Finite Group Databasebeta

← #14534361

Subgroups — #14534361

Order 2176.

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

normal subgroups only

This group has 492 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×C4×Dic34C211yesmaximal, characteristic, complemented
2#11762739C211yesmaximal, characteristic, complemented
3#11762734C211yesmaximal, characteristic
4#11762729C211yesmaximal, characteristic, complemented
5#11762721C211yesmaximal, characteristic
6#11762697C212yesmaximal
7#11762693C212yesmaximal, complemented
8#11762690C212yesmaximal, complemented
9#11762682C212yesmaximal
10#11762677C211yesmaximal, characteristic, complemented
11C2×C17×#117C211yesmaximal, characteristic
12C23×C4×C17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C17×#117—24no—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C22⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C42×C17C2211yescharacteristic
20C2×C42×C17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2212yes—
28C4×Dic34—24no—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17—22no—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×Dic17C2312yes—
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2312yes—
51C2×Dic34—22no—
52C2×Dic34—24no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—22no—
55C22⋊C4×C17—22no—
56C22⋊C4×C17—22no—
57C42×C17—22no—
58C42×C17D414yes—
59C22⋊Dic17—24no—
60C22⋊Dic17—24no—
61C4⋊Dic17—22no—
62#63225—24no—
63#63225—24no—
64#63225—24no—
65C4×Dic17—24no—
66C23×C17—21no—
67C23×C17—22no—
68C23×C17—21no—
69C23×C17—22no—
70C23×C17#4112yes—
71C23×C17C2411yescharacteristic
72C2×C4×C17—28no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17C2×D412yes—
84C2×C4×C17C2×D412yes—
85C2×C4×C17C2×D412yes—
86C2×C4×C17#4112yes—
87C2×Dic17—22no—
88C2×Dic17—22no—
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—22no—
98C2×Dic17#4114yes—
99Dic34—28no—
100#2599—171nomaximal, Sylow
101C22×C17—22no—
102C22×C17—24no—
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—42no—
106C22×C17C22×D411yescharacteristic
107C22×C17C2×#4111yescharacteristic
108C22×C17C2×#4111yescharacteristic
109C22×C17C2×#4112yes—
110C22×C172−1+412yes—
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—22no—
114C4×C17—22no—
115C4×C17—41no—
116C4×C17—41no—
117Dic17—24no—
118Dic17—42no—
119Dic17—42no—
120Dic17—42no—
121C2×C4×Q8—171no—
122C2×#117D1711yescharacteristic
123#398—171no—
124#397—171no—
125#395—171no—
126#393—171no—
127#388—172no—
128#387—172no—
129#387—172no—
130#385—172no—
131#383—171no—
132C2×C17—42no—
133C2×C17#52511yescharacteristic
134C2×C17#52511yescharacteristic
135C2×C17#53611yescharacteristic
136C2×C17#55212yes—
137C2×C17#55612yes—
138C22×Q8—171no—
139C23×C4C2×D1711yescharacteristic
140C4×Q8—344no—
141#117—24no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—171no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C4⋊C4C2×D1711yescharacteristic
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4C2×D1711yescharacteristic
151C2×C22⋊C4C2×D1711yescharacteristic
152C2×C42—172no—
153C2×C42C2×D1711yescharacteristic
154C2×C42C2×D1711yescharacteristic
155#95—172no—
156#95—172no—
157#95—171no—
158#95—171no—
159#95—172no—
160#95—172no—
161#95—172no—
162#95—172no—
163C17#259911yescharacteristic, complemented, Sylow
164C24C22×D1711yescharacteristic
165C2×Q8—344no—
166C2×Q8—342no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—172no—
171C22×C4—172no—
172C22×C4—172no—
173C22×C4—172no—
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1711yescharacteristic
176C22×C4C22×D1711yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C22×C4C22×D1711yescharacteristic
179C22×C4C22×D1711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—344no—
183C4⋊C4—344no—
184C4⋊C4—344no—
185C4⋊C4—342no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—344no—
189C22⋊C4—344no—
190C42—22no—
191C42—344no—
192C42C22⋊D1714yes—
193C23—21no—
194C23—22no—
195C23—22no—
196C23—21no—
197C23#6325212yes—
198C23C23×D1711yescharacteristic
199Q8—348no—
200C2×C4—28no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—174no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4#6325212yes—
224C2×C4C2×C22⋊D1712yes—
225C2×C4C2×C22⋊D1712yes—
226C2×C4C2×C22⋊D1712yes—
227C22—24no—
228C22—22no—
229C22—42no—
230C22—41no—
231C22—41no—
232C22C2×#6325211yescharacteristic
233C22C2×#6325211yescharacteristic
234C22#41×D1712yes—
235C22#1058769712yes—
236C22C22×C22⋊D1711yescharacteristic
237C4—22no—
238C4—22no—
239C4—22no—
240C4—22no—
241C4—41no—
242C4—41no—
243C4—344no—
244C4—682no—
245C4—682no—
246C4—682no—
247C2—42no—
248C2#1176331611yescharacteristic
249C2#1176332011yescharacteristic
250C2#1176338011yescharacteristic
251C2#1176349212yes—
252C2#1176352712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#117C2
prime radicalC24×C17C23
hypercenterC2×#117D17
hypocenterC17#2599
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic34C2semidirect
2. #11762739C2semidirect
4. #11762729C2semidirect
7. #11762693C2semidirect
8. #11762690C2semidirect
10. #11762677C2semidirect
163. C17#2599semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic342
#117627392
#117627342
#117627292
#117627212
#117626972
#117626932
#117626902
#117626822
#117626772
C2×C17×#1172
#259917