Finite Group Databasebeta

← #14536823

Subgroups — #14536823

Order 2176.

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

normal subgroups only

This group has 630 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×#10587626C211yesmaximal, characteristic, complemented
2C2×#10587565C211yesmaximal, characteristic, complemented
3#11762727C211yesmaximal, characteristic, complemented
4#11762725C212yesmaximal, complemented
5#11762724C212yesmaximal, complemented
6#11762723C211yesmaximal, characteristic, complemented
7#11762718C211yesmaximal, characteristic, complemented
8#11762717C212yesmaximal, complemented
9#11762717C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#392C211yesmaximal, characteristic, complemented
12C22×D4×C17C2211yescharacteristic
13C22×D4×C17C2212yes—
14C22×C22⋊D17C2212yes—
15C22×C22⋊D17C2212yes—
16C22×C22⋊D17C2212yes—
17C22×D68C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2212yes—
20C2×C22⋊Dic17C2211yescharacteristic
21C2×C22⋊Dic17C2211yescharacteristic
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24#10587626—24no—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×C4⋊Dic17C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×C4×Dic17C2211yescharacteristic
32#10587565—24no—
33C17×#95C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C24×C17C2312yes—
39C23×D17C2312yes—
40C2×D4×C17—22no—
41C2×D4×C17—42no—
42C2×D4×C17—42no—
43C2×D4×C17—42no—
44C2×D4×C17—42no—
45C22×C4×C17C2312yes—
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—24no—
49C2×C22⋊D17—24no—
50C2×C22⋊D17—42no—
51C2×C22⋊D17—42no—
52C2×C22⋊D17—42no—
53C2×C22⋊D17—42no—
54C2×C22⋊D17—42no—
55C2×C22⋊D17—42no—
56C2×C22⋊D17—42no—
57C22×Dic17C2312yes—
58C22×Dic17C2311yescharacteristic
59C22×Dic17C2311yescharacteristic
60C22×Dic17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C2×D68—22no—
63C2×D68—42no—
64C22⋊C4×C17—22no—
65C22⋊C4×C17—42no—
66C22⋊Dic17—22no—
67C22⋊Dic17—22no—
68C22⋊Dic17—42no—
69C22⋊Dic17—42no—
70#63227—42no—
71#63227—42no—
72#63227—42no—
73C4⋊Dic17—22no—
74#63225—22no—
75#63225—22no—
76C4×Dic17D414yescomplemented
77C23×C17—21no—
78C23×C17—22no—
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—21no—
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—22no—
93C22×D17—22no—
94C22×D17—22no—
95C22×D17—22no—
96C22×D17—22no—
97D4×C17—44no—
98D4×C17—48no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17C2×D412yes—
106C22⋊D17—48no—
107C22⋊D17—48no—
108C22⋊D17—48no—
109C2×Dic17—22no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—22no—
116C2×Dic17—22no—
117C2×Dic17—22no—
118C2×Dic17—22no—
119C2×Dic17—21no—
120C2×Dic17C2×D412yes—
121C2×Dic17C2×D412yes—
122D68—44no—
123#2818—171nomaximal, Sylow
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—42no—
127C22×C17—41no—
128C22×C17—41no—
129C22×C17—41no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—42no—
133C22×C17—42no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17C22×D411yescharacteristic
137C22×C17H(5,𝔽2)12yes—
138C22×C17H(5,𝔽2)12yes—
139C22×C17H(5,𝔽2)11yescharacteristic
140C22×C17H(5,𝔽2)11yescharacteristic
141C2×D17—42no—
142C2×D17—42no—
143C2×D17—42no—
144C2×D17—42no—
145C2×D17—42no—
146C2×D17—42no—
147C2×D17—42no—
148C4×C17—22no—
149C4×C17—41no—
150C4×C17—42no—
151Dic17—22no—
152Dic17—22no—
153Dic17—41no—
154Dic17—41no—
155Dic17—42no—
156C2×#127—171no—
157C2×#126—171no—
158#394—171no—
