Finite Group Databasebeta

← #14534404

Subgroups — #14534404

Order 2176.

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

normal subgroups only

This group has 582 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×#10587565C211yesmaximal, characteristic, complemented
2C2×D4×Dic17C211yesmaximal, characteristic, complemented
3#11762741C211yesmaximal, characteristic, complemented
4#11762734C211yesmaximal, characteristic, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762712C212yesmaximal, complemented
7#11762697C212yesmaximal, complemented
8#11762689C212yesmaximal, complemented
9#11762684C212yesmaximal, complemented
10#11762654C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C23×Dic17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊Dic17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23D4×Dic17—24no—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2212yes—
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×C4×Dic17C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587565—24no—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C24×C17C2312yes—
39C2×D4×C17—24no—
40C2×D4×C17—24no—
41C2×D4×C17—22no—
42C2×D4×C17—24no—
43C2×D4×C17—42no—
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C22×Dic17—24no—
48C22×Dic17—42no—
49C22×Dic17—42no—
50C22×Dic17C2312yes—
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C22⋊C4×C17—22no—
57C22⋊C4×C17—24no—
58C22⋊Dic17—24no—
59C22⋊Dic17—22no—
60C22⋊Dic17—22no—
61C22⋊Dic17—24no—
62C22⋊Dic17—42no—
63C22⋊Dic17—42no—
64C4⋊Dic17—22no—
65C4⋊Dic17—22no—
66C4⋊Dic17—42no—
67#63225—22no—
68#63225—22no—
69C4×Dic17—24no—
70C4×Dic17D414yescomplemented
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—21no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—21no—
81C23×C17—21no—
82C23×C17#4114yes—
83C23×C17C2411yescharacteristic
84D4×C17—28no—
85D4×C17—48no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17C2×D412yes—
92C2×C4×C17#4114yes—
93C2×Dic17—22no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—22no—
97C2×Dic17—21no—
98C2×Dic17—21no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—48no—
108C2×Dic17C2×D412yes—
109C2×Dic17C2×D412yes—
110#2603—171nomaximal, Sylow
111C22×C17—24no—
112C22×C17—24no—
113C22×C17—24no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—41no—
119C22×C17—41no—
120C22×C17—41no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17C22×D411yescharacteristic
124C22×C17C2×#4112yes—
125C22×C17C2×#4112yes—
126C22×C17H(5,𝔽2)11yescharacteristic
127C22×C17H(5,𝔽2)11yescharacteristic
128C4×C17—22no—
129C4×C17—24no—
130C4×C17—41no—
131Dic17—22no—
132Dic17—22no—
133Dic17—41no—
134Dic17—41no—
135Dic17—42no—
136Dic17—42no—
137C2×#126—171no—
138C2×C4×D4—171no—
139#398—171no—
140#397—171no—
141#394—171no—
142#392D1711yescharacteristic
143#390—172no—
144#388—172no—
145#386—172no—
146#385—172no—
147#376—171no—
148C2×C17—44no—
149C2×C17—81no—
150C2×C17#52511yescharacteristic
151C2×C17#53211yescharacteristic
152C2×C17#53511yescharacteristic
153C2×C17#55112yes—
154C2×C17#55312yes—
155C22×D4C2×D1711yescharacteristic
156C22×D4C2×D1712yes—
157C23×C4—172no—
158#126—344no—
159C4×D4—344no—
160C2×C4⋊C4—171no—
161C2×C4⋊C4—171no—
162C2×C4⋊C4—171no—
163C2×C4⋊C4—171no—
164C2×C4⋊C4—172no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—171no—
167C2×C22⋊C4—171no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4—172no—
170C2×C22⋊C4—172no—
171C2×C22⋊C4C2×D1711yescharacteristic
172C2×C22⋊C4C2×D1712yes—
173C2×C42—171no—
174C2×C42—172no—
175#95—172no—
176#95—171no—
177#95—172no—
178#95—172no—
179#95C2×D1711yescharacteristic
180C17#260311yescharacteristic, complemented, Sylow
181C24C22×D1711yescharacteristic
182C24C22×D1712yes—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—22no—
186C2×D4—24no—
187C2×D4—42no—
188C22×C4—171no—
189C22×C4—171no—
190C22×C4—172no—
191C22×C4—171no—
192C22×C4—172no—
193C22×C4—171no—
194C22×C4—344no—
195C22×C4—682no—
196C22×C4—682no—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C22×C4C22×D1712yes—
200C4⋊C4—342no—
201C4⋊C4—342no—
202C4⋊C4—342no—
203C4⋊C4—342no—
204C4⋊C4—682no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—344no—
208C22⋊C4—342no—
209C22⋊C4—342no—
210C22⋊C4—344no—
211C22⋊C4—682no—
212C22⋊C4—682no—
213C42—174no—
214C42—344no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23#6325414yes—
227C23C23×D1711yescharacteristic
228D4—28no—
229D4—48no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—172no—
236C2×C4—172no—
237C2×C4—342no—
238C2×C4—341no—
239C2×C4—341no—
240C2×C4—341no—
241C2×C4—341no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—341no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—341no—
250C2×C4—342no—
251C2×C4—688no—
252C2×C4D4×D1712yes—
253C2×C4#6325414yes—
254C22—24no—
255C22—24no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22C2×D4×D1711yescharacteristic
267C22C2×#6325412yes—
268C22C2×#6325412yes—
269C22#1058769111yescharacteristic
270C22#1058769611yescharacteristic
271C4—24no—
272C4—22no—
273C4—41no—
274C4—342no—
275C4—342no—
276C4—682no—
277C4—681no—
278C4—682no—
279C4—681no—
280C2—44no—
281C2—81no—
282C2#1176331511yescharacteristic
283C2#1176335711yescharacteristic
284C2#1176337511yescharacteristic
285C2#1176349012yes—
286C2#1176351112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC17×#392C2
hypercenter#392D17
hypocenterC17#2603
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587565C2semidirect
2. C2×D4×Dic17C2semidirect
3. #11762741C2semidirect
4. #11762734C2semidirect
5. #11762723C2semidirect
6. #11762712C2semidirect
7. #11762697C2semidirect
8. #11762689C2semidirect
9. #11762684C2semidirect
10. #11762654C2semidirect
70. C4×Dic17D4semidirect
180. C17#2603semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
C2×D4×Dic172
#117627412
#117627342
#117627232
#117627122
#117626972
#117626892
#117626842
#117626542
C17×#3922
#260317