Finite Group Databasebeta

← #17959094

Subgroups — #17959094

Order 4352.

456 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 90 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#14545013C211yesmaximal, characteristic
2#14544470C211yesmaximal, characteristic
3#14543772C214yesmaximal, complemented
4C2×#11763047C211yesmaximal, characteristic
5C2×#10587677—21no—
6C2×#10587538—21no—
7#11763704—21no—
8#11763704C2212yes—
9#11763683—21no—
10#11763683C2212yes—
11C22×C8×D17C2211yescharacteristic
12C2×#10587632—21no—
13#11763047—21no—
14#11763047—21no—
15#11763047C412yescomplemented
16#11763047C2212yes—
17C2×#10587489C411yescharacteristic, complemented
18C2×#10587506—21no—
19C22×Q16×C17C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×C4×D17D411yescharacteristic
22C2×C22⋊#595—41no—
23#10587677—24no—
24#10587677—42no—
25C2×C4⋊#595—22no—
26C2×C4×#595—41no—
27C2×Q16×C17—21no—
28C2×Q16×C17—42no—
29C2×Q16×C17C2×C412yes—
30C22×C8×C17C2×C411yescharacteristic
31#10587632—22no—
32#10587632—22no—
33#10587632—41no—
34#10587632—41no—
35C2×#63227—41no—
36C2×C4⋊Dic17—21no—
37C2×#63225—41no—
38C22×#3622D411yescharacteristic
39C2×C8×D17—42no—
40C2×C8×D17C2×C411yescharacteristic
41C2×C8×D17D411yescharacteristic
42C2×C8×D17D411yescharacteristic
43C2×C8×D17C2311yescharacteristic
44#10587538—42no—
45C2×#63216—22no—
46#10587506—22no—
47#10587506—22no—
48#10587489—21no—
49#10587489C2×C412yes—
50C23×D17D811yescharacteristic
51C22×#595—42no—
52C2×Q8×C17—22no—
53C2×Q8×C17—41no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C22×C4×C17—41no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×Dic17—41no—
59C22×Dic17SD1611yescharacteristic
60C2×C4×D17—22no—
61C2×C4×D17—22no—
62C2×C4×D17C22⋊C411yescharacteristic
63C2×C4×D17D811yescharacteristic
64C2×C4×D17SD1611yescharacteristic
65C2×C4×D17C2×D411yescharacteristic
66C22⋊#595—44no—
67C4⋊#595—24no—
68C4⋊#595—42no—
69C4×#595—44no—
70Q16×C17—42no—
71Q16×C17—42no—
72C2×C8×C17—22no—
73C2×C8×C17C22⋊C411yescharacteristic
74C2×C8×C17C22×C411yescharacteristic
75#63227—42no—
76#63227—42no—
77C4⋊Dic17—22no—
78C4⋊Dic17—41no—
79#63225—42no—
80#63225—42no—
81C2×#3622—22no—
82C2×#3622C22⋊C411yescharacteristic
83C2×#3622C2×D411yescharacteristic
84C8×D17—41no—
85C8×D17—41no—
86C8×D17—41no—
87C8×D17—41no—
88#63216—42no—
89#11706—171nomaximal, Sylow
90C23×C17#10211yescharacteristic
91C22×D17—22no—
92C22×D17—22no—
93C22×D17—41no—
94C22×D17#10211yescharacteristic
95C22×D17C2×D811yescharacteristic
96C2×#595—44no—
97C2×#595—82no—
98C2×#595—82no—
99Q8×C17—42no—
100Q8×C17—42no—
101Q8×C17—82no—
102C2×C4×C17—42no—
103C2×C4×C17—82no—
104C2×C4×C17#9912yes—
105C2×C4×C17#10211yescharacteristic
106C2×C4×C17#10412yes—
107C2×C4×C17C2×C22⋊C411yescharacteristic
108C2×Dic17—22no—
109C2×Dic17—42no—
110C2×Dic17—82no—
111C2×Dic17#10211yescharacteristic
112C2×Dic17C2×SD1611yescharacteristic
113C4×D17—22no—
114C4×D17—22no—
115C4×D17—21no—
116C4×D17—21no—
117C4×D17—21no—
118C4×D17—21no—
119C8×C17—21no—
120C8×C17—21no—
121#3622—21no—
122#3622—21no—
123C2×#467—171no—
124#2112—171no—
125#1735—171no—
126#1348—174no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17#32712yes—
130C22×C17C2×#10211yescharacteristic
131C2×D17—21no—
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—44no—
137C2×D17#51012yes—
138#595—84no—
139C4×C17—82no—
140C4×C17#32712yes—
141C4×C17C2×#9911yescharacteristic
142C4×C17C2×#10411yescharacteristic
