Finite Group Databasebeta

← #14535745

Subgroups — #14535745

Order 2176.

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

normal subgroups only

This group has 476 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×#10587584C212yesmaximal, complemented
2#11762746C212yesmaximal, complemented
3#11762731C212yesmaximal, complemented
4#11762726C212yesmaximal
5#11762711C211yesmaximal, characteristic
6#11762708C212yesmaximal, complemented
7#11762707C211yesmaximal, characteristic, complemented
8#11762701C212yesmaximal, complemented
9C17×#389C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C2×C4⋊C4×C17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2212yes—
28#10587584—28no—
29C17×#95C2212yes—
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—22no—
42C2×D68—41no—
43C2×D68—41no—
44C4⋊C4×C17—24no—
45C4⋊C4×C17—41no—
46C4⋊C4×C17D414yescomplemented
47C42×C17—41no—
48#63227—24no—
49#63227—24no—
50#63227—24no—
51#63227—24no—
52#63227—42no—
53#63227—42no—
54C4⋊Dic17—24no—
55C4⋊Dic17—22no—
56C4⋊Dic17—41no—
57C4⋊Dic17—42no—
58C4⋊Dic17—41no—
59#63225—24no—
60C4×Dic17—24no—
61C23×C17C2411yescharacteristic
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
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×C17C2×D414yes—
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—
94#2727—171nomaximal, Sylow
95C22×C17C22×D411yescharacteristic
96C22×C17C2×#4111yescharacteristic
97C22×C17C2×#4111yescharacteristic
98C22×C17H(5,𝔽2)11yescharacteristic
99C22×C17H(5,𝔽2)11yescharacteristic
100C22×C172−1+411yescharacteristic
101C22×C172−1+411yescharacteristic
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C2×D17—42no—
106C2×D17—42no—
107C2×D17—42no—
108C2×D17—42no—
109C4×C17—22no—
110C4×C17—22no—
111C4×C17—22no—
112C4×C17—24no—
113C4×C17—42no—
114Dic17—42no—
115Dic17—42no—
116Dic17—42no—
117C2×#126—172no—
118#399—172no—
119#396—172no—
120#394—172no—
121#390—171no—
122#389—172no—
123#389—171no—
124#389D1711yescharacteristic, complemented
125#388—172no—
126C2×C17#53711yescharacteristic
127C2×C17#53811yescharacteristic
128C2×C17#53911yescharacteristic
129C2×C17#54711yescharacteristic
130C2×C17#55011yescharacteristic
131C2×C17#55611yescharacteristic
132C2×C17#56311yescharacteristic
133D17—82no—
134C22×D4—171no—
135#126—348no—
136C2×C4⋊C4—172no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—171no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4C2×D1712yes—
143C2×C4⋊C4C2×D1711yescharacteristic
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C42—172no—
152C2×C42C2×D1711yescharacteristic
153#95—172no—
154#95—172no—
155#95C2×D1712yes—
156C17#272711yescharacteristic, complemented, Sylow
157C24—172no—
158C2×D4—342no—
159C2×D4—681no—
160C2×D4—681no—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—172no—
164C22×C4C22×D1712yes—
165C22×C4C22×D1712yes—
166C22×C4C22×D1711yescharacteristic
167C22×C4C22×D1711yescharacteristic
168C22×C4C22×D1711yescharacteristic
169C4⋊C4—24no—
170C4⋊C4—41no—
171C4⋊C4—344no—
172C4⋊C4—344no—
173C4⋊C4—342no—
174C4⋊C4—681no—
175C4⋊C4—681no—
176C4⋊C4—682no—
177C4⋊C4D6814yescomplemented
178C22⋊C4—344no—
179C22⋊C4—344no—
180C22⋊C4—344no—
181C22⋊C4—344no—
182C22⋊C4—682no—
183C22⋊C4—682no—
184C42—41no—
185C42—344no—
186C23—342no—
187C23—342no—
188C23—342no—
189C23—342no—
190C23—342no—
191C23—342no—
192C23—342no—
193C23C23×D1711yescharacteristic
194D4—684no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
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×C4C2×D6814yes—
216C2×C4C2×D6812yes—
217C2×C4#6325412yes—
218C2×C4#6325612yes—
219C22—682no—
220C22—682no—
221C22—682no—
222C22—682no—
223C22—682no—
224C22—682no—
225C22—682no—
226C22C22×D6811yescharacteristic
227C22C2×#6325411yescharacteristic
228C22#1058769111yescharacteristic
229C22C2×#6325611yescharacteristic
230C22#1058769411yescharacteristic
231C22#1058769611yescharacteristic
232C22#1058769711yescharacteristic
233C4—24no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—42no—
238C4—682no—
239C4—682no—
240C4—682no—
241C2—1362no—
242C2#1176339011yescharacteristic
243C2#1176339611yescharacteristic
244C2#1176340811yescharacteristic
245C2#1176346311yescharacteristic
246C2#1176348111yescharacteristic
247C2#1176353311yescharacteristic
248C2#1176356311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#389C2
prime radicalC22×D68C22
hypercenter#389D17
hypocenterC17#2727
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#10587584C2semidirect
2. #11762746C2semidirect
3. #11762731C2semidirect
6. #11762708C2semidirect
7. #11762707C2semidirect
8. #11762701C2semidirect
9. C17×#389C2semidirect
46. C4⋊C4×C17D4semidirect
124. #389D17semidirect
156. C17#2727semidirect
177. C4⋊C4D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875842
#117627462
#117627312
#117627262
#117627112
#117627082
#117627072
#117627012
C17×#3892
#272717