Finite Group Databasebeta

← #17953621

Subgroups — #17953621

Order 4352.

514 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 124 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×#11763654C212yesmaximal
2#14543627C214yesmaximal
3#14531140C211yesmaximal, characteristic
4C2×C22⋊#3624C2214yes—
5#11763654—22no—
6#11763654—22no—
7C2×#10587612C411yescharacteristic
8C22×C4⋊Dic17C2211yescharacteristic
9C4⋊C4×Dic17—22no—
10C4×C4⋊Dic17—22no—
11C22⋊C4×Dic17C412yes—
12C22⋊C4×Dic17C2212yes—
13C2×C17×#117C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C23×Dic17C2×C411yescharacteristic
16C23×Dic17C2311yescharacteristic
17C22×#3624—24no—
18C17×#117—22no—
19C17×#117C814yescomplemented
20C2×C4⋊C4×C17—21no—
21C2×C22⋊C4×C17C2×C412yes—
22C2×C42×C17—21no—
23C2×C22⋊Dic17C2×C412yes—
24#10587612—24no—
25C2×C4⋊Dic17—21no—
26C2×C4⋊Dic17—24no—
27C2×C4⋊Dic17D412yes—
28C2×C4⋊Dic17D412yes—
29C2×C4×Dic17—21no—
30C2×C4×Dic17—22no—
31C2×C4×Dic17—22no—
32C22⋊#3624—24no—
33C22⋊#3624—24no—
34#10587502—22no—
35#10587502—22no—
36C24×C17C22×C411yescharacteristic
37C22×C4×C17—22no—
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17C22⋊C411yescharacteristic
41C22×C4×C17C22⋊C411yescharacteristic
42C22×C4×C17C2×C812yes—
43C22×C4×C17M1612yes—
44C22×Dic17—22no—
45C22×Dic17—22no—
46C22×Dic17—81no—
47C22×Dic17—81no—
48C22×Dic17C22⋊C412yes—
49C22×Dic17C2×D412yes—
50C22×Dic17#4114yes—
51C2×#3624—44no—
52C2×#3624—84no—
53C4⋊C4×C17—24no—
54C22⋊C4×C17—22no—
55C22⋊C4×C17C2×C814yes—
56C42×C17—24no—
57C22⋊Dic17—22no—
58C22⋊Dic17M1614yes—
59C4⋊Dic17—24no—
60C4⋊Dic17—24no—
61C4⋊Dic17—24no—
62C4×Dic17—24no—
63C4×Dic17—82no—
64C4×Dic17—82no—
65#8735—171nomaximal, Sylow
66C23×C17—41no—
67C23×C17—41no—
68C23×C17C2×C22⋊C411yescharacteristic
69C23×C17#11712yes—
70C23×C17C22×C811yescharacteristic
71C23×C17C2×M1612yes—
72C23×C17C2×M1611yescharacteristic
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—24no—
78C2×C4×C17—24no—
79C2×C4×C17—24no—
80C2×C4×C17—42no—
81C2×C4×C17—42no—
82C2×C4×C17C22⋊C812yes—
83C2×C4×C17C22⋊C812yes—
84C2×Dic17—22no—
85C2×Dic17—24no—
86C2×Dic17—44no—
87C2×Dic17—42no—
88C2×Dic17—42no—
89C2×Dic17—42no—
90C2×Dic17—42no—
91C2×Dic17—82no—
92C2×Dic17#12314yes—
93#3624—84no—
94#2274—171no—
95C2×#324—172no—
96#1282—174no—
97C22×C17—21no—
98C22×C17—22no—
99C22×C17—21no—
100C22×C17—22no—
101C22×C17—41no—
102C22×C17—42no—
103C22×C17—41no—
104C22×C17#38111yescharacteristic
105C22×C17C2×C22⋊C811yescharacteristic
106C22×C17#40711yescharacteristic
107C22×C17#43112yes—
108C22×C17#43212yes—
109C4×C17—22no—
110C4×C17—22no—
111C4×C17—44no—
112Dic17—24no—
113Dic17—44no—
114Dic17—42no—
115C2×#117—171no—
116C2×#117#59511yescharacteristic
117C22×C4⋊C4—171no—
118C2×C22⋊C8—174no—
119C4×C4⋊C4—342no—
120C4×C4⋊C4—342no—
121C4×C22⋊C4—172no—
122C4×C22⋊C4—172no—
123#324—342no—
124#324—342no—
125C2×C17—21no—
126C2×C17—21no—
127C2×C17—22no—
128C2×C17—22no—
129C2×C17#181711yescharacteristic
130C2×C17#182811yescharacteristic
131C2×C17#183011yescharacteristic
132C23×C4—171no—
133C23×C4—171no—
134C23×C4C2×#59511yescharacteristic
135C22×C8—344no—
136#117—22no—
137#117—344no—
138#117#362414yescomplemented
139C2×C4⋊C4—21no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—344no—
143C2×C4⋊C4—341no—
144C2×C22⋊C4—172no—
145C2×C22⋊C4C2×#59512yes—
146C2×C42—21no—
147C2×C42—341no—
148C2×C42—342no—
149C2×C42—342no—
150C22⋊C8—344no—
151C22⋊C8—344no—
152#95—342no—
153#95—342no—
154C17#873511yescharacteristic, complemented, Sylow
155C24C22×#59511yescharacteristic
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—174no—
160C22×C4—172no—
161C22×C4—172no—
162C22×C4—342no—
163C22×C4—342no—
164C22×C4—1361no—
165C22×C4—1361no—
166C22×C4C2×#362412yes—
167C22×C4#6324712yes—
168C22×C4C22⋊#59511yescharacteristic
169C22×C4C22⋊#59511yescharacteristic
170C2×C8—684no—
171C2×C8—1364no—
172C4⋊C4—24no—
173C4⋊C4—344no—
174C4⋊C4—344no—
175C4⋊C4—344no—
176C22⋊C4—22no—
177C22⋊C4—174no—
178C22⋊C4—342no—
179C22⋊C4C2×#362414yes—
180C42—24no—
181C42—344no—
182C42—1362no—
183C42—1362no—
184C23—41no—
185C23—41no—
186C23D17⋊M1612yes—
187C23#1058767712yes—
188C23C22×#362411yescharacteristic
189C23C2×#6324711yescharacteristic
190C23C2×C22⋊#59511yescharacteristic
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—174no—
201C2×C4—342no—
202C2×C4—344no—
203C2×C4—682no—
204C2×C4—684no—
205C2×C4—682no—
206C2×C4—682no—
207C2×C4—682no—
208C2×C4—1362no—
209C2×C4C22⋊#362412yes—
210C2×C4C22⋊#362412yes—
211C8—1364no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—22no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22#1176375211yescharacteristic
220C22C2×C22⋊#362411yescharacteristic
221C22#1176378111yescharacteristic
222C22#1176382512yes—
223C22#1176382912yes—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—344no—
228C4—684no—
229C4—682no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2#1454467411yescharacteristic
235C2#1454469611yescharacteristic
236C2#1454469811yescharacteristic
subgroupG/H
centerC22#11763752
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC2×C17×#117C4
socleC22×C17#381
prime radicalC24×C17C22×C4
cosocleC23×Dic17C23
hypercenterC2×#117#595
hypocenterC17#8735
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
19. C17×#117C8semidirect
138. #117#3624semidirect
154. C17#8735semidirect

Maximal subgroups

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

subgroupindex
C2×#117636542
#145436272
#145311402
#873517