Finite Group Databasebeta

← #17952121

Subgroups — #17952121

Order 4352.

516 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 132 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#14544569C212yesmaximal
2C2×#11763669C212yesmaximal
3#14543662C212yesmaximal
4#14531353C211yesmaximal, characteristic
5C2×#10587544C2214yes—
6C2×#10587542—22no—
7#11763675—22no—
8#11763669C418yescomplemented
9C2×#10587630C2211yescharacteristic
10C2×Q8×Dic17C411yescharacteristic
11C2×#10587573C411yescharacteristic
12#11762715C411yescharacteristic
13#11762706—22no—
14#11762678C2211yescharacteristic
15#11762660—22no—
16C42×Dic17C2211yescharacteristic
17C17×#391C411yescharacteristic
18C22×Q8×C17C2×C411yescharacteristic
19C22×Dic34C2×C411yescharacteristic
20C22×#3624—24no—
21C2×C4⋊C4×C17C2×C411yescharacteristic
22C2×C42×C17C2×C411yescharacteristic
23Q8×Dic17—42no—
24#10587630—22no—
25#10587630C2×C414yes—
26C2×C4⋊Dic17—21no—
27C2×C4⋊Dic17C2×C411yescharacteristic
28C2×#63225—21no—
29C2×#63225—22no—
30C2×#63225—22no—
31C2×C4×Dic17—21no—
32C2×C4×Dic17—21no—
33C2×C4×Dic17C2×C411yescharacteristic
34C2×C4×Dic17D412yes—
35C2×C4×Dic17D412yes—
36C2×C4×Dic17C2311yescharacteristic
37#10587573—42no—
38#10587544—24no—
39#10587544C2×C418yes—
40#10587542—44no—
41C17×#95—22no—
42#10587502—22no—
43C2×Q8×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17C4212yes—
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17C22⋊C412yes—
50C22×C4×C17C22×C411yescharacteristic
51C22×Dic17—21no—
52C22×Dic17—21no—
53C22×Dic17C22⋊C411yescharacteristic
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D412yes—
56C2×Dic34—21no—
57C2×Dic34—41no—
58C2×Dic34—41no—
59C2×Dic34C4212yes—
60C2×#3624—28no—
61C2×#3624—44no—
62C2×#3624—44no—
63C4⋊C4×C17—41no—
64C42×C17—41no—
65C4⋊Dic17—42no—
66C4⋊Dic17—41no—
67#63225—28no—
68#63225—42no—
69#63225—42no—
70#63225—42no—
71C4×Dic17—22no—
72C4×Dic17—22no—
73C4×Dic17—24no—
74C4×Dic17—22no—
75C4×Dic17—42no—
76C4×Dic17—42no—
77C4×Dic17C22×C412yes—
78C4×Dic17#4112yes—
79#8232—171nomaximal, Sylow
80C23×C17C2×C22⋊C411yescharacteristic
81Q8×C17—42no—
82Q8×C17—42no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—41no—
90C2×C4×C17—41no—
91C2×C4×C17—41no—
92C2×C4×C17—42no—
93C2×C4×C17C2×C4211yescharacteristic
94C2×C4×C17#11711yescharacteristic
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—21no—
98C2×Dic17—22no—
99C2×Dic17—41no—
100C2×Dic17—41no—
101C2×Dic17—41no—
102C2×Dic17—42no—
103C2×Dic17#10012yes—
104C2×Dic17C4×D414yes—
105C2×Dic17AGL(1,ℤ/8)12yes—
106C2×Dic17#13714yes—
107C2×Dic17#13712yes—
108Dic34—42no—
109Dic34—42no—
110#3624—48no—
111C4×C4⋊Q8—171no—
112#1767—172no—
113C2×#333—172no—
114#1296—172no—
115C22×C17C4×C22⋊C411yescharacteristic
116C22×C17C2×#10011yescharacteristic
117C22×C17#41311yescharacteristic
118C22×C17#41712yes—
119C22×C17#41912yes—
120C4×C17—21no—
121C4×C17—21no—
122C4×C17—42no—
