Finite Group Databasebeta

← #17959747

Subgroups — #17959747

Order 4352.

412 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 82 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#14545012C211yesmaximal, characteristic
2#14544660C211yesmaximal, characteristic
3#14543772C214yesmaximal, complemented
4C2×#11763047C211yesmaximal, characteristic
5C2×#10587530—21no—
6C2×#10587533—21no—
7#11763751—21no—
8#11763704C2212yes—
9C22⋊#63216—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—
23C2×C4⋊#595—22no—
24C2×Q16×C17—21no—
25C2×Q16×C17—42no—
26C2×Q16×C17C2×C412yes—
27C22×C8×C17C2×C411yescharacteristic
28#10587632—22no—
29#10587632—22no—
30#10587632—41no—
31#10587632—41no—
32C2×#63227—41no—
33C2×C4⋊Dic17—21no—
34C2×#63225—41no—
35C22×#3622D411yescharacteristic
36C2×C8×D17—42no—
37C2×C8×D17C2×C411yescharacteristic
38C2×C8×D17D411yescharacteristic
39C2×C8×D17D411yescharacteristic
40C2×C8×D17C2311yescharacteristic
41(C2×#595)⋊C4—41no—
42C2×#63216—22no—
43#10587533—42no—
44#10587530—42no—
45#10587506—22no—
46#10587506—22no—
47#10587489—21no—
48#10587489C2×C412yes—
49C23×D17D811yescharacteristic
50C22×#595—42no—
51C2×Q8×C17—22no—
52C2×Q8×C17—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×Dic17—41no—
58C22×Dic17SD1611yescharacteristic
59C2×C4×D17—41no—
60C2×C4×D17—41no—
61C2×C4×D17C22⋊C411yescharacteristic
62C2×C4×D17D811yescharacteristic
63C2×C4×D17SD1611yescharacteristic
64C2×C4×D17C2×D411yescharacteristic
65C22⋊#595—82no—
66C4⋊#595—42no—
67C4⋊#595—42no—
68Q16×C17—42no—
69Q16×C17—42no—
70C2×C8×C17—22no—
71C2×C8×C17C22⋊C411yescharacteristic
72C2×C8×C17C22×C411yescharacteristic
73#63227—42no—
74#63227—42no—
75C4⋊Dic17—22no—
76C4⋊Dic17—41no—
77#63225—42no—
78#63225—42no—
79C2×#3622—41no—
80C2×#3622C22⋊C411yescharacteristic
81C2×#3622C2×D411yescharacteristic
82C8×D17—22no—
83C8×D17—22no—
84C8×D17—41no—
85C8×D17—41no—
86#63216—42no—
87#11889—171nomaximal, Sylow
88C23×C17#10211yescharacteristic
89C22×D17—22no—
90C22×D17—22no—
91C22×D17—41no—
92C22×D17#10211yescharacteristic
93C22×D17C2×D811yescharacteristic
94C2×#595—82no—
95C2×#595—82no—
96C2×#595—82no—
97Q8×C17—42no—
98Q8×C17—42no—
99Q8×C17—82no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—42no—
103C2×C4×C17—82no—
104C2×C4×C17#10211yescharacteristic
105C2×C4×C17C2×C22⋊C411yescharacteristic
106C2×Dic17—41no—
107C2×Dic17—42no—
108C2×Dic17—82no—
109C2×Dic17#10211yescharacteristic
110C2×Dic17C2×SD1611yescharacteristic
111C4×D17—21no—
112C4×D17—21no—
113C4×D17—21no—
114C4×D17—21no—
115C4×D17—41no—
116C4×D17—41no—
117C8×C17—21no—
118C8×C17#9912yes—
119#3622—21no—
120#3622—21no—
121C2×#467—171no—
122#2112—171no—
123#1807—171no—
124#1348—174no—
125C22×C17—21no—
126C22×C17—21no—
127C22×C17#36012yes—
128C22×C17C2×#10211yescharacteristic
129C2×D17—21no—
130C2×D17—21no—
131C2×D17—21no—
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—44no—
135C2×D17#51012yes—
136#595—162no—
137C4×C17—21no—
138C4×C17—82no—
139C4×C17C2×#9911yescharacteristic
140C4×C17#42111yescharacteristic
141Dic17—21no—
142Dic17—21no—
