Finite Group Databasebeta

← #17960866

Subgroups — #17960866

Order 4352.

500 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#11763720C212yesmaximal
2#14544537C212yesmaximal
3#14543882C212yesmaximal
4#14531351C211yesmaximal, characteristic
5C2×#10587545C2214yes—
6C4×#63247—22no—
7#11763720C418yescomplemented
8#11763689—22no—
9C2×#10587555C2211yescharacteristic
10C2×#10587573C411yescharacteristic
11C2×#10587573C2211yescharacteristic
12#11762714C411yescharacteristic
13#11762714C2211yescharacteristic
14#11762705—22no—
15C4⋊C4×Dic17—22no—
16C42×Dic17C411yescharacteristic
17C17×#384C411yescharacteristic
18C22×Dic34C2×C411yescharacteristic
19C22×Dic34C2311yescharacteristic
20C2×#63247—24no—
21C2×C4⋊C4×C17—22no—
22C2×C4⋊C4×C17C2×C411yescharacteristic
23C2×C4⋊C4×C17C2×C411yescharacteristic
24C2×C42×C17C2×C411yescharacteristic
25C2×C4⋊Dic17—22no—
26C2×#63225—21no—
27C2×#63225D412yes—
28C2×C4×Dic17—21no—
29C2×C4×Dic17—21no—
30C2×C4×Dic17—21no—
31C2×C4×Dic17C2×C411yescharacteristic
32C2×C4×Dic17C2×C411yescharacteristic
33C2×C4×Dic17D412yes—
34#10587573—41no—
35#10587573—41no—
36#10587573—41no—
37#10587573—41no—
38#10587555—22no—
39#10587555C2×C414yes—
40#10587545—24no—
41#10587545C2×C418yes—
42#10587542—42no—
43C4×#3624—42no—
44#10587502—22no—
45#10587502—22no—
46C22×C4×C17—21no—
47C22×C4×C17—21no—
48C22×C4×C17C22⋊C412yes—
49C22×C4×C17C22×C411yescharacteristic
50C22×Dic17—21no—
51C22×Dic17—21no—
52C22×Dic17C22⋊C411yescharacteristic
53C22×Dic17C22⋊C411yescharacteristic
54C22×Dic17C2×D412yes—
55C2×Dic34—21no—
56C2×Dic34—41no—
57C2×Dic34—41no—
58C2×Dic34—41no—
59C2×Dic34—41no—
60C2×Dic34C4212yes—
61C2×Dic34C22×C412yes—
62C2×Dic34#4112yes—
63#63247—28no—
64#63247—44no—
65C2×#3624—44no—
66C4⋊C4×C17—41no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—44no—
69C42×C17—21no—
70C42×C17C4212yes—
71C4⋊Dic17—28no—
72#63225—42no—
73#63225—42no—
74C4×Dic17—24no—
75C4×Dic17—22no—
76C4×Dic17—22no—
77C4×Dic17—21no—
78C4×Dic17—21no—
79C4×Dic17—42no—
80C4×Dic17—41no—
81C4×Dic17—41no—
82C4×Dic17—41no—
83C4×#356—171nomaximal, Sylow
84C23×C17C2×C22⋊C411yescharacteristic
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17—41no—
94C2×C4×C17C2×C4211yescharacteristic
95C2×C4×C17#11711yescharacteristic
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—41no—
103C2×Dic17—41no—
104C2×Dic17—41no—
105C2×Dic17—41no—
106C2×Dic17—41no—
107C2×Dic17#9911yescharacteristic
108C2×Dic17#9911yescharacteristic
109C2×Dic17C4×D414yes—
110Dic34—42no—
111Dic34—42no—
112Dic34—42no—
113Dic34—42no—
114#3624—48no—
115C4×C4⋊Q8—171no—
116C2×#356—172no—
117C4×#101—172no—
118#1394—172no—
119C22×C17—21no—
120C22×C17—21no—
121C22×C17C4×C22⋊C411yescharacteristic
122C22×C17C2×#9911yescharacteristic
123C22×C17#41011yescharacteristic
124C4×C17—24no—
125C4×C17—21no—
126C4×C17—21no—
127C4×C17—42no—
128C4×C17—81no—
129Dic17—21no—
130Dic17—24no—
131Dic17—42no—
132Dic17—41no—