123C4×C17—42no—
124C4×C17—81no—
125Dic17—22no—
126Dic17—41no—
127Dic17—42no—
128Dic17—81no—
129Dic17#44014yes—
130Dic17#44114yes—
131C2×C4⋊Q8—171no—
132C2×C4×Q8—171no—
133C2×C4×Q8—171no—
134C2×C4⋊C8—174no—
135C2×#97—342no—
136#391—171no—
137#391#59511yescharacteristic
138#389—342no—
139#384—171no—
140C4×C4⋊C4—342no—
141C43—171no—
142#336—342no—
143#333—178no—
144C2×C17#155811yescharacteristic
145C2×C17#155912yes—
146C2×C17#156011yescharacteristic
147C2×C17C4×#10011yescharacteristic
148C2×C17#176312yes—
149C22×Q8—171no—
150C22×Q8C2×#59511yescharacteristic
151C22×C8—344no—
152C4⋊Q8—174no—
153C4⋊Q8—342no—
154C4×Q8—682no—
155C4×Q8—682no—
156C2×C4⋊C4—171no—
157C2×C4⋊C4—342no—
158C2×C4⋊C4—341no—
159C2×C4⋊C4—341no—
160C2×C4⋊C4—342no—
161C2×C4⋊C4C2×#59511yescharacteristic
162C2×C42—171no—
163C2×C42—172no—
164C2×C42—172no—
165C2×C42—171no—
166C2×C42—341no—
167C2×C42—341no—
168C2×C42C2×#59511yescharacteristic
169C4⋊C8—178no—
170C4⋊C8—344no—
171#97—684no—
172#95—22no—
173#95—342no—
174C17#823211yescharacteristic, complemented, Sylow
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—172no—
179C2×Q8—341no—
180C2×Q8—681no—
181C2×Q8—681no—
182C2×Q8C4×#59512yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—172no—
186C22×C4—171no—
187C22×C4—171no—
188C22×C4—341no—
189C22×C4—341no—
190C22×C4C22⋊#59512yes—
191C22×C4C22×#59511yescharacteristic
192C2×C8—348no—
193C2×C8—684no—
194C2×C8—684no—
195C4⋊C4—41no—
196C4⋊C4—348no—
197C4⋊C4—681no—
198C4⋊C4—682no—
199C4⋊C4—682no—
200C4⋊C4—682no—
201C4⋊C4—682no—
202C42—41no—
203C42—172no—
204C42—172no—
205C42—342no—
206C42—342no—
207C42—342no—
208C42—344no—
209C42—682no—
210C42—682no—
211C23C2×C22⋊#59511yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—682no—
215Q8—682no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—174no—
227C2×C4—172no—
228C2×C4—172no—
229C2×C4—174no—
230C2×C4—172no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—681no—
236C2×C4—681no—
237C2×C4—681no—
238C2×C4—682no—
239C2×C4C2×C4×#59511yescharacteristic
240C2×C4#1058767711yescharacteristic
241C8—688no—
242C22C4×C22⋊#59511yescharacteristic
243C22D17⋊#10011yescharacteristic
244C22#1176379611yescharacteristic
245C22#1176380012yes—
246C22#1176380212yes—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—81no—
252C4—174no—
253C4—174no—
254C4—342no—
255C4—682no—
256C4—681no—
257C4—1361no—
258C2#1454431811yescharacteristic
259C2#1454432012yes—
260C2#1454432211yescharacteristic
261C2#1454453411yescharacteristic
262C2#1454455612yes—
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#391C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#8232
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
8. #11763669C4semidirect
174. C17#8232semidirect

Maximal subgroups

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

subgroupindex
#145445692
C2×#117636692
#145436622
#145313532
#823217