Finite Group Databasebeta

← #14532069

Subgroups — #14532069

Order 2176.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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×#10587573C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic
3#11762750C212yesmaximal, complemented
4#11762735C211yesmaximal, characteristic, complemented
5#11762724C212yesmaximal
6#11762722C211yesmaximal, characteristic, complemented
7#11762721C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic17C212yesmaximal, complemented
9#11762656C212yesmaximal, complemented
10#11762654C211yesmaximal, characteristic, complemented
11C17×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×C4×Dic17C2212yes—
26C2×C4×Dic17C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587573—24no—
29#10587567—24no—
30C17×#95C2211yescharacteristic
31C17×#95C2212yes—
32#10587502C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C2×C22⋊D17—22no—
44C2×C22⋊D17—22no—
45C2×C22⋊D17—22no—
46C22×Dic17—22no—
47C22×Dic17—24no—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2312yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C2×C4×D17—22no—
54C2×C4×D17—24no—
55C2×Dic34—22no—
56C2×Dic34—24no—
57C4⋊C4×C17—22no—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—22no—
60C22⋊Dic17—24no—
61#63227—24no—
62#63227—22no—
63#63227—24no—
64#63225—22no—
65#63225—24no—
66C4×Dic17—24no—
67C4×Dic17—24no—
68C4×Dic17D414yes—
69C23×C17—21no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—21no—
73C23×C17#4112yes—
74C23×C17C2411yescharacteristic
75C22×D17—21no—
76C22×D17—22no—
77C22×D17—21no—
78C22×D17—22no—
79C22×D17#4112yes—
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—28no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—44no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103C2×Dic17#4114yes—
104C2×Dic17#4114yes—
105C4×D17—44no—
106Dic34—28no—
107#2398—171nomaximal, Sylow
108C22×C17—22no—
109C22×C17—24no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—42no—
113C22×C17C22×D411yescharacteristic
114C22×C17C2×#4111yescharacteristic
115C22×C17C2×#4111yescharacteristic
116C22×C17C2×#4112yes—
117C22×C17C2×#4112yes—
118C2×D17—22no—
119C2×D17—24no—
120C2×D17—41no—
121C2×D17—41no—
122C2×D17—42no—
123C4×C17—22no—
124C4×C17—42no—
125C4×C17—41no—
126C4×C17—42no—
127Dic17—24no—
128Dic17—22no—
129Dic17—41no—
130Dic17—42no—
131Dic17#12414yes—
132C2×C4×Q8—171no—
133C2×C4×D4—171no—
134#400—172no—
135#397—171no—
136#397D1711yescharacteristic, complemented
137#394—172no—
138#393—171no—
139#393—171no—
140C4×C22⋊C4—172no—
141C4×C22⋊C4—172no—
142#376—171no—
143C2×C17—42no—
144C2×C17#52511yescharacteristic
145C2×C17#52511yescharacteristic
146C2×C17C2×#12411yescharacteristic
147C2×C17#52912yes—
148C2×C17#52912yes—
149D17—42no—
150C22×Q8—171no—
151C22×D4—171no—
152C23×C4—171no—
153C23×C4—171no—
154C4×Q8—344no—
155C4×D4—344no—
156C2×C4⋊C4—171no—
157C2×C4⋊C4—172no—
158C2×C4⋊C4C2×D1711yescharacteristic
159C2×C22⋊C4—172no—
160C2×C22⋊C4—171no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4C2×D1711yescharacteristic
164C2×C22⋊C4C2×D1712yes—
165C2×C42—171no—
166C2×C42—172no—
167C2×C42—172no—
168#95—171no—
169#95—172no—
170#95—172no—
171#95—172no—
172#95—172no—
173#95C2×D1711yescharacteristic
174#95C2×D1712yes—
175C17#239811yescharacteristic, complemented, Sylow
176C24—171no—
177C24C22×D1711yescharacteristic
178C2×Q8—344no—
179C2×Q8—342no—
180C2×D4—342no—
181C2×D4—342no—
182C2×D4—342no—
183C22×C4—171no—
184C22×C4—172no—
185C22×C4—171no—
186C22×C4—172no—
187C22×C4—171no—
188C22×C4—342no—
189C22×C4—344no—
190C22×C4—342no—
191C22×C4—344no—
192C22×C4C22×D1712yes—
193C22×C4C22×D1711yescharacteristic
194C22×C4C22×D1712yes—
195C22×C4C22×D1711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—342no—
198C4⋊C4—344no—
199C22⋊C4—24no—
200C22⋊C4—22no—
201C22⋊C4—344no—
202C22⋊C4—344no—
203C22⋊C4—344no—
204C22⋊C4—342no—
205C42—174no—
206C42—344no—
207C42—344no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—172no—
213C23—341no—
214C23—341no—
215C23—342no—
216C23—342no—
217C23#6325412yes—
218C23C23×D1711yescharacteristic
219Q8—348no—
220D4—348no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
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—174no—
234C2×C4—342no—
235C2×C4—341no—
236C2×C4—341no—
237C2×C4—342no—
238C2×C4—341no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—684no—
244C2×C4—684no—
245C2×C4D4×D1712yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—342no—
252C22—344no—
253C22—681no—
254C22—681no—
255C22—682no—
256C22C2×D4×D1711yescharacteristic
257C22C2×#6325411yescharacteristic
258C22#41×D1712yes—
259C22#41×D1711yescharacteristic
260C22#41×D1712yes—
261C4—22no—
262C4—42no—
263C4—41no—
264C4—42no—
265C4—174no—
266C4—344no—
267C4—342no—
268C4—681no—
269C4—682no—
270C2—42no—
271C2—682no—
272C2#1176331711yescharacteristic
273C2#1176331911yescharacteristic
274C2#124×D1711yescharacteristic
275C2#1176333612yes—
276C2#1176334212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC22×C22⋊D17C22
hypercenter#397D17
hypocenterC17#2398
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#10587573C2semidirect
3. #11762750C2semidirect
4. #11762735C2semidirect
6. #11762722C2semidirect
7. #11762721C2semidirect
8. C22⋊C4×Dic17C2semidirect
9. #11762656C2semidirect
10. #11762654C2semidirect
11. C17×#397C2semidirect
136. #397D17semidirect
175. C17#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#105875732
C2×#105875672
#117627502
#117627352
#117627242
#117627222
#117627212
C22⋊C4×Dic172
#117626562
#117626542
C17×#3972
#239817