Finite Group Databasebeta

← #14533858

Subgroups — #14533858

Order 2176.

696 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 145 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×#10587571C212yesmaximal, complemented
2C2×#10587570C212yesmaximal, complemented
3#11762729C211yesmaximal, characteristic, complemented
4#11762726C211yesmaximal, characteristic, complemented
5#11762717C212yesmaximal, complemented
6#11762697C212yesmaximal, complemented
7#11762685C212yesmaximal, complemented
8C4×#63227C212yesmaximal, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C22×C4×D17C2212yes—
15C22×Dic34C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C42×C17C2211yescharacteristic
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2212yes—
27#10587571—28no—
28#10587570—28no—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2211yescharacteristic
33C24×C17C2312yes—
34C23×D17C2312yes—
35C2×D4×C17—22no—
36C2×D4×C17—41no—
37C2×D4×C17—41no—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C2×C22⋊D17—24no—
43C2×C22⋊D17—24no—
44C2×C22⋊D17—24no—
45C2×C22⋊D17—24no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17—42no—
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C2×D68—24no—
52C2×D68—41no—
53C2×C4×D17—24no—
54C2×C4×D17—24no—
55C2×C4×D17—42no—
56C2×Dic34—24no—
57C2×Dic34—41no—
58C4⋊C4×C17—41no—
59C22⋊C4×C17—24no—
60C22⋊C4×C17—24no—
61C42×C17—41no—
62C22⋊Dic17—24no—
63C22⋊Dic17—24no—
64#63227—24no—
65#63227—24no—
66#63227D418yescomplemented
67#63225—24no—
68C4×Dic17—24no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17C2411yescharacteristic
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17C2×D414yes—
84D4×C17—44no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17C2×D414yes—
94C2×C4×C17#4112yes—
95C22⋊D17—48no—
96C22⋊D17—48no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17C2×D414yes—
105C2×Dic17#4114yes—
106D68—44no—
107C4×D17—48no—
108Dic34—44no—
109#2555—171nomaximal, Sylow
110C22×C17—42no—
111C22×C17—42no—
112C22×C17—42no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17C22×D412yes—
118C22×C17C2×#4111yescharacteristic
119C22×C17C2×#4112yes—
120C22×C17H(5,𝔽2)11yescharacteristic
121C22×C17H(5,𝔽2)11yescharacteristic
122C2×D17—24no—
123C2×D17—24no—
124C2×D17—24no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C4×C17—22no—
130C4×C17—24no—
131C4×C17—41no—
132C4×C17—41no—
133Dic17—24no—
134Dic17—24no—
135Dic17—42no—
136C2×#124—172no—
137C2×#121—172no—
138#395—171no—
139#394—171no—
140#392—172no—
141#390D1711yescharacteristic, complemented
142#388—172no—
143#386—172no—
144C4×C22⋊C4—172no—
145C2×C17—82no—
146C2×C17#53212yes—
147C2×C17#53811yescharacteristic
148C2×C17D4211yescharacteristic
149C2×C17#54612yes—
150C2×C17#55511yescharacteristic
151D17—44no—
152C22×Q8—171no—
153C22×D4—172no—
154C22×D4—172no—
155C22×D4—171no—
156C22×D4C2×D1711yescharacteristic
157C23×C4—172no—
158#124—348no—
159#121—348no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4C2×D1711yescharacteristic
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4C2×D1712yes—
168C2×C22⋊C4C2×D1712yes—
169C2×C42—172no—
170C2×C42C2×D1711yescharacteristic
171#95—172no—
172#95—172no—
173#95—171no—
174#95—171no—
175C17#255511yescharacteristic, complemented, Sylow
176C24—172no—
177C24C22×D1712yes—
178C2×Q8—344no—
179C2×Q8—681no—
180C2×D4—22no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—344no—
184C2×D4—344no—
185C2×D4—344no—
186C2×D4—344no—
187C2×D4—344no—
188C2×D4—681no—
189C2×D4—682no—
190C2×D4—682no—
191C22×C4—172no—
192C22×C4—172no—
193C22×C4—172no—
194C22×C4—344no—
195C22×C4—344no—
196C22×C4—682no—
197C22×C4C22×D1711yescharacteristic
198C22×C4C22×D1711yescharacteristic
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1712yes—
201C4⋊C4—41no—
202C4⋊C4—344no—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4—178no—
206C22⋊C4—344no—
207C22⋊C4—344no—
208C22⋊C4—344no—
209C22⋊C4—344no—
210C42—41no—
211C42—344no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—174no—
220C23—342no—
221C23—342no—
222C23—342no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23C23×D1711yescharacteristic
227Q8—684no—
228D4—44no—
229D4—684no—
230D4—688no—
231D4—688no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—174no—
241C2×C4—174no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—688no—
250C2×C4#6325212yes—
251C2×C4D4×D1714yes—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—344no—
260C22—344no—
261C22—344no—
262C22—682no—
263C22—682no—
264C22—682no—
265C22—682no—
266C22C2×#6325211yescharacteristic
267C22C2×D4×D1712yes—
268C22#1058769111yescharacteristic
269C22#1058769111yescharacteristic
270C22#41×D1712yes—
271C4—22no—
272C4—24no—
273C4—41no—
274C4—41no—
275C4—344no—
276C4—344no—
277C4—682no—
278C2—82no—
279C2—684no—
280C2#1176335612yes—
281C2#1176339711yescharacteristic
282C2#1176344411yescharacteristic
283C2#1176344812yes—
284C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
hypercenter#390D17
hypocenterC17#2555
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587571C2semidirect
2. C2×#10587570C2semidirect
3. #11762729C2semidirect
4. #11762726C2semidirect
5. #11762717C2semidirect
6. #11762697C2semidirect
7. #11762685C2semidirect
8. C4×#63227C2semidirect
9. C17×#390C2semidirect
66. #63227D4semidirect
141. #390D17semidirect
175. C17#2555semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
C2×#105875702
#117627292
#117627262
#117627172
#117626972
#117626852
C4×#632272
C17×#3902
#255517