133Dic17—42no—
134Dic17#35612yescomplemented
135C2×C4⋊Q8—171no—
136C2×C4×Q8—171no—
137C2×C4×Q8—171no—
138C2×#101—174no—
139C4×M16—342no—
140#391—171no—
141#391—171no—
142#389—342no—
143#384#59511yescharacteristic
144C4×C4⋊C4—342no—
145C43—171no—
146#356—178no—
147#343—342no—
148C2×C17—21no—
149C2×C17—21no—
150C2×C17C4×#9911yescharacteristic
151C2×C17C2×#35611yescharacteristic
152C2×C17#213211yescharacteristic
153C22×Q8—171no—
154C22×Q8—171no—
155C2×M16—344no—
156C4⋊Q8—174no—
157C4⋊Q8—342no—
158C4×Q8—681no—
159C4×Q8—681no—
160C4×Q8—681no—
161C4×Q8—681no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—172no—
164C2×C4⋊C4—342no—
165C2×C4⋊C4—341no—
166C2×C4⋊C4C2×#59511yescharacteristic
167C2×C4⋊C4C2×#59511yescharacteristic
168C2×C42—172no—
169C2×C42—171no—
170C2×C42—171no—
171C2×C42—341no—
172C2×C42—341no—
173C2×C42—341no—
174C2×C42C2×#59511yescharacteristic
175#101—178no—
176#101—344no—
177#97—682no—
178C4×C8—682no—
179#95—342no—
180#95—342no—
181C17C4×#35611yescharacteristic, complemented, Sylow
182C2×Q8—172no—
183C2×Q8—172no—
184C2×Q8—172no—
185C2×Q8—341no—
186C2×Q8—681no—
187C2×Q8—681no—
188C2×Q8—681no—
189C2×Q8—681no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—171no—
193C22×C4—172no—
194C22×C4—171no—
195C22×C4—341no—
196C22×C4—341no—
197C22×C4C22⋊#59512yes—
198C22×C4C22×#59511yescharacteristic
199M16—348no—
200M16—684no—
201C2×C8—684no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—44no—
205C4⋊C4—348no—
206C4⋊C4—682no—
207C4⋊C4—682no—
208C42—21no—
209C42—342no—
210C42—342no—
211C42—341no—
212C42—341no—
213C42—344no—
214C42—681no—
215C42—682no—
216C42—681no—
217C42—681no—
218C42C4×#59512yes—
219C23C2×C22⋊#59511yescharacteristic
220Q8—682no—
221Q8—682no—
222Q8—682no—
223Q8—682no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—171no—
234C2×C4—174no—
235C2×C4—171no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—341no—
242C2×C4—681no—
243C2×C4—681no—
244C2×C4—681no—
245C2×C4—681no—
246C2×C4—681no—
247C2×C4C2×C4×#59511yescharacteristic
248C2×C4#1058767711yescharacteristic
249C8—688no—
250C22—21no—
251C22—21no—
252C22C4×C22⋊#59511yescharacteristic
253C22D17⋊#9911yescharacteristic
254C22#1176379011yescharacteristic
255C4—24no—
256C4—21no—
257C4—21no—
258C4—42no—
259C4—81no—
260C4—172no—
261C4—344no—
262C4—341no—
263C4—682no—
264C4—682no—
265C4—681no—
266C2—21no—
267C2—21no—
268C2#1454452911yescharacteristic
269C2D17⋊#35611yescharacteristic
270C2#1454507611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C2×C42
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC22×Dic34C23
hypercenter#384#595
hypocenterC17C4×#356
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
7. #11763720C4semidirect
134. Dic17#356semidirect
181. C17C4×#356semidirect

Maximal subgroups

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

subgroupindex
C2×#117637202
#145445372
#145438822
#145313512
C4×#35617