Finite Group Databasebeta

← #14534356

Subgroups — #14534356

Order 2176.

492 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 139 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×Q8×Dic17C211yesmaximal, characteristic, complemented
2C2×#10587578C211yesmaximal, characteristic
3#11762739C211yesmaximal, characteristic, complemented
4#11762736C211yesmaximal, characteristic
5#11762720C211yesmaximal, characteristic
6#11762699C212yesmaximal
7#11762696C212yesmaximal, complemented
8#11762695C212yesmaximal, complemented
9#11762683C212yesmaximal
10#11762676C211yesmaximal, characteristic, complemented
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16Q8×Dic17—24no—
17C2×#63227C2211yescharacteristic
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2211yescharacteristic
21C2×C4⋊Dic17C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26C2×C4×Dic17C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587578—24no—
29C17×#95C2212yes—
30C17×#95C2211yescharacteristic
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C23×D17C2311yescharacteristic
38C2×Q8×C17—22no—
39C2×Q8×C17—24no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2312yes—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×C4×D17—22no—
51C2×C4×D17—22no—
52C2×C4×D17—22no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—24no—
55#63227—24no—
56#63227—22no—
57#63227—24no—
58#63227—22no—
59C4⋊Dic17—22no—
60C4⋊Dic17—22no—
61C4⋊Dic17—24no—
62#63225—24no—
63C4×Dic17—22no—
64C4×Dic17—24no—
65C4×Dic17D414yes—
66C23×C17C2411yescharacteristic
67C22×D17—21no—
68C22×D17—21no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17#4112yes—
72Q8×C17—28no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D412yes—
84C2×C4×C17#4114yes—
85C2×C4×C17#4112yes—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17C2×D412yes—
98C2×Dic17C2×D412yes—
99C4×D17—28no—
100#2599—171nomaximal, Sylow
101C22×C17C22×D411yescharacteristic
102C22×C17C2×#4112yes—
103C22×C17C2×#4111yescharacteristic
104C22×C17C2×#4111yescharacteristic
105C22×C172−1+412yes—
106C2×D17—22no—
107C2×D17—24no—
108C2×D17—41no—
109C2×D17—41no—
110C2×D17—42no—
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—24no—
114C4×C17—41no—
115C4×C17—42no—
116Dic17—22no—
117Dic17—22no—
118Dic17—42no—
119Dic17—41no—
120Dic17—42no—
121C2×C4×Q8—171no—
122C2×#117—171no—
123#398—171no—
124#397—171no—
125#395D1711yescharacteristic, complemented
126#393—171no—
127#388—172no—
128#387—172no—
129#387—172no—
130#385—172no—
131#383—171no—
132C2×C17#52511yescharacteristic
133C2×C17#52511yescharacteristic
134C2×C17#53611yescharacteristic
135C2×C17#55212yes—
136C2×C17#55612yes—
137D17—42no—
138C22×Q8C2×D1711yescharacteristic
139C23×C4—171no—
140C4×Q8—344no—
141#117—344no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—171no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C4⋊C4C2×D1712yes—
148C2×C22⋊C4—171no—
149C2×C22⋊C4—171no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C42—171no—
153C2×C42—171no—
154C2×C42—172no—
155#95—172no—
156#95—172no—
157#95—171no—
158#95—172no—
159#95—172no—
160#95—172no—
161#95C2×D1711yescharacteristic
162#95C2×D1712yes—
163C17#259911yescharacteristic, complemented, Sylow
164C24—171no—
165C2×Q8—22no—
166C2×Q8—24no—
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—172no—
170C22×C4—171no—
171C22×C4—172no—
172C22×C4—342no—
173C22×C4—342no—
174C22×C4—342no—
175C22×C4C22×D1712yes—
176C22×C4C22×D1711yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C22×C4C22×D1712yes—
179C22×C4C22×D1711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—342no—
183C4⋊C4—344no—
184C4⋊C4—342no—
185C4⋊C4—344no—
186C22⋊C4—342no—
187C22⋊C4—344no—
188C22⋊C4—342no—
189C22⋊C4—344no—
190C42—174no—
191C42—342no—
192C42—344no—
193C23—172no—
194C23—341no—
195C23—342no—
196C23—341no—
197C23—342no—
198C23C23×D1711yescharacteristic
199Q8—28no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—172no—
211C2×C4—172no—
212C2×C4—348no—
213C2×C4—341no—
214C2×C4—342no—
215C2×C4—341no—
216C2×C4—341no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4D4×D1712yes—
225C2×C4#6325414yes—
226C2×C4#6325612yes—
227C22—344no—
228C22—342no—
229C22—682no—
230C22—681no—
231C22—681no—
232C22C2×D4×D1711yescharacteristic
233C22C2×#6325412yes—
234C22C2×#6325611yescharacteristic
235C22#1058769412yes—
236C22#41×D1711yescharacteristic
237C4—22no—
238C4—24no—
239C4—22no—
240C4—42no—
241C4—41no—
242C4—342no—
243C4—342no—
244C4—682no—
245C4—682no—
246C4—681no—
247C2—682no—
248C2#1176331511yescharacteristic
249C2#1176332211yescharacteristic
250C2#1176338411yescharacteristic
251C2#1176349312yes—
252C2#1176353312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC23×D17C23
hypercenter#395D17
hypocenterC17#2599
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic17C2semidirect
3. #11762739C2semidirect
7. #11762696C2semidirect
8. #11762695C2semidirect
10. #11762676C2semidirect
11. C17×#395C2semidirect
125. #395D17semidirect
163. C17#2599semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic172
C2×#105875782
#117627392
#117627362
#117627202
#117626992
#117626962
#117626952
#117626832
#117626762
C17×#3952
#259917