Finite Group Databasebeta

← #17958316

Subgroups — #17958316

Order 4352.

524 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 98 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#14544020C212yesmaximal, complemented
2#14543724C212yesmaximal, complemented
3#14543712C212yesmaximal, complemented
4#14531510C211yesmaximal, characteristic
5C2×C22⋊#3624—22no—
6#11763654C2214yes—
7#11763652—22no—
8C2×D4×Dic17C411yescharacteristic
9C2×D4×Dic17C2211yescharacteristic
10#11762712C411yescharacteristic
11#11762712C2211yescharacteristic
12#11762689—22no—
13#11762689—22no—
14#11762675C2211yescharacteristic
15#11762655C411yescharacteristic
16C2×C17×#127C411yescharacteristic
17C22×D4×C17—22no—
18C22×D4×C17C2×C411yescharacteristic
19C22×D4×C17C2×C411yescharacteristic
20C23×Dic17—21no—
21C23×Dic17D412yes—
22C22×#3624—42no—
23C17×#127—22no—
24C17×#127C814yescomplemented
25C2×C42×C17C2×C411yescharacteristic
26C2×C22⋊Dic17—21no—
27C2×C22⋊Dic17—22no—
28C2×C22⋊Dic17—22no—
29C2×C22⋊Dic17D412yes—
30D4×Dic17—42no—
31D4×Dic17—42no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4⋊Dic17C2311yescharacteristic
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17C2×C411yescharacteristic
36C22⋊#3624—24no—
37C22⋊#3624—42no—
38#10587502—21no—
39#10587502—21no—
40#10587502—22no—
41C24×C17—21no—
42C24×C17C22⋊C412yes—
43C2×D4×C17—28no—
44C2×D4×C17—21no—
45C2×D4×C17—41no—
46C2×D4×C17—41no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17—82no—
50C2×D4×C17—82no—
51C2×D4×C17C2×C812yes—
52C2×D4×C17C2×C812yescomplemented
53C2×D4×C17M1612yes—
54C22×C4×C17—21no—
55C22×C4×C17C22×C411yescharacteristic
56C22×Dic17—22no—
57C22×Dic17—22no—
58C22×Dic17—21no—
59C22×Dic17—21no—
60C22×Dic17—42no—
61C22×Dic17—82no—
62C22×Dic17—82no—
63C22×Dic17C22⋊C411yescharacteristic
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C2×D412yes—
66C2×#3624—44no—
67C2×#3624—84no—
68C42×C17—21no—
69C42×C17C2×C812yescomplemented
70C22⋊Dic17—42no—
71C22⋊Dic17—42no—
72C22⋊Dic17—82no—
73C22⋊Dic17—82no—
74C4⋊Dic17—41no—
75C4⋊Dic17—41no—
76C4×Dic17—22no—
77C4×Dic17—41no—
78#11479—171nomaximal, Sylow
79C23×C17—22no—
80C23×C17—22no—
81C23×C17—22no—
82C23×C17—42no—
83C23×C17—42no—
84C23×C17—42no—
85C23×C17—42no—
86C23×C17—41no—
87C23×C17—41no—
88C23×C17C22⋊C814yes—
89C23×C17C2×C22⋊C411yescharacteristic
90D4×C17—42no—
91D4×C17—42no—
92D4×C17—42no—
93D4×C17—42no—
94D4×C17—88no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—41no—
99C2×C4×C17—41no—
100C2×C4×C17C22×C811yescharacteristic
101C2×C4×C17C2×M1611yescharacteristic
102C2×Dic17—21no—
103C2×Dic17—22no—
104C2×Dic17—41no—
105C2×Dic17—44no—
106C2×Dic17—42no—
107C2×Dic17—41no—
108C2×Dic17—41no—
109C2×Dic17—41no—
110C2×Dic17—42no—
111C2×Dic17—41no—
112C2×Dic17—41no—
113C2×Dic17—82no—
114C2×Dic17—84no—
115C2×Dic17#9912yes—
116#3624—84no—
117#2328—171no—
118#1459—172no—
119#1324—172no—
120#1318—172no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17—24no—
124C22×C17—44no—
125C22×C17—44no—
126C22×C17—42no—
127C22×C17—42no—
128C22×C17—42no—
129C22×C17—82no—
130C22×C17—82no—
131C22×C17—81no—
132C22×C17—81no—
133C22×C17C2×C22⋊C811yescharacteristic
134C22×C17C2×#9911yescharacteristic
135C22×C17#41311yescharacteristic
136C4×C17—21no—
137C4×C17—21no—
138C4×C17—42no—
139Dic17—22no—
140Dic17—42no—
141Dic17—41no—
