Finite Group Databasebeta

← C2×#14543711 #17958126

Subgroups — C2×#14543711 #17958126

Order 4352.

726 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 120 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×#11763652C212yesmaximal
2#14543711C214yesmaximal, direct product factor
3C2×#11762670C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5#11763652—22no—
6#11763652C2214yes—
7C24×Dic17C411yescharacteristic
8C22×C22⋊Dic17C2211yescharacteristic
9#11762670—22no—
10#11762670C412yes—
11#11762670C2212yes—
12C2×#10587502—22no—
13C22×C22⋊C4×C17C411yescharacteristic
14C25×C17C2×C411yescharacteristic
15C23×C4×C17—21no—
16C23×Dic17—21no—
17C23×Dic17—21no—
18C23×Dic17—22no—
19C23×Dic17—41no—
20C23×Dic17—41no—
21C23×Dic17—42no—
22C23×Dic17C814yescomplemented
23C23×Dic17C2×C412yes—
24C23×Dic17D412yes—
25C22×#3624—42no—
26C2×C22⋊C4×C17—21no—
27C2×C22⋊C4×C17—42no—
28C2×C22⋊C4×C17C2×C412yes—
29C2×C22⋊Dic17—41no—
30C2×C22⋊Dic17—41no—
31C2×C22⋊Dic17C2×C411yescharacteristic
32C2×C22⋊Dic17D412yes—
33C2×C22⋊Dic17C2311yescharacteristic
34C22⋊#3624—24no—
35C22⋊#3624—42no—
36#10587502—24no—
37#10587502—24no—
38C24×C17—21no—
39C24×C17—21no—
40C24×C17—22no—
41C24×C17—44no—
42C24×C17C22⋊C411yescharacteristic
43C24×C17C22⋊C411yescharacteristic
44C24×C17C2×C812yes—
45C24×C17M1612yes—
46C24×C17C22×C411yescharacteristic
47C22×C4×C17—22no—
48C22×C4×C17—44no—
49C22×C4×C17—42no—
50C22×Dic17—24no—
51C22×Dic17—24no—
52C22×Dic17—22no—
53C22×Dic17—22no—
54C22×Dic17—22no—
55C22×Dic17—22no—
56C22×Dic17—44no—
57C22×Dic17—44no—
58C22×Dic17—42no—
59C22×Dic17—42no—
60C22×Dic17—41no—
61C22×Dic17—41no—
62C22×Dic17—42no—
63C22×Dic17—44no—
64C22×Dic17—44no—
65C22×Dic17—42no—
66C22×Dic17—42no—
67C22×Dic17—44no—
68C22×Dic17C22⋊C412yes—
69C22×Dic17C2×C814yescomplemented
70C22×Dic17C2×D412yes—
71C2×#3624—44no—
72C2×#3624—84no—
73C22⋊C4×C17—42no—
74C22⋊C4×C17—42no—
75C22⋊Dic17—41no—
76C22⋊Dic17—42no—
77C22⋊Dic17—41no—
78C2×#1317—171nomaximal, Sylow
79C23×C17—24no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—22no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—42no—
89C23×C17—44no—
90C23×C17—44no—
91C23×C17—42no—
92C23×C17—48no—
93C23×C17—44no—
94C23×C17—44no—
95C23×C17C22⋊C812yes—
96C23×C17C22⋊C812yes—
97C23×C17#9911yescharacteristic
98C23×C17#9911yescharacteristic
99C23×C17#10012yes—
100C23×C17C2×C22⋊C411yescharacteristic
101C23×C17C22×C811yescharacteristic
102C23×C17C2×M1611yescharacteristic
103C2×C4×C17—44no—
104C2×C4×C17—42no—
105C2×C4×C17—84no—
106C2×Dic17—24no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—48no—
110C2×Dic17—44no—
111C2×Dic17—44no—
112C2×Dic17—44no—
113C2×Dic17—44no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—44no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—82no—
120#3624—84no—
121C2×#381—171no—
122C2×#323—172no—
123#1317—174no—
124C22×C17—24no—
125C22×C17—24no—
126C22×C17—22no—
127C22×C17—22no—
128C22×C17—22no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—42no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—42no—
135C22×C17—42no—
136C22×C17—44no—
137C22×C17—44no—
138C22×C17—44no—
139C22×C17—44no—
140C22×C17—42no—
141C22×C17—42no—
142C22×C17#32312yes—
143C22×C17#32312yes—
144C22×C17#35214yes—
145C22×C17C2×C22⋊C811yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10011yescharacteristic
