Finite Group Databasebeta

← #17953528

Subgroups — #17953528

Order 4352.

642 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 144 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#14544674C212yesmaximal
2#14543622C214yesmaximal
3C2×C4⋊C4×Dic17C211yesmaximal, characteristic
4C2×C22⋊#3624C2214yes—
5#11763675—22no—
6#11763675—22no—
7C22×C4⋊Dic17C411yescharacteristic
8C22×C4×Dic17C412yes—
9C22×C4×Dic17C2211yescharacteristic
10C4⋊C4×Dic17—22no—
11C4⋊C4×Dic17—22no—
12C2×#10587502C2212yes—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C23×C4×C17C2×C412yes—
16C23×Dic17C2×C411yescharacteristic
17C23×Dic17C2×C412yes—
18C23×Dic17C2311yescharacteristic
19C22×#3624—24no—
20C2×C4⋊C4×C17—22no—
21C2×C4⋊C4×C17—22no—
22C2×C4⋊C4×C17C814yescomplemented
23C2×C4⋊Dic17—22no—
24C2×C4⋊Dic17—24no—
25C2×C4×Dic17—22no—
26C2×C4×Dic17—22no—
27C2×C4×Dic17—24no—
28C2×C4×Dic17—44no—
29C2×C4×Dic17D412yes—
30C2×C4×Dic17D412yes—
31C22⋊#3624—28no—
32#10587502—24no—
33C24×C17C22×C411yescharacteristic
34C22×C4×C17—22no—
35C22×C4×C17—22no—
36C22×C4×C17—22no—
37C22×C4×C17—22no—
38C22×C4×C17—42no—
39C22×C4×C17—42no—
40C22×C4×C17C22⋊C412yes—
41C22×C4×C17C2×C812yes—
42C22×C4×C17C2×C814yes—
43C22×C4×C17M1612yes—
44C22×Dic17—22no—
45C22×Dic17—22no—
46C22×Dic17—22no—
47C22×Dic17—22no—
48C22×Dic17—22no—
49C22×Dic17—41no—
50C22×Dic17—41no—
51C22×Dic17—42no—
52C22×Dic17—42no—
53C22×Dic17C22⋊C411yescharacteristic
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17M1614yes—
56C22×Dic17C2×D412yes—
57C22×Dic17#4114yes—
58C2×#3624—44no—
59C2×#3624—44no—
60C2×#3624—44no—
61C4⋊C4×C17—28no—
62C4⋊Dic17—28no—
63C4×Dic17—24no—
64C4×Dic17—24no—
65C4×Dic17—44no—
66C4×Dic17—44no—
67#8701—171nomaximal, Sylow
68C23×C17—21no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17C2×C22⋊C411yescharacteristic
73C23×C17#11712yes—
74C23×C17C22×C811yescharacteristic
75C23×C17C2×M1612yes—
76C23×C17C2×M1611yescharacteristic
77C2×C4×C17—22no—
78C2×C4×C17—22no—
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⋊C814yes—
87C2×Dic17—28no—
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—24no—
95C2×Dic17—44no—
96C2×Dic17—44no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—44no—
100C2×Dic17#10012yes—
101C2×Dic17#10112yes—
102C2×Dic17#12314yes—
103#3624—84no—
104C2×C4×C4⋊C4—171no—
105#1817—172no—
106#1280—174no—
107C22×C17—22no—
108C22×C17—22no—
109C22×C17—21no—
110C22×C17—21no—
111C22×C17—21no—
112C22×C17—21no—
113C22×C17—22no—
114C22×C17—22no—
115C22×C17#38111yescharacteristic
116C22×C17C2×C22⋊C811yescharacteristic
117C22×C17#40711yescharacteristic
118C22×C17#41011yescharacteristic
119C22×C17#41011yescharacteristic
120C22×C17C2×#10011yescharacteristic
121C22×C17C2×#10111yescharacteristic
122C22×C17#43112yes—
123C22×C17#43212yes—
124C4×C17—24no—
125C4×C17—44no—
126Dic17—24no—
127Dic17—22no—
128Dic17—22no—
129Dic17—44no—
130C22×C4⋊C4—171no—
131C22×C4⋊C4#59511yescharacteristic
132C22×C42—171no—
133C22×C42—172no—
134C2×C22⋊C8—174no—
135C4×C4⋊C4—342no—
