Finite Group Databasebeta

← #14536198

Subgroups — #14536198

Order 2176.

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

normal subgroups only

This group has 424 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#11762751C212yesmaximal, complemented
2#11762729C212yesmaximal, complemented
3#11762721C211yesmaximal, characteristic
4#11762721C211yesmaximal, characteristic
5#11762713C212yesmaximal, complemented
6#11762702C212yesmaximal
7#11762682C212yesmaximal
8#11762674C212yesmaximal, complemented
9C17×#385C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C4⋊Dic17C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×C4×Dic17C2212yes—
22C17×#95C2212yes—
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C22×C4×C17—42no—
33C22×C4×C17—41no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×Dic34—24no—
43C2×Dic34—41no—
44C2×Dic34—42no—
45C2×Dic34—41no—
46C4⋊C4×C17—42no—
47C22⋊C4×C17—41no—
48C22⋊C4×C17Q814yescomplemented
49C22⋊Dic17—24no—
50C22⋊Dic17—42no—
51C4⋊Dic17—42no—
52#63225—42no—
53#63225—42no—
54C4×Dic17—24no—
55C23×C17—22no—
56C23×C17—22no—
57C23×C17—21no—
58C23×C17—21no—
59C23×C17C2×Q812yes—
60C23×C17C2411yescharacteristic
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—21no—
65C2×C4×C17—21no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—44no—
71C2×C4×C17C2×Q814yes—
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#4114yes—
84Dic34—44no—
85Dic34—44no—
86#2763—171nomaximal, Sylow
87C22×C17—22no—
88C22×C17—24no—
89C22×C17—42no—
90C22×C17—41no—
91C22×C17—41no—
92C22×C17C22×Q811yescharacteristic
93C22×C17C2×#4112yes—
94C22×C17H(5,𝔽2)12yes—
95C22×C17H(5,𝔽2)11yescharacteristic
96C22×C172−1+411yescharacteristic
97C4×C17—24no—
98C4×C17—41no—
99C4×C17—41no—
100C4×C17—42no—
101Dic17—24no—
102Dic17—42no—
103Dic17—42no—
104Dic17—42no—
105#400—172no—
106#395—172no—
107#393—171no—
108#393—171no—
109#391—172no—
110#388—172no—
111#385—172no—
112#385D1711yescharacteristic
113#382—172no—
114C2×C17—42no—
115C2×C17#54212yes—
116C2×C17#54411yescharacteristic
117C2×C17#55111yescharacteristic
118C2×C17#55412yes—
119C2×C17H(3,𝔽4)11yescharacteristic
120C22×Q8—171no—
121C22×Q8—171no—
122C23×C4C2×D1711yescharacteristic
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4C2×D1712yes—
127C2×C22⋊C4—172no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4C2×D1711yescharacteristic
130C2×C22⋊C4C2×D1711yescharacteristic
131C2×C42—172no—
132#95—171no—
133#95—172no—
134#95—172no—
135#95—172no—
136#95—172no—
137#95—172no—
138#95—172no—
139#95—171no—
140#95C2×D1712yes—
141C17#276311yescharacteristic, complemented, Sylow
142C24C22×D1711yescharacteristic
143C2×Q8—344no—
144C2×Q8—681no—
145C2×Q8—681no—
146C2×Q8—682no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1712yes—
156C22×C4C22×D1712yes—
157C4⋊C4—42no—
158C4⋊C4—682no—
159C4⋊C4—682no—
160C4⋊C4—682no—
161C22⋊C4—41no—
162C22⋊C4—344no—
163C22⋊C4—682no—
164C22⋊C4Dic3414yescomplemented
165C42—344no—
166C23—21no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23C2×Dic3412yes—
171C23C23×D1711yescharacteristic
172Q8—684no—
173Q8—684no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—44no—
184C2×C4—174no—
185C2×C4—342no—
186C2×C4—342no—
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×C4C2×Dic3414yes—
197C22—24no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22C22×Dic3411yescharacteristic
203C22#1058769111yescharacteristic
204C22#1058769112yes—
205C22#1058769411yescharacteristic
206C22#41×D1712yes—
207C4—24no—
208C4—41no—
209C4—42no—
210C4—41no—
211C4—344no—
212C4—682no—
213C4—682no—
214C4—682no—
215C2—42no—
216C2#1176342912yes—
217C2#1176344011yescharacteristic
218C2#1176348711yescharacteristic
219C2#1176351812yes—
220C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2763
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #11762751C2semidirect
2. #11762729C2semidirect
5. #11762713C2semidirect
8. #11762674C2semidirect
48. C22⋊C4×C17Q8semidirect
141. C17#2763semidirect
164. C22⋊C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627512
#117627292
#117627212
#117627132
#117627022
#117626822
#117626742
C17×#3852
#276317