Finite Group Databasebeta

← #17958138

Subgroups — #17958138

Order 4352.

526 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#14544015C212yesmaximal
2#14543710C214yesmaximal
3#14531370C211yesmaximal, characteristic
4#11763825—22no—
5#11763654—22no—
6#11763652C2214yes—
7C2×#10587612C411yescharacteristic
8C2×#10587612C2211yescharacteristic
9#11762716—22no—
10#11762716—22no—
11#11762672C412yes—
12#11762672C2212yes—
13C23×Q8×C17C411yescharacteristic
14C22×Q8×C17—21no—
15C22×Q8×C17—21no—
16C22×Q8×C17—22no—
17C22×Q8×C17—42no—
18C22×Q8×C17—42no—
19C22×Q8×C17C814yescomplemented
20C23×C4×C17C2×C411yescharacteristic
21C23×C4×C17C2×C412yes—
22C2×C22⋊Dic17C2×C412yes—
23C2×C22⋊Dic17C2×C411yescharacteristic
24C2×C22⋊Dic17C2311yescharacteristic
25#10587612—42no—
26#10587612—42no—
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17D412yes—
29C2×C4×Dic17—21no—
30C2×C4×Dic17D412yes—
31C22⋊#3624—24no—
32C4⋊#3624—42no—
33C4×#3624—42no—
34#10587502—24no—
35#10587502—24no—
36C24×C17C22×C411yescharacteristic
37C2×Q8×C17—24no—
38C2×Q8×C17—24no—
39C2×Q8×C17—48no—
40C2×Q8×C17—42no—
41C2×Q8×C17—42no—
42C2×Q8×C17—48no—
43C2×Q8×C17—42no—
44C2×Q8×C17—42no—
45C22×C4×C17—22no—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17—42no—
50C22×C4×C17—42no—
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C2×C812yes—
54C22×C4×C17C2×C814yes—
55C22×C4×C17M1612yes—
56C22×Dic17—22no—
57C22×Dic17C22⋊C412yes—
58C22×Dic17C2×D412yes—
59C2×#3624—44no—
60C22⋊Dic17—41no—
61C22⋊Dic17—41no—
62C22⋊Dic17—42no—
63C4⋊Dic17—42no—
64C4⋊Dic17—42no—
65C4×Dic17—22no—
66C4×Dic17—22no—
67C4×Dic17—42no—
68#11428—171nomaximal, Sylow
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—42no—
72C23×C17C2×C22⋊C411yescharacteristic
73C23×C17C22×C811yescharacteristic
74C23×C17C2×M1611yescharacteristic
75Q8×C17—48no—
76Q8×C17—48no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—24no—
80C2×C4×C17—24no—
81C2×C4×C17—24no—
82C2×C4×C17—44no—
83C2×C4×C17—42no—
84C2×C4×C17—42no—
85C2×C4×C17—44no—
86C2×C4×C17C22⋊C812yes—
87C2×C4×C17C22⋊C812yes—
88C2×C4×C17#9912yes—
89C2×C4×C17#10012yes—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—42no—
93C2×Dic17—42no—
94C2×Dic17—44no—
95C2×Dic17—42no—
96#3624—84no—
97#2310—171no—
98#1458—172no—
99#1317—174no—
100C22×C17—22no—
101C22×C17—21no—
102C22×C17—21no—
103C22×C17—41no—
104C22×C17—41no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17C2×C22⋊C811yescharacteristic
108C22×C17C2×#9911yescharacteristic
109C22×C17C2×#10011yescharacteristic
110C4×C17—22no—
111C4×C17—44no—
112C4×C17#32314yes—
113Dic17—42no—
114Dic17—42no—
115Dic17—82no—
116C23×Q8#59511yescharacteristic
117C2×#117—171no—
118C2×#117—171no—
119#431—342no—
120#391—342no—
121#391—342no—
122#381—172no—
123#381—172no—
124#324—342no—
125#323—174no—
126C2×C17—21no—
127C2×C17—21no—
128C2×C17—41no—
129C2×C17—41no—
130C2×C17C2×#32311yescharacteristic
131C2×C17#212312yes—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8#362414yescomplemented
138C23×C4C2×#59512yes—
139C23×C4C2×#59511yescharacteristic
140#117—682no—
141#117—682no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—341no—
144C2×C22⋊C4—171no—
145C2×C22⋊C4—171no—
146C2×C22⋊C4—172no—
147C2×C42—172no—
148C2×C42—341no—
149C4⋊C8—682no—
150C22⋊C8—344no—
151C4×C8—682no—
152#95—344no—
153#95—344no—
154C17#1142811yescharacteristic, complemented, Sylow
155C24C22×#59511yescharacteristic
156C2×Q8—24no—
157C2×Q8—24no—
158C2×Q8—48no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—172no—
171C22×C4—172no—
172C22×C4—342no—
173C22×C4C2×#362412yes—
174C22×C4C2×#362414yes—
175C22×C4#6324712yes—
176C22×C4C22⋊#59511yescharacteristic
177C22×C4C22⋊#59511yescharacteristic
178C2×C8—684no—
179C4⋊C4—682no—
180C4⋊C4—682no—
181C22⋊C4—681no—
182C22⋊C4—681no—
183C22⋊C4—682no—
184C42—342no—
185C42—342no—
186C42—682no—
187C23—21no—
188C23—21no—
189C23—42no—
190C23C22×#362411yescharacteristic
191C23C2×#6324711yescharacteristic
192C23C2×C22⋊#59511yescharacteristic
193Q8—48no—
194Q8—48no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—682no—
207C2×C4—682no—
208C2×C4—684no—
209C2×C4—682no—
210C2×C4#1058755112yes—
211C2×C4C22⋊#362412yes—
212C2×C4C22⋊#362412yes—
213C2×C4#1058755312yes—
214C8—1364no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—42no—
222C22C2×C22⋊#362411yescharacteristic
223C22C2×#1058755111yescharacteristic
224C22C2×#1058755311yescharacteristic
225C4—22no—
226C4—44no—
227C4—682no—
228C4—682no—
229C4—1362no—
230C4#1176365214yes—
231C2—21no—
232C2—21no—
233C2—41no—
234C2—41no—
235C2C2×#1176365211yescharacteristic
236C2#1454504612yes—
subgroupG/H
centerC22C2×#10587551
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC23×Q8×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22×C4
cosocleC2×C22⋊Dic17C23
hypercenterC23×Q8#595
hypocenterC17#11428
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
19. C22×Q8×C17C8semidirect
137. C22×Q8#3624semidirect
154. C17#11428semidirect

Maximal subgroups

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

subgroupindex
#145440152
#145437102
#145313702
#1142817