Finite Group Databasebeta

← #17959916

Subgroups — #17959916

Order 4352.

316 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 80 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#14545142C211yesmaximal, characteristic, complemented
2C2×#11763671C211yesmaximal, characteristic, complemented
3#14543800C212yesmaximal, complemented
4#14543796C212yesmaximal, complemented
5#14539349C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7#11763730C2212yes—
8C22⋊#63216—21no—
9#11763695—21no—
10#11763671—21no—
11#11763671C2212yes—
12#11763340—21no—
13#11763210C2211yescharacteristic
14#11763192C2211yescharacteristic
15#11763162—21no—
16#11763050C411yescharacteristic
17#11763036—21no—
18#11762903C411yescharacteristic
19#11762900—21no—
20D8×Dic17C411yescharacteristic
21C2×C8×Dic17C2211yescharacteristic
22C17×#466C411yescharacteristic
23C22×#3624—41no—
24C2×D8×C17C2×C411yescharacteristic
25C22×C8×C17C2×C411yescharacteristic
26C17×#121—21no—
27#10587622—21no—
28#10587621—41no—
29D4×Dic17—21no—
30D4×Dic17—41no—
31C2×#63229—21no—
32#10587610—21no—
33C2×C4×Dic17D411yescharacteristic
34C22×#3622D411yescharacteristic
35C2×Dic68C2×C411yescharacteristic
36#10587576—21no—
37#10587573—21no—
38#10587565—41no—
39#10587563—41no—
40C4⋊#3624—22no—
41C4⋊#3624—41no—
42C2×#63216—22no—
43C17×#107C2×C411yescharacteristic
44C17×#102—21no—
45#10587503—21no—
46#10587489C2×C411yescharacteristic
47#10587487—21no—
48C8×Dic17C2×C411yescharacteristic
49C8×Dic17D411yescharacteristic
50C8×Dic17D411yescharacteristic
51C8×Dic17C2311yescharacteristic
52#10587481—21no—
53#10587479—21no—
54C2×D4×C17—21no—
55C2×D4×C17—41no—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×Dic17—41no—
58C22×Dic17D811yescharacteristic
59C22×Dic17SD1611yescharacteristic
60C2×Dic34—21no—
61C2×#3624—42no—
62C2×#3624—82no—
63D8×C17—41no—
64C2×C8×C17—41no—
65C2×C8×C17C22⋊C411yescharacteristic
66C2×C8×C17C22×C411yescharacteristic
67C4⋊C4×C17—21no—
68C22⋊C4×C17—41no—
69C22⋊Dic17—41no—
70C22⋊Dic17—41no—
71C22⋊Dic17—41no—
72C22⋊Dic17—41no—
73#63229—42no—
74C4⋊Dic17—21no—
75C4⋊Dic17—41no—
76#63225—41no—
77#63225—41no—
78#63225—41no—
79C4×Dic17—41no—
80C4×Dic17C22⋊C411yescharacteristic
81C4×Dic17D811yescharacteristic
82C4×Dic17SD1611yescharacteristic
83C4×Dic17C2×D411yescharacteristic
84C2×#3622—22no—
85C2×#3622C22⋊C411yescharacteristic
86C2×#3622C2×D411yescharacteristic
87Dic68—41no—
88#63216—42no—
89#11944—171nomaximal, Sylow
90C23×C17—41no—
91C23×C17#10211yescharacteristic
92D4×C17—41no—
93D4×C17—41no—
94D4×C17—82no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—41no—
98C2×C4×C17#10211yescharacteristic
99C2×C4×C17C2×C22⋊C411yescharacteristic
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—82no—
105C2×Dic17#10211yescharacteristic
106C2×Dic17#10211yescharacteristic
107C2×Dic17C2×D811yescharacteristic
108C2×Dic17C2×SD1611yescharacteristic
109Dic34—41no—
110Dic34—41no—
111#3624—82no—
112C8×C17—21no—
113C8×C17—21no—
114#3622—21no—
115#3622#13512yes—
116#3168—171no—
117#2162—171no—
118C2×#334—171no—
119#1363—172no—
120#1360—172no—
121C22×C17—21no—
122C22×C17—21no—
123C22×C17—81no—
124C22×C17—81no—
125C22×C17—81no—
126C22×C17#35912yes—
127C22×C17C2×#10211yescharacteristic
128C4×C17—21no—
129C4×C17—81no—
