Finite Group Databasebeta

← #16949514

Subgroups — #16949514

Order 4352.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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×#11762606C212yesmaximal
2C4×#10587507C212yesmaximal, complemented
3#14526911C212yesmaximal, complemented
4C17×#2326C211yesmaximal, characteristic
5C2×#10587507C2214yes—
6C4×#63223—22no—
7#11762606C418yescomplemented
8#11762578—22no—
9C2×C17×#124C2211yescharacteristic
10C2×C4×Q8×C17C2211yescharacteristic
11C2×C4×D4×C17C411yescharacteristic
12C17×#391C2211yescharacteristic
13C17×#390C411yescharacteristic
14C17×#388—22no—
15C17×#385—22no—
16C17×#384C411yescharacteristic
17C17×#376C411yescharacteristic
18C22×Q8×C17C2311yescharacteristic
19C22×D4×C17C2×C411yescharacteristic
20C23×C4×C17—21no—
21C17×#124—22no—
22C17×#124C2×C414yes—
23C4×Q8×C17—42no—
24C4×D4×C17—42no—
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C4⋊C4×C17D412yes—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17—21no—
31C2×C22⋊C4×C17—22no—
32C2×C42×C17C2×C411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17D412yes—
36C2×#63223—24no—
37C17×#95—21no—
38C17×#95—21no—
39C17×#95—22no—
40#10587507—24no—
41#10587507C2×C418yescomplemented
42#10587474—42no—
43C4×#3622—42no—
44C24×C17—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17C22×C412yes—
48C2×Q8×C17#4112yes—
49C2×D4×C17—21no—
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17C4212yes—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—42no—
57C22×C4×C17—82no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C412yes—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D412yes—
63C4⋊C4×C17—41no—
64C4⋊C4×C17—42no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—82no—
67C22⋊C4×C17—28no—
68C22⋊C4×C17—42no—
69C22⋊C4×C17—82no—
70C42×C17—22no—
71C42×C17—22no—
72C42×C17—22no—
73C42×C17—21no—
74C42×C17—41no—
75C42×C17C4212yes—
76#63223—28no—
77#63223—44no—
78C2×#3622—44no—
79#12632—171nomaximal, Sylow
80C23×C17—22no—
81C23×C17—42no—
82C23×C17—42no—
83C23×C17—41no—
84C23×C17—41no—
85C23×C17C2×C22⋊C411yescharacteristic
86Q8×C17—42no—
87Q8×C17—42no—
88D4×C17—42no—
89D4×C17—42no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17—42no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—42no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—84no—
108C2×C4×C17#9912yes—
109C2×C4×C17C2×C4211yescharacteristic
110C2×C4×C17#11711yescharacteristic
111C2×C4×C17C4×D414yes—
112#3622—48no—
113#2326D1711yescharacteristic
114C2×#355—172no—
115C4×#101—172no—
116#1394—172no—
117C22×C17—21no—
118C22×C17—21no—
119C22×C17—44no—
120C22×C17—42no—
121C22×C17—82no—
122C22×C17—81no—
123C22×C17—81no—
124C22×C17C4×C22⋊C411yescharacteristic
125C22×C17C2×#9911yescharacteristic
126C22×C17#41011yescharacteristic
127C4×C17—22no—
128C4×C17—24no—
129C4×C17—21no—
130C4×C17—21no—
131C4×C17—42no—
132C4×C17—42no—
133C4×C17—41no—
134C4×C17—81no—
135C4×C17—81no—
136C2×#124C2×D1711yescharacteristic
137C2×C4×Q8C2×D1711yescharacteristic
138C2×C4×D4Dic1711yescharacteristic
139C2×#101—174no—
140C4×M16—342no—
141#391C2×D1711yescharacteristic
142#390Dic1711yescharacteristic
143#388—22no—
144#385—22no—
145#384Dic1711yescharacteristic
146#376Dic1711yescharacteristic
147#355—178no—
148#343—342no—
149C2×C17—21no—
150C2×C17—21no—
151C2×C17—82no—
152C2×C17C4×#9911yescharacteristic
153C2×C17#213311yescharacteristic
154C2×C17#213411yescharacteristic
155C22×Q8C22×D1711yescharacteristic
156C22×D4C2×Dic1711yescharacteristic
157C23×C4—21no—
158C2×M16—344no—
159#124—22no—
160#124C4×D1714yes—
161C4×Q8—42no—
162C4×D4—42no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4C2×Dic1711yescharacteristic
165C2×C4⋊C4C2×Dic1711yescharacteristic
166C2×C4⋊C4C22⋊D1712yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C42C2×Dic1711yescharacteristic
171C2×C42C2×Dic1711yescharacteristic
172C2×C42C2×Dic1711yescharacteristic
173C2×C42C22⋊D1712yes—
174#101—178no—
175#101—344no—
176#97—682no—
177C4×C8—682no—
178#95—21no—
179#95—21no—
180#95—22no—
181C17#1263211yescharacteristic, complemented, Sylow
182C24—21no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8C2×C4×D1712yes—
186C2×Q8#6325212yes—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4C4×Dic1712yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—82no—
196C22×C4C22⋊Dic1711yescharacteristic
197C22×C4C22⋊Dic1712yes—
198C22×C4C22⋊Dic1711yescharacteristic
199C22×C4C22×Dic1711yescharacteristic
200C22×C4C2×C22⋊D1712yes—
201M16—348no—
202M16—684no—
203C2×C8—684no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—82no—
208C22⋊C4—28no—
209C22⋊C4—42no—
210C22⋊C4—82no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—21no—
215C42—41no—
216C42C4×Dic1712yes—
217C23—22no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23C2×C22⋊Dic1711yescharacteristic
223Q8—42no—
224Q8—42no—
225D4—42no—
226D4—42no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—84no—
245C2×C4#1058750512yes—
246C2×C4C2×C4×Dic1711yescharacteristic
247C2×C4#1058761211yescharacteristic
248C2×C4C4×C22⋊D1714yes—
249C8—688no—
250C22—21no—
251C22—21no—
252C22—44no—
253C22—42no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C4×C22⋊Dic1711yescharacteristic
258C22C2×#1058750511yescharacteristic
259C22#1176278311yescharacteristic
260C4—24no—
261C4—21no—
262C4—21no—
263C4—22no—
264C4—42no—
265C4—41no—
266C4—42no—
267C4—81no—
268C4—81no—
269C2—21no—
270C2—21no—
271C2—82no—
272C2C4×#1058750511yescharacteristic
273C2#1453047311yescharacteristic
274C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC2×C4×C17C2×C42
FrattiniC22×Q8C22×D17
FittingC17×#2326C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC22×Q8×C17C23
hypercenter#2326D17
hypocenterC17#12632
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C4×#10587507C2semidirect
3. #14526911C2semidirect
7. #11762606C4semidirect
41. #10587507C2×C4semidirect
181. C17#12632semidirect

Maximal subgroups

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

subgroupindex
C2×#117626062
C4×#105875072
#145269112
C17×#23262
#1263217