Finite Group Databasebeta

← #14535713

Subgroups — #14535713

Order 2176.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587580C212yesmaximal
2#11762742C212yesmaximal, complemented
3#11762736C212yesmaximal, complemented
4#11762735C212yesmaximal, complemented
5#11762711C211yesmaximal, characteristic
6#11762707C211yesmaximal, characteristic, complemented
7#11762683C212yesmaximal, complemented
8#11762680C212yesmaximal, complemented
9C17×#389C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C42×C17C2211yescharacteristic
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26#10587580—28no—
27C17×#95C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C23×D17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C2×D68—41no—
42C2×D68—41no—
43C2×D68D414yes—
44C2×C4×D17—24no—
45C2×C4×D17—42no—
46C2×C4×D17—42no—
47C4⋊C4×C17—24no—
48C4⋊C4×C17—22no—
49C4⋊C4×C17—41no—
50C42×C17—41no—
51#63227—24no—
52#63227—24no—
53#63227—24no—
54#63227—24no—
55#63227—24no—
56C4⋊Dic17—22no—
57C4⋊Dic17—41no—
58C4⋊Dic17—41no—
59#63225—24no—
60#63225—42no—
61C23×C17C2411yescharacteristic
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17C2×D414yes—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17#4112yes—
83C2×C4×C17#4112yes—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93D68—44no—
94C4×D17—48no—
95#2724—171nomaximal, Sylow
96C22×C17C22×D411yescharacteristic
97C22×C17C2×#4111yescharacteristic
98C22×C17C2×#4111yescharacteristic
99C22×C17H(5,𝔽2)11yescharacteristic
100C22×C17H(5,𝔽2)11yescharacteristic
101C22×C172−1+411yescharacteristic
102C22×C172−1+411yescharacteristic
103C2×D17—24no—
104C2×D17—24no—
105C2×D17—24no—
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—42no—
109C2×D17—42no—
110C4×C17—22no—
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—42no—
114C4×C17—42no—
115Dic17—42no—
116Dic17—42no—
117Dic17—42no—
118C2×#123—172no—
119#398—172no—
120#397—172no—
121#397—172no—
122#390—171no—
123#389—171no—
124#389D1711yescharacteristic, complemented
125#385—172no—
126#385—172no—
127C2×C17#53411yescharacteristic
128C2×C17#53911yescharacteristic
129C2×C17#53911yescharacteristic
130C2×C17#54711yescharacteristic
131C2×C17#54711yescharacteristic
132C2×C17#55911yescharacteristic
133C2×C17#56311yescharacteristic
134D17—44no—
135C22×D4—171no—
136C23×C4—172no—
137#123—348no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4C2×D1712yes—
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C42C2×D1711yescharacteristic
152#95—172no—
153#95—172no—
154#95—172no—
155#95—172no—
156#95C2×D1712yes—
157C17#272411yescharacteristic, complemented, Sylow
158C24—172no—
159C2×D4—174no—
160C2×D4—681no—
161C2×D4—681no—
162C22×C4—172no—
163C22×C4—172no—
164C22×C4—172no—
165C22×C4—344no—
166C22×C4—682no—
167C22×C4—682no—
168C22×C4C22×D1712yes—
169C22×C4C22×D1712yes—
170C22×C4C22×D1711yescharacteristic
171C22×C4C22×D1711yescharacteristic
172C22×C4C22×D1711yescharacteristic
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—41no—
176C4⋊C4—344no—
177C4⋊C4—342no—
178C4⋊C4—682no—
179C4⋊C4—681no—
180C4⋊C4—681no—
181C22⋊C4—344no—
182C22⋊C4—344no—
183C22⋊C4—344no—
184C22⋊C4—344no—
185C22⋊C4—344no—
186C42—41no—
187C23—174no—
188C23—342no—
189C23—342no—
190C23—342no—
191C23—342no—
192C23—342no—
193C23—342no—
194C23C23×D1711yescharacteristic
195D4—684no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—688no—
218C2×C4#6325212yes—
219C2×C4D4×D1712yes—
220C2×C4#6325612yes—
221C22—344no—
222C22—344no—
223C22—344no—
224C22—682no—
225C22—682no—
226C22—682no—
227C22—682no—
228C22C2×#6325211yescharacteristic
229C22C2×D4×D1711yescharacteristic
230C22#1058769111yescharacteristic
231C22C2×#6325611yescharacteristic
232C22#1058769411yescharacteristic
233C22#1058769611yescharacteristic
234C22#1058769711yescharacteristic
235C4—22no—
236C4—22no—
237C4—22no—
238C4—42no—
239C4—42no—
240C4—682no—
241C4—682no—
242C4—682no—
243C2—684no—
244C2#1176336711yescharacteristic
245C2#1176340611yescharacteristic
246C2#1176340811yescharacteristic
247C2#1176346211yescharacteristic
248C2#1176346411yescharacteristic
249C2#1176354411yescharacteristic
250C2#1176356311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#389C2
prime radicalC22×D68C22
hypercenter#389D17
hypocenterC17#2724
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
2. #11762742C2semidirect
3. #11762736C2semidirect
4. #11762735C2semidirect
6. #11762707C2semidirect
7. #11762683C2semidirect
8. #11762680C2semidirect
9. C17×#389C2semidirect
124. #389D17semidirect
157. C17#2724semidirect

Maximal subgroups

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

subgroupindex
C2×#105875802
#117627422
#117627362
#117627352
#117627112
#117627072
#117626832
#117626802
C17×#3892
#272417