148C4×C17—82no—
149Dic17—42no—
150Dic17—42no—
151Dic17—44no—
152C24×C4—171no—
153C22×C22⋊C4—171no—
154C22×C22⋊C4#59511yescharacteristic
155C2×C22⋊C8—342no—
156#381—172no—
157#381—172no—
158#381—342no—
159C2×#95—342no—
160#323—174no—
161#323—342no—
162C2×C17—21no—
163C2×C17—22no—
164C2×C17—21no—
165C2×C17—42no—
166C2×C17—41no—
167C2×C17—41no—
168C2×C17#131714yescomplemented
169C2×C17C2×#32311yescharacteristic
170C2×C17C2×#35212yes—
171C25C2×#59511yescharacteristic
172C23×C4—21no—
173C23×C4—174no—
174C23×C4—172no—
175C23×C4—172no—
176C23×C4—342no—
177C23×C4—341no—
178C23×C4—341no—
179C23×C4—681no—
180C23×C4—681no—
181C23×C4—682no—
182C22×C8—682no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—172no—
186C2×C22⋊C4—171no—
187C2×C22⋊C4—171no—
188C2×C22⋊C4—681no—
189C2×C22⋊C4—681no—
190C2×C22⋊C4C2×#59512yes—
191C22⋊C8—344no—
192C22⋊C8—682no—
193#95—344no—
194#95—344no—
195C17C2×#131711yescharacteristic, complemented, Sylow
196C24—21no—
197C24—21no—
198C24—22no—
199C24—44no—
200C24#6324212yes—
201C24#6324312yes—
202C24C22⋊#59511yescharacteristic
203C24C22⋊#59511yescharacteristic
204C24C22×#59511yescharacteristic
205C22×C4—22no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—174no—
209C22×C4—172no—
210C22×C4—172no—
211C22×C4—342no—
212C22×C4—342no—
213C22×C4—342no—
214C22×C4—344no—
215C22×C4—344no—
216C22×C4—342no—
217C22×C4—682no—
218C22×C4—682no—
219C22×C4—684no—
220C22×C4—684no—
221C22×C4—681no—
222C22×C4—681no—
223C22×C4—682no—
224C22×C4—682no—
225C22×C4—682no—
226C22×C4—684no—
227C22×C4—684no—
228C22×C4—684no—
229C2×C8—684no—
230C2×C8—1364no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—681no—
234C22⋊C4—682no—
235C22⋊C4—681no—
236C23—24no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—44no—
246C23—44no—
247C23—42no—
248C23—42no—
249C23—48no—
250C23—44no—
251C23—44no—
252C23#1058753911yescharacteristic
253C23#1058753911yescharacteristic
254C23#1058754312yes—
255C23#1058754312yes—
256C23#1058755312yes—
257C23C2×#6324211yescharacteristic
258C23C2×#6324311yescharacteristic
259C23C2×C22⋊#59511yescharacteristic
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—84no—
263C2×C4—344no—
264C2×C4—342no—
265C2×C4—342no—
266C2×C4—684no—
267C2×C4—688no—
268C2×C4—684no—
269C2×C4—682no—
270C2×C4—682no—
271C2×C4—684no—
272C2×C4—684no—
273C2×C4—684no—
274C2×C4—682no—
275C2×C4—682no—
276C2×C4—1362no—
277C8—1364no—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22#1176365312yes—
297C22#1176365312yes—
298C22#1176371214yes—
299C22C2×#1058754311yescharacteristic
300C22C2×#1058753911yescharacteristic
301C22C2×#1058755311yescharacteristic
302C4—82no—
303C4—682no—
304C4—682no—
305C4—684no—
306C2—21no—
307C2—21no—
308C2—22no—
309C2—41no—
310C2—41no—
311C2—42no—
312C2#1454371114yesdirect product factor
313C2C2×#1176365311yescharacteristic
314C2C2×#1176371212yes—
subgroupG/H
centerC23#10587539
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC23×C17#99
prime radicalC25×C17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17C2×#1317
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543711C2direct (N × H)
22. C23×Dic17C8semidirect
69. C22×Dic17C2×C8semidirect
168. C2×C17#1317semidirect
195. C17C2×#1317semidirect
312. C2#14543711direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117636522
#145437112
C2×#117626702
C2×#131717