Finite Group Databasebeta

← #14533238

Subgroups — #14533238

Order 2176.

484 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587612C211yesmaximal, characteristic
2#11762737C214yesmaximal
3#11762716C211yesmaximal, characteristic, complemented
4#11762716C211yesmaximal, characteristic, complemented
5#11762690C214yesmaximal, complemented
6#11762678C211yesmaximal, characteristic, complemented
7#11762674C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic17C211yesmaximal, characteristic
9C2×C17×C22⋊Q8C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C17×C22⋊Q8—24no—
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18#10587612—24no—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2214yes—
22C2×C4×Dic17C2212yes—
23C2×C4×Dic17C2211yescharacteristic
24C2×C4×Dic17C2211yescharacteristic
25#10587502C2212yes—
26#10587502C2214yes—
27#10587502C2214yes—
28#10587502C2214yes—
29C24×C17C2311yescharacteristic
30C2×Q8×C17—22no—
31C2×Q8×C17—22no—
32C2×Q8×C17—22no—
33C22×C4×C17—22no—
34C22×C4×C17—22no—
35C22×C4×C17D414yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C4⋊C4×C17—22no—
45C4⋊C4×C17—24no—
46C22⋊C4×C17—24no—
47C22⋊Dic17—24no—
48C22⋊Dic17—24no—
49C4⋊Dic17—22no—
50C4⋊Dic17—22no—
51#63225—28no—
52C4×Dic17—24no—
53C4×Dic17—22no—
54C4×Dic17—22no—
55C23×C17—22no—
56C23×C17—21no—
57C23×C17—21no—
58C23×C17—22no—
59C23×C17C2×D412yes—
60C23×C17C2411yescharacteristic
61Q8×C17—44no—
62C2×C4×C17—21no—
63C2×C4×C17—21no—
64C2×C4×C17—28no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—24no—
70C2×C4×C17C2×D412yes—
71C2×C4×C17C2×D412yes—
72C2×C4×C17#4112yes—
73C2×C4×C17#4112yes—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—24no—
81C2×Dic17—24no—
82C2×Dic17—22no—
83C2×Dic17#4114yes—
84#2499—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—22no—
87C22×C17—22no—
88C22×C17—42no—
89C22×C17—42no—
90C22×C17C22×D411yescharacteristic
91C22×C17C2×#4111yescharacteristic
92C22×C17C2×#4111yescharacteristic
93C22×C17C2×#4112yes—
94C22×C172−1+412yes—
95C4×C17—22no—
96C4×C17—42no—
97C4×C17—42no—
98C4×C17#12314yes—
99Dic17—24no—
100Dic17—42no—
101Dic17—42no—
102Dic17—42no—
103C2×C22⋊Q8D1711yescharacteristic
104C2×#117—171no—
105#397—174no—
106#391—171no—
107#391—171no—
108#387—174no—
109#384—171no—
110#382—171no—
111C4×C22⋊C4—171no—
112C2×C17—42no—
113C2×C17C2×#12311yescharacteristic
114C2×C17#53211yescharacteristic
115C2×C17#53611yescharacteristic
116C2×C17#55212yes—
117C2×C17#55612yes—
118C22×Q8C2×D1711yescharacteristic
119C23×C4C2×D1711yescharacteristic
120C22⋊Q8—24no—
121#117—344no—
122C2×C4⋊C4—171no—
123C2×C4⋊C4—171no—
124C2×C4⋊C4—174no—
125C2×C4⋊C4C2×D1711yescharacteristic
126C2×C4⋊C4C2×D1712yes—
127C2×C22⋊C4—172no—
128C2×C22⋊C4—172no—
129C2×C22⋊C4C2×D1712yes—
130C2×C42—172no—
131C2×C42—171no—
132C2×C42—171no—
133#95—172no—
134#95—174no—
135#95—174no—
136#95—174no—
137C17#249911yescharacteristic, complemented, Sylow
138C24C22×D1711yescharacteristic
139C2×Q8—22no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—172no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4C22⋊D1714yes—
149C22×C4C22×D1712yes—
150C22×C4C22×D1712yes—
151C22×C4C22×D1711yescharacteristic
152C22×C4C22×D1711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—24no—
155C4⋊C4—348no—
156C4⋊C4—342no—
157C4⋊C4—342no—
158C22⋊C4—24no—
159C22⋊C4—344no—
160C22⋊C4—344no—
161C42—342no—
162C42—342no—
163C42—344no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—22no—
168C23C2×C22⋊D1712yes—
169C23C23×D1711yescharacteristic
170Q8—44no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—28no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—174no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—344no—
185C2×C4—342no—
186C2×C4—344no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4#6325412yes—
190C2×C4#6325412yes—
191C2×C4C2×C22⋊D1712yes—
192C2×C4C2×C22⋊D1712yes—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C2×#6325411yescharacteristic
199C22C2×#6325411yescharacteristic
200C22#1058769412yes—
201C22#41×D1712yes—
202C22C22×C22⋊D1711yescharacteristic
203C4—22no—
204C4—42no—
205C4—42no—
206C4—344no—
207C4—682no—
208C4—682no—
209C4—682no—
210C4#1058762114yes—
211C2—42no—
212C2C2×#1058762111yescharacteristic
213C2#1176335511yescharacteristic
214C2#1176338311yescharacteristic
215C2#1176349312yes—
216C2#1176353012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C22⋊Q8C2
prime radicalC24×C17C23
hypercenterC2×C22⋊Q8D17
hypocenterC17#2499
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #11762716C2semidirect
4. #11762716C2semidirect
5. #11762690C2semidirect
6. #11762678C2semidirect
7. #11762674C2semidirect
137. C17#2499semidirect

Maximal subgroups

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

subgroupindex
C2×#105876122
#117627372
#117627162
#117626902
#117626782
#117626742
C4×C22⋊Dic172
C2×C17×C22⋊Q82
#249917