Finite Group Databasebeta

← #17958195

Subgroups — #17958195

Order 4352.

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

normal subgroups only

This group has 598 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#14544013C212yesmaximal
2#14543727C212yesmaximal, complemented
3#14543726C212yesmaximal, complemented
4D17×#384C211yesmaximal, characteristic
5#11763777—22no—
6#11763655—22no—
7#11763654C2214yes—
8C2×C4⋊C4×D17—22no—
9C2×C4⋊C4×D17C411yescharacteristic
10C2×C4⋊C4×D17C2211yescharacteristic
11C2×C42×D17C411yescharacteristic
12#11762679C2211yescharacteristic
13#11762678—22no—
14#11762678C411yescharacteristic
15#11762678C2211yescharacteristic
16C17×#384C411yescharacteristic
17C22×C4×D17—21no—
18C22×C4×D17—21no—
19C22×C4×D17C2×C411yescharacteristic
20C22×C4×D17D412yes—
21C2×#63242—42no—
22C2×C4⋊C4×C17—22no—
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C42×C17C2×C411yescharacteristic
26C2×C4⋊Dic17—22no—
27C2×C4⋊Dic17C2×C411yescharacteristic
28C2×C4⋊Dic17C2311yescharacteristic
29C2×#63225—22no—
30C2×#63225—21no—
31C2×#63225—22no—
32C2×#63225D412yes—
33C2×C4×Dic17—21no—
34C2×C4×Dic17C2×C411yescharacteristic
35C4⋊C4×D17—41no—
36C4⋊C4×D17—41no—
37C4⋊C4×D17—48no—
38C4⋊C4×D17—41no—
39C4⋊C4×D17—41no—
40C42×D17—22no—
41C42×D17C814yescomplemented
42C22⋊#3624—24no—
43#10587543—42no—
44C23×D17C22⋊C411yescharacteristic
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C22⋊C412yes—
48C22×C4×C17C22×C411yescharacteristic
49C22×Dic17—21no—
50C22×Dic17—21no—
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C2×D412yes—
53C2×C4×D17—24no—
54C2×C4×D17—24no—
55C2×C4×D17—24no—
56C2×C4×D17—21no—
57C2×C4×D17—21no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—44no—
62C2×C4×D17—44no—
63C2×C4×D17—41no—
64C2×C4×D17—41no—
65C2×C4×D17C2×C812yes—
66C2×C4×D17M1612yes—
67C2×#3624—44no—
68#63242—84no—
69C4⋊C4×C17—41no—
70C4⋊C4×C17—41no—
71C4⋊C4×C17—44no—
72C42×C17—21no—
73C42×C17C2×C812yescomplemented
74C4⋊Dic17—41no—
75C4⋊Dic17—44no—
76C4⋊Dic17—41no—
77#63225—42no—
78#63225—44no—
79#63225—44no—
80#63225—42no—
81C4×Dic17—21no—
82C4×Dic17—24no—
83C4×Dic17C2×C812yescomplemented
84C2×#1325—171nomaximal, Sylow
85C23×C17C2×C22⋊C411yescharacteristic
86C22×D17—22no—
87C22×D17—22no—
88C22×D17C22⋊C812yes—
89C22×D17#9911yescharacteristic
90C22×D17#9911yescharacteristic
91C22×D17#10112yes—
92C2×C4×C17—22no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—22no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—41no—
101C2×C4×C17C22×C811yescharacteristic
102C2×C4×C17C2×M1611yescharacteristic
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—22no—
109C2×Dic17—42no—
110C2×Dic17—42no—
111C2×Dic17—41no—
112C2×Dic17C22⋊C812yes—
113C4×D17—216no—
114C4×D17—22no—
115C4×D17—22no—
116C4×D17—22no—
117C4×D17—22no—
118C4×D17—44no—
119C4×D17—82no—
120C4×D17—82no—
121C4×D17—82no—
122#3624—84no—
123C2×#384—171no—
124C2×#324—172no—
125#1325—172no—
126#1325—172no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17C2×C22⋊C811yescharacteristic
130C22×C17C2×#9911yescharacteristic
131C22×C17C2×#10111yescharacteristic
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—24no—
135C2×D17—24no—
136C2×D17#32412yes—
137C2×D17#32412yes—
138C2×D17#35312yes—
139C2×D17#35612yes—
140C4×C17—24no—
141C4×C17—21no—
142C4×C17—21no—
143C4×C17—42no—
144C4×C17—81no—
145Dic17—24no—
146Dic17—22no—
147Dic17—42no—
148Dic17—81no—
149C22×C4⋊C4—171no—