136C4×C4⋊C4—342no—
137C2×#95—172no—
138#336—342no—
139#336—342no—
140C2×C17—22no—
141C2×C17—22no—
142C2×C17#146011yescharacteristic
143C2×C17#146111yescharacteristic
144C2×C17#146611yescharacteristic
145C2×C17#146711yescharacteristic
146C2×C17#181711yescharacteristic
147C2×C17#182011yescharacteristic
148C2×C17#182111yescharacteristic
149C23×C4—171no—
150C23×C4—171no—
151C23×C4—172no—
152C23×C4C2×#59512yes—
153C23×C4C2×#59511yescharacteristic
154C22×C8—344no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—344no—
158C2×C4⋊C4—342no—
159C2×C4⋊C4#362414yescomplemented
160C2×C42—172no—
161C2×C42—172no—
162C2×C42—344no—
163C2×C42—342no—
164C2×C42—342no—
165C2×C42—684no—
166C22⋊C8—348no—
167#95—344no—
168C17#870111yescharacteristic, complemented, Sylow
169C24C22×#59511yescharacteristic
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—171no—
177C22×C4—171no—
178C22×C4—172no—
179C22×C4—174no—
180C22×C4—174no—
181C22×C4—342no—
182C22×C4—342no—
183C22×C4—342no—
184C22×C4—342no—
185C22×C4—342no—
186C22×C4—681no—
187C22×C4—681no—
188C22×C4—682no—
189C22×C4—682no—
190C22×C4C2×#362412yes—
191C22×C4C2×#362414yes—
192C22×C4#6324712yes—
193C22×C4C22⋊#59512yes—
194C2×C8—684no—
195C2×C8—684no—
196C2×C8—684no—
197C4⋊C4—28no—
198C4⋊C4—348no—
199C42—344no—
200C42—344no—
201C42—684no—
202C42—684no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23D17⋊M1612yes—
208C23#1058767712yes—
209C23C22×#362411yescharacteristic
210C23C2×#6324711yescharacteristic
211C23C2×C22⋊#59511yescharacteristic
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—24no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—174no—
222C2×C4—172no—
223C2×C4—172no—
224C2×C4—348no—
225C2×C4—341no—
226C2×C4—341no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—344no—
232C2×C4—684no—
233C2×C4—684no—
234C2×C4—682no—
235C2×C4—682no—
236C2×C4—684no—
237C2×C4C22⋊#362414yes—
238C8—1364no—
239C22—22no—
240C22—22no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22#1176375211yescharacteristic
248C22C2×C22⋊#362411yescharacteristic
249C22#1176378111yescharacteristic
250C22#1176378911yescharacteristic
251C22#1176379011yescharacteristic
252C22D17⋊#10011yescharacteristic
253C22D17⋊#10111yescharacteristic
254C22#1176382512yes—
255C22#1176382912yes—
256C4—24no—
257C4—44no—
258C4—344no—
259C4—342no—
260C4—342no—
261C4—684no—
262C2—22no—
263C2—22no—
264C2#1454402411yescharacteristic
265C2#1454402711yescharacteristic
266C2#1454404511yescharacteristic
267C2#1454404811yescharacteristic
268C2#1454467411yescharacteristic
269C2#1454468211yescharacteristic
270C2#1454468411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22×C4
cosocleC23×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#8701
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
22. C2×C4⋊C4×C17C8semidirect
159. C2×C4⋊C4#3624semidirect
168. C17#8701semidirect

Maximal subgroups

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

subgroupindex
#145446742
#145436222
C2×C4⋊C4×Dic172
#870117