Finite Group Databasebeta

← #17958308

Subgroups — #17958308

Order 4352.

480 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544020C212yesmaximal
2#14543724C212yesmaximal, complemented
3#14543714C212yesmaximal, complemented
4#124×Dic17C211yesmaximal, characteristic
5C2×C22⋊#3624—22no—
6#11763654C2214yes—
7#11763652—22no—
8C2×Q8×Dic17C411yescharacteristic
9C2×D4×Dic17C2211yescharacteristic
10#11762716C2211yescharacteristic
11#11762712C411yescharacteristic
12#11762702—22no—
13#11762675C2211yescharacteristic
14C22⋊C4×Dic17—22no—
15C42×Dic17C411yescharacteristic
16C2×C17×#124C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×Dic17D412yes—
20C22×#3624—42no—
21C17×#124—22no—
22C17×#124C814yescomplemented
23C2×C22⋊C4×C17—22no—
24C2×C42×C17C2×C411yescharacteristic
25C2×C22⋊Dic17—22no—
26C2×C22⋊Dic17D412yes—
27Q8×Dic17—41no—
28Q8×Dic17—41no—
29D4×Dic17—41no—
30D4×Dic17—41no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4⋊Dic17C2311yescharacteristic
34C2×C4×Dic17—21no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17C2×C411yescharacteristic
39C22⋊#3624—24no—
40C22⋊#3624—42no—
41#10587502—22no—
42#10587502—22no—
43C24×C17C22⋊C412yes—
44C2×Q8×C17—21no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17C2×C812yescomplemented
48C2×D4×C17—41no—
49C2×D4×C17—41no—
50C2×D4×C17C2×C812yes—
51C2×D4×C17M1612yes—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17C22×C411yescharacteristic
55C22×Dic17—21no—
56C22×Dic17—22no—
57C22×Dic17—22no—
58C22×Dic17—21no—
59C22×Dic17—44no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C22⋊C411yescharacteristic
62C22×Dic17C2×D412yes—
63C2×#3624—44no—
64C2×#3624—84no—
65C22⋊C4×C17—28no—
66C42×C17—21no—
67C42×C17C2×C812yescomplemented
68C22⋊Dic17—42no—
69C22⋊Dic17—44no—
70C4⋊Dic17—41no—
71C4⋊Dic17—42no—
72C4⋊Dic17—41no—
73C4×Dic17—21no—
74C4×Dic17—21no—
75C4×Dic17—42no—
76C4×Dic17—42no—
77C4×Dic17—41no—
78C4×Dic17—41no—
79C4×Dic17—41no—
80#11477—171nomaximal, Sylow
81C23×C17—22no—
82C23×C17—22no—
83C23×C17—42no—
84C23×C17—42no—
85C23×C17C22⋊C814yes—
86C23×C17C2×C22⋊C411yescharacteristic
87Q8×C17—42no—
88Q8×C17—42no—
89D4×C17—42no—
90D4×C17—42no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17C22×C811yescharacteristic
100C2×C4×C17C2×M1611yescharacteristic
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—21no—
104C2×Dic17—22no—
105C2×Dic17—41no—
106C2×Dic17—41no—
107C2×Dic17—44no—
108C2×Dic17—42no—
109C2×Dic17—41no—
110C2×Dic17—42no—
111C2×Dic17—41no—
112C2×Dic17—82no—
113C2×Dic17#9911yescharacteristic
114C2×Dic17#9911yescharacteristic
115#3624—84no—
116C4×#124—171no—
117#1459—172no—
118#1324—172no—
119#1319—172no—
120C22×C17—21no—
121C22×C17—21no—
122C22×C17—24no—
123C22×C17—44no—
124C22×C17—42no—
125C22×C17—42no—
126C22×C17—82no—
127C22×C17C2×C22⋊C811yescharacteristic
128C22×C17C2×#9911yescharacteristic
129C22×C17#41311yescharacteristic
130C4×C17—21no—
131C4×C17—21no—
132C4×C17—42no—
133C4×C17—42no—
134Dic17—21no—
135Dic17—42no—
136Dic17—42no—
137Dic17—41no—
138Dic17—81no—
139Dic17#35412yes—
140C2×#124#59511yescharacteristic
141C2×C4×Q8—171no—
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—44no—
155C2×C17#145911yescharacteristic
156C2×C17C2×#35411yescharacteristic
157C2×C17#213211yescharacteristic
158C22×Q8C2×#59511yescharacteristic
159C22×D4C2×#59511yescharacteristic
160C23×C4—172no—
161C22×C8—682no—
162#124—22no—
163#124#362414yescomplemented
164C4×Q8—681no—
165C4×Q8—681no—
166C4×D4—681no—
167C4×D4—681no—
168C2×C4⋊C4—171no—
169C2×C4⋊C4—171no—
170C2×C4⋊C4—341no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—172no—
173C2×C22⋊C4—342no—
174C2×C42—171no—
175C2×C42—171no—
176C2×C42—341no—
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
185C24C22⋊#59512yes—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8C2×#362412yescomplemented
190C2×D4—41no—
191C2×D4—41no—
192C2×D4C2×#362412yes—
193C2×D4#6324712yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—171no—
197C22×C4—171no—
198C22×C4—172no—
199C22×C4—342no—
200C22×C4—342no—
201C22×C4—341no—
202C22×C4—341no—
203C22×C4—684no—
204C22×C4C22×#59511yescharacteristic
205C2×C8—684no—
206C2×C8—1364no—
207C4⋊C4—681no—
208C4⋊C4—681no—
209C4⋊C4—682no—
210C22⋊C4—28no—
211C22⋊C4—682no—
212C22⋊C4—684no—
213C42—21no—
214C42—341no—
215C42—341no—
216C42—681no—
217C42—681no—
218C42—681no—
219C42—682no—
220C42—682no—
221C42C2×#362412yescomplemented
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23C22⋊#362414yes—
227C23C2×C22⋊#59511yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—42no—
231D4—42no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—171no—
241C2×C4—171no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—341no—
245C2×C4—342no—
246C2×C4—684no—
247C2×C4—681no—
248C2×C4—681no—
249C2×C4—681no—
250C2×C4—682no—
251C2×C4—682no—
252C2×C4—681no—
253C2×C4—1362no—
254C2×C4C22×#362411yescharacteristic
255C2×C4C2×#6324711yescharacteristic
256C8—1364no—
257C22—21no—
258C22—21no—
259C22—24no—
260C22—44no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22C2×C22⋊#362411yescharacteristic
265C22D17⋊#9911yescharacteristic
266C22#1176379611yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—172no—
272C4—341no—
273C4—682no—
274C4—682no—
275C4—681no—
276C4—1361no—
277C2—21no—
278C2—21no—
279C2—44no—
280C2#1454402011yescharacteristic
281C2D17⋊#35411yescharacteristic
282C2#1454507811yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×#124#595
hypocenterC17#11477
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #14543724C2semidirect
3. #14543714C2semidirect
22. C17×#124C8semidirect
47. C2×Q8×C17C2×C8semidirect
67. C42×C17C2×C8semidirect
163. #124#3624semidirect
184. C17#11477semidirect
189. C2×Q8C2×#3624semidirect
221. C42C2×#3624semidirect

Maximal subgroups

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

subgroupindex
#145440202
#145437242
#145437142
#124×Dic172
#1147717