130C4×C17#41311yescharacteristic
131C4×C17#41611yescharacteristic
132Dic17—21no—
133Dic17—21no—
134Dic17—21no—
135Dic17—81no—
136Dic17—81no—
137Dic17#36012yes—
138#529—341no—
139#499—171no—
140#495—171no—
141#488—341no—
142#467—171no—
143#466#59511yescharacteristic
144#465—341no—
145C4×Q16—171no—
146C4×SD16—341no—
147C4×D8—171no—
148C2×C4⋊C8—341no—
149C2×C4×C8—171no—
150C4⋊C16—172no—
151C22⋊C16—341no—
152C4×C16—341no—
153#334—172no—
154#334—341no—
155C2×C17—22no—
156C2×C17—161no—
157C2×C17#154011yescharacteristic
158C2×C17C2×#35911yescharacteristic
159C2×C17C2×#36011yescharacteristic
160C2×Q16—171no—
161C2×SD16—341no—
162C2×D8C2×#59511yescharacteristic
163C22×C8—171no—
164C22×C8—681no—
165C22×C8C2×#59511yescharacteristic
166#126—681no—
167#125—341no—
168#124—341no—
169#123—681no—
170C22⋊Q8—341no—
171#121—21no—
172C4×Q8—341no—
173C4×D4—341no—
174C4×D4—681no—
175#117—681no—
176C2×C42—171no—
177C2×C16—342no—
178#107—171no—
179#107C2×#59511yescharacteristic
180#106—341no—
181C4⋊C8—342no—
182C4⋊C8—681no—
183#103—341no—
184#103—341no—
185#102—21no—
186#102—341no—
187C4×C8—171no—
188C4×C8—171no—
189C4×C8—171no—
190C4×C8—171no—
191C17#1194411yescharacteristic, complemented, Sylow
192C2×Q8—341no—
193C2×D4—21no—
194C2×D4—41no—
195C22×C4—171no—
196C22×C4—171no—
197C22×C4—681no—
198C22×C4C22⋊#59511yescharacteristic
199Q16—681no—
200SD16—682no—
201D8—41no—
202C2×C8—41no—
203C2×C8—171no—
204C2×C8—171no—
205C2×C8—342no—
206C2×C8—682no—
207C2×C8—1362no—
208C2×C8C22⋊#59511yescharacteristic
209C2×C8C22×#59511yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—341no—
212C4⋊C4—681no—
213C4⋊C4—681no—
214C4⋊C4—681no—
215C4⋊C4—681no—
216C22⋊C4—41no—
217C22⋊C4—681no—
218C22⋊C4—681no—
219C22⋊C4—681no—
220C22⋊C4—681no—
221C42—171no—
222C42—171no—
223C42—171no—
224C42—171no—
225C42—681no—
226C16—682no—
227C23—41no—
228C23#1058754711yescharacteristic
229Q8—681no—
230Q8—681no—
231D4—41no—
232D4—41no—
233D4—82no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—171no—
238C2×C4—171no—
239C2×C4—171no—
240C2×C4—171no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—681no—
244C2×C4—681no—
245C2×C4—1362no—
246C2×C4#1058754711yescharacteristic
247C2×C4C2×C22⋊#59511yescharacteristic
248C8—21no—
249C8—21no—
250C8—172no—
251C8—341no—
252C8—1362no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#1176372312yes—
259C22C2×#1058754711yescharacteristic
260C4—21no—
261C4—81no—
262C4—172no—
263C4—341no—
264C4—341no—
265C4—341no—
266C4—1361no—
267C4—1361no—
268C4#1176379611yescharacteristic
269C4#1176379911yescharacteristic
270C2—22no—
271C2—161no—
272C2#1454428211yescharacteristic
273C2C2×#1176372311yescharacteristic
274C2D17⋊#36011yescharacteristic
subgroupG/H
centerC22C2×#10587547
commutatorC2×C8×C17C22×C4
FrattiniC2×C8C22×#595
Fitting, prime radicalC17×#466C4
socleC22×C17C2×#102
cosocleC8×Dic17C23
hypercenter#466#595
hypocenterC17#11944
LCSG ▷ C2×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #14545142C2semidirect
2. C2×#11763671C2semidirect
3. #14543800C2semidirect
4. #14543796C2semidirect
191. C17#11944semidirect

Maximal subgroups

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

subgroupindex
#145451422
C2×#117636712
#145438002
#145437962
#145393492
#1194417