Finite Group Databasebeta

← #14533942

Subgroups — #14533942

Order 2176.

672 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 157 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×#10587610C211yesmaximal, characteristic, complemented
2#11762734C212yesmaximal, complemented
3#11762721C212yesmaximal, complemented
4#11762703C214yesmaximal, complemented
5#11762672C212yesmaximal, complemented
6#11762665C212yesmaximal
7C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C17C211yesmaximal, characteristic
9C25×C17C2211yescharacteristic
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C22×Dic34C2211yescharacteristic, complemented
13C2×C22⋊C4×C17—24no—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2214yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2214yes—
20C2×C4⋊Dic17C2211yescharacteristic, complemented
21#10587610—24no—
22C2×#63225C2212yescomplemented
23C2×C4×Dic17C2212yescomplemented
24#10587502C2214yescomplemented
25#10587502C2214yescomplemented
26#10587502C2212yescomplemented
27C24×C17—22no—
28C24×C17—24no—
29C24×C17—24no—
30C24×C17—42no—
31C24×C17C2312yes—
32C24×C17C2311yescharacteristic
33C22×C4×C17—22no—
34C22×C4×C17—24no—
35C22×C4×C17—42no—
36C22×C4×C17D414yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×Dic17C2312yes—
41C22×Dic17C2314yes—
42C22×Dic17C2312yes—
43C2×Dic34—22no—
44C2×Dic34—42no—
45C22⋊C4×C17—24no—
46C22⋊C4×C17—24no—
47C22⋊Dic17—24no—
48C22⋊Dic17—28no—
49C22⋊Dic17—24no—
50C22⋊Dic17—44no—
51C4⋊Dic17—22no—
52#63225—24no—
53C4×Dic17—42no—
54C23×C17—22no—
55C23×C17—22no—
56C23×C17—28no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—24no—
60C23×C17—24no—
61C23×C17—416no—
62C23×C17—44no—
63C23×C17—44no—
64C23×C17C2×D412yes—
65C23×C17#4114yes—
66C23×C17#4114yes—
67C23×C17C2411yescharacteristic
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—24no—
72C2×C4×C17—44no—
73C2×C4×C17—44no—
74C2×C4×C17C2×D412yes—
75C2×C4×C17C2×D412yes—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—24no—
79C2×Dic17—24no—
80C2×Dic17—24no—
81C2×Dic17—24no—
82C2×Dic17—24no—
83Dic34—44no—
84#2561—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—22no—
87C22×C17—28no—
88C22×C17—24no—
89C22×C17—28no—
90C22×C17—416no—
91C22×C17—42no—
92C22×C17—42no—
93C22×C17—44no—
94C22×C17#12414yes—
95C22×C17C22×D411yescharacteristic
96C22×C17C2×#4112yes—
97C22×C17C2×#4112yes—
98C22×C17H(5,𝔽2)12yes—
99C4×C17—22no—
100C4×C17—22no—
101C4×C17—42no—
102Dic17—42no—
103Dic17—42no—
104Dic17—44no—
105C2×C22⋊Q8—171no—
106C22×C22⋊C4D1711yescharacteristic
107#397—172no—
108#393—172no—
109#388—174no—
110#381—172no—
111#379—172no—
112C4×C22⋊C4—171no—
113C2×C17—24no—
114C2×C17—44no—
115C2×C17#52511yescharacteristic
116C2×C17C2×#12411yescharacteristic
117C2×C17#53411yescharacteristic
118C2×C17#55114yes—
119C25C2×D1711yescharacteristic
120C22×Q8—171no—
121C23×C4C2×D1711yescharacteristic
122C23×C4C2×D1711yescharacteristic
123C22⋊Q8—344no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—171no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—172no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4—174no—
130C2×C22⋊C4—174no—
131C2×C22⋊C4C2×D1712yes—
132C2×C22⋊C4C2×D1712yes—
133C2×C42—172no—
134#95—172no—
135#95—174no—
136#95—174no—
137C17#256111yescharacteristic, complemented, Sylow
138C24—24no—
139C24—22no—
140C24—24no—
141C24—42no—
142C24C22×D1712yes—
143C24C22×D1711yescharacteristic
144C2×Q8—342no—
145C2×Q8—682no—
146C22×C4—24no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—174no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4C22⋊D1714yes—
153C22×C4C22×D1712yes—
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1711yescharacteristic
156C4⋊C4—344no—
157C4⋊C4—342no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—348no—
161C22⋊C4—344no—
162C22⋊C4—344no—
163C22⋊C4—684no—
164C42—682no—
165C23—22no—
166C23—22no—
167C23—28no—
168C23—22no—
169C23—24no—
170C23—24no—
171C23—22no—
172C23—416no—
173C23—44no—
174C23—44no—
175C23#6325214yes—
176C23#6325414yes—
177C23C2×C22⋊D1712yes—
178C23C23×D1711yescharacteristic
179Q8—684no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—344no—
187C2×C4—344no—
188C2×C4—344no—
189C2×C4—344no—
190C2×C4—344no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4C2×C22⋊D1712yes—
194C2×C4C2×C22⋊D1712yes—
195C22—28no—
196C22—22no—
197C22—22no—
198C22—28no—
199C22—24no—
200C22—416no—
201C22—44no—
202C22—42no—
203C22—42no—
204C22#1058762214yes—
205C22C2×#6325212yes—
206C22C2×#6325412yes—
207C22#1058769112yes—
208C22C22×C22⋊D1711yescharacteristic
209C4—22no—
210C4—22no—
211C4—42no—
212C4—684no—
213C4—682no—
214C4—682no—
215C2—24no—
216C2—44no—
217C2#1176332011yescharacteristic
218C2C2×#1058762211yescharacteristic
219C2#1176336811yescharacteristic
220C2#1176348514yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C22⋊C4×C17C2
prime radicalC25×C17C22
hypercenterC22×C22⋊C4D17
hypocenterC17#2561
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587610C2semidirect
2. #11762734C2semidirect
3. #11762721C2semidirect
4. #11762703C2semidirect
5. #11762672C2semidirect
7. C4×C22⋊Dic17C2semidirect
12. C22×Dic34C22semidirect
20. C2×C4⋊Dic17C22semidirect
22. C2×#63225C22semidirect
23. C2×C4×Dic17C22semidirect
24. #10587502C22semidirect
25. #10587502C22semidirect
26. #10587502C22semidirect
137. C17#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#105876102
#117627342
#117627212
#117627032
#117626722
#117626652
C4×C22⋊Dic172
C22×C22⋊C4×C172
#256117