143Dic17—82no—
144C22×Q16#59511yescharacteristic, complemented
145C23×C8—171no—
146C2×C22⋊Q8—341no—
147C2×M32—341no—
148#467—172no—
149#467—172no—
150#467—341no—
151#467—341no—
152C2×#107—171no—
153C2×#106—341no—
154C2×#103—341no—
155#380—341no—
156C22⋊C16—172no—
157C22⋊C16—341no—
158#340—172no—
159C2×C17—22no—
160C2×C17#151911yescharacteristic
161C2×C17#214111yescharacteristic
162C2×C17C2×#36011yescharacteristic
163D17—22no—
164D17—41no—
165C22×Q8—21no—
166C23×C4—171no—
167C2×Q16—21no—
168C2×Q16—42no—
169C2×Q16C2×#59512yes—
170C22×C8—171no—
171C22×C8—171no—
172C22×C8—171no—
173C22×C8—171no—
174C22×C8—171no—
175C22×C8—682no—
176C22×C8C2×#59511yescharacteristic
177C22⋊Q8—342no—
178C22⋊Q8—342no—
179C22⋊Q8—681no—
180C22⋊Q8—681no—
181C2×C4⋊C4—341no—
182C2×C4⋊C4—342no—
183C2×C4⋊C4—681no—
184C2×C22⋊C4—681no—
185C2×C22⋊C4—681no—
186M32—682no—
187C2×C16—342no—
188#107—172no—
189#107—341no—
190#106—682no—
191#103—342no—
192#103—342no—
193#95—681no—
194C17#1188911yescharacteristic, complemented, Sylow
195C24—171no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—41no—
200C22×C4—41no—
201C22×C4—171no—
202C22×C4—171no—
203C22×C4—171no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4—681no—
207C22×C4—681no—
208C22×C4—681no—
209C22×C4—682no—
210C22×C4C22⋊#59511yescharacteristic
211Q16—42no—
212Q16—42no—
213C2×C8—22no—
214C2×C8—171no—
215C2×C8—171no—
216C2×C8—342no—
217C2×C8—342no—
218C2×C8—681no—
219C2×C8—681no—
220C2×C8—681no—
221C2×C8C22⋊#59511yescharacteristic
222C2×C8C22×#59511yescharacteristic
223C4⋊C4—342no—
224C4⋊C4—682no—
225C4⋊C4—682no—
226C4⋊C4—682no—
227C4⋊C4—682no—
228C4⋊C4—681no—
229C22⋊C4—682no—
230C22⋊C4—682no—
231C22⋊C4—1362no—
232C16—682no—
233C23—171no—
234C23—171no—
235C23—342no—
236C23—342no—
237C23—681no—
238C23#1058754711yescharacteristic
239Q8—42no—
240Q8—42no—
241Q8—82no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—171no—
247C2×C4—171no—
248C2×C4—341no—
249C2×C4—341no—
250C2×C4—341no—
251C2×C4—341no—
252C2×C4—681no—
253C2×C4—681no—
254C2×C4—681no—
255C2×C4—682no—
256C2×C4—1362no—
257C2×C4—1362no—
258C2×C4—1362no—
259C2×C4—1362no—
260C2×C4#1058754711yescharacteristic
261C2×C4C2×C22⋊#59511yescharacteristic
262C8—21no—
263C8—341no—
264C8—341no—
265C8#1058755112yes—
266C22—21no—
267C22—21no—
268C22—172no—
269C22—341no—
270C22—341no—
271C22—341no—
272C22—341no—
273C22—341no—
274C22—684no—
275C22#1176372612yes—
276C22C2×#1058754711yescharacteristic
277C4—21no—
278C4—82no—
279C4—2722no—
280C4—341no—
281C4—341no—
282C4—1362no—
283C4C2×#1058755111yescharacteristic
284C4#1176380711yescharacteristic
285C2—22no—
286C2—342no—
287C2—681no—
288C2#1454422011yescharacteristic
289C2#1454509411yescharacteristic
290C2C2×#1176372611yescharacteristic
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#11889
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
144. C22×Q16#595semidirect
194. C17#11889semidirect

Maximal subgroups

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

subgroupindex
#145450122
#145446602
#145437722
C2×#117630472
#1188917