Finite Group Databasebeta

← #14533267

Subgroups — #14533267

Order 2176.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587564C211yesmaximal, characteristic
2C22×#63225C211yesmaximal, characteristic
3#11762751C212yesmaximal, complemented
4#11762750C212yesmaximal, complemented
5#11762713C212yesmaximal, complemented
6#11762673C212yesmaximal, complemented
7#11762672C211yesmaximal, characteristic
8#11762670C211yesmaximal, characteristic
9#11762670C211yesmaximal, characteristic
10C22⋊C4×Dic17C211yesmaximal, characteristic
11C17×#381C211yesmaximal, characteristic
12C23×C4×C17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C23×Dic17C2211yescharacteristic
15C22×Dic34C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊Dic17C2211yescharacteristic
19C2×C22⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225—24no—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×C4×Dic17C2212yes—
27#10587564—24no—
28C17×#95C2212yes—
29C17×#95C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17—22no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2312yes—
44C22×C4×C17C2311yescharacteristic
45C22×Dic17—24no—
46C22×Dic17—22no—
47C22×Dic17—22no—
48C22×Dic17—22no—
49C22×Dic17D414yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22×Dic17C2311yescharacteristic
56C2×Dic34—22no—
57C2×Dic34—24no—
58C22⋊C4×C17—22no—
59C22⋊C4×C17—22no—
60C22⋊Dic17—22no—
61C22⋊Dic17—22no—
62C4⋊Dic17—22no—
63#63225—22no—
64#63225—22no—
65#63225—24no—
66#63225—24no—
67C4×Dic17—24no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17C2×D412yes—
71C23×C17#4112yes—
72C23×C17#4112yes—
73C23×C17C2411yescharacteristic
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—44no—
85C2×Dic17—22no—
86C2×Dic17—28no—
87C2×Dic17—21no—
88C2×Dic17—21no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—44no—
98C2×Dic17C2×D412yes—
99C2×Dic17C2×D412yes—
100C2×Dic17#4114yes—
101Dic34—44no—
102#2503—171nomaximal, Sylow
103C22×C17—22no—
104C22×C17—22no—
105C22×C17—24no—
106C22×C17—24no—
107C22×C17#12414yes—
108C22×C17C22×D411yescharacteristic
109C22×C17C2×#4111yescharacteristic
110C22×C17C2×#4111yescharacteristic
111C22×C17C2×#4112yes—
112C22×C172−1+412yes—
113C4×C17—41no—
114C4×C17—42no—
115C4×C17—41no—
116C4×C17—42no—
117Dic17—22no—
118Dic17—41no—
119Dic17—42no—
120Dic17—41no—
121Dic17—42no—
122Dic17#12314yes—
123C2×C22⋊Q8—171no—
124C22×C4⋊C4—171no—
125#400—172no—
126#400—172no—
127#391—172no—
128#382—172no—
129#381—171no—
130#381—171no—
131#381—171no—
132#381D1711yescharacteristic
133C4×C22⋊C4—171no—
134C2×C17—24no—
135C2×C17C2×#12311yescharacteristic
136C2×C17C2×#12411yescharacteristic
137C2×C17#53611yescharacteristic
138C2×C17#55212yes—
139C2×C17#55212yes—
140C22×Q8—171no—
141C23×C4—171no—
142C23×C4—171no—
143C23×C4C2×D1711yescharacteristic
144C22⋊Q8—344no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4—344no—
151C2×C22⋊C4—171no—
152C2×C22⋊C4—171no—
153C2×C22⋊C4C2×D1711yescharacteristic
154C2×C22⋊C4C2×D1711yescharacteristic
155C2×C42—172no—
156#95—172no—
157#95—172no—
158#95—172no—
159#95—172no—
160#95—172no—
161#95—172no—
162#95—172no—
163#95C2×D1712yes—
164#95C2×D1712yes—
165C17#250311yescharacteristic, complemented, Sylow
166C24C22×D1711yescharacteristic
167C2×Q8—342no—
168C2×Q8—344no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—171no—
173C22×C4—174no—
174C22×C4—171no—
175C22×C4—172no—
176C22×C4—171no—
177C22×C4—171no—
178C22×C4—172no—
179C22×C4—344no—
180C22×C4—342no—
181C22×C4—342no—
182C22×C4—342no—
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C22×C4C22×D1711yescharacteristic
186C22×C4C22×D1712yes—
187C4⋊C4—344no—
188C4⋊C4—344no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—342no—
195C22⋊C4—342no—
196C42—344no—
197C23—22no—
198C23—22no—
199C23#6325212yes—
200C23D4×D1712yes—
201C23#6325412yes—
202C23C23×D1711yescharacteristic
203Q8—684no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—172no—
216C2×C4—174no—
217C2×C4—172no—
218C2×C4—348no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—341no—
223C2×C4—341no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—684no—
231C22—24no—
232C22—24no—
233C22—22no—
234C22—22no—
235C22#1058757114yes—
236C22C2×#6325211yescharacteristic
237C22C2×D4×D1711yescharacteristic
238C22C2×#6325411yescharacteristic
239C22#41×D1712yes—
240C22#1058769712yes—
241C4—42no—
242C4—41no—
243C4—42no—
244C4—41no—
245C4—174no—
246C4—342no—
247C4—682no—
248C4—681no—
249C4—682no—
250C4—681no—
251C2—24no—
252C2#123×D1711yescharacteristic
253C2C2×#1058757111yescharacteristic
254C2#1176338111yescharacteristic
255C2#1176349212yes—
256C2#1176349612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC24×C17C23
hypercenter#381D17
hypocenterC17#2503
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #11762751C2semidirect
4. #11762750C2semidirect
5. #11762713C2semidirect
6. #11762673C2semidirect
165. C17#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
C22×#632252
#117627512
#117627502
#117627132
#117626732
#117626722
#117626702
C22⋊C4×Dic172
C17×#3812
#250317