Finite Group Databasebeta

← #14531502

Subgroups — #14531502

Order 2176.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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
2C2×Q8×Dic17C211yesmaximal, characteristic
3C2×#10587573C211yesmaximal, characteristic
4#11762715C211yesmaximal, characteristic
5#11762696C214yesmaximal
6#11762691C214yesmaximal
7#11762674C211yesmaximal, characteristic
8#11762655C211yesmaximal, characteristic
9C17×#391C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C42×C17C2211yescharacteristic
14Q8×Dic17—24no—
15#10587630C418yescomplemented
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×#63225C2212yes—
19C2×#63225C2214yes—
20C2×C4×Dic17C2212yes—
21C2×C4×Dic17C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C2×C4×Dic17C2211yescharacteristic
24#10587573—24no—
25C17×#95C2214yes—
26#10587502C2212yes—
27#10587502C2214yes—
28#10587502C2214yes—
29#10587502C2212yes—
30C2×Q8×C17—22no—
31C2×Q8×C17—22no—
32C2×Q8×C17C2×C414yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2311yescharacteristic
41C22×Dic17C2311yescharacteristic
42C2×Dic34—22no—
43C2×Dic34—22no—
44C2×Dic34C2×C414yes—
45C4⋊C4×C17—22no—
46C42×C17—22no—
47C4⋊Dic17—22no—
48C4⋊Dic17—24no—
49#63225—24no—
50#63225C2×C4116yes—
51C4×Dic17—24no—
52C4×Dic17—24no—
53C4×Dic17C2×C414yes—
54C4×Dic17D414yes—
55C23×C17C2411yescharacteristic
56Q8×C17—28no—
57C2×C4×C17—22no—
58C2×C4×C17—22no—
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—24no—
63C2×C4×C17—22no—
64C2×C4×C17C22×C414yes—
65C2×C4×C17C22×C412yes—
66C2×C4×C17C2×D412yes—
67C2×C4×C17#4112yes—
68C2×Dic17—22no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—24no—
76C2×Dic17C22×C414yes—
77C2×Dic17C22×C414yes—
78C2×Dic17C2×D412yes—
79C2×Dic17C2×D412yes—
80Dic34—28no—
81#2327—171nomaximal, Sylow
82C22×C17C23×C411yescharacteristic
83C22×C17C22×D411yescharacteristic
84C22×C17C2×#4111yescharacteristic
85C22×C172−1+412yes—
86C22×C172−1+412yes—
87C4×C17—24no—
88C4×C17—42no—
89C4×C17—42no—
90C4×C17C4×D414yes—
91Dic17—24no—
92Dic17—24no—
93Dic17—22no—
94Dic17—22no—
95Dic17—42no—
96C2×C4⋊Q8—171no—
97C2×C4×Q8—171no—
98C2×C4×Q8—171no—
99#391—171no—
100#391D1711yescharacteristic
101#387—174no—
102#387—174no—
103#382—171no—
104#376—171no—
105C2×C17C2×C4×D411yescharacteristic
106C2×C17#51912yes—
107C2×C17#53611yescharacteristic
108C2×C17#53611yescharacteristic
109C2×C17#54112yes—
110C22×Q8—171no—
111C22×Q8C2×D1711yescharacteristic
112C4⋊Q8—178no—
113C4×Q8—344no—
114C4×Q8—344no—
115C2×C4⋊C4—174no—
116C2×C4⋊C4—171no—
117C2×C4⋊C4—172no—
118C2×C4⋊C4—172no—
119C2×C4⋊C4C2×D1711yescharacteristic
120C2×C42—171no—
121C2×C42—171no—
122C2×C42—172no—
123C2×C42—172no—
124C2×C42C2×D1711yescharacteristic
125#95—172no—
126#95—172no—
127#95—174no—
128#95—174no—
129#95C2×D1714yes—
130C17#232711yescharacteristic, complemented, Sylow
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—174no—
134C2×Q8—342no—
135C2×Q8—342no—
136C2×Q8C4×D1714yes—
137C22×C4—172no—
138C22×C4—171no—
139C22×C4—171no—
140C22×C4—172no—
141C22×C4—172no—
142C22×C4C22×D1712yes—
143C22×C4C22×D1712yes—
144C22×C4C22×D1712yes—
145C22×C4C22×D1711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—1716no—
148C4⋊C4—344no—
149C4⋊C4—344no—
150C4⋊C4—342no—
151C42—22no—
152C42—174no—
153C42—174no—
154C42—344no—
155C42—344no—
156C23C23×D1711yescharacteristic
157Q8—28no—
158Q8—348no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—174no—
167C2×C4—174no—
168C2×C4—172no—
169C2×C4—172no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—342no—
174C2×C4—342no—
175C2×C4—342no—
176C2×C4—342no—
177C2×C4—344no—
178C2×C4C2×C4×D1714yes—
179C2×C4C2×C4×D1712yes—
180C2×C4D4×D1712yes—
181C2×C4#6325412yes—
182C22C22×C4×D1711yescharacteristic
183C22C2×D4×D1711yescharacteristic
184C22C2×#6325411yescharacteristic
185C22#1058769412yes—
186C22#1058769712yes—
187C4—24no—
188C4—42no—
189C4—42no—
190C4—344no—
191C4—344no—
192C4—342no—
193C4—342no—
194C4—682no—
195C4#1058756714yes—
196C2C2×#1058756711yescharacteristic
197C2#1176328512yes—
198C2#1176337911yescharacteristic
199C2#1176338411yescharacteristic
200C2#1176341912yes—
subgroupG/H
center, FrattiniC23C23×D17
commutatorC22×C17C23×C4
FittingC17×#391C2
socle, prime radical, cosocleC23×C17C24
hypercenter#391D17
hypocenterC17#2327
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
15. #10587630C4semidirect
130. C17#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×Q8×Dic172
C2×#105875732
#117627152
#117626962
#117626912
#117626742
#117626552
C17×#3912
#232717