Finite Group Databasebeta

← #14534635

Subgroups — #14534635

Order 2176.

628 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587571C211yesmaximal, characteristic, complemented
2C2×D4×Dic17C211yesmaximal, characteristic, complemented
3#11762736C211yesmaximal, characteristic, complemented
4#11762734C211yesmaximal, characteristic, complemented
5#11762721C212yesmaximal, complemented
6#11762717C212yesmaximal, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762689C212yesmaximal, complemented
9#11762686C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C23×Dic17C2212yes—
17C22×Dic34C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24D4×Dic17—24no—
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587571—24no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C24×C17C2312yes—
39C23×D17C2311yescharacteristic
40C2×D4×C17—24no—
41C2×D4×C17—24no—
42C2×D4×C17—22no—
43C2×D4×C17—42no—
44C2×D4×C17—42no—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—24no—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—42no—
51C2×C22⋊D17—42no—
52C2×C22⋊D17—42no—
53C22×Dic17—24no—
54C22×Dic17—42no—
55C22×Dic17—42no—
56C22×Dic17C2311yescharacteristic
57C22×Dic17C2312yes—
58C22×Dic17C2312yes—
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C2×Dic34—22no—
62C2×Dic34—42no—
63C22⋊C4×C17—22no—
64C22⋊C4×C17—42no—
65C22⋊Dic17—22no—
66C22⋊Dic17—24no—
67C22⋊Dic17—42no—
68C22⋊Dic17—42no—
69#63227—22no—
70#63227—22no—
71#63227—42no—
72C4⋊Dic17—22no—
73C4×Dic17D414yescomplemented
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—21no—
81C23×C17—22no—
82C23×C17—21no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17#4114yes—
86C23×C17C2411yescharacteristic
87C22×D17—21no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—21no—
91C22×D17—21no—
92D4×C17—28no—
93D4×C17—48no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17C2×D412yes—
101C22⋊D17—44no—
102C22⋊D17—48no—
103C2×Dic17—22no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—48no—
115C2×Dic17C2×D412yes—
116C2×Dic17C2×D412yes—
117Dic34—44no—
118#2631—171nomaximal, Sylow
119C22×C17—24no—
120C22×C17—24no—
121C22×C17—24no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—41no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17C22×D411yescharacteristic
132C22×C17C2×#4112yes—
133C22×C17H(5,𝔽2)12yes—
134C22×C17H(5,𝔽2)11yescharacteristic
135C22×C17H(5,𝔽2)11yescharacteristic
136C2×D17—41no—
137C2×D17—41no—
138C2×D17—41no—
139C2×D17—42no—
140C2×D17—42no—
141C4×C17—22no—
142C4×C17—41no—
143C4×C17—42no—
144Dic17—22no—
145Dic17—22no—
146Dic17—41no—
147Dic17—42no—
148Dic17—42no—
149C2×#124—171no—
150C2×C4×D4—171no—
151#397—171no—
152#397—171no—
153#393—172no—
154#392—172no—
155#392—171no—
156#392D1711yescharacteristic, complemented
157#386—172no—
158#386—172no—
159#383—171no—
160C2×C17—44no—
161C2×C17—81no—
162C2×C17#52511yescharacteristic
163C2×C17#53511yescharacteristic
164C2×C17#53511yescharacteristic
165C2×C17#53812yes—
166C2×C17#53812yes—
167D17—81no—
168C22×Q8—171no—
169C22×D4—171no—
170C22×D4—172no—
171C22×D4C2×D1711yescharacteristic
172C22×D4C2×D1712yes—
173C23×C4—172no—
174#124—344no—
175C4×D4—344no—
176C2×C4⋊C4—171no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—171no—
179C2×C22⋊C4—171no—
180C2×C22⋊C4—171no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—172no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4C2×D1711yescharacteristic
185C2×C22⋊C4C2×D1712yes—
186C2×C42—171no—
187#95—172no—
188#95—172no—
189#95—171no—
190#95—172no—
191#95—172no—
192#95C2×D1711yescharacteristic
193C17#263111yescharacteristic, complemented, Sylow
194C24—171no—
195C24C22×D1711yescharacteristic
196C24C22×D1712yes—
197C2×Q8—342no—
198C2×Q8—682no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—342no—
205C2×D4—344no—
206C2×D4—682no—
207C2×D4—682no—
208C2×D4—682no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—172no—
212C22×C4—171no—
213C22×C4—172no—
214C22×C4—344no—
215C22×C4—682no—
216C22×C4—682no—
217C22×C4C22×D1712yes—
218C22×C4C22×D1711yescharacteristic
219C22×C4C22×D1711yescharacteristic
220C4⋊C4—342no—
221C22⋊C4—22no—
222C22⋊C4—42no—
223C22⋊C4—342no—
224C22⋊C4—344no—
225C22⋊C4—342no—
226C22⋊C4—342no—
227C22⋊C4—682no—
228C22⋊C4—682no—
229C22⋊C4—682no—
230C42—174no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—21no—
240C23—22no—
241C23—21no—
242C23—341no—
243C23—342no—
244C23—341no—
245C23—342no—
246C23—341no—
247C23#6325414yes—
248C23C23×D1711yescharacteristic
249Q8—684no—
250D4—28no—
251D4—48no—
252D4—684no—
253D4—688no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—172no—
261C2×C4—172no—
262C2×C4—341no—
263C2×C4—342no—
264C2×C4—341no—
265C2×C4—341no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—342no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—688no—
274C2×C4D4×D1712yes—
275C22—24no—
276C22—24no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—681no—
288C22—681no—
289C22—681no—
290C22—682no—
291C22—682no—
292C22C2×D4×D1711yescharacteristic
293C22C2×#6325412yes—
294C22#1058769112yes—
295C22#1058769111yescharacteristic
296C22#1058769611yescharacteristic
297C4—22no—
298C4—41no—
299C4—42no—
300C4—342no—
301C4—342no—
302C4—681no—
303C4—682no—
304C4—682no—
305C2—44no—
306C2—81no—
307C2—1361no—
308C2#1176331511yescharacteristic
309C2#1176337211yescharacteristic
310C2#1176337511yescharacteristic
311C2#1176339412yes—
312C2#1176339612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2631
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587571C2semidirect
2. C2×D4×Dic17C2semidirect
3. #11762736C2semidirect
4. #11762734C2semidirect
5. #11762721C2semidirect
6. #11762717C2semidirect
7. #11762717C2semidirect
8. #11762689C2semidirect
9. #11762686C2semidirect
10. #11762676C2semidirect
11. C17×#392C2semidirect
73. C4×Dic17D4semidirect
156. #392D17semidirect
193. C17#2631semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
C2×D4×Dic172
#117627362
#117627342
#117627212
#117627172
#117626892
#117626862
#117626762
C17×#3922
#263117