
177C2×Q8—348no—
178C2×Q8—684no—
179C2×Q8—682no—
180C2×Q8—682no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—172no—
186C22×C4—172no—
187C22×C4—342no—
188C22×C4—344no—
189C22×C4—344no—
190C22×C4—342no—
191C22×C4—342no—
192C22×C4C22⋊#59511yescharacteristic
193C22×C4C22×#59511yescharacteristic
194C2×C8—684no—
195C2×C8—684no—
196C2×C8—684no—
197C4⋊C4—22no—
198C4⋊C4—174no—
199C4⋊C4—348no—
200C4⋊C4—348no—
201C4⋊C4—348no—
202C4⋊C4—342no—
203C4⋊C4C22⋊#59514yes—
204C22⋊C4—24no—
205C22⋊C4—344no—
206C42—24no—
207C42—174no—
208C42—344no—
209C42—344no—
210C42—342no—
211C23—21no—
212C23—21no—
213C23—22no—
214C23—22no—
215C23#1058754512yes—
216C23C2×C22⋊#59511yescharacteristic
217Q8—688no—
218Q8—688no—
219C2×C4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—24no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—174no—
228C2×C4—174no—
229C2×C4—172no—
230C2×C4—172no—
231C2×C4—342no—
232C2×C4—344no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—684no—
236C2×C4—684no—
237C2×C4—682no—
238C2×C4—682no—
239C2×C4#1058754512yes—
240C2×C4C2×C22⋊#59512yes—
241C8—1364no—
242C22—24no—
243C22—22no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22#1176375011yescharacteristic
248C22C2×#1058754511yescharacteristic
249C22C2×#1058754511yescharacteristic
250C22#1176379912yes—
251C22#1176380012yes—
252C4—22no—
253C4—22no—
254C4—44no—
255C4—174no—
256C4—174no—
257C4—682no—
258C4—684no—
259C2—42no—
260C2#1454428911yescharacteristic
261C2#1454430512yes—
262C2#1454432311yescharacteristic
263C2#1454461811yescharacteristic
264C2#1454462412yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#117#595
hypocenterC17#8345
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#11763668C2semidirect
175. C17#8345semidirect

Maximal subgroups

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

subgroupindex
#145446402
C2×#117636682
C2×#117633622
#834517