Finite Group Databasebeta

← #14533788

Subgroups — #14533788

Order 2176.

676 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 157 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×#10587626C211yesmaximal, characteristic
2C2×#10587561C211yesmaximal, characteristic, complemented
3C2×#10587622C211yesmaximal, characteristic, complemented
4C2×#10587573C211yesmaximal, characteristic, complemented
5#11762724C214yesmaximal, complemented
6#11762710C211yesmaximal, characteristic, complemented
7#11762709C214yesmaximal, complemented
8C4×#63225C211yesmaximal, characteristic, complemented
9C17×#390C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2214yes—
17C2×C42×C17C2211yescharacteristic
18C2×C22⋊Dic17C2214yes—
19#10587626—24no—
20#10587622—24no—
21C2×#63227C2214yes—
22C2×#63227C2214yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×C4×Dic17C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2211yescharacteristic
28#10587573—24no—
29#10587561—24no—
30#10587502C2212yes—
31C24×C17C2312yes—
32C23×D17C2312yes—
33C2×D4×C17—22no—
34C2×D4×C17—22no—
35C2×D4×C17—22no—
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C2×C22⋊D17—24no—
40C2×C22⋊D17—24no—
41C2×C22⋊D17—28no—
42C2×C22⋊D17—44no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C2×D68—22no—
48C2×D68—22no—
49C2×D68—22no—
50C2×Dic34—22no—
51C2×Dic34—22no—
52C2×Dic34D414yescomplemented
53C4⋊C4×C17—22no—
54C22⋊C4×C17—44no—
55C42×C17—22no—
56C22⋊Dic17—28no—
57#63227—28no—
58#63227—44no—
59#63225—24no—
60#63225—24no—
61C4×Dic17—22no—
62C4×Dic17—24no—
63C4×Dic17D414yes—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—24no—
67C23×C17—22no—
68C23×C17—24no—
69C23×C17C2411yescharacteristic
70C22×D17—22no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17—24no—
74C22×D17—24no—
75D4×C17—44no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—24no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2×D412yes—
83C2×C4×C17#4112yes—
84C22⋊D17—48no—
85C22⋊D17—48no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17C2×D414yes—
93C2×Dic17C2×D412yes—
94C2×Dic17C2×D412yes—
95C2×Dic17#4114yes—
96D68—44no—
97Dic34—28no—
98#2548—171nomaximal, Sylow
99C22×C17—42no—
100C22×C17—42no—
101C22×C17—42no—
102C22×C17—44no—
103C22×C17—44no—
104C22×C17C22×D411yescharacteristic
105C22×C17C22×D411yescharacteristic
106C22×C17C2×#4111yescharacteristic
107C22×C17C2×#4112yes—
108C22×C17H(5,𝔽2)12yes—
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—42no—
112C2×D17—44no—
113C2×D17—44no—
114C4×C17—42no—
115C4×C17—42no—
116C4×C17#12114yes—
117Dic17—24no—
118Dic17—22no—
119Dic17—42no—
120Dic17#12414yescomplemented
121C2×#127—171no—
122C2×#124—171no—
123C2×#124—171no—
124C2×C4×Q8—171no—
125#394—174no—
126#390—171no—
127#390—174no—
128#390D1711yescharacteristic, complemented
129C4×C4⋊C4—171no—
130C2×C17—82no—
131C2×C17C2×#12111yescharacteristic
132C2×C17C2×#12411yescharacteristic
133C2×C17#53511yescharacteristic
134C2×C17#55012yes—
135C2×C17#55912yes—
136D17—82no—
137C22×Q8—171no—
138C22×D4—172no—
139C22×D4—172no—
140C22×D4—171no—
141C22×D4C2×D1711yescharacteristic
142#127—344no—
143#124—344no—
144#124—344no—
145C4×Q8—344no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4C2×D1711yescharacteristic
149C2×C22⋊C4—174no—
150C2×C22⋊C4—174no—
151C2×C22⋊C4—174no—
152C2×C22⋊C4C2×D1714yes—
153C2×C42—171no—
154C2×C42—172no—
155C2×C42—171no—
156C2×C42C2×D1711yescharacteristic
157#95—172no—
158C17#254811yescharacteristic, complemented, Sylow
159C24—172no—
160C24C22×D1712yes—
161C2×Q8—174no—
162C2×Q8—342no—
163C2×Q8—342no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—344no—
168C2×D4—344no—
169C2×D4—348no—
170C2×D4—342no—
171C2×D4—342no—
172C2×D4—342no—
173C2×D4—684no—
174C22×C4—171no—
175C22×C4—171no—
176C22×C4—172no—
177C22×C4—172no—
178C22×C4C22×D1712yes—
179C22×C4C22×D1712yes—
180C22×C4C22×D1711yescharacteristic
181C4⋊C4—22no—
182C4⋊C4—344no—
183C4⋊C4—344no—
184C22⋊C4—44no—
185C22⋊C4—348no—
186C22⋊C4—348no—
187C22⋊C4—684no—
188C42—22no—
189C42—174no—
190C42—344no—
191C42—342no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—342no—
198C23—342no—
199C23—342no—
200C23—344no—
201C23—344no—
202C23C23×D1711yescharacteristic
203Q8—348no—
204D4—44no—
205D4—688no—
206D4—688no—
207D4—684no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—172no—
214C2×C4—174no—
215C2×C4—172no—
216C2×C4—174no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4#6325212yes—
224C2×C4D4×D1712yes—
225C2×C4D4×D1712yes—
226C22—44no—
227C22—44no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—684no—
232C22—684no—
233C22—682no—
234C22—682no—
235C22—682no—
236C22C2×#6325211yescharacteristic
237C22C2×D4×D1711yescharacteristic
238C22C2×D4×D1711yescharacteristic
239C22#1058769112yes—
240C22#41×D1712yes—
241C4—42no—
242C4—42no—
243C4—174no—
244C4—344no—
245C4—342no—
246C4—682no—
247C4#1058757014yes—
248C2—82no—
249C2—1362no—
250C2C2×#1058757011yescharacteristic
251C2#124×D1711yescharacteristic
252C2#1176337211yescharacteristic
253C2#1176347712yes—
254C2#1176354212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC2×#10587626C2
hypercenter#390D17
hypocenterC17#2548
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#10587561C2semidirect
3. C2×#10587622C2semidirect
4. C2×#10587573C2semidirect
5. #11762724C2semidirect
6. #11762710C2semidirect
7. #11762709C2semidirect
8. C4×#63225C2semidirect
9. C17×#390C2semidirect
52. C2×Dic34D4semidirect
120. Dic17#124semidirect
128. #390D17semidirect
158. C17#2548semidirect

Maximal subgroups

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

subgroupindex
C2×#105876262
C2×#105875612
C2×#105876222
C2×#105875732
#117627242
#117627102
#117627092
C4×#632252
C17×#3902
#254817