Finite Group Databasebeta

← #17954200

Subgroups — #17954200

Order 4352.

534 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 126 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#14545677C211yesmaximal, characteristic
2#14544086C212yesmaximal
3#14544027C211yesmaximal, characteristic
4#14543612C212yesmaximal
5C4⋊Q8×Dic17C211yesmaximal, characteristic
6Q8×#3624—24no—
7#11763825—22no—
8#11763825C2212yes—
9#11763825C2212yes—
10Q8⋊#3624—24no—
11#11763654—22no—
12C2×Q8×Dic17C2211yescharacteristic
13C2×Q8×Dic17C2211yescharacteristic
14#11762716C411yescharacteristic
15#11762716C411yescharacteristic
16#11762708—22no—
17#11762679C411yescharacteristic
18C4⋊C4×Dic17—22no—
19C42×Dic17C2211yescharacteristic
20C2×C17×C4⋊Q8C411yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C22×Q8×C17C2×C411yescharacteristic
23C17×C4⋊Q8—22no—
24C17×C4⋊Q8C814yescomplemented
25C2×C4⋊C4×C17—22no—
26C2×C42×C17C2×C411yescharacteristic
27Q8×Dic17—22no—
28Q8×Dic17—22no—
29Q8×Dic17—22no—
30Q8×Dic17—22no—
31C2×C4⋊Dic17—22no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17D412yes—
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D412yes—
41C2×C4×Dic17C2311yescharacteristic
42C22⋊#3624—22no—
43C22⋊#3624—22no—
44C4⋊#3624—24no—
45C4⋊#3624—22no—
46C4⋊#3624D414yes—
47C4×#3624—24no—
48C4×#3624—22no—
49C4×#3624—22no—
50#10587502—22no—
51#10587502—22no—
52C2×Q8×C17—22no—
53C2×Q8×C17—21no—
54C2×Q8×C17—22no—
55C2×Q8×C17—22no—
56C2×Q8×C17—22no—
57C2×Q8×C17C2×C812yes—
58C2×Q8×C17C2×C812yes—
59C2×Q8×C17M1612yes—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17C22⋊C412yes—
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17C22⋊C412yes—
67C22×Dic17C2×D411yescharacteristic
68C22×Dic17C2×D411yescharacteristic
69C2×#3624—24no—
70C2×#3624—42no—
71C4⋊C4×C17—28no—
72C42×C17—21no—
73C42×C17C2×C812yes—
74C4⋊Dic17—22no—
75C4⋊Dic17—24no—
76C4⋊Dic17—22no—
77C4⋊Dic17—44no—
78C4⋊Dic17—42no—
79C4×Dic17—24no—
80C4×Dic17—22no—
81C4×Dic17—22no—
82C4×Dic17—22no—
83C4×Dic17—42no—
84C4×Dic17—41no—
85C4×Dic17—41no—
86C4×Dic17C2×D412yes—
87C4×Dic17#4112yes—
88#8958—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90Q8×C17—24no—
91Q8×C17—24no—
92Q8×C17—24no—
93Q8×C17—42no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17—22no—
97C2×C4×C17—22no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17C22⋊C814yes—
103C2×C4×C17C4×D412yes—
104C2×C4×C17C22×C811yescharacteristic
105C2×C4×C17C2×M1611yescharacteristic
106C2×C4×C17#13112yes—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17—22no—
110C2×Dic17—41no—
111C2×Dic17—41no—
112C2×Dic17—41no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—41no—
116C2×Dic17#12011yescharacteristic
117C2×Dic17#12112yes—
118C2×Dic17#12311yescharacteristic
119#3624—44no—
120#3624—82no—
121#2982—171no—
122C4×C4⋊Q8—171no—
123#1479—172no—
124#1461—171no—
125#1276—172no—
126C22×C17—21no—
127C22×C17—21no—
128C22×C17#38611yescharacteristic
129C22×C17C2×C22⋊C811yescharacteristic
130C22×C17#40811yescharacteristic
131C4×C17—24no—
132C4×C17—24no—
133C4×C17—42no—
134C4×C17C8×D412yes—
135C4×C17#43512yes—
136Dic17—22no—
137Dic17—21no—
138Dic17—42no—
139Dic17—42no—
140Dic17—81no—
141Dic17#45612yes—
142C2×C4⋊Q8#59511yescharacteristic
143C2×C4×Q8—171no—
144C2×C4×Q8—171no—
145C8×Q8—344no—
146#431—172no—
147#431—172no—
148#431—342no—
149#391—171no—
150#391—171no—
151#389—342no—
152#384—171no—
153C4×C4⋊C4—342no—
154C43—171no—
155Q8⋊C8—344no—
156#324—342no—
157C2×C17—21no—
158C2×C17—21no—
159C2×C17#186911yescharacteristic
160C2×C17#188411yescharacteristic
161C2×C17#188611yescharacteristic
162C22×Q8C2×#59511yescharacteristic
163C22×Q8C2×#59511yescharacteristic
164C4⋊Q8—22no—
165C4⋊Q8#362414yescomplemented
166C4×Q8—342no—
167C4×Q8—342no—
168C4×Q8—342no—
169C4×Q8—342no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—172no—
172C2×C4⋊C4—171no—
173C2×C4⋊C4—171no—
174C2×C4⋊C4—341no—
175C2×C4⋊C4—342no—
176C2×C42—171no—
177C2×C42—172no—
178C2×C42—171no—
179C2×C42—341no—
180C2×C42—341no—
181C2×C42—341no—
182C2×C42C2×#59511yescharacteristic
183C4⋊