Finite Group Databasebeta

← #14536805

Subgroups — #14536805

Order 2176.

644 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587570C211yesmaximal, characteristic, complemented
2C2×#10587624C211yesmaximal, characteristic, complemented
3#11762737C212yesmaximal, complemented
4#11762736C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic, complemented
6#11762718C212yesmaximal, complemented
7#11762717C212yesmaximal, complemented
8#11762717C212yesmaximal, complemented
9#11762717C211yesmaximal, characteristic, complemented
10#11762671C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C22×C22⋊D17C2211yescharacteristic
15C22×C22⋊D17C2211yescharacteristic
16C22×C22⋊D17C2212yes—
17C22×C22⋊D17C2212yes—
18C22×D68C2211yescharacteristic
19C22×C4×D17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24C2×C22⋊Dic17C2212yes—
25#10587624—24no—
26C2×#63227C2211yescharacteristic
27C2×#63227C2212yes—
28C2×#63227C2212yes—
29C2×C4⋊Dic17C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31#10587570—24no—
32C17×#95C2211yescharacteristic
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2312yes—
38C24×C17C2311yescharacteristic
39C23×D17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C2×D4×C17—22no—
42C2×D4×C17—42no—
43C2×D4×C17—42no—
44C2×D4×C17—42no—
45C2×D4×C17—42no—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2312yes—
48C22×C4×C17C2311yescharacteristic
49C2×C22⋊D17—22no—
50C2×C22⋊D17—24no—
51C2×C22⋊D17—22no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C2×C22⋊D17—42no—
55C2×C22⋊D17—42no—
56C2×C22⋊D17—42no—
57C2×C22⋊D17—42no—
58C2×C22⋊D17—42no—
59C22×Dic17C2312yes—
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2312yes—
62C22×Dic17C2311yescharacteristic
63C2×D68—22no—
64C2×D68—42no—
65C2×C4×D17—42no—
66C2×C4×D17D414yescomplemented
67C22⋊C4×C17—22no—
68C22⋊C4×C17—42no—
69C22⋊Dic17—24no—
70C22⋊Dic17—42no—
71C22⋊Dic17—42no—
72#63227—22no—
73#63227—42no—
74#63227—42no—
75C4⋊Dic17—22no—
76#63225—22no—
77C23×C17—22no—
78C23×C17—21no—
79C23×C17—21no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—22no—
85C23×C17—22no—
86C23×C17—22no—
87C23×C17—22no—
88C23×C17—21no—
89C23×C17C2411yescharacteristic
90C22×D17—22no—
91C22×D17—22no—
92C22×D17—21no—
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—22no—
96C22×D17—22no—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17C2×D412yes—
100D4×C17—44no—
101D4×C17—48no—
102C2×C4×C17—22no—
103C2×C4×C17—21no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17C2×D412yes—
109C22⋊D17—44no—
110C22⋊D17—44no—
111C22⋊D17—48no—
112C22⋊D17—48no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—21no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—22no—
122C2×Dic17C2×D412yes—
123D68—44no—
124C4×D17—44no—
125#2815—171nomaximal, Sylow
126C22×C17—41no—
127C22×C17—42no—
128C22×C17—41no—
129C22×C17—42no—
130C22×C17—42no—
131C22×C17—41no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17C22×D411yescharacteristic
139C22×C17H(5,𝔽2)12yes—
140C22×C17H(5,𝔽2)12yes—
141C22×C17H(5,𝔽2)11yescharacteristic
142C22×C17H(5,𝔽2)11yescharacteristic
143C2×D17—22no—
144C2×D17—22no—
145C2×D17—22no—
146C2×D17—42no—
147C2×D17—42no—
148C2×D17—41no—
149C2×D17—41no—
150C2×D17—41no—
151C2×D17—42no—
152C2×D17—42no—
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—42no—
156Dic17—22no—
157Dic17—41no—
158Dic17—42no—
159Dic17—42no—
160C2×#121—171no—
161C2×#121—171no—
162#397—171no—
163#397—172no—
