Finite Group Databasebeta

← #14534913

Subgroups — #14534913

Order 2176.

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

normal subgroups only

This group has 598 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×#10587615C211yesmaximal, characteristic, complemented
2#11762740C211yesmaximal, characteristic, complemented
3#11762735C211yesmaximal, characteristic, complemented
4#11762723C212yesmaximal, complemented
5#11762723C211yesmaximal, characteristic, complemented
6#11762717C211yesmaximal, characteristic, complemented
7#11762717C212yesmaximal, complemented
8#11762709C212yesmaximal, complemented
9#11762703C212yesmaximal, complemented
10C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
11C2×C17×#121C211yesmaximal, characteristic, complemented
12C22×D4×C17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C23×C4×C17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C22×C22⋊D17C2211yescharacteristic
17C17×#121—24no—
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2211yescharacteristic
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25#10587615—24no—
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2212yes—
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2212yes—
32C2×C4×Dic17C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2312yes—
38C24×C17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C2×D4×C17—24no—
41C2×D4×C17—22no—
42C2×D4×C17—42no—
43C2×D4×C17—42no—
44C2×D4×C17—42no—
45C22×C4×C17—42no—
46C22×C4×C17D414yescomplemented
47C22×C4×C17C2312yes—
48C22×C4×C17C2311yescharacteristic
49C22×C4×C17C2311yescharacteristic
50C2×C22⋊D17—22no—
51C2×C22⋊D17—24no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C2×C22⋊D17—42no—
55C22×Dic17C2311yescharacteristic
56C22×Dic17C2312yes—
57C22×Dic17C2311yescharacteristic
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2312yes—
60C4⋊C4×C17—22no—
61C22⋊C4×C17—24no—
62C22⋊Dic17—22no—
63C22⋊Dic17—24no—
64C22⋊Dic17—24no—
65C22⋊Dic17—42no—
66C22⋊Dic17—42no—
67#63227—22no—
68#63227—22no—
69#63227—42no—
70#63225—22no—
71#63225—22no—
72#63225—42no—
73C4×Dic17—42no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—21no—
77C23×C17—21no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17C2×D412yes—
86C23×C17C2411yescharacteristic
87C22×D17—21no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—21no—
91C22×D17—21no—
92D4×C17—44no—
93D4×C17—48no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—44no—
100C2×C4×C17C2×D412yes—
101C2×C4×C17C2×D412yes—
102C22⋊D17—48no—
103C22⋊D17—44no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—21no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—21no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—21no—
116C2×Dic17#4114yes—
117#2661—171nomaximal, Sylow
118C22×C17—22no—
119C22×C17—22no—
120C22×C17—22no—
121C22×C17—42no—
122C22×C17—42no—
123C22×C17—42no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17C22×D411yescharacteristic
131C22×C17C2×#4112yes—
132C22×C17H(5,𝔽2)12yes—
133C22×C17H(5,𝔽2)11yescharacteristic
134C22×C17H(5,𝔽2)11yescharacteristic
135C2×D17—41no—
136C2×D17—41no—
137C2×D17—41no—
138C2×D17—42no—
139C2×D17—42no—
140C4×C17—22no—
141C4×C17—22no—
142C4×C17—42no—
143Dic17—24no—
144Dic17—41no—
145Dic17—41no—
146Dic17—41no—
147Dic17—42no—
148C2×#123—171no—
149C2×#121D1711yescharacteristic, complemented
150#398—171no—
151#397—171no—
152#394—171no—
153#394—172no—
154#392—171no—
155#392—172no—
156#390—172no—
157#388—172no—
158C4×C22⋊C4—171no—
159C2×C17—42no—
160C2×C17—82no—
161C2×C17#53211yescharacteristic
162C2×C17#53411yescharacteristic
163C2×C17#53511yescharacteristic
164C2×C17#53812yes—
165C2×C17#54012yes—
166D17—81no—
167C22×D4—171no—
168C22×D4—172no—
169C22×D4C2×D1712yes—
170C22×D4C2×D1711yescharacteristic
171C23×C4C2×D1711yescharacteristic
172#123—344no—
173#121—24no—
174C2×C4⋊C4—171no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—172no—
177C2×C4⋊C4C2×D1711yescharacteristic
178C2×C22⋊C4—171no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—171no—
182C2×C22⋊C4—171no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4C2×D1712yes—
187C2×C42—172no—
188#95—171no—
189#95—171no—
190#95—172no—
191#95—172no—
192C17#266111yescharacteristic, complemented, Sylow
193C24—171no—
194C24C22×D1712yes—
195C24C22×D1711yescharacteristic
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—344no—
202C2×D4—342no—
203C2×D4—682no—
204C2×D4—682no—
205C2×D4—682no—
206C22×C4—42no—
207C22×C4—172no—
208C22×C4—172no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—171no—
212C22×C4C22⋊D1714yescomplemented
213C22×C4C22×D1712yes—
214C22×C4C22×D1711yescharacteristic
215C22×C4C22×D1711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—342no—
218C4⋊C4—342no—
219C4⋊C4—682no—
220C22⋊C4—24no—
221C22⋊C4—344no—
222C22⋊C4—344no—
223C22⋊C4—342no—
224C22⋊C4—342no—
225C22⋊C4—342no—
226C22⋊C4—682no—
227C22⋊C4—682no—
228C22⋊C4—682no—
229C42—682no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—341no—
242C23—342no—
243C23—341no—
244C23—342no—
245C23—341no—
246C23C2×C22⋊D1712yes—
247C23C23×D1711yescharacteristic
248D4—44no—
249D4—48no—
250D4—688no—
251D4—684no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—174no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—342no—
262C2×C4—341no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—341no—
266C2×C4—342no—
267C2×C4—342no—
268C2×C4—341no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4C2×C22⋊D1712yes—
272C2×C4C2×C22⋊D1712yes—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—681no—
286C22—681no—
287C22—681no—
288C22—682no—
289C22—682no—
290C22#1058769112yes—
291C22#1058769111yescharacteristic
292C22#1058769111yescharacteristic
293C22#41×D1712yes—
294C22C22×C22⋊D1711yescharacteristic
295C4—22no—
296C4—22no—
297C4—42no—
298C4—344no—
299C4—681no—
300C4—681no—
301C4—681no—
302C4—682no—
303C2—42no—
304C2—82no—
305C2—1361no—
306C2#1176335511yescharacteristic
307C2#1176336811yescharacteristic
308C2#1176337611yescharacteristic
309C2#1176339112yes—
310C2#1176340912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2661
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
2. #11762740C2semidirect
3. #11762735C2semidirect
4. #11762723C2semidirect
5. #11762723C2semidirect
6. #11762717C2semidirect
7. #11762717C2semidirect
8. #11762709C2semidirect
9. #11762703C2semidirect
10. C4×C22⋊Dic17C2semidirect
11. C2×C17×#121C2semidirect
46. C22×C4×C17D4semidirect
149. C2×#121D17semidirect
192. C17#2661semidirect
212. C22×C4C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
#117627402
#117627352
#117627232
#117627172
#117627092
#117627032
C4×C22⋊Dic172
C2×C17×#1212
#266117