Finite Group Databasebeta

← #14535127

Subgroups — #14535127

Order 2176.

568 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587621C212yesmaximal
2C2×#10587564C212yesmaximal, complemented
3#11762743C212yesmaximal, complemented
4#11762737C212yesmaximal, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762697C212yesmaximal, complemented
7#11762684C212yesmaximal, complemented
8#11762678C211yesmaximal, characteristic, complemented
9C17×#394C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19#10587621—28no—
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27#10587564—28no—
28C17×#95C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32#10587502C2212yes—
33C24×C17C2312yes—
34C2×D4×C17—24no—
35C2×D4×C17—41no—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C22×Dic17—24no—
41C22×Dic17—24no—
42C22×Dic17—42no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C2×Dic34—24no—
48C2×Dic34—41no—
49C4⋊C4×C17—41no—
50C22⋊C4×C17—24no—
51C22⋊C4×C17—24no—
52C22⋊Dic17—24no—
53C22⋊Dic17—24no—
54C22⋊Dic17D418yes—
55C4⋊Dic17—24no—
56C4⋊Dic17—41no—
57#63225—24no—
58#63225—24no—
59#63225—24no—
60#63225—24no—
61C4×Dic17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17C2×D414yes—
69C23×C17C2411yescharacteristic
70D4×C17—44no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17#4112yes—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—48no—
92C2×Dic17C2×D414yes—
93C2×Dic17C2×D414yes—
94Dic34—44no—
95#2680—171nomaximal, Sylow
96C22×C17—24no—
97C22×C17—24no—
98C22×C17—24no—
99C22×C17—42no—
100C22×C17—42no—
101C22×C17—42no—
102C22×C17—42no—
103C22×C17C22×D412yes—
104C22×C17C2×#4111yescharacteristic
105C22×C17H(5,𝔽2)11yescharacteristic
106C22×C172−1+411yescharacteristic
107C22×C172−1+412yes—
108C4×C17—22no—
109C4×C17—41no—
110C4×C17—41no—
111C4×C17—42no—
112Dic17—24no—
113Dic17—24no—
114Dic17—42no—
115Dic17—42no—
116C2×#123—172no—
117C2×C22⋊Q8—172no—
118#398—172no—
119#397—172no—
120#395—171no—
121#394D1711yescharacteristic
122#388—172no—
123#385—172no—
124#384—171no—
125C2×C17—44no—
126C2×C17#53612yes—
127C2×C17#53911yescharacteristic
128C2×C17D4211yescharacteristic
129C2×C17#54712yes—
130C2×C17#56311yescharacteristic
131C22×Q8—171no—
132C22×D4C2×D1711yescharacteristic
133C23×C4—172no—
134#123—348no—
135C22⋊Q8—348no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4C2×D1711yescharacteristic
143C2×C22⋊C4—172no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4—172no—
146C2×C22⋊C4C2×D1712yes—
147C2×C22⋊C4C2×D1712yes—
148C2×C42—171no—
149#95—172no—
150#95—172no—
151#95—172no—
152#95—171no—
153#95C2×D1711yescharacteristic
154C17#268011yescharacteristic, complemented, Sylow
155C24C22×D1712yes—
156C2×Q8—344no—
157C2×Q8—681no—
158C2×D4—24no—
159C2×D4—41no—
160C22×C4—172no—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—172no—
164C22×C4—344no—
165C22×C4—344no—
166C22×C4—682no—
167C22×C4C22×D1712yes—
168C22×C4C22×D1711yescharacteristic
169C22×C4C22×D1711yescharacteristic
170C22×C4C22×D1711yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—344no—
173C4⋊C4—344no—
174C4⋊C4—344no—
175C4⋊C4—344no—
176C4⋊C4—344no—
177C4⋊C4—681no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—178no—
181C22⋊C4—344no—
182C22⋊C4—344no—
183C42—342no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23D4×D1714yes—
191C23C23×D1711yescharacteristic
192Q8—684no—
193D4—44no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—174no—
204C2×C4—174no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—688no—
216C2×C4#6325412yes—
217C22—24no—
218C22—24no—
219C22—24no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22C2×D4×D1712yes—
225C22C2×#6325411yescharacteristic
226C22#1058769111yescharacteristic
227C22#1058769712yes—
228C22#1058769711yescharacteristic
229C4—22no—
230C4—42no—
231C4—41no—
232C4—41no—
233C4—344no—
234C4—344no—
235C4—682no—
236C4—682no—
237C2—44no—
238C2#1176338112yes—
239C2#1176340411yescharacteristic
240C2#1176344111yescharacteristic
241C2#1176345912yes—
242C2#1176355911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2680
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10587564C2semidirect
3. #11762743C2semidirect
4. #11762737C2semidirect
5. #11762729C2semidirect
6. #11762697C2semidirect
7. #11762684C2semidirect
8. #11762678C2semidirect
154. C17#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#105876212
C2×#105875642
#117627432
#117627372
#117627292
#117626972
#117626842
#117626782
C17×#3942
#268017