Finite Group Databasebeta

← #14534444

Subgroups — #14534444

Order 2176.

444 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 137 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#11762745C212yesmaximal
2#11762721C211yesmaximal, characteristic
3#11762716C212yesmaximal, complemented
4#11762713C212yesmaximal, complemented
5#11762694C212yesmaximal, complemented
6#11762691C212yesmaximal, complemented
7#11762670C211yesmaximal, characteristic
8C22⋊C4×Dic17C212yesmaximal
9C17×#393C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C23×Dic17C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C4⋊Dic17C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×C4×Dic17C2212yes—
21C2×C4×Dic17C2212yes—
22C17×#95C2211yescharacteristic
23C17×#95C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2211yescharacteristic
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C2×Q8×C17—24no—
33C2×Q8×C17—41no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×Dic17—41no—
39C22×Dic17—41no—
40C22×Dic17Q814yescomplemented
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C2×Dic34—24no—
46C2×Dic34—41no—
47C22⋊C4×C17—24no—
48C22⋊C4×C17—41no—
49C22⋊Dic17—24no—
50C22⋊Dic17—41no—
51C4⋊Dic17—24no—
52#63225—24no—
53#63225—42no—
54C4×Dic17—42no—
55C4×Dic17—42no—
56C23×C17—22no—
57C23×C17—22no—
58C23×C17—21no—
59C23×C17—21no—
60C23×C17C2×Q812yes—
61C23×C17C2411yescharacteristic
62Q8×C17—44no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
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#4114yes—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—44no—
84C2×Dic17C2×Q814yes—
85C2×Dic17#4114yes—
86Dic34—44no—
87#2607—171nomaximal, Sylow
88C22×C17—24no—
89C22×C17—22no—
90C22×C17—41no—
91C22×C17—41no—
92C22×C17—42no—
93C22×C17C22×Q811yescharacteristic
94C22×C17C2×#4112yes—
95C22×C17C2×#4112yes—
96C22×C17H(5,𝔽2)11yescharacteristic
97C22×C17H(5,𝔽2)11yescharacteristic
98C4×C17—24no—
99C4×C17—41no—
100C4×C17—41no—
101C4×C17—42no—
102Dic17—24no—
103Dic17—24no—
104Dic17—42no—
105Dic17—42no—
106#399—172no—
107#393—171no—
108#393D1711yescharacteristic
109#391—172no—
110#391—172no—
111#387—172no—
112#387—172no—
113#381—171no—
114C4×C22⋊C4—172no—
115C2×C17—42no—
116C2×C17#53312yes—
117C2×C17#54311yescharacteristic
118C2×C17#55111yescharacteristic
119C2×C17#55111yescharacteristic
120C2×C17#55912yes—
121C22×Q8—171no—
122C22×Q8C2×D1711yescharacteristic
123C23×C4—171no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C22⋊C4—172no—
128C2×C22⋊C4—171no—
129C2×C22⋊C4C2×D1712yes—
130C2×C22⋊C4C2×D1711yescharacteristic
131C2×C42—172no—
132C2×C42—172no—
133#95—172no—
134#95—172no—
135#95—172no—
136#95—172no—
137#95—171no—
138#95—172no—
139#95—172no—
140#95C2×D1711yescharacteristic
141#95C2×D1712yes—
142C17#260711yescharacteristic, complemented, Sylow
143C24C22×D1711yescharacteristic
144C2×Q8—24no—
145C2×Q8—41no—
146C2×Q8—344no—
147C2×Q8—681no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—174no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—681no—
154C22×C4—681no—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1711yescharacteristic
157C22×C4C22×D1712yes—
158C22×C4C22×D1712yes—
159C4⋊C4—344no—
160C4⋊C4—344no—
161C4⋊C4—682no—
162C22⋊C4—24no—
163C22⋊C4—41no—
164C22⋊C4—344no—
165C22⋊C4—681no—
166C42—682no—
167C42—682no—
168C23—22no—
169C23—21no—
170C23—22no—
171C23—21no—
172C23Q8×D1712yes—
173C23C23×D1711yescharacteristic
174Q8—44no—
175Q8—684no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—174no—
186C2×C4—174no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—684no—
198C2×C4#6325414yes—
199C22—24no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—42no—
204C22C2×#6325412yes—
205C22#1058769111yescharacteristic
206C22C2×Q8×D1711yescharacteristic
207C22#41×D1712yes—
208C22#1058769611yescharacteristic
209C4—24no—
210C4—41no—
211C4—41no—
212C4—42no—
213C4—344no—
214C4—344no—
215C4—682no—
216C4—682no—
217C2—42no—
218C2#1176336112yes—
219C2#1176343411yescharacteristic
220C2#1176348911yescharacteristic
221C2#1176349011yescharacteristic
222C2#1176354112yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC24×C17C23
hypercenter#393D17
hypocenterC17#2607
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #11762716C2semidirect
4. #11762713C2semidirect
5. #11762694C2semidirect
6. #11762691C2semidirect
40. C22×Dic17Q8semidirect
142. C17#2607semidirect

Maximal subgroups

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

subgroupindex
#117627452
#117627212
#117627162
#117627132
#117626942
#117626912
#117626702
C22⋊C4×Dic172
C17×#3932
#260717