Finite Group Databasebeta

← #14534674

Subgroups — #14534674

Order 2176.

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

normal subgroups only

This group has 620 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×#10587581C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic, complemented
3#11762747C211yesmaximal, characteristic, complemented
4#11762727C211yesmaximal, characteristic, complemented
5#11762726C212yesmaximal, complemented
6#11762718C212yesmaximal, complemented
7#11762718C211yesmaximal, characteristic, complemented
8#11762699C212yesmaximal, complemented
9#11762686C212yesmaximal, complemented
10#11762671C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C22×C22⋊D17C2212yes—
14C23×Dic17C2211yescharacteristic
15C22×D68C2212yes—
16C22×D68C2211yescharacteristic
17C22×C4×D17C2211yescharacteristic
18C2×C4⋊C4×C17C2211yescharacteristic
19C2×C4⋊C4×C17C2212yes—
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2211yescharacteristic
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×#63227C2212yes—
28C2×#63225C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587581—24no—
31#10587567—24no—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2312yes—
39C23×D17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—22no—
45C2×C22⋊D17—22no—
46C2×C22⋊D17—24no—
47C2×C22⋊D17—22no—
48C2×C22⋊D17—42no—
49C2×C22⋊D17—42no—
50C22×Dic17—22no—
51C22×Dic17—42no—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2312yes—
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C2×D68—24no—
57C2×D68—22no—
58C2×D68—42no—
59C2×D68—42no—
60C2×D68—42no—
61C2×C4×D17—42no—
62C2×C4×D17D414yescomplemented
63C4⋊C4×C17—22no—
64C4⋊C4×C17—42no—
65C22⋊C4×C17—22no—
66C22⋊C4×C17—42no—
67#63227—22no—
68#63227—24no—
69#63227—24no—
70#63227—42no—
71#63227—42no—
72#63227—42no—
73#63225—22no—
74C4×Dic17—22no—
75C23×C17—21no—
76C23×C17—21no—
77C23×C17—22no—
78C23×C17—22no—
79C23×C17#4112yes—
80C23×C17C2411yescharacteristic
81C22×D17—21no—
82C22×D17—21no—
83C22×D17—22no—
84C22×D17—22no—
85C22×D17—22no—
86C22×D17—22no—
87C22×D17—22no—
88C22×D17—22no—
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—22no—
92C22×D17C2×D412yes—
93C2×C4×C17—22no—
94C2×C4×C17—21no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17C2×D412yes—
103C22⋊D17—28no—
104C22⋊D17—48no—
105C2×Dic17—21no—
106C2×Dic17—21no—
107C2×Dic17—22no—
108C2×Dic17—21no—
109C2×Dic17—22no—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—44no—
114C2×Dic17C2×D412yes—
115C2×Dic17#4112yes—
116D68—44no—
117D68—48no—
118C4×D17—44no—
119#2635—171nomaximal, Sylow
120C22×C17—22no—
121C22×C17—24no—
122C22×C17—41no—
123C22×C17—41no—
124C22×C17—42no—
125C22×C17C22×D411yescharacteristic
126C22×C17C2×#4111yescharacteristic
127C22×C17C2×#4111yescharacteristic
128C22×C17H(5,𝔽2)12yes—
129C22×C17H(5,𝔽2)12yes—
130C2×D17—22no—
131C2×D17—22no—
132C2×D17—22no—
133C2×D17—42no—
134C2×D17—42no—
135C2×D17—42no—
136C2×D17—42no—
137C2×D17—42no—
138C2×D17—42no—
139C2×D17—42no—
140C2×D17—42no—
141C2×D17—42no—
142C4×C17—22no—
143C4×C17—42no—
144C4×C17—41no—
145C4×C17—42no—
146Dic17—22no—
147Dic17—22no—
148Dic17—41no—
149Dic17—42no—
150C2×#121—171no—
151C2×C4×D4—171no—
152#400—171no—
153#399D1711yescharacteristic, complemented
154#394—172no—
155#394—171no—
156#392—171no—
157#392—172no—
