Finite Group Databasebeta

← #14531872

Subgroups — #14531872

Order 2176.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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×#10587630C211yesmaximal, characteristic
2#11762716C211yesmaximal, characteristic
3#11762713C212yesmaximal
4#11762706C212yesmaximal
5#11762704C212yesmaximal
6C4×#63225C212yesmaximal
7#11762660C212yesmaximal
8#11762660C212yesmaximal
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14#10587630C418yescomplemented
15C2×C4⋊Dic17C2211yescharacteristic
16C2×#63225C2212yes—
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×C4×Dic17C2212yes—
22C2×C4×Dic17C2212yes—
23C2×C4×Dic17C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25C2×C4×Dic17C2211yescharacteristic
26C17×#95C2212yes—
27C17×#95C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C2×Q8×C17—41no—
33C2×Q8×C17—41no—
34C2×Q8×C17C2×C414yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×Dic34—41no—
45C2×Dic34—41no—
46C2×Dic34C2×C414yes—
47C4⋊C4×C17—41no—
48C42×C17—41no—
49C4⋊Dic17—41no—
50#63225—24no—
51#63225—42no—
52#63225C2×C418yes—
53#63225C2×C418yes—
54#63225Q818yescomplemented
55C4×Dic17—24no—
56C4×Dic17—24no—
57C4×Dic17—42no—
58C4×Dic17—41no—
59C4×Dic17C2×C414yes—
60C23×C17C2411yescharacteristic
61Q8×C17—44no—
62C2×C4×C17—22no—
63C2×C4×C17—21no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17C22×C412yes—
74C2×C4×C17C22×C412yes—
75C2×C4×C17C22×C412yes—
76C2×C4×C17C2×Q814yes—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17C22×C414yes—
85C2×Dic17C22×C414yes—
86C2×Dic17C2×Q814yes—
87C2×Dic17C2×Q814yes—
88C2×Dic17#4114yes—
89Dic34—44no—
90#2369—171nomaximal, Sylow
91C22×C17C23×C411yescharacteristic
92C22×C17C22×Q812yes—
93C22×C17C2×#4112yes—
94C22×C17H(5,𝔽2)11yescharacteristic
95C22×C17H(5,𝔽2)11yescharacteristic
96C4×C17—22no—
97C4×C17—22no—
98C4×C17—22no—
99C4×C17—24no—
100C4×C17—42no—
101Dic17—24no—
102Dic17—24no—
103Dic17—42no—
104Dic17C4×Q818yes—
105C2×C4⋊Q8—171no—
106#391—171no—
107#391—172no—
108#391D1711yescharacteristic
109#389—172no—
110#389—172no—
111C4×C4⋊C4—172no—
112C4×C4⋊C4—172no—
113C4×C4⋊C4—172no—
114C2×C17C2×C4×Q812yes—
115C2×C17#51811yescharacteristic
116C2×C17#55412yes—
117C2×C17#55511yescharacteristic
118C2×C17Q8211yescharacteristic
119C22×Q8—171no—
120C22×Q8C2×D1711yescharacteristic
121C4⋊Q8—178no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4C2×D1711yescharacteristic
129C2×C42—171no—
130C2×C42—172no—
131C2×C42—172no—
132C2×C42—171no—
133C2×C42—172no—
134C2×C42C2×D1711yescharacteristic
135#95—172no—
136#95—172no—
137#95—172no—
138#95—172no—
139#95C2×D1712yes—
140#95C2×D1712yes—
141C17#236911yescharacteristic, complemented, Sylow
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—174no—
145C2×Q8—681no—
146C2×Q8—681no—
147C2×Q8C4×D1714yes—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4C22×D1712yes—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1711yescharacteristic
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1712yes—
157C4⋊C4—41no—
158C4⋊C4—178no—
159C4⋊C4—178no—
160C4⋊C4—178no—
161C4⋊C4—344no—
162C4⋊C4—682no—
163C4⋊C4—681no—
164C42—41no—
165C42—174no—
166C42—344no—
167C42—344no—
168C42—682no—
169C42—681no—
170C23C23×D1711yescharacteristic
171Q8—44no—
172Q8—684no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—174no—
185C2×C4—174no—
186C2×C4—174no—
187C2×C4—174no—
188C2×C4—174no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4C2×C4×D1712yes—
197C2×C4C2×C4×D1712yes—
198C2×C4C2×C4×D1712yes—
199C2×C4Q8×D1714yes—
200C22C22×C4×D1711yescharacteristic
201C22#1058769111yescharacteristic
202C22#1058769111yescharacteristic
203C22C2×Q8×D1712yes—
204C22#41×D1712yes—
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—178no—
211C4—344no—
212C4—344no—
213C4—682no—
214C2C4×Q8×D1712yes—
215C2#1176328211yescharacteristic
216C2#1176351312yes—
217C2#1176352111yescharacteristic
218C2#1176353911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutatorC22×C17C23×C4
FittingC17×#391C2
socle, prime radical, cosocleC23×C17C24
hypercenter#391D17
hypocenterC17#2369
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
14. #10587630C4semidirect
54. #63225Q8semidirect
141. C17#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
#117627162
#117627132
#117627062
#117627042
C4×#632252
#117626602
C17×#3912
#236917