164#397—171no—
165#392—172no—
166#392—171no—
167#392—172no—
168#392—172no—
169#392D1711yescharacteristic, complemented
170#381—171no—
171C2×C17—81no—
172C2×C17—82no—
173C2×C17#53411yescharacteristic
174C2×C17#53511yescharacteristic
175C2×C17#53511yescharacteristic
176C2×C17#56012yes—
177C2×C17#56012yes—
178D17—42no—
179D17—81no—
180C22×D4—171no—
181C22×D4—172no—
182C22×D4—171no—
183C22×D4—172no—
184C22×D4—171no—
185C22×D4C2×D1711yescharacteristic
186C22×D4C2×D1712yes—
187C23×C4—171no—
188#121—344no—
189#121—344no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C22⋊C4—172no—
193C2×C22⋊C4—171no—
194C2×C22⋊C4—172no—
195C2×C22⋊C4—172no—
196C2×C22⋊C4—172no—
197C2×C22⋊C4—172no—
198C2×C22⋊C4C2×D1711yescharacteristic
199C2×C22⋊C4C2×D1712yes—
200#95—172no—
201#95—171no—
202#95—172no—
203#95—172no—
204#95C2×D1711yescharacteristic
205C17#281511yescharacteristic, complemented, Sylow
206C24—171no—
207C24—171no—
208C24C22×D1711yescharacteristic
209C24C22×D1712yes—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—342no—
216C2×D4—342no—
217C2×D4—342no—
218C2×D4—344no—
219C2×D4—682no—
220C2×D4—682no—
221C2×D4—682no—
222C2×D4—682no—
223C2×D4—682no—
224C2×D4—682no—
225C2×D4—682no—
226C2×D4—682no—
227C22×C4—171no—
228C22×C4—174no—
229C22×C4—172no—
230C22×C4—171no—
231C22×C4—172no—
232C22×C4—682no—
233C22×C4C22×D1712yes—
234C22×C4C22×D1711yescharacteristic
235C22×C4C22×D1711yescharacteristic
236C4⋊C4—342no—
237C4⋊C4—342no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—344no—
241C22⋊C4—342no—
242C22⋊C4—682no—
243C22⋊C4—682no—
244C22⋊C4—682no—
245C22⋊C4—682no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—172no—
259C23—342no—
260C23—342no—
261C23—341no—
262C23—341no—
263C23—341no—
264C23—341no—
265C23—342no—
266C23—342no—
267C23—341no—
268C23C23×D1711yescharacteristic
269D4—44no—
270D4—48no—
271D4—684no—
272D4—684no—
273D4—684no—
274D4—688no—
275D4—688no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—172no—
283C2×C4—342no—
284C2×C4—341no—
285C2×C4—342no—
286C2×C4—342no—
287C2×C4—342no—
288C2×C4—342no—
289C2×C4—342no—
290C2×C4—342no—
291C2×C4—342no—
292C2×C4—684no—
293C2×C4D4×D1712yes—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—342no—
307C22—342no—
308C22—342no—
309C22—682no—
310C22—682no—
311C22—682no—
312C22—682no—
313C22—681no—
314C22—681no—
315C22—681no—
316C22C2×D4×D1711yescharacteristic
317C22#1058769112yes—
318C22#1058769112yes—
319C22#1058769111yescharacteristic
320C22#1058769611yescharacteristic
321C4—22no—
322C4—42no—
323C4—41no—
324C4—342no—
325C4—682no—
326C4—681no—
327C4—682no—
328C2—82no—
329C2—81no—
330C2—682no—
331C2—1361no—
332C2#1176336911yescharacteristic
333C2#1176337211yescharacteristic
334C2#1176337511yescharacteristic
335C2#1176354712yes—
336C2#1176354812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2815
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587570C2semidirect
2. C2×#10587624C2semidirect
3. #11762737C2semidirect
4. #11762736C2semidirect
5. #11762734C2semidirect
6. #11762718C2semidirect
7. #11762717C2semidirect
8. #11762717C2semidirect
9. #11762717C2semidirect
10. #11762671C2semidirect
11. C17×#392C2semidirect
66. C2×C4×D17D4semidirect
169. #392D17semidirect
205. C17#2815semidirect

Maximal subgroups

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

subgroupindex
C2×#105875702
C2×#105876242
#117627372
#117627362
#117627342
#117627182
#117627172
#117626712
C17×#3922
#281517