Finite Group Databasebeta

← #17981889

Subgroups — #17981889

Order 4352.

526 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 83 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#14545896C212yesmaximal
2#14545812C214yesmaximal, complemented
3#14545258C211yesmaximal, characteristic
4#11763922—22no—
5#11763895C2214yes—
6#11763894—22no—
7C2×#10587677C411yescharacteristic
8C2×#10587677C2211yescharacteristic
9(C2×#595)⋊Q8—22no—
10(C2×#595)⋊Q8C412yes—
11(C2×#595)⋊Q8C2212yes—
12#11763752—22no—
13C22×Q8×D17C411yescharacteristic
14C22×Q8×C17C811yescharacteristic, complemented
15C2×Q8×D17—21no—
16C2×Q8×D17—21no—
17C2×Q8×D17—42no—
18C2×Q8×D17—42no—
19C2×Q8×D17C812yescomplemented
20C2×Q8×D17C2×C412yes—
21C22×C4×D17—21no—
22C22×C4×D17C2×C411yescharacteristic
23C22×Dic34—21no—
24C22×Dic34C811yescharacteristic, complemented
25C2×C22⋊#595—21no—
26C2×C22⋊#595D412yes—
27#10587677—42no—
28#10587677—42no—
29C2×C4⋊#595—21no—
30C2×C4⋊#595C2×C411yescharacteristic
31C2×C4⋊#595C2311yescharacteristic
32C2×C4×#595C2×C412yes—
33C2×C4×#595D412yes—
34C22⋊#3633—24no—
35C4⋊#3633—42no—
36C4×#3633—42no—
37(C2×#595)⋊C4—24no—
38(C2×#595)⋊C4—24no—
39C23×D17C22⋊C411yescharacteristic
40C22×#595—22no—
41C22×#595C22⋊C412yes—
42C22×#595C2×D412yes—
43C2×#3633—44no—
44C2×Q8×C17—21no—
45C2×Q8×C17—42no—
46C2×Q8×C17C2×C812yes—
47C22×C4×C17—21no—
48C22×C4×C17C2×C811yescharacteristic
49C22×Dic17—21no—
50C22×Dic17M1611yescharacteristic
51Q8×D17—44no—
52Q8×D17—42no—
53Q8×D17—42no—
54Q8×D17—42no—
55Q8×D17—42no—
56Q8×D17—44no—
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—42no—
62C2×C4×D17—42no—
63C2×C4×D17C22⋊C411yescharacteristic
64C2×C4×D17C2×C811yescharacteristic
65C2×C4×D17M1611yescharacteristic
66C2×C4×D17C22×C411yescharacteristic
67C2×Dic34—21no—
68C2×Dic34—22no—
69C2×Dic34—22no—
70C2×Dic34—44no—
71C2×Dic34—42no—
72C2×Dic34C2×C812yes—
73C22⋊#595—42no—
74C22⋊#595—42no—
75C4⋊#595—41no—
76C4⋊#595—41no—
77C4⋊#595—42no—
78C4×#595—22no—
79C4×#595—22no—
80C4×#595—42no—
81#11434—171nomaximal, Sylow
82C23×C17C22⋊C811yescharacteristic
83C22×D17—21no—
84C22×D17—21no—
85C22×D17—41no—
86C22×D17—41no—
87C22×D17C22⋊C811yescharacteristic
88C22×D17C2×C22⋊C411yescharacteristic
89C2×#595—22no—
90C2×#595—22no—
91C2×#595—42no—
92C2×#595—42no—
93C2×#595—42no—
94C2×#595—42no—
95C2×#595—42no—
96#3633—84no—
97Q8×C17—42no—
98Q8×C17—42no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—42no—
102C2×C4×C17C22⋊C811yescharacteristic
103C2×C4×C17C22×C811yescharacteristic
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—42no—
107C2×Dic17C22⋊C811yescharacteristic
108C2×Dic17C2×M1611yescharacteristic
109C4×D17—21no—
110C4×D17—21no—
111C4×D17—22no—
112C4×D17—22no—
113C4×D17—44no—
114C4×D17—42no—
115C4×D17—42no—
116C4×D17#9912yes—
117C4×D17#10012yes—
118Dic34—44no—
119Dic34—42no—
120Dic34—44no—
121Dic34—42no—
122#2310—171no—
123#1458—172no—
124#1319—174no—
125C22×C17—21no—
126C22×C17—41no—
127C22×C17C2×C22⋊C811yescharacteristic
128C2×D17—21no—
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—42no—
132C2×D17—41no—
133C2×D17—41no—
134C2×D17—41no—
135C2×D17C2×#9911yescharacteristic
136C2×D17C2×#10011yescharacteristic
137#595—42no—
138#595—42no—
139#595—82no—
140C4×C17—21no—
141C4×C17—42no—
142C4×C17#32312yes—