143Dic17—21no—
144Dic17—21no—
145Dic17—82no—
146C22×Q16#59511yescharacteristic, complemented
147C23×C8—171no—
148C2×C22⋊Q8—341no—
149C2×#117—341no—
150C2×M32—341no—
151#467—172no—
152#467—172no—
153#467—341no—
154#467—341no—
155C2×#107—171no—
156C2×#103—341no—
157C22⋊C16—172no—
158C22⋊C16—341no—
159#340—172no—
160#340—341no—
161C2×C17—22no—
162C2×C17C2×#32711yescharacteristic
163C2×C17#214111yescharacteristic
164C2×C17#214511yescharacteristic
165D17—22no—
166D17—41no—
167C22×Q8—21no—
168C23×C4—171no—
169C2×Q16—21no—
170C2×Q16—42no—
171C2×Q16C2×#59512yes—
172C22×C8—171no—
173C22×C8—171no—
174C22×C8—171no—
175C22×C8—171no—
176C22×C8—171no—
177C22×C8—682no—
178C22×C8C2×#59511yescharacteristic
179C22⋊Q8—342no—
180C22⋊Q8—342no—
181C22⋊Q8—681no—
182C22⋊Q8—681no—
183#117—344no—
184#117—682no—
185C2×C4⋊C4—342no—
186C2×C4⋊C4—341no—
187C2×C4⋊C4—681no—
188C2×C22⋊C4—681no—
189C2×C22⋊C4—681no—
190C2×C42—681no—
191M32—682no—
192C2×C16—342no—
193#107—172no—
194#107—341no—
195#103—342no—
196#103—342no—
197C17#1170611yescharacteristic, complemented, Sylow
198C24—171no—
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—42no—
203C22×C4—41no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4—171no—
207C22×C4—171no—
208C22×C4—171no—
209C22×C4—342no—
210C22×C4—342no—
211C22×C4—682no—
212C22×C4—681no—
213C22×C4C22⋊#59511yescharacteristic
214Q16—42no—
215Q16—42no—
216C2×C8—22no—
217C2×C8—171no—
218C2×C8—171no—
219C2×C8—342no—
220C2×C8—681no—
221C2×C8—681no—
222C2×C8—681no—
223C2×C8—681no—
224C2×C8C22⋊#59511yescharacteristic
225C2×C8C22×#59511yescharacteristic
226C4⋊C4—344no—
227C4⋊C4—342no—
228C4⋊C4—682no—
229C4⋊C4—682no—
230C4⋊C4—682no—
231C4⋊C4—681no—
232C22⋊C4—682no—
233C22⋊C4—682no—
234C22⋊C4—684no—
235C42—684no—
236C16—682no—
237C23—171no—
238C23—171no—
239C23—342no—
240C23—342no—
241C23—681no—
242C23#1058754711yescharacteristic
243Q8—42no—
244Q8—42no—
245Q8—82no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—171no—
249C2×C4—171no—
250C2×C4—342no—
251C2×C4—341no—
252C2×C4—341no—
253C2×C4—341no—
254C2×C4—341no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—684no—
258C2×C4—682no—
259C2×C4—1362no—
260C2×C4—1362no—
261C2×C4—1362no—
262C2×C4#1058754711yescharacteristic
263C2×C4#1058754812yes—
264C2×C4#1058755112yes—
265C2×C4C2×C22⋊#59511yescharacteristic
266C8—21no—
267C8—21no—
268C8—341no—
269C8—341no—
270C22—21no—
271C22—21no—
272C22—172no—
273C22—341no—
274C22—341no—
275C22—341no—
276C22—341no—
277C22—341no—
278C22—684no—
279C22#1176366012yes—
280C22C2×#1058754711yescharacteristic
281C4—82no—
282C4—341no—
283C4—341no—
284C4—1364no—
285C4—1362no—
286C4#1176366012yes—
287C4C2×#1058755111yescharacteristic
288C4C2×#1058754811yescharacteristic
289C2—22no—
290C2—342no—
291C2—681no—
292C2C2×#1176366011yescharacteristic
293C2#1454509411yescharacteristic
294C2#1454510611yescharacteristic
subgroupG/H
centerC22C2×#10587547
commutatorC2×C8×C17C22×C4
FrattiniC2×C8C22×#595
FittingC22×Q16×C17C4
socleC22×C17C2×#102
prime radicalC23×D17D8
cosocleC2×C8×D17C23
hypercenterC22×Q16#595
hypocenterC17#11706
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #14543772C2semidirect
15. #11763047C4semidirect
17. C2×#10587489C4semidirect
19. C22×Q16×C17C4semidirect
146. C22×Q16#595semidirect
197. C17#11706semidirect

Maximal subgroups

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

subgroupindex
#145450132
#145444702
#145437722
C2×#117630472
#1170617