Finite Group Databasebeta

← #16947142

Subgroups — #16947142

Order 4352.

460 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 59 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#14535156C211yesmaximal, characteristic
2#14527758C212yesmaximal, complemented
3#14526813C212yesmaximal, complemented
4#14526716C211yesmaximal, characteristic
5C17×#2027C211yesmaximal, characteristic
6C2×#10587564—22no—
7#11762964—22no—
8#11762732—22no—
9#11762729C2212yes—
10#11762707—21no—
11#11762689—21no—
12#11762546—22no—
13#11762535C2212yes—
14C17×#544C2211yescharacteristic
15C17×#418—21no—
16C17×#390C2211yescharacteristic
17C17×#343C2211yescharacteristic
18C17×#339—21no—
19C22×D4×C17D411yescharacteristic
20C23×Dic17—21no—
21C22×Dic34D412yes—
22C2×M16×C17—21no—
23C17×C4⋊Q8—41no—
24C17×#125—41no—
25C17×C22⋊Q8—22no—
26C17×#117—21no—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C42×C17D411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—21no—
32C2×#63231—42no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×#63225—22no—
36C2×#63225—22no—
37#10587564—48no—
38C17×#102—41no—
39C17×#102—41no—
40#10587502—22no—
41#10587502D412yes—
42C22⋊#3622—22no—
43#10587481—42no—
44C24×C17—21no—
45C2×Q8×C17—22no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17Q1612yescomplemented
50C22×C4×C17—21no—
51C22×C4×C17C2×D412yes—
52C22×C4×C17C2×D411yescharacteristic
53C22×Dic17—22no—
54C22×Dic17—22no—
55C22×Dic17—44no—
56C22×Dic17—81no—
57C2×Dic34—42no—
58C2×Dic34—44no—
59M16×C17—41no—
60M16×C17—41no—
61C2×C8×C17—41no—
62C4⋊C4×C17—22no—
63C4⋊C4×C17—21no—
64C4⋊C4×C17—21no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—42no—
67C22⋊C4×C17—44no—
68C22⋊C4×C17—41no—
69C42×C17—41no—
70C42×C17—41no—
71C22⋊Dic17—44no—
72C22⋊Dic17—81no—
73#63231—84no—
74C4⋊Dic17—24no—
75C4⋊Dic17—81no—
76#63225—44no—
77#63225—44no—
78C2×#3622—42no—
79#12220—171nomaximal, Sylow
80C23×C17—22no—
81C23×C17—42no—
82C23×C17—42no—
83C23×C17—41no—
84C23×C17—41no—
85C23×C17#12011yescharacteristic
86Q8×C17—82no—
87D4×C17—42no—
88D4×C17—81no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17C2×Q1611yescharacteristic
99C2×C4×C17#13711yescharacteristic
100C2×Dic17—24no—
101C2×Dic17—42no—
102C2×Dic17—42no—
103C2×Dic17—42no—
104C2×Dic17—42no—
105C2×Dic17—42no—
106C2×Dic17—84no—
107Dic34—44no—
108Dic34—82no—
109C8×C17—81no—
110#3622—82no—
111#2683—171no—
112#2027D1711yescharacteristic
113#1605—172no—
114#1354—172no—
115#1318—171no—
116C22×C17—21no—
117C22×C17—21no—
118C22×C17—44no—
119C22×C17—42no—
120C22×C17—82no—
121C22×C17—81no—
122C22×C17—81no—
123C22×C17#45111yescharacteristic
124C22×C17#45511yescharacteristic
125C22×C17B(4,𝔽2)11yescharacteristic
126C4×C17—21no—
127C4×C17—21no—
128C4×C17—42no—
129C4×C17—42no—
130C4×C17—81no—
131C4×C17—81no—
132Dic17—44no—
133Dic17—82no—
134#544C2×D1711yescharacteristic
135C2×C22⋊Q8—342no—
136#451—342no—
137#418—21no—
138#396—342no—
139#395—172no—
140#390C2×D1711yescharacteristic
141#389—341no—
142#386—341no—
143#343C2×D1711yescharacteristic
144#339—21no—
145#328—342no—
146#324—172no—
147C2×C17—21no—
148C2×C17—21no—
149C2×C17—82no—
150C2×C17#160211yescharacteristic
151C2×C17#219611yescharacteristic
152C2×C17#220311yescharacteristic
153C22×Q8—172no—
154C22×D4C22⋊D1711yescharacteristic
155C23×C4—341no—
156C2×Q16—682no—
157C2×M16—21no—
158C4⋊Q8—41no—
159#125—41no—
160C22⋊Q8—22no—
161C22⋊Q8—688no—
162#117—21no—
163C2×C4⋊C4—341no—
164C2×C4⋊C4—342no—
165C2×C4⋊C4—342no—
166C2×C4⋊C4—341no—
167C2×C4⋊C4C22×D1711yescharacteristic
168C2×C22⋊C4—22no—
169C2×C22⋊C4—342no—
170C2×C22⋊C4—341no—
171C2×C42C22⋊D1711yescharacteristic
172#103—682no—
173#102—41no—
174#102—41no—
175C22⋊C8—342no—
176#95—172no—
177#95—342no—
178C17#1222011yescharacteristic, complemented, Sylow
179C24—21no—
180C2×Q8—22no—
181C2×Q8—682no—
182C2×Q8—684no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4#6323112yescomplemented
187C22×C4—21no—
188C22×C4—342no—
189C22×C4—342no—
190C22×C4—684no—
191C22×C4—1361no—
192C22×C4D4×D1712yes—
193C22×C4C2×C22⋊D1711yescharacteristic
194Q16—1364no—
195M16—41no—
196M16—41no—
197C2×C8—41no—
198C2×C8—682no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—344no—
205C4⋊C4—684no—
206C4⋊C4—684no—
207C4⋊C4—1361no—
208C22⋊C4—44no—
209C22⋊C4—41no—
210C22⋊C4—684no—
211C22⋊C4—1361no—
212C42—41no—
213C42—41no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23—41no—
218C23—41no—
219C23#1058762311yescharacteristic
220Q8—82no—
221Q8—684no—
222Q8—1362no—
223D4—42no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—344no—
235C2×C4—682no—
236C2×C4—682no—
237C2×C4—682no—
238C2×C4—682no—
239C2×C4—682no—
240C2×C4—1364no—
241C2×C4C2×#6323111yescharacteristic
242C2×C4#1058763711yescharacteristic
243C8—81no—
244C8—1362no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22#1176296411yescharacteristic
253C22#1176298611yescharacteristic
254C22#1176299511yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—81no—
260C4—81no—
261C4—684no—
262C4—1362no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1452774011yescharacteristic
267C2#1453076711yescharacteristic
268C2#1453080411yescharacteristic
subgroupG/H
centerC22#11762995
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2027C2
socleC22×C17B(4,𝔽2)
prime radicalC22×D4×C17D4
hypercenter#2027D17
hypocenterC17#12220
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2027

Complements

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

NH ≅ G/Ntype
2. #14527758C2semidirect
3. #14526813C2semidirect
49. C2×D4×C17Q16semidirect
178. C17#12220semidirect
186. C2×D4#63231semidirect

Maximal subgroups

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

subgroupindex
#145351562
#145277582
#145268132
#145267162
C17×#20272
#1222017