Finite Group Databasebeta

← #14536859

Subgroups — #14536859

Order 2176.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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#11762743C212yesmaximal, complemented
3#11762733C212yesmaximal, complemented
4#11762731C212yesmaximal, complemented
5#11762731C212yesmaximal, complemented
6#11762727C212yesmaximal
7#11762726C211yesmaximal, characteristic
8#11762678C211yesmaximal, characteristic, complemented
9C17×#398C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C2×C4⋊C4×C17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×#63227C2212yes—
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27#10587584—28no—
28C17×#95C2212yes—
29C17×#95C2211yescharacteristic
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C23×D17C2312yes—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C2×D68—41no—
42C2×D68—42no—
43C4⋊C4×C17—24no—
44C4⋊C4×C17—41no—
45C4⋊C4×C17—41no—
46#63227—24no—
47#63227—24no—
48#63227—24no—
49#63227—42no—
50#63227—42no—
51#63227—42no—
52C4⋊Dic17—24no—
53C4⋊Dic17—41no—
54#63225—24no—
55#63225—42no—
56#63225—42no—
57#63225—42no—
58C4×Dic17D414yescomplemented
59C23×C17C2411yescharacteristic
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
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—22no—
78C2×C4×C17—22no—
79C2×C4×C17C2×D412yes—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17C2×D414yes—
89D68—44no—
90#2823—171nomaximal, Sylow
91C22×C17C22×D411yescharacteristic
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C17H(5,𝔽2)12yes—
94C22×C172−1+412yes—
95C22×C172−1+411yescharacteristic
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—42no—
101C2×D17—42no—
102C2×D17—42no—
103C4×C17—22no—
104C4×C17—42no—
105C4×C17—42no—
106C4×C17—41no—
107C4×C17—41no—
108Dic17—24no—
109Dic17—42no—
110Dic17—42no—
111C2×#126—172no—
112#398—172no—
113#398D1711yescharacteristic, complemented
114#396—172no—
115#396—172no—
116#396—172no—
117#394—171no—
118#394—172no—
119#384—171no—
120C2×C17#53711yescharacteristic
121C2×C17#53712yes—
122C2×C17#56011yescharacteristic
123C2×C17#56311yescharacteristic
124C2×C17#56312yes—
125D17—82no—
126C22×D4—171no—
127#126—348no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4—172no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—171no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4C2×D1712yes—
135C2×C4⋊C4C2×D1711yescharacteristic
136C2×C4⋊C4C2×D1711yescharacteristic
137C2×C22⋊C4—172no—
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—172no—
141C2×C22⋊C4—172no—
142C2×C22⋊C4—172no—
143C2×C42—171no—
144#95—171no—
145#95—172no—
146#95C2×D1712yes—
147#95C2×D1711yescharacteristic
148C17#282311yescharacteristic, complemented, Sylow
149C24—172no—
150C2×D4—681no—
151C2×D4—682no—
152C22×C4—172no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1712yes—
157C22×C4C22×D1712yes—
158C22×C4C22×D1711yescharacteristic
159C22×C4C22×D1711yescharacteristic
160C4⋊C4—24no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—344no—
164C4⋊C4—344no—
165C4⋊C4—682no—
166C4⋊C4—682no—
167C4⋊C4—682no—
168C4⋊C4—681no—
169C22⋊C4—344no—
170C22⋊C4—344no—
171C22⋊C4—344no—
172C22⋊C4—682no—
173C22⋊C4—682no—
174C22⋊C4—682no—
175C42—174no—
176C23—342no—
177C23—342no—
178C23—342no—
179C23—342no—
180C23—342no—
181C23—342no—
182C23—342no—
183C23C23×D1711yescharacteristic
184D4—684no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—174no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4D4×D1712yes—
207C22—682no—
208C22—682no—
209C22—682no—
210C22—682no—
211C22—682no—
212C22—682no—
213C22—682no—
214C22C2×D4×D1711yescharacteristic
215C22#1058769111yescharacteristic
216C22#1058769412yes—
217C22#1058769612yes—
218C22#1058769711yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—42no—
224C4—344no—
225C4—682no—
226C4—682no—
227C2—1362no—
228C2#1176338611yescharacteristic
229C2#1176338912yes—
230C2#1176354911yescharacteristic
231C2#1176355811yescharacteristic
232C2#1176356312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC22×D68C22
hypercenter#398D17
hypocenterC17#2823
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#10587584C2semidirect
2. #11762743C2semidirect
3. #11762733C2semidirect
4. #11762731C2semidirect
5. #11762731C2semidirect
8. #11762678C2semidirect
9. C17×#398C2semidirect
58. C4×Dic17D4semidirect
113. #398D17semidirect
148. C17#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#105875842
#117627432
#117627332
#117627312
#117627272
#117627262
#117626782
C17×#3982
#282317