Finite Group Databasebeta

← #14534641

Subgroups — #14534641

Order 2176.

628 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 131 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×#10587622C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic, complemented
3#11762737C212yesmaximal, complemented
4#11762722C211yesmaximal, characteristic, complemented
5#11762721C211yesmaximal, characteristic, complemented
6#11762717C212yesmaximal, complemented
7#11762717C211yesmaximal, characteristic, complemented
8#11762689C212yesmaximal, complemented
9#11762685C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic, complemented
12C22×D4×C17C2212yes—
13C22×D4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2212yes—
16C23×Dic17C2211yescharacteristic
17C22×C4×D17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25#10587622—24no—
26C2×#63227C2211yescharacteristic
27C2×#63227C2212yes—
28C2×#63225C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587567—24no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
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×C17C2311yescharacteristic
46C22×C4×C17C2312yes—
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—22no—
49C2×C22⋊D17—22no—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C22×Dic17—22no—
55C22×Dic17—42no—
56C22×Dic17C2312yes—
57C22×Dic17C2311yescharacteristic
58C22×Dic17C2312yes—
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C2×C4×D17—22no—
62C2×C4×D17—42no—
63C2×Dic34—22no—
64C2×Dic34—42no—
65C22⋊C4×C17—22no—
66C22⋊C4×C17—42no—
67C22⋊Dic17—24no—
68C22⋊Dic17—24no—
69C22⋊Dic17—42no—
70C22⋊Dic17—42no—
71#63227—22no—
72#63227—42no—
73#63225—22no—
74C4×Dic17D414yescomplemented
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17#4112yes—
87C23×C17C2411yescharacteristic
88C22×D17—21no—
89C22×D17—21no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17#4112yes—
93D4×C17—44no—
94D4×C17—48no—
95C2×C4×C17—22no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17C2×D412yes—
102C22⋊D17—28no—
103C22⋊D17—48no—
104C2×Dic17—21no—
105C2×Dic17—21no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—21no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—44no—
116C2×Dic17C2×D412yes—
117C2×Dic17C2×D412yes—
118C4×D17—44no—
119Dic34—44no—
120#2631—171nomaximal, Sylow
121C22×C17—22no—
122C22×C17—24no—
123C22×C17—41no—
124C22×C17—41no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17C22×D411yescharacteristic
134C22×C17C2×#4111yescharacteristic
135C22×C17C2×#4111yescharacteristic
136C22×C17H(5,𝔽2)12yes—
137C22×C17H(5,𝔽2)12yes—
138C2×D17—22no—
139C2×D17—24no—
140C2×D17—41no—
141C2×D17—41no—
142C2×D17—42no—
143C4×C17—22no—
144C4×C17—41no—
145C4×C17—42no—
146Dic17—22no—
147Dic17—22no—
148Dic17—42no—
149Dic17—41no—
150Dic17—42no—
151C2×#124—171no—
152C2×C4×D4—171no—
153#397—172no—
154#393—171no—
155#393—171no—
156#392—171no—
157#392—172no—
158#392D1711yescharacteristic, complemented
159#386—172no—
160#386—172no—
161#383—171no—
162C2×C17—42no—
163C2×C17—82no—
164C2×C17#52511yescharacteristic
165C2×C17#53511yescharacteristic
166C2×C17#53511yescharacteristic
167C2×C17#53812yes—
168C2×C17#53812yes—
169D17—42no—
170C22×Q8—171no—
171C22×D4—171no—
172C22×D4—172no—
173C22×D4C2×D1711yescharacteristic
174C22×D4C2×D1712yes—
175C23×C4—171no—
176C23×C4—171no—
177#124—344no—
178C4×D4—344no—
179C2×C4⋊C4—171no—
180C2×C22⋊C4—172no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—171no—
183C2×C22⋊C4—172no—
184C2×C22⋊C4—172no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4C2×D1711yescharacteristic
187C2×C22⋊C4C2×D1712yes—
188C2×C42—171no—
189#95—172no—
190#95—171no—
191#95—172no—
192#95—172no—
193#95—172no—
194#95C2×D1711yescharacteristic
195C17#263111yescharacteristic, complemented, Sylow
196C24—171no—
197C24C22×D1712yes—
198C24C22×D1711yescharacteristic
199C2×Q8—342no—
200C2×Q8—682no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—342no—
207C2×D4—342no—
208C2×D4—342no—
209C2×D4—344no—
210C2×D4—682no—
211C2×D4—682no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—172no—
215C22×C4—172no—
216C22×C4—171no—
217C22×C4—342no—
218C22×C4—342no—
219C22×C4—682no—
220C22×C4—682no—
221C22×C4C22×D1712yes—
222C22×C4C22×D1711yescharacteristic
223C22×C4C22×D1711yescharacteristic
224C4⋊C4—342no—
225C22⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—344no—
228C22⋊C4—344no—
229C22⋊C4—342no—
230C22⋊C4—682no—
231C22⋊C4—682no—
232C22⋊C4—682no—
233C42—174no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—172no—
246C23—341no—
247C23—341no—
248C23—342no—
249C23—342no—
250C23#6325412yes—
251C23C23×D1711yescharacteristic
252Q8—684no—
253D4—44no—
254D4—48no—
255D4—348no—
256D4—688no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—172no—
264C2×C4—172no—
265C2×C4—341no—
266C2×C4—342no—
267C2×C4—341no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4—684no—
277C2×C4—684no—
278C2×C4D4×D1712yes—
279C22—22no—
280C22—24no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—342no—
292C22—344no—
293C22—681no—
294C22—681no—
295C22—682no—
296C22C2×D4×D1711yescharacteristic
297C22C2×#6325411yescharacteristic
298C22#1058769112yes—
299C22#1058769112yes—
300C22#41×D1711yescharacteristic
301C4—22no—
302C4—41no—
303C4—42no—
304C4—342no—
305C4—342no—
306C4—681no—
307C4—682no—
308C4—682no—
309C2—42no—
310C2—82no—
311C2—682no—
312C2#1176331911yescharacteristic
313C2#1176337211yescharacteristic
314C2#1176337411yescharacteristic
315C2#1176339112yes—
316C2#1176339412yes—
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×#10587622C2semidirect
2. C2×#10587567C2semidirect
3. #11762737C2semidirect
4. #11762722C2semidirect
5. #11762721C2semidirect
6. #11762717C2semidirect
7. #11762717C2semidirect
8. #11762689C2semidirect
9. #11762685C2semidirect
10. #11762676C2semidirect
11. C17×#392C2semidirect
74. C4×Dic17D4semidirect
158. #392D17semidirect
195. C17#2631semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
C2×#105875672
#117627372
#117627222
#117627212
#117627172
#117626892
#117626852
#117626762
C17×#3922
#263117