Finite Group Databasebeta

← #14534929

Subgroups — #14534929

Order 2176.

534 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 129 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×#10587584C211yesmaximal, characteristic, complemented
2C2×#10587622C211yesmaximal, characteristic, complemented
3#11762732C212yesmaximal, complemented
4#11762729C211yesmaximal, characteristic, complemented
5#11762724C212yesmaximal, complemented
6#11762722C211yesmaximal, characteristic, complemented
7#11762717C211yesmaximal, characteristic
8#11762709C212yesmaximal, complemented
9#11762697C212yesmaximal, complemented
10#11762660C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21#10587622—24no—
22C2×#63227C2211yescharacteristic
23C2×#63227C2211yescharacteristic
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2212yes—
31C2×C4×Dic17C2211yescharacteristic
32#10587584—24no—
33C17×#95C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2312yes—
38C23×D17C2311yescharacteristic
39C2×D4×C17—22no—
40C2×D4×C17—42no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17—42no—
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C2×Dic34—22no—
54C2×Dic34—42no—
55C4⋊C4×C17—22no—
56C22⋊C4×C17—42no—
57C22⋊C4×C17—42no—
58C22⋊Dic17—24no—
59C22⋊Dic17—24no—
60C22⋊Dic17—24no—
61#63227—22no—
62#63227—22no—
63#63227—22no—
64#63227—42no—
65C4⋊Dic17—22no—
66#63225—22no—
67#63225—24no—
68#63225—42no—
69C4×Dic17—42no—
70C4×Dic17D414yescomplemented
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17C2411yescharacteristic
79C22×D17—21no—
80C22×D17—22no—
81C22×D17—21no—
82C22×D17—22no—
83C22×D17—21no—
84D4×C17—44no—
85C2×C4×C17—22no—
86C2×C4×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17C2×D412yes—
95C22⋊D17—48no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—21no—
101C2×Dic17—22no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17C2×D412yes—
107C2×Dic17C2×D412yes—
108C2×Dic17#4114yes—
109Dic34—44no—
110#2662—171nomaximal, Sylow
111C22×C17—42no—
112C22×C17—42no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4112yes—
120C22×C17H(5,𝔽2)12yes—
121C22×C17H(5,𝔽2)11yescharacteristic
122C22×C172−1+411yescharacteristic
123C2×D17—41no—
124C2×D17—41no—
125C2×D17—41no—
126C2×D17—42no—
127C2×D17—42no—
128C4×C17—22no—
129C4×C17—41no—
130C4×C17—41no—
131C4×C17—42no—
132Dic17—22no—
133Dic17—22no—
134Dic17—24no—
135Dic17—41no—
136Dic17—42no—
137C2×#126—171no—
138C2×#124—171no—
139#396—172no—
140#395—171no—
141#394—172no—
142#394D1711yescharacteristic, complemented
143#393—171no—
144#392—171no—
145#390—172no—
146#388—172no—
147C4×C4⋊C4—171no—
148C2×C17—82no—
149C2×C17#53211yescharacteristic
150C2×C17#53511yescharacteristic
151C2×C17#53711yescharacteristic
152C2×C17#53812yes—
153C2×C17#54212yes—
154D17—81no—
155C22×Q8—171no—
156C22×D4—172no—
157C22×D4C2×D1711yescharacteristic
158#126—344no—
159#124—344no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—171no—
162C2×C4⋊C4—172no—
163C2×C4⋊C4—172no—
164C2×C4⋊C4C2×D1711yescharacteristic
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4—171no—
169C2×C22⋊C4—171no—
170C2×C22⋊C4—171no—
171C2×C22⋊C4—172no—
172C2×C22⋊C4C2×D1712yes—
173C2×C22⋊C4C2×D1712yes—
174C2×C42—171no—
175C2×C42—172no—
176#95—171no—
177#95—172no—
178#95—172no—
179#95C2×D1711yescharacteristic
180C17#266211yescharacteristic, complemented, Sylow
181C24—171no—
182C24C22×D1712yes—
183C2×Q8—342no—
184C2×Q8—682no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—344no—
188C2×D4—682no—
189C2×D4—682no—
190C22×C4—172no—
191C22×C4—171no—
192C22×C4—171no—
193C22×C4—171no—
194C22×C4—172no—
195C22×C4C22×D1711yescharacteristic
196C22×C4C22×D1712yes—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—342no—
201C4⋊C4—342no—
202C4⋊C4—344no—
203C4⋊C4—682no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—342no—
207C22⋊C4—342no—
208C22⋊C4—344no—
209C22⋊C4—344no—
210C22⋊C4—344no—
211C22⋊C4—342no—
212C22⋊C4—682no—
213C42—174no—
214C42—682no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—341no—
223C23—342no—
224C23—341no—
225C23—342no—
226C23—341no—
227C23C23×D1711yescharacteristic
228Q8—684no—
229D4—44no—
230D4—688no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—172no—
241C2×C4—172no—
242C2×C4—174no—
243C2×C4—341no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—341no—
248C2×C4—342no—
249C2×C4—341no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4D4×D1712yes—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—681no—
262C22—681no—
263C22—681no—
264C22—682no—
265C22—682no—
266C22C2×D4×D1711yescharacteristic
267C22#1058769112yes—
268C22#1058769111yescharacteristic
269C22#41×D1712yes—
270C22#1058769711yescharacteristic
271C4—22no—
272C4—41no—
273C4—41no—
274C4—42no—
275C4—342no—
276C4—342no—
277C4—344no—
278C4—681no—
279C4—682no—
280C2—82no—
281C2—1361no—
282C2#1176335811yescharacteristic
283C2#1176337411yescharacteristic
284C2#1176338811yescharacteristic
285C2#1176339112yes—
286C2#1176342712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radical#11762717C2
hypercenter#394D17
hypocenterC17#2662
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587584C2semidirect
2. C2×#10587622C2semidirect
3. #11762732C2semidirect
4. #11762729C2semidirect
5. #11762724C2semidirect
6. #11762722C2semidirect
8. #11762709C2semidirect
9. #11762697C2semidirect
10. #11762660C2semidirect
11. C17×#394C2semidirect
70. C4×Dic17D4semidirect
142. #394D17semidirect
180. C17#2662semidirect

Maximal subgroups

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

subgroupindex
C2×#105875842
C2×#105876222
#117627322
#117627292
#117627242
#117627222
#117627172
#117627092
#117626972
#117626602
C17×#3942
#266217