142Dic17—81no—
143Dic17—81no—
144C2×#127#59511yescharacteristic
145C2×C4×D4—171no—
146C2×C4×D4—171no—
147C2×C22⋊C8—342no—
148#390—171no—
149#390—171no—
150#386—342no—
151#386—342no—
152#382—171no—
153#376—171no—
154#324—174no—
155#323—342no—
156C2×C17—21no—
157C2×C17—21no—
158C2×C17—44no—
159C2×C17—82no—
160C2×C17#145911yescharacteristic
161C2×C17#212711yescharacteristic
162C2×C17#213411yescharacteristic
163C22×D4—22no—
164C22×D4C2×#59511yescharacteristic
165C22×D4C2×#59511yescharacteristic
166C23×C4—172no—
167C23×C4—341no—
168C22×C8—682no—
169#127—22no—
170#127#362414yescomplemented
171C4×D4—682no—
172C4×D4—682no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4—171no—
175C2×C22⋊C4—172no—
176C2×C22⋊C4—341no—
177C2×C22⋊C4—342no—
178C2×C22⋊C4—342no—
179C2×C42—171no—
180C2×C42—171no—
181C2×C42C2×#59511yescharacteristic
182C22⋊C8—344no—
183C22⋊C8—682no—
184#95—341no—
185#95—341no—
186#95—342no—
187C17#1147911yescharacteristic, complemented, Sylow
188C24—21no—
189C24C22⋊#59512yes—
190C2×D4—28no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—82no—
197C2×D4—82no—
198C2×D4C2×#362412yes—
199C2×D4C2×#362412yescomplemented
200C2×D4#6324712yes—
201C22×C4—21no—
202C22×C4—171no—
203C22×C4—171no—
204C22×C4—172no—
205C22×C4—342no—
206C22×C4—342no—
207C22×C4—341no—
208C22×C4—341no—
209C22×C4—682no—
210C22×C4—1362no—
211C22×C4—1362no—
212C22×C4C22×#59511yescharacteristic
213C2×C8—684no—
214C2×C8—1364no—
215C4⋊C4—681no—
216C4⋊C4—681no—
217C22⋊C4—682no—
218C22⋊C4—682no—
219C22⋊C4—1362no—
220C22⋊C4—1362no—
221C42—21no—
222C42—342no—
223C42—681no—
224C42C2×#362412yescomplemented
225C23—22no—
226C23—22no—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—41no—
233C23—41no—
234C23C22⋊#362414yes—
235C23C2×C22⋊#59511yescharacteristic
236D4—42no—
237D4—42no—
238D4—42no—
239D4—42no—
240D4—88no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—172no—
247C2×C4—341no—
248C2×C4—342no—
249C2×C4—681no—
250C2×C4—684no—
251C2×C4—682no—
252C2×C4—681no—
253C2×C4—681no—
254C2×C4—681no—
255C2×C4—682no—
256C2×C4—681no—
257C2×C4—681no—
258C2×C4—1362no—
259C2×C4—1364no—
260C2×C4C22×#362411yescharacteristic
261C2×C4C2×#6324711yescharacteristic
262C8—1364no—
263C22—21no—
264C22—21no—
265C22—24no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—82no—
272C22—82no—
273C22—81no—
274C22—81no—
275C22C2×C22⋊#362411yescharacteristic
276C22D17⋊#9911yescharacteristic
277C22#1176379611yescharacteristic
278C4—21no—
279C4—21no—
280C4—42no—
281C4—342no—
282C4—682no—
283C4—681no—
284C4—1361no—
285C4—1361no—
286C2—21no—
287C2—21no—
288C2—44no—
289C2—82no—
290C2#1454402011yescharacteristic
291C2#1454506011yescharacteristic
292C2#1454508411yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
Fitting, prime radicalC2×C17×#127C4
socleC22×C17C2×#99
cosocleC2×C4⋊Dic17C23
hypercenterC2×#127#595
hypocenterC17#11479
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #14544020C2semidirect
2. #14543724C2semidirect
3. #14543712C2semidirect
24. C17×#127C8semidirect
52. C2×D4×C17C2×C8semidirect
69. C42×C17C2×C8semidirect
170. #127#3624semidirect
187. C17#11479semidirect
199. C2×D4C2×#3624semidirect
224. C42C2×#3624semidirect

Maximal subgroups

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

subgroupindex
#145440202
#145437242
#145437122
#145315102
#1147917