Finite Group Databasebeta

← #14534721

Subgroups — #14534721

Order 2176.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587574C211yesmaximal, characteristic, complemented
2C2×#10587622C211yesmaximal, characteristic
3#11762737C214yesmaximal, complemented
4#11762721C214yesmaximal, complemented
5#11762715C211yesmaximal, characteristic, complemented
6#11762714C211yesmaximal, characteristic, complemented
7#11762712C211yesmaximal, characteristic
8#11762655C211yesmaximal, characteristic, complemented
9C17×#390C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2214yes—
15C2×C42×C17C2211yescharacteristic
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2214yes—
19#10587622—24no—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C2×C4×Dic17C2211yescharacteristic
24#10587574—24no—
25#10587502C2214yes—
26#10587502C2214yes—
27#10587502C2212yes—
28#10587502C2212yes—
29C24×C17C2312yes—
30C2×D4×C17—22no—
31C2×D4×C17—22no—
32C2×D4×C17—22no—
33C22×C4×C17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2311yescharacteristic
40C22×Dic17C2311yescharacteristic
41C2×Dic34—22no—
42C2×Dic34—22no—
43C2×Dic34—22no—
44C2×Dic34—24no—
45C2×Dic34—44no—
46C4⋊C4×C17—22no—
47C22⋊C4×C17—44no—
48C42×C17—22no—
49C22⋊Dic17—24no—
50C22⋊Dic17—24no—
51C22⋊Dic17—44no—
52C4⋊Dic17—22no—
53#63225—24no—
54C4×Dic17—22no—
55C4×Dic17D414yes—
56C23×C17—22no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—24no—
60C23×C17—24no—
61C23×C17C2411yescharacteristic
62D4×C17—44no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—24no—
67C2×C4×C17—22no—
68C2×C4×C17C2×D412yes—
69C2×C4×C17#4112yes—
70C2×C4×C17#4112yes—
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—24no—
80C2×Dic17—22no—
81C2×Dic17C2×D412yes—
82C2×Dic17C2×D412yes—
83Dic34—48no—
84Dic34—44no—
85#2640—171nomaximal, Sylow
86C22×C17—42no—
87C22×C17—42no—
88C22×C17—42no—
89C22×C17—44no—
90C22×C17—44no—
91C22×C17C22×D411yescharacteristic
92C22×C17C2×#4111yescharacteristic
93C22×C17C2×#4111yescharacteristic
94C22×C17H(5,𝔽2)12yes—
95C22×C172−1+412yes—
96C4×C17—42no—
97C4×C17—42no—
98C4×C17#12414yes—
99Dic17—22no—
100Dic17—22no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104C2×C4⋊Q8—171no—
105C2×#124—171no—
106#397—174no—
107#393—174no—
108#391—171no—
109#391—171no—
110#390—171no—
111#390D1711yescharacteristic
112#376—171no—
113C2×C17—82no—
114C2×C17C2×#12411yescharacteristic
115C2×C17#53511yescharacteristic
116C2×C17#53611yescharacteristic
117C2×C17#54212yes—
118C2×C17#54212yes—
119C22×Q8—171no—
120C22×Q8—172no—
121C22×D4C2×D1711yescharacteristic
122C4⋊Q8—344no—
123#124—344no—
124C2×C4⋊C4—171no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4C2×D1711yescharacteristic
127C2×C22⋊C4—172no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4—174no—
130C2×C22⋊C4C2×D1714yes—
131C2×C42—171no—
132C2×C42—171no—
133C2×C42C2×D1711yescharacteristic
134#95—172no—
135#95—172no—
136#95—174no—
137#95—174no—
138C17#264011yescharacteristic, complemented, Sylow
139C24C22×D1712yes—
140C2×Q8—344no—
141C2×Q8—342no—
142C2×Q8—342no—
143C2×Q8—342no—
144C2×Q8—684no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4C22×D1712yes—
154C22×C4C22×D1712yes—
155C22×C4C22×D1711yescharacteristic
156C4⋊C4—22no—
157C4⋊C4—344no—
158C4⋊C4—342no—
159C22⋊C4—44no—
160C22⋊C4—344no—
161C22⋊C4—344no—
162C22⋊C4—684no—
163C42—22no—
164C42—174no—
165C42—342no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—24no—
170C23—24no—
171C23C23×D1711yescharacteristic
172Q8—688no—
173Q8—684no—
174D4—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—172no—
181C2×C4—172no—
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—344no—
191C2×C4—342no—
192C2×C4#6325212yes—
193C2×C4D4×D1712yes—
194C2×C4#6325412yes—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—44no—
199C22—44no—
200C22C2×#6325211yescharacteristic
201C22C2×D4×D1711yescharacteristic
202C22C2×#6325411yescharacteristic
203C22#1058769112yes—
204C22#1058769712yes—
205C4—42no—
206C4—42no—
207C4—342no—
208C4—342no—
209C4—682no—
210C4—682no—
211C4—682no—
212C4#1058757114yes—
213C2—82no—
214C2C2×#1058757111yescharacteristic
215C2#1176337211yescharacteristic
216C2#1176337911yescharacteristic
217C2#1176342412yes—
218C2#1176342612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2640
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
3. #11762737C2semidirect
4. #11762721C2semidirect
5. #11762715C2semidirect
6. #11762714C2semidirect
8. #11762655C2semidirect
138. C17#2640semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105876222
#117627372
#117627212
#117627152
#117627142
#117627122
#117626552
C17×#3902
#264017