158#388—172no—
159#386—172no—
160#381—171no—
161C2×C17—42no—
162C2×C17#52511yescharacteristic
163C2×C17#53411yescharacteristic
164C2×C17#53511yescharacteristic
165C2×C17#53812yes—
166C2×C17#54012yes—
167D17—42no—
168D17—82no—
169C22×D4—171no—
170C22×D4—172no—
171C22×D4—172no—
172C22×D4—171no—
173C23×C4—171no—
174C23×C4—171no—
175#121—344no—
176C4×D4—344no—
177C2×C4⋊C4—171no—
178C2×C4⋊C4C2×D1711yescharacteristic
179C2×C4⋊C4C2×D1712yes—
180C2×C22⋊C4—171no—
181C2×C22⋊C4—172no—
182C2×C22⋊C4—172no—
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#95C2×D1711yescharacteristic
194C17#263511yescharacteristic, complemented, Sylow
195C24—171no—
196C24—172no—
197C24C22×D1711yescharacteristic
198C2×D4—344no—
199C2×D4—342no—
200C2×D4—342no—
201C2×D4—342no—
202C2×D4—344no—
203C2×D4—342no—
204C2×D4—682no—
205C2×D4—682no—
206C2×D4—682no—
207C2×D4—682no—
208C2×D4—682no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—174no—
212C22×C4—172no—
213C22×C4—171no—
214C22×C4—342no—
215C22×C4—682no—
216C22×C4—682no—
217C22×C4C22×D1712yes—
218C22×C4C22×D1712yes—
219C22×C4C22×D1711yescharacteristic
220C22×C4C22×D1711yescharacteristic
221C4⋊C4—22no—
222C4⋊C4—42no—
223C4⋊C4—342no—
224C22⋊C4—22no—
225C22⋊C4—42no—
226C22⋊C4—344no—
227C22⋊C4—344no—
228C22⋊C4—342no—
229C22⋊C4—682no—
230C22⋊C4—682no—
231C22⋊C4—682no—
232C42—342no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—22no—
237C23—172no—
238C23—342no—
239C23—341no—
240C23—341no—
241C23—342no—
242C23—342no—
243C23—342no—
244C23—342no—
245C23—342no—
246C23—342no—
247C23—342no—
248C23—342no—
249C23#6325612yes—
250C23C23×D1711yescharacteristic
251D4—348no—
252D4—684no—
253D4—688no—
254D4—688no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—172no—
265C2×C4—172no—
266C2×C4—341no—
267C2×C4—341no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—342no—
271C2×C4—342no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4—684no—
275C2×C4—684no—
276C2×C4D4×D1712yes—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—342no—
283C22—342no—
284C22—342no—
285C22—682no—
286C22—682no—
287C22—682no—
288C22—682no—
289C22—682no—
290C22—682no—
291C22—682no—
292C22—682no—
293C22—682no—
294C22C2×D4×D1711yescharacteristic
295C22#1058769112yes—
296C22C2×#6325611yescharacteristic
297C22#41×D1711yescharacteristic
298C22#1058769612yes—
299C4—22no—
300C4—42no—
301C4—41no—
302C4—42no—
303C4—342no—
304C4—342no—
305C4—681no—
306C4—682no—
307C2—42no—
308C2—682no—
309C2—1362no—
310C2#1176332211yescharacteristic
311C2#1176337011yescharacteristic
312C2#1176337211yescharacteristic
313C2#1176339912yes—
314C2#1176341112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
hypercenter#399D17
hypocenterC17#2635
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587581C2semidirect
2. C2×#10587567C2semidirect
3. #11762747C2semidirect
4. #11762727C2semidirect
5. #11762726C2semidirect
6. #11762718C2semidirect
7. #11762718C2semidirect
8. #11762699C2semidirect
9. #11762686C2semidirect
10. #11762671C2semidirect
11. C17×#399C2semidirect
62. C2×C4×D17D4semidirect
153. #399D17semidirect
194. C17#2635semidirect

Maximal subgroups

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

subgroupindex
C2×#105875812
C2×#105875672
#117627472
#117627272
#117627262
#117627182
#117626992
#117626862
#117626712
C17×#3992
#263517