159#394—172no—
160#394—172no—
161#394—171no—
162#392—172no—
163#392—172no—
164#392—171no—
165#392D1711yescharacteristic, complemented
166#383—171no—
167C2×C17—82no—
168C2×C17—81no—
169C2×C17#53511yescharacteristic
170C2×C17#53511yescharacteristic
171C2×C17#53511yescharacteristic
172C2×C17#56012yes—
173C2×C17#56012yes—
174D17—82no—
175C22×D4—172no—
176C22×D4—172no—
177C22×D4—172no—
178C22×D4—171no—
179C22×D4C2×D1711yescharacteristic
180C22×D4C2×D1712yes—
181#127—344no—
182#126—344no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—171no—
186C2×C22⋊C4—171no—
187C2×C22⋊C4—171no—
188C2×C22⋊C4—172no—
189C2×C22⋊C4—172no—
190C2×C22⋊C4—172no—
191C2×C22⋊C4—172no—
192C2×C22⋊C4—172no—
193C2×C22⋊C4C2×D1711yescharacteristic
194C2×C22⋊C4C2×D1712yes—
195C2×C42—171no—
196#95—172no—
197#95—172no—
198#95—171no—
199#95C2×D1711yescharacteristic
200C17#281811yescharacteristic, complemented, Sylow
201C24—172no—
202C24C22×D1711yescharacteristic
203C24C22×D1712yes—
204C2×D4—22no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—344no—
210C2×D4—344no—
211C2×D4—342no—
212C2×D4—682no—
213C2×D4—682no—
214C2×D4—682no—
215C2×D4—682no—
216C2×D4—682no—
217C2×D4—682no—
218C2×D4—682no—
219C2×D4—682no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—172no—
223C22×C4—171no—
224C22×C4—171no—
225C22×C4C22×D1712yes—
226C22×C4C22×D1711yescharacteristic
227C22×C4C22×D1711yescharacteristic
228C4⋊C4—342no—
229C4⋊C4—342no—
230C4⋊C4—342no—
231C22⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—342no—
234C22⋊C4—342no—
235C22⋊C4—682no—
236C22⋊C4—682no—
237C22⋊C4—682no—
238C22⋊C4—682no—
239C22⋊C4—682no—
240C42—174no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—342no—
254C23—342no—
255C23—342no—
256C23—342no—
257C23—342no—
258C23—342no—
259C23—342no—
260C23C23×D1711yescharacteristic
261D4—44no—
262D4—48no—
263D4—684no—
264D4—688no—
265D4—688no—
266D4—688no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—172no—
274C2×C4—172no—
275C2×C4—342no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—342no—
281C2×C4—341no—
282C2×C4—342no—
283C2×C4—342no—
284C2×C4—341no—
285C2×C4—342no—
286C2×C4D4×D1712yes—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—682no—
300C22—682no—
301C22—682no—
302C22—682no—
303C22—682no—
304C22—682no—
305C22—682no—
306C22C2×D4×D1711yescharacteristic
307C22#1058769112yes—
308C22#1058769112yes—
309C22#1058769111yescharacteristic
310C22#1058769611yescharacteristic
311C4—22no—
312C4—42no—
313C4—41no—
314C4—342no—
315C4—342no—
316C4—682no—
317C4—681no—
318C4—681no—
319C2—81no—
320C2—82no—
321C2—1362no—
322C2#1176337211yescharacteristic
323C2#1176337411yescharacteristic
324C2#1176337511yescharacteristic
325C2#1176354712yes—
326C2#1176354812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#392C2
hypercenter#392D17
hypocenterC17#2818
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#10587626C2semidirect
2. C2×#10587565C2semidirect
3. #11762727C2semidirect
4. #11762725C2semidirect
5. #11762724C2semidirect
6. #11762723C2semidirect
7. #11762718C2semidirect
8. #11762717C2semidirect
9. #11762717C2semidirect
10. #11762676C2semidirect
11. C17×#392C2semidirect
76. C4×Dic17D4semidirect
165. #392D17semidirect
200. C17#2818semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
C2×#105875652
#117627272
#117627252
#117627242
#117627232
#117627182
#117627172
#117626762
C17×#3922
#281817