Finite Group Databasebeta

← #14536618

Subgroups — #14536618

Order 2176.

420 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#11762731C212yesmaximal, complemented
2#11762727C212yesmaximal
3#11762726C211yesmaximal, characteristic
4#11762701C212yesmaximal, complemented
5#11762701C212yesmaximal, complemented
6#11762698C212yesmaximal, complemented
7#11762692C212yesmaximal, complemented
8#11762678C211yesmaximal, characteristic, complemented
9C17×#400C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×#63227C2212yes—
14C2×#63227C2212yes—
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×C4⋊Dic17C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24C2×C4×Dic17C2212yes—
25C17×#95C2212yes—
26C17×#95C2211yescharacteristic
27C17×#95C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2212yes—
31C23×D17C2312yes—
32C22×C4×C17C2311yescharacteristic
33C22×C4×C17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C2×D68—41no—
41C2×D68—42no—
42C4⋊C4×C17—41no—
43C4⋊C4×C17—41no—
44#63227—24no—
45#63227—24no—
46#63227—24no—
47#63227—42no—
48#63227—42no—
49#63227—42no—
50C4⋊Dic17—24no—
51C4⋊Dic17—41no—
52#63225—42no—
53#63225—42no—
54C4×Dic17—22no—
55C4×Dic17—42no—
56C23×C17C2411yescharacteristic
57C22×D17—22no—
58C22×D17—22no—
59C22×D17—22no—
60C22×D17—22no—
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17#4112yes—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17#4114yes—
86D68—44no—
87#2795—171nomaximal, Sylow
88C22×C17C2×#4111yescharacteristic
89C22×C17C2×#4112yes—
90C22×C17H(5,𝔽2)11yescharacteristic
91C22×C17H(5,𝔽2)12yes—
92C22×C172−1+411yescharacteristic
93C2×D17—42no—
94C2×D17—42no—
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C2×D17—42no—
100C4×C17—22no—
101C4×C17—42no—
102C4×C17—42no—
103C4×C17—41no—
104C4×C17—41no—
105Dic17—24no—
106Dic17—42no—
107Dic17—42no—
108#400D1711yescharacteristic, complemented
109#396—172no—
110#394—171no—
111#394—172no—
112#388—172no—
113#388—172no—
114#388—172no—
115#387—172no—
116#384—171no—
117C2×C17#53911yescharacteristic
118C2×C17#54212yes—
119C2×C17#55312yes—
120C2×C17#55511yescharacteristic
121C2×C17#56011yescharacteristic
122D17—82no—
123C22×D4—171no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—171no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4C2×D1711yescharacteristic
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C22⋊C4—172no—
131C2×C22⋊C4—172no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4—172no—
135C2×C22⋊C4—172no—
136C2×C42—171no—
137C2×C42—172no—
138#95—171no—
139#95—172no—
140#95—172no—
141#95C2×D1712yes—
142#95C2×D1711yescharacteristic
143#95C2×D1712yes—
144C17#279511yescharacteristic, complemented, Sylow
145C24—172no—
146C2×D4—681no—
147C2×D4—682no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4C22×D1712yes—
152C22×C4C22×D1712yes—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1711yescharacteristic
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—344no—
159C4⋊C4—682no—
160C4⋊C4—682no—
161C4⋊C4—681no—
162C22⋊C4—344no—
163C22⋊C4—344no—
164C22⋊C4—344no—
165C22⋊C4—682no—
166C22⋊C4—682no—
167C22⋊C4—682no—
168C42—342no—
169C42—682no—
170C23—342no—
171C23—342no—
172C23—342no—
173C23—342no—
174C23—342no—
175C23—342no—
176C23—342no—
177C23C23×D1711yescharacteristic
178D4—684no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—174no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4#6325612yes—
201C22—682no—
202C22—682no—
203C22—682no—
204C22—682no—
205C22—682no—
206C22—682no—
207C22—682no—
208C22#1058769111yescharacteristic
209C22C2×#6325611yescharacteristic
210C22#41×D1712yes—
211C22#1058769612yes—
212C22#1058769711yescharacteristic
213C4—22no—
214C4—42no—
215C4—41no—
216C4—41no—
217C4—42no—
218C4—344no—
219C4—682no—
220C4—682no—
221C2—1362no—
222C2#1176340811yescharacteristic
223C2#1176342512yes—
224C2#1176351212yes—
225C2#1176352111yescharacteristic
226C2#1176354911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#400C2
prime radicalC22×D68C22
hypercenter#400D17
hypocenterC17#2795
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #11762731C2semidirect
4. #11762701C2semidirect
5. #11762701C2semidirect
6. #11762698C2semidirect
7. #11762692C2semidirect
8. #11762678C2semidirect
9. C17×#400C2semidirect
108. #400D17semidirect
144. C17#2795semidirect

Maximal subgroups

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

subgroupindex
#117627312
#117627272
#117627262
#117627012
#117626982
#117626922
#117626782
C17×#4002
#279517