Finite Group Databasebeta

← #17958307

Subgroups — #17958307

Order 4352.

480 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 90 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#14544021C212yesmaximal
2#14543725C212yesmaximal, complemented
3#14543714C212yesmaximal, complemented
4C4×#10587633C211yesmaximal, characteristic
5#11763777—22no—
6#11763654C2214yes—
7#11763653—22no—
8C2×#10587633C411yescharacteristic
9C2×C4×D68C2211yescharacteristic
10#11762716C2211yescharacteristic
11#11762710C411yescharacteristic
12#11762699—22no—
13#11762674C2211yescharacteristic
14C4×#63227—22no—
15C42×Dic17C411yescharacteristic
16C2×C4×Q8×C17C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D68C2×C411yescharacteristic
19C22×C4×D17D412yes—
20C2×#63242—42no—
21C4×Q8×C17—41no—
22C4×Q8×C17—41no—
23C2×C4⋊C4×C17—21no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C42×C17—21no—
26C2×C42×C17C2×C411yescharacteristic
27#10587633—22no—
28#10587633C814yescomplemented
29C2×#63227—22no—
30C2×#63227—22no—
31C2×#63227D412yes—
32C2×C4⋊Dic17C2311yescharacteristic
33C2×C4×Dic17—21no—
34C2×C4×Dic17—21no—
35C2×C4×Dic17C2×C411yescharacteristic
36C2×C4×Dic17C2×C411yescharacteristic
37C4×D68—41no—
38C4×D68—41no—
39C22⋊#3624—24no—
40#10587543—42no—
41#10587502—22no—
42#10587502—22no—
43C23×D17C22⋊C412yes—
44C2×Q8×C17—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17C2×C812yescomplemented
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17C22⋊C411yescharacteristic
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17—21no—
55C22×Dic17C2×D412yes—
56C2×D68—41no—
57C2×D68—41no—
58C2×D68C2×C812yes—
59C2×D68M1612yes—
60C2×C4×D17—22no—
61C2×C4×D17—22no—
62C2×C4×D17—44no—
63C2×#3624—44no—
64#63242—84no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—42no—
67C42×C17—21no—
68C42×C17—21no—
69C42×C17—41no—
70C42×C17—41no—
71#63227—28no—
72#63227—42no—
73#63227—44no—
74C4⋊Dic17—41no—
75C4×Dic17—21no—
76C4×Dic17—42no—
77C4×Dic17—42no—
78C4×Dic17—41no—
79C4×Dic17C2×C812yescomplemented
80#11477—171nomaximal, Sylow
81C23×C17C2×C22⋊C411yescharacteristic
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—42no—
85C22×D17—42no—
86C22×D17C22⋊C814yes—
87Q8×C17—42no—
88Q8×C17—42no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—21no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—42no—
98C2×C4×C17—41no—
99C2×C4×C17#9911yescharacteristic
100C2×C4×C17#9911yescharacteristic
101C2×C4×C17C22×C811yescharacteristic
102C2×C4×C17C2×M1611yescharacteristic
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—41no—
107C2×Dic17—41no—
108C2×Dic17—41no—
109C2×Dic17—41no—
110C2×Dic17—42no—
111D68—42no—
112D68—42no—
113C4×D17—44no—
114C4×D17—82no—
115#3624—84no—
116C4×#124—171no—
117#1459—172no—
118#1324—172no—
119#1319—172no—
120C22×C17—21no—
121C22×C17—21no—
122C22×C17C2×C22⋊C811yescharacteristic
123C22×C17C2×#9911yescharacteristic
124C22×C17#41311yescharacteristic
125C2×D17—24no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—44no—
129C2×D17—82no—
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—21no—
133C4×C17—41no—
134C4×C17—42no—
135C4×C17—81no—
136C4×C17#35412yes—
137Dic17—42no—
138Dic17—42no—
139Dic17—42no—
140C2×#124—171no—
141C2×C4×Q8#59511yescharacteristic
142C2×C4×D4—171no—
143C2×C22⋊C8—342no—
144#391—171no—
145#390—171no—
146#388—342no—
147#382—171no—
148C4×C22⋊C4—342no—
149C43—171no—
150#324—174no—
151#323—342no—
152C2×C17—21no—
153C2×C17—21no—
154C2×C17#145911yescharacteristic
155C2×C17C2×#35411yescharacteristic
156C2×C17#213211yescharacteristic
157D17—44no—
158C22×Q8C2×#59511yescharacteristic
159C22×D4—171no—
160C23×C4—172no—
161C22×C8—682no—
162#124—174no—
163#124—342no—
164C4×Q8—41no—
165C4×Q8—41no—
166C4×D4—681no—
167C4×D4—681no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4C2×#59511yescharacteristic
171C2×C22⋊C4—172no—
172C2×C22⋊C4—342no—
173C2×C22⋊C4—342no—
174C2×C42—21no—
175C2×C42—171no—
176C2×C42—171no—
177C2×C42—341no—
178C2×C42—341no—
179C2×C42C2×#59511yescharacteristic
180C22⋊C8—344no—
181C22⋊C8—682no—
182#95—342no—
183#95—342no—
184C17#1147711yescharacteristic, complemented, Sylow
185C24—172no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8#6324212yescomplemented
190C2×D4—172no—
191C2×D4—172no—
192C2×D4—681no—
193C2×D4—681no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—172no—
197C22×C4—341no—
198C22×C4—341no—
199C22×C4—342no—
200C22×C4—342no—
201C22×C4—684no—
202C22×C4C22⋊#59511yescharacteristic
203C22×C4C22⋊#59511yescharacteristic
204C22×C4C22×#59511yescharacteristic
205C2×C8—684no—
206C2×C8—1364no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—681no—
210C22⋊C4—348no—
211C22⋊C4—682no—
212C22⋊C4—684no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—172no—
218C42—341no—
219C42—681no—
220C42—682no—
221C42—682no—
222C23—174no—
223C23—342no—
224C23—342no—
225C23—682no—
226C23—682no—
227C23C2×C22⋊#59511yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—682no—
231D4—682no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—342no—
245C2×C4—681no—
246C2×C4—681no—
247C2×C4—681no—
248C2×C4—681no—
249C2×C4—684no—
250C2×C4—682no—
251C2×C4—1362no—
252C2×C4#1058755111yescharacteristic
253C2×C4#1058755111yescharacteristic
254C2×C4C2×#6324211yescharacteristic
255C2×C4D17⋊M1611yescharacteristic
256C8—1364no—
257C22—21no—
258C22—21no—
259C22—344no—
260C22—684no—
261C22—682no—
262C22—682no—
263C22—1362no—
264C22#1176377711yescharacteristic
265C22C2×#1058755111yescharacteristic
266C22#1176379511yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—41no—
272C4—81no—
273C4—682no—
274C4—682no—
275C4—682no—
276C4#1176371512yes—
277C2—21no—
278C2—21no—
279C2—684no—
280C2#1454401911yescharacteristic
281C2C2×#1176371511yescharacteristic
282C2#1454507511yescharacteristic
subgroupG/H
centerC2×C4#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C4×Q8×C17C4
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×C4×Q8#595
hypocenterC17#11477
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #14543725C2semidirect
3. #14543714C2semidirect
28. #10587633C8semidirect
47. C2×Q8×C17C2×C8semidirect
79. C4×Dic17C2×C8semidirect
184. C17#11477semidirect
189. C2×Q8#63242semidirect

Maximal subgroups

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

subgroupindex
#145440212
#145437252
#145437142
C4×#105876332
#1147717