Finite Group Databasebeta

← #14534704

Subgroups — #14534704

Order 2176.

520 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 137 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
2C2×C4×C22⋊D17C211yesmaximal, characteristic
3#11762741C211yesmaximal, characteristic, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762734C212yesmaximal, complemented
6#11762730C211yesmaximal, characteristic, complemented
7#11762725C212yesmaximal
8#11762722C211yesmaximal, characteristic, complemented
9#11762694C212yesmaximal, complemented
10#11762683C212yesmaximal, complemented
11C17×#381C211yesmaximal, characteristic, complemented
12C23×C4×C17C2211yescharacteristic
13C22×C22⋊D17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2211yescharacteristic
19C4×C22⋊D17—24no—
20C2×#63227C2211yescharacteristic
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2212yes—
26#10587610—24no—
27C2×#63225C2212yes—
28C2×#63225C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17—42no—
40C22×C4×C17D414yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17—22no—
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C2×C4×D17—22no—
54C2×C4×D17—42no—
55C2×Dic34—22no—
56C2×Dic34—42no—
57C22⋊C4×C17—42no—
58C22⋊Dic17—24no—
59C22⋊Dic17—22no—
60#63227—24no—
61#63227—22no—
62#63227—42no—
63C4⋊Dic17—22no—
64C4⋊Dic17—24no—
65C4⋊Dic17—42no—
66#63225—24no—
67#63225—22no—
68C4×Dic17—22no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C2×D412yes—
74C23×C17C2411yescharacteristic
75C22×D17—21no—
76C22×D17—21no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17#4112yes—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
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—44no—
89C2×C4×C17C2×D412yes—
90C2×C4×C17C2×D412yes—
91C22⋊D17—28no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—22no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17#4112yes—
105C4×D17—44no—
106Dic34—44no—
107#2638—171nomaximal, Sylow
108C22×C17—22no—
109C22×C17—22no—
110C22×C17—22no—
111C22×C17—42no—
112C22×C17—42no—
113C22×C17C22×D411yescharacteristic
114C22×C17C2×#4111yescharacteristic
115C22×C17C2×#4111yescharacteristic
116C22×C17H(5,𝔽2)12yes—
117C22×C172−1+412yes—
118C2×D17—22no—
119C2×D17—24no—
120C2×D17—41no—
121C2×D17—41no—
122C2×D17—42no—
123C4×C17—22no—
124C4×C17—22no—
125C4×C17—42no—
126C4×C17—42no—
127Dic17—22no—
128Dic17—42no—
129Dic17—42no—
130Dic17—41no—
131Dic17—41no—
132C2×C22⋊Q8—171no—
133C2×C4×D4—171no—
134#398—171no—
135#397—172no—
136#397—171no—
137#395—171no—
138#394—172no—
139#393—171no—
140#387—172no—
141#385—172no—
142#381D1711yescharacteristic, complemented
143C2×C17—42no—
144C2×C17#52511yescharacteristic
145C2×C17#53411yescharacteristic
146C2×C17#53611yescharacteristic
147C2×C17#53912yes—
148C2×C17#54212yes—
149D17—42no—
150C22×Q8—171no—
151C22×D4—171no—
152C23×C4—171no—
153C23×C4C2×D1711yescharacteristic
154C22⋊Q8—344no—
155C4×D4—344no—
156C2×C4⋊C4—171no—
157C2×C4⋊C4—172no—
158C2×C4⋊C4—171no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—172no—
161C2×C22⋊C4—171no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4—171no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4C2×D1712yes—
167C2×C42—171no—
168#95—172no—
169#95—172no—
170#95—172no—
171#95—172no—
172#95C2×D1712yes—
173#95C2×D1711yescharacteristic
174#95C2×D1711yescharacteristic
175C17#263811yescharacteristic, complemented, Sylow
176C24—171no—
177C24C22×D1711yescharacteristic
178C2×Q8—342no—
179C2×Q8—682no—
180C2×D4—342no—
181C2×D4—342no—
182C2×D4—342no—
183C22×C4—42no—
184C22×C4—171no—
185C22×C4—171no—
186C22×C4—172no—
187C22×C4—172no—
188C22×C4—171no—
189C22×C4—342no—
190C22×C4—682no—
191C22×C4D6814yes—
192C22×C4C22×D1712yes—
193C22×C4C22×D1712yes—
194C22×C4C22×D1711yescharacteristic
195C22×C4C22×D1711yescharacteristic
196C4⋊C4—344no—
197C4⋊C4—342no—
198C4⋊C4—344no—
199C4⋊C4—342no—
200C4⋊C4—682no—
201C22⋊C4—42no—
202C22⋊C4—344no—
203C22⋊C4—342no—
204C22⋊C4—344no—
205C22⋊C4—342no—
206C22⋊C4—682no—
207C42—342no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—172no—
213C23—341no—
214C23—342no—
215C23—341no—
216C23—342no—
217C23C2×D6812yes—
218C23C23×D1711yescharacteristic
219Q8—684no—
220D4—348no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—44no—
230C2×C4—172no—
231C2×C4—341no—
232C2×C4—341no—
233C2×C4—341no—
234C2×C4—342no—
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×C4—684no—
244C2×C4C2×D6812yes—
245C2×C4C2×D6812yes—
246C22—22no—
247C22—22no—
248C22—22no—
249C22—42no—
250C22—42no—
251C22—342no—
252C22—344no—
253C22—681no—
254C22—681no—
255C22—682no—
256C22C22×D6811yescharacteristic
257C22#1058769112yes—
258C22#41×D1711yescharacteristic
259C22#41×D1711yescharacteristic
260C22#1058769712yes—
261C4—22no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—342no—
266C4—682no—
267C4—681no—
268C4—682no—
269C4—681no—
270C2—42no—
271C2—682no—
272C2#1176332111yescharacteristic
273C2#1176337111yescharacteristic
274C2#1176338511yescharacteristic
275C2#1176340212yes—
276C2#1176342712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC22×C22⋊D17C22
hypercenter#381D17
hypocenterC17#2638
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#10587610C2semidirect
3. #11762741C2semidirect
4. #11762736C2semidirect
5. #11762734C2semidirect
6. #11762730C2semidirect
8. #11762722C2semidirect
9. #11762694C2semidirect
10. #11762683C2semidirect
11. C17×#381C2semidirect
142. #381D17semidirect
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
C2×C4×C22⋊D172
#117627412
#117627362
#117627342
#117627302
#117627252
#117627222
#117626942
#117626832
C17×#3812
#263817