Finite Group Databasebeta

← #14534722

Subgroups — #14534722

Order 2176.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587630C211yesmaximal, characteristic, complemented
2C2×#10587571C211yesmaximal, characteristic
3#11762738C212yesmaximal, complemented
4#11762737C212yesmaximal, complemented
5#11762722C211yesmaximal, characteristic, complemented
6#11762721C211yesmaximal, characteristic, complemented
7#11762720C211yesmaximal, characteristic, complemented
8#11762713C212yesmaximal, complemented
9#11762709C212yesmaximal
10#11762654C211yesmaximal, characteristic, complemented
11C17×#393C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20#10587630—24no—
21C2×#63227C2211yescharacteristic
22C2×#63227C2211yescharacteristic
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587571—24no—
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C2×Q8×C17—22no—
40C2×Q8×C17—42no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—22no—
46C2×C22⋊D17—24no—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C2×Dic34—22no—
53C2×Dic34—22no—
54C2×Dic34—24no—
55C2×Dic34—42no—
56C22⋊C4×C17—22no—
57C22⋊C4×C17—42no—
58C22⋊Dic17—22no—
59C22⋊Dic17—42no—
60#63227—22no—
61#63227—22no—
62#63227—42no—
63#63227—42no—
64#63225—24no—
65#63225—24no—
66C4×Dic17—24no—
67C4×Dic17D414yes—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—21no—
72C23×C17—21no—
73C23×C17C2411yescharacteristic
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—21no—
78C22×D17—21no—
79Q8×C17—44no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17C2×D412yes—
90C22⋊D17—44no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103C2×Dic17#4114yes—
104Dic34—44no—
105Dic34—44no—
106#2640—171nomaximal, Sylow
107C22×C17—41no—
108C22×C17—41no—
109C22×C17—41no—
110C22×C17—42no—
111C22×C17—42no—
112C22×C17C22×D411yescharacteristic
113C22×C17C2×#4112yes—
114C22×C17H(5,𝔽2)12yes—
115C22×C172−1+411yescharacteristic
116C22×C172−1+411yescharacteristic
117C2×D17—41no—
118C2×D17—41no—
119C2×D17—41no—
120C2×D17—42no—
121C2×D17—42no—
122C4×C17—22no—
123C4×C17—41no—
124C4×C17—42no—
125C4×C17—42no—
126Dic17—22no—
127Dic17—41no—
128Dic17—42no—
129Dic17—42no—
130Dic17#12414yes—
131C2×C4⋊Q8—171no—
132C2×#124—171no—
133#397—172no—
134#397—172no—
135#393—171no—
136#393—171no—
137#393—171no—
138#393D1711yescharacteristic, complemented
139#391—172no—
140#390—172no—
141#376—171no—
142C2×C17—81no—
143C2×C17C2×#12411yescharacteristic
144C2×C17#53511yescharacteristic
145C2×C17#53611yescharacteristic
146C2×C17#54212yes—
147C2×C17#54212yes—
148D17—81no—
149C22×Q8—171no—
150C22×Q8—171no—
151C22×Q8C2×D1711yescharacteristic
152C22×D4—171no—
153C4⋊Q8—344no—
154#124—344no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—172no—
157C2×C22⋊C4—171no—
158C2×C22⋊C4—171no—
159C2×C22⋊C4—171no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4C2×D1711yescharacteristic
164C2×C22⋊C4C2×D1712yes—
165C2×C42—171no—
166C2×C42—172no—
167#95—171no—
168#95—172no—
169#95—172no—
170#95—172no—
171#95—172no—
172#95C2×D1711yescharacteristic
173#95C2×D1712yes—
174C17#264011yescharacteristic, complemented, Sylow
175C24—171no—
176C24C22×D1711yescharacteristic
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—342no—
180C2×Q8—344no—
181C2×Q8—342no—
182C2×Q8—682no—
183C2×D4—342no—
184C2×D4—344no—
185C22×C4—171no—
186C22×C4—171no—
187C22×C4—172no—
188C22×C4—171no—
189C22×C4—172no—
190C22×C4C22×D1712yes—
191C22×C4C22×D1711yescharacteristic
192C22×C4C22×D1712yes—
193C22×C4C22×D1711yescharacteristic
194C4⋊C4—344no—
195C4⋊C4—344no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—342no—
199C22⋊C4—342no—
200C22⋊C4—342no—
201C22⋊C4—682no—
202C22⋊C4—682no—
203C22⋊C4—682no—
204C42—174no—
205C42—344no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—341no—
212C23—342no—
213C23—341no—
214C23—342no—
215C23—341no—
216C23C23×D1711yescharacteristic
217Q8—44no—
218Q8—684no—
219Q8—684no—
220D4—684no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—172no—
231C2×C4—174no—
232C2×C4—172no—
233C2×C4—341no—
234C2×C4—341no—
235C2×C4—341no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4D4×D1712yes—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—42no—
248C22—42no—
249C22—681no—
250C22—681no—
251C22—681no—
252C22—682no—
253C22—682no—
254C22C2×D4×D1711yescharacteristic
255C22#1058769112yes—
256C22#1058769411yescharacteristic
257C22#41×D1712yes—
258C22#1058769711yescharacteristic
259C4—22no—
260C4—41no—
261C4—42no—
262C4—42no—
263C4—174no—
264C4—342no—
265C4—681no—
266C4—682no—
267C4—682no—
268C2—81no—
269C2—1361no—
270C2#124×D1711yescharacteristic
271C2#1176337411yescharacteristic
272C2#1176338211yescharacteristic
273C2#1176342312yes—
274C2#1176342912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC22×C22⋊D17C22
hypercenter#393D17
hypocenterC17#2640
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×#10587630C2semidirect
3. #11762738C2semidirect
4. #11762737C2semidirect
5. #11762722C2semidirect
6. #11762721C2semidirect
7. #11762720C2semidirect
8. #11762713C2semidirect
10. #11762654C2semidirect
11. C17×#393C2semidirect
138. #393D17semidirect
174. C17#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875712
#117627382
#117627372
#117627222
#117627212
#117627202
#117627132
#117627092
#117626542
C17×#3932
#264017