150C22×C4⋊C4—171no—
151C22×C4⋊C4—342no—
152C22×C42—171no—
153C2×C22⋊C8—342no—
154#384—171no—
155#384—171no—
156#384—171no—
157#384—342no—
158#384#59511yescharacteristic
159#324—174no—
160#324—342no—
161C2×C17—21no—
162C2×C17—21no—
163C2×C17C2×#32411yescharacteristic
164C2×C17C2×#35311yescharacteristic
165C2×C17C2×#35611yescharacteristic
166D17—22no—
167D17#132514yescomplemented
168C23×C4—171no—
169C23×C4—172no—
170C23×C4—341no—
171C23×C4—341no—
172C22×C8—682no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—172no—
175C2×C4⋊C4—171no—
176C2×C4⋊C4—171no—
177C2×C4⋊C4—341no—
178C2×C4⋊C4—342no—
179C2×C4⋊C4—342no—
180C2×C4⋊C4—342no—
181C2×C4⋊C4—681no—
182C2×C4⋊C4—681no—
183C2×C4⋊C4—681no—
184C2×C4⋊C4—681no—
185C2×C4⋊C4—688no—
186C2×C4⋊C4C2×#59511yescharacteristic
187C2×C4⋊C4C2×#59511yescharacteristic
188C2×C42—171no—
189C2×C42—174no—
190C2×C42—342no—
191C2×C42—341no—
192C2×C42C2×#59511yescharacteristic
193C22⋊C8—344no—
194C22⋊C8—682no—
195C17C2×#132511yescharacteristic, complemented, Sylow
196C24—171no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—171no—
200C22×C4—172no—
201C22×C4—172no—
202C22×C4—172no—
203C22×C4—344no—
204C22×C4—341no—
205C22×C4—341no—
206C22×C4—342no—
207C22×C4—344no—
208C22×C4—344no—
209C22×C4—342no—
210C22×C4—342no—
211C22×C4—341no—
212C22×C4—341no—
213C22×C4—684no—
214C22×C4—681no—
215C22×C4—681no—
216C22×C4—684no—
217C22×C4C22⋊#59512yes—
218C22×C4C22×#59511yescharacteristic
219C2×C8—684no—
220C2×C8—1364no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—44no—
224C4⋊C4—682no—
225C4⋊C4—681no—
226C4⋊C4—681no—
227C4⋊C4—684no—
228C4⋊C4—684no—
229C4⋊C4—684no—
230C4⋊C4—682no—
231C42—21no—
232C42—172no—
233C42—344no—
234C42—341no—
235C42#6324212yescomplemented
236C23—171no—
237C23—172no—
238C23—171no—
239C23—172no—
240C23—342no—
241C23—342no—
242C23C2×C22⋊#59511yescharacteristic
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—172no—
253C2×C4—342no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—3416no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—342no—
261C2×C4—341no—
262C2×C4—341no—
263C2×C4—342no—
264C2×C4—684no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4—681no—
268C2×C4—1362no—
269C2×C4—1362no—
270C2×C4—1362no—
271C2×C4C2×#6324211yescharacteristic
272C2×C4D17⋊M1611yescharacteristic
273C8—1364no—
274C22—21no—
275C22—21no—
276C22—172no—
277C22—172no—
278C22—172no—
279C22—172no—
280C22—344no—
281C22—344no—
282C22—341no—
283C22—341no—
284C22#1176377711yescharacteristic
285C22D17⋊#9911yescharacteristic
286C22D17⋊#10111yescharacteristic
287C4—24no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—81no—
292C4—344no—
293C4—342no—
294C4—682no—
295C4—1361no—
296C2—21no—
297C2—21no—
298C2—174no—
299C2—342no—
300C2#1454401311yescharacteristic
301C2D17⋊#35311yescharacteristic
302C2D17⋊#35611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC22×C17C2×#99
prime radicalC23×D17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenter#384#595
hypocenterC17C2×#1325
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #14543727C2semidirect
3. #14543726C2semidirect
41. C42×D17C8semidirect
73. C42×C17C2×C8semidirect
83. C4×Dic17C2×C8semidirect
167. D17#1325semidirect
195. C17C2×#1325semidirect
235. C42#63242semidirect

Maximal subgroups

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

subgroupindex
#145440132
#145437272
#145437262
D17×#3842
C2×#132517