Finite Group Databasebeta

← #14536210

Subgroups — #14536210

Order 2176.

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

normal subgroups only

This group has 400 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
1#11762751C212yesmaximal, complemented
2#11762741C212yesmaximal, complemented
3#11762734C211yesmaximal, characteristic
4#11762716C212yesmaximal, complemented
5#11762697C212yesmaximal
6#11762684C212yesmaximal
7#11762676C212yesmaximal, complemented
8#11762670C211yesmaximal, characteristic
9C17×#393C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C23×Dic17C2211yescharacteristic
12C2×C22⋊C4×C17C2212yes—
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2211yescharacteristic
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×C4⋊Dic17C2212yes—
19C2×#63225C2212yes—
20C2×C4×Dic17C2212yes—
21C17×#95C2212yes—
22C17×#95C2211yescharacteristic
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2211yescharacteristic
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—22no—
39C22×Dic17—41no—
40C22×Dic17—41no—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22⋊C4×C17—24no—
46C22⋊C4×C17—41no—
47C22⋊Dic17—24no—
48C22⋊Dic17—41no—
49C4⋊Dic17—42no—
50C4⋊Dic17—42no—
51C4⋊Dic17—41no—
52#63225—42no—
53C4×Dic17—24no—
54C23×C17—22no—
55C23×C17—21no—
56C23×C17—21no—
57C23×C17—22no—
58C23×C17#4112yes—
59C23×C17C2411yescharacteristic
60Q8×C17—44no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17#4114yes—
71C2×Dic17—22no—
72C2×Dic17—22no—
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—
84#2764—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—24no—
87C22×C17—42no—
88C22×C17—41no—
89C22×C17—41no—
90C22×C17C2×#4112yes—
91C22×C17C2×#4111yescharacteristic
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C172−1+412yes—
95C4×C17—24no—
96C4×C17—42no—
97C4×C17—41no—
98C4×C17—41no—
99Dic17—42no—
100Dic17—42no—
101Dic17—42no—
102Dic17—42no—
103#400—172no—
104#398—172no—
105#397—171no—
106#393D1711yescharacteristic
107#391—172no—
108#388—172no—
109#385—172no—
110#383—172no—
111#381—171no—
112C2×C17—42no—
113C2×C17#53811yescharacteristic
114C2×C17#54212yes—
115C2×C17#55111yescharacteristic
116C2×C17#55212yes—
117C2×C17#56311yescharacteristic
118C22×Q8C2×D1711yescharacteristic
119C23×C4—171no—
120C2×C4⋊C4—172no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—171no—
124C2×C22⋊C4—172no—
125C2×C22⋊C4—171no—
126C2×C22⋊C4C2×D1712yes—
127C2×C22⋊C4C2×D1711yescharacteristic
128C2×C42—172no—
129#95—172no—
130#95—172no—
131#95—171no—
132#95—172no—
133#95—172no—
134#95—172no—
135#95—172no—
136#95—172no—
137#95C2×D1712yes—
138#95C2×D1711yescharacteristic
139C17#276411yescharacteristic, complemented, Sylow
140C24C22×D1711yescharacteristic
141C2×Q8—24no—
142C2×Q8—41no—
143C22×C4—172no—
144C22×C4—172no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—342no—
148C22×C4—681no—
149C22×C4—681no—
150C22×C4C22×D1711yescharacteristic
151C22×C4C22×D1711yescharacteristic
152C22×C4C22×D1712yes—
153C22×C4C22×D1712yes—
154C4⋊C4—682no—
155C4⋊C4—682no—
156C4⋊C4—681no—
157C4⋊C4—682no—
158C22⋊C4—24no—
159C22⋊C4—41no—
160C22⋊C4—344no—
161C22⋊C4—681no—
162C42—344no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23#6325612yes—
168C23C23×D1711yescharacteristic
169Q8—44no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—684no—
192C2×C4#6325414yes—
193C22—24no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—42no—
198C22C2×#6325412yes—
199C22#1058769111yescharacteristic
200C22C2×#6325611yescharacteristic
201C22#1058769611yescharacteristic
202C22#1058769712yes—
203C4—24no—
204C4—41no—
205C4—41no—
206C4—42no—
207C4—682no—
208C4—682no—
209C4—682no—
210C4—682no—
211C2—42no—
212C2#1176339911yescharacteristic
213C2#1176342812yes—
214C2#1176349011yescharacteristic
215C2#1176350112yes—
216C2#1176355911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#393C2
prime radicalC24×C17C23
hypercenter#393D17
hypocenterC17#2764
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762741C2semidirect
4. #11762716C2semidirect
7. #11762676C2semidirect
139. C17#2764semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627412
#117627342
#117627162
#117626972
#117626842
#117626762
#117626702
C17×#3932
#276417