Finite Group Databasebeta

← #14533967

Subgroups — #14533967

Order 2176.

584 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 147 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×#10587583C211yesmaximal, characteristic, complemented
2C2×#10587579C211yesmaximal, characteristic
3#11762744C211yesmaximal, characteristic, complemented
4#11762729C211yesmaximal, characteristic, complemented
5#11762722C211yesmaximal, characteristic, complemented
6#11762696C212yesmaximal, complemented
7#11762687C212yesmaximal
8#11762683C212yesmaximal, complemented
9#11762668C212yesmaximal, complemented
10#95×D17C211yesmaximal, characteristic, complemented
11C17×#400C211yesmaximal, characteristic, complemented
12C22×D68C2211yescharacteristic
13C22×C4×D17C2212yes—
14C22×C4×D17C2211yescharacteristic
15C22×C4×D17C2211yescharacteristic
16C22×Dic34C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2211yescharacteristic
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587583—24no—
29#10587579—24no—
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32C17×#95C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2312yes—
48C2×D68—22no—
49C2×D68—22no—
50C2×D68Q814yescomplemented
51C2×C4×D17—24no—
52C2×C4×D17—22no—
53C2×C4×D17—24no—
54C2×C4×D17—42no—
55C2×C4×D17—42no—
56C2×C4×D17—42no—
57C2×C4×D17D414yes—
58C2×Dic34—22no—
59C2×Dic34—42no—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—22no—
62#63227—22no—
63#63227—24no—
64#63227—22no—
65#63227—24no—
66#63227—42no—
67C4⋊Dic17—24no—
68#63225—24no—
69#63225—42no—
70C4×Dic17—22no—
71C23×C17C2411yescharacteristic
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—21no—
75C22×D17—21no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17C2×D412yes—
79C22×D17C2×Q812yes—
80C22×D17C2×Q812yes—
81C22×D17#4112yes—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17C2×D412yes—
94C2×C4×C17C2×Q812yes—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17C2×D412yes—
104C2×Dic17#4114yes—
105D68—28no—
106C4×D17—44no—
107C4×D17—48no—
108C4×D17—44no—
109Dic34—44no—
110#2564—171nomaximal, Sylow
111C22×C17C22×D411yescharacteristic
112C22×C17C22×Q811yescharacteristic
113C22×C17C2×#4111yescharacteristic
114C22×C17C2×#4112yes—
115C22×C17H(5,𝔽2)12yes—
116C2×D17—22no—
117C2×D17—22no—
118C2×D17—24no—
119C2×D17—24no—
120C2×D17—22no—
121C2×D17—24no—
122C2×D17—41no—
123C2×D17—41no—
124C2×D17—42no—
125C2×D17C22⋊Q814yes—
126C4×C17—22no—
127C4×C17—22no—
128C4×C17—42no—
129C4×C17—41no—
130C4×C17—42no—
131Dic17—22no—
132Dic17—24no—
133Dic17—41no—
134Dic17—42no—
135C2×C22⋊Q8—171no—
136C2×C4×D4—171no—
137#400D1711yescharacteristic, complemented
138#399—171no—
139#395—171no—
140#393—171no—
141#387—172no—
142#386—172no—
143#385—172no—
144#380—172no—
145C2×#95—171no—
146C2×C17C2×C22⋊Q811yescharacteristic
147C2×C17#53211yescharacteristic
148C2×C17#53411yescharacteristic
149C2×C17#55112yes—
150C2×C17#55412yes—
151D17—24no—
152D17—42no—
153C22×Q8—171no—
154C22×D4—171no—
155C23×C4—171no—
156C23×C4—172no—
157C23×C4—171no—
158C22⋊Q8—344no—
159C4×D4—344no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4—172no—
163C2×C4⋊C4C2×D1711yescharacteristic
164C2×C4⋊C4C2×D1711yescharacteristic
165C2×C22⋊C4—171no—
166C2×C22⋊C4—171no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C42—171no—
171#95—172no—
172#95—171no—
173#95—172no—
174#95—172no—
175#95C2×D1711yescharacteristic
176#95C2×D1712yes—
177#95C2×D1712yes—
178C17#256411yescharacteristic, complemented, Sylow
179C24—171no—
180C24—171no—
181C2×Q8—342no—
182C2×Q8—682no—
183C2×D4—174no—
184C2×D4—342no—
185C2×D4—342no—
186C22×C4—172no—
187C22×C4—171no—
188C22×C4—174no—
189C22×C4—171no—
190C22×C4—172no—
191C22×C4—342no—
192C22×C4—344no—
193C22×C4—344no—
194C22×C4—682no—
195C22×C4—682no—
196C22×C4—682no—
197C22×C4C22×D1712yes—
198C22×C4C22×D1712yes—
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1711yescharacteristic
201C22×C4C22×D1711yescharacteristic
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—344no—
205C4⋊C4—344no—
206C4⋊C4—682no—
207C22⋊C4—344no—
208C22⋊C4—342no—
209C22⋊C4—344no—
210C22⋊C4—342no—
211C22⋊C4—682no—
212C42—342no—
213C23—172no—
214C23—172no—
215C23—172no—
216C23—172no—
217C23—342no—
218C23—342no—
219C23—341no—
220C23—342no—
221C23—341no—
222C23—342no—
223C23C23×D1711yescharacteristic
224Q8—684no—
225D4—348no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—172no—
238C2×C4—174no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—341no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—684no—
248C2×C4—688no—
249C2×C4—684no—
250C2×C4D4×D1712yes—
251C2×C4Q8×D1712yes—
252C22—174no—
253C22—344no—
254C22—344no—
255C22—342no—
256C22—342no—
257C22—344no—
258C22—342no—
259C22—682no—
260C22—681no—
261C22—681no—
262C22C2×D4×D1711yescharacteristic
263C22C2×Q8×D1711yescharacteristic
264C22#41×D1711yescharacteristic
265C22#41×D1712yes—
266C22#1058769612yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—344no—
273C4—342no—
274C4—681no—
275C4—682no—
276C2—344no—
277C2—682no—
278C2C22⋊Q8×D1711yescharacteristic
279C2#1176335811yescharacteristic
280C2#1176337011yescharacteristic
281C2#1176348412yes—
282C2#1176351512yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC22×D68C22
hypercenter#400D17
hypocenterC17#2564
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#10587583C2semidirect
3. #11762744C2semidirect
4. #11762729C2semidirect
5. #11762722C2semidirect
6. #11762696C2semidirect
8. #11762683C2semidirect
9. #11762668C2semidirect
10. #95×D17C2semidirect
11. C17×#400C2semidirect
50. C2×D68Q8semidirect
137. #400D17semidirect
178. C17#2564semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×#105875792
#117627442
#117627292
#117627222
#117626962
#117626872
#117626832
#117626682
#95×D172
C17×#4002
#256417