143Dic17—21no—
144Dic17—42no—
145Dic17#32312yes—
146C23×Q8—171no—
147C2×#117—171no—
148C2×#117—171no—
149#431—342no—
150#391—172no—
151#391—172no—
152#391—342no—
153#381—342no—
154#324—174no—
155#323—342no—
156C2×C17—21no—
157C2×C17—41no—
158C2×C17C2×#32311yescharacteristic
159D17—21no—
160D17—41no—
161D17#212911yescharacteristic
162D17#213511yescharacteristic
163C22×Q8—172no—
164C22×Q8—172no—
165C22×Q8—171no—
166C22×Q8—341no—
167C22×Q8—341no—
168C22×Q8—341no—
169C22×Q8—682no—
170C22×Q8—682no—
171C22×Q8#363311yescharacteristic, complemented
172C23×C4—171no—
173C23×C4—341no—
174#117—682no—
175#117—682no—
176C2×C4⋊C4—171no—
177C2×C4⋊C4—171no—
178C2×C4⋊C4—341no—
179C2×C22⋊C4—172no—
180C2×C22⋊C4—341no—
181C2×C42—172no—
182C2×C42—172no—
183C4⋊C8—682no—
184C22⋊C8—344no—
185C4×C8—682no—
186#95—344no—
187#95—344no—
188C17#1143411yescharacteristic, complemented, Sylow
189C24—171no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—172no—
193C2×Q8—342no—
194C2×Q8—342no—
195C2×Q8—341no—
196C2×Q8—682no—
197C2×Q8—684no—
198C2×Q8—682no—
199C2×Q8—682no—
200C2×Q8—684no—
201C2×Q8—684no—
202C2×Q8—682no—
203C2×Q8—682no—
204C2×Q8C2×#363312yes—
205C22×C4—21no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—171no—
210C22×C4—171no—
211C22×C4—172no—
212C22×C4—172no—
213C22×C4—342no—
214C22×C4—342no—
215C22×C4—342no—
216C22×C4—342no—
217C22×C4—341no—
218C22×C4—342no—
219C22×C4—682no—
220C22×C4—682no—
221C22×C4C2×#363311yescharacteristic
222C2×C8—684no—
223C4⋊C4—682no—
224C4⋊C4—681no—
225C4⋊C4—681no—
226C22⋊C4—682no—
227C22⋊C4—682no—
228C42—342no—
229C42—342no—
230C42—682no—
231C23—171no—
232C23—171no—
233C23—341no—
234C23—341no—
235C23—681no—
236C23—681no—
237C23C22⋊#363311yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—682no—
241Q8—684no—
242Q8—684no—
243Q8—682no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—171no—
248C2×C4—172no—
249C2×C4—172no—
250C2×C4—171no—
251C2×C4—342no—
252C2×C4—341no—
253C2×C4—341no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—342no—
259C2×C4—682no—
260C2×C4—682no—
261C2×C4—684no—
262C2×C4—682no—
263C2×C4—682no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4—682no—
268C2×C4C22⋊#363311yescharacteristic
269C2×C4C22×#363311yescharacteristic
270C8—1364no—
271C22—21no—
272C22—41no—
273C22—171no—
274C22—171no—
275C22—341no—
276C22—341no—
277C22—341no—
278C22—681no—
279C22—681no—
280C22—681no—
281C22—682no—
282C22C2×C22⋊#363311yescharacteristic
283C4—21no—
284C4—42no—
285C4—172no—
286C4—341no—
287C4—682no—
288C4—682no—
289C4—682no—
290C4—1362no—
291C4#1176389412yes—
292C2—21no—
293C2—41no—
294C2—171no—
295C2—171no—
296C2—341no—
297C2—681no—
298C2C2×#1176389411yescharacteristic
subgroupG/H
centerC2C2×#11763894
commutatorC2×C4×C17C22×C8
FrattiniC2×C4C22×#3633
FittingC22×Q8×C17C8
socleC2×C17C2×#323
prime radicalC23×D17C22⋊C4
cosocleC2×C4⋊#595C23
hypercenterC22×Q8#3633
hypocenterC17#11434
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. #14545812C2semidirect
14. C22×Q8×C17C8semidirect
19. C2×Q8×D17C8semidirect
24. C22×Dic34C8semidirect
171. C22×Q8#3633semidirect
188. C17#11434semidirect

Maximal subgroups

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

subgroupindex
#145458962
#145458122
#145452582
#1143417