Finite Group Databasebeta

← #14534701

Subgroups — #14534701

Order 2176.

520 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587610C211yesmaximal, characteristic, complemented
2#11762739C211yesmaximal, characteristic, complemented
3#11762737C212yesmaximal, complemented
4#11762734C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762723C212yesmaximal
7#11762721C211yesmaximal, characteristic, complemented
8#11762693C212yesmaximal, complemented
9#11762682C212yesmaximal, complemented
10#11762672C211yesmaximal, characteristic, complemented
11C2×C4×D4×C17C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C4×D4×C17—24no—
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C42×C17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25C2×C4⋊Dic17C2211yescharacteristic
26#10587610—24no—
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C24×C17C2311yescharacteristic
39C2×D4×C17—22no—
40C2×D4×C17—22no—
41C2×D4×C17—22no—
42C22×C4×C17—22no—
43C22×C4×C17—42no—
44C22×C4×C17—42no—
45C22×C4×C17D414yes—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C22×C4×C17C2311yescharacteristic
49C22×C4×C17C2311yescharacteristic
50C22×C4×C17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C4⋊C4×C17—22no—
58C22⋊C4×C17—22no—
59C22⋊C4×C17—22no—
60C42×C17—22no—
61C22⋊Dic17—24no—
62C22⋊Dic17—24no—
63C22⋊Dic17—42no—
64C22⋊Dic17—42no—
65C4⋊Dic17—22no—
66#63225—24no—
67#63225—24no—
68#63225—42no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17—21no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17C2×D412yes—
78C23×C17#4112yes—
79C23×C17C2411yescharacteristic
80D4×C17—28no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—44no—
90C2×C4×C17—44no—
91C2×C4×C17C2×D412yes—
92C2×C4×C17C2×D412yes—
93C2×C4×C17#4112yes—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106Dic34—44no—
107#2638—171nomaximal, Sylow
108C22×C17—22no—
109C22×C17—22no—
110C22×C17—22no—
111C22×C17—22no—
112C22×C17—24no—
113C22×C17—41no—
114C22×C17—41no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4111yescharacteristic
120C22×C17C2×#4111yescharacteristic
121C22×C17H(5,𝔽2)12yes—
122C22×C172−1+412yes—
123C4×C17—22no—
124C4×C17—22no—
125C4×C17—22no—
126C4×C17—41no—
127C4×C17—41no—
128Dic17—42no—
129Dic17—42no—
130Dic17—42no—
131Dic17—42no—
132C2×C22⋊Q8—171no—
133C2×C4×D4D1711yescharacteristic
134#398—171no—
135#397—171no—
136#397—172no—
137#395—171no—
138#394—172no—
139#393—171no—
140#387—172no—
141#385—172no—
142#381—171no—
143C2×C17—42no—
144C2×C17—42no—
145C2×C17#52511yescharacteristic
146C2×C17#53411yescharacteristic
147C2×C17#53611yescharacteristic
148C2×C17#53912yes—
149C2×C17#54212yes—
150C22×Q8—171no—
151C22×D4C2×D1711yescharacteristic
152C23×C4C2×D1711yescharacteristic
153C23×C4C2×D1711yescharacteristic
154C22⋊Q8—344no—
155C4×D4—24no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—171no—
158C2×C4⋊C4—172no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4C2×D1711yescharacteristic
161C2×C22⋊C4—172no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4C2×D1711yescharacteristic
166C2×C22⋊C4C2×D1711yescharacteristic
167C2×C42C2×D1711yescharacteristic
168#95—172no—
169#95—171no—
170#95—171no—
171#95—172no—
172#95—172no—
173#95—172no—
174#95—172no—
175C17#263811yescharacteristic, complemented, Sylow
176C24C22×D1711yescharacteristic
177C24C22×D1711yescharacteristic
178C2×Q8—342no—
179C2×Q8—682no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—172no—
187C22×C4—172no—
188C22×C4—172no—
189C22×C4—172no—
190C22×C4C22⋊D1714yes—
191C22×C4C22×D1711yescharacteristic
192C22×C4C22×D1711yescharacteristic
193C22×C4C22×D1711yescharacteristic
194C22×C4C22×D1711yescharacteristic
195C22×C4C22×D1711yescharacteristic
196C4⋊C4—22no—
197C4⋊C4—344no—
198C4⋊C4—344no—
199C4⋊C4—342no—
200C4⋊C4—682no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—344no—
204C22⋊C4—344no—
205C22⋊C4—682no—
206C22⋊C4—682no—
207C42—22no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23#6325212yes—
217C23C2×C22⋊D1712yes—
218C23C23×D1711yescharacteristic
219Q8—684no—
220D4—28no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4#6325212yes—
244C2×C4C2×C22⋊D1712yes—
245C2×C4C2×C22⋊D1712yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—22no—
250C22—24no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22C2×#6325211yescharacteristic
257C22C2×#6325211yescharacteristic
258C22#1058769112yes—
259C22#1058769712yes—
260C22C22×C22⋊D1711yescharacteristic
261C4—22no—
262C4—22no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—682no—
267C4—682no—
268C4—682no—
269C4—682no—
270C2—42no—
271C2—42no—
272C2#1176332011yescharacteristic
273C2#1176336811yescharacteristic
274C2#1176338011yescharacteristic
275C2#1176340112yes—
276C2#1176342412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C4×D4×C17C2
prime radicalC22×D4×C17C22
hypercenterC2×C4×D4D17
hypocenterC17#2638
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10587610C2semidirect
2. #11762739C2semidirect
3. #11762737C2semidirect
4. #11762734C2semidirect
5. #11762729C2semidirect
7. #11762721C2semidirect
8. #11762693C2semidirect
9. #11762682C2semidirect
10. #11762672C2semidirect
175. C17#2638semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
#117627392
#117627372
#117627342
#117627292
#117627232
#117627212
#117626932
#117626822
#117626722
C2×C4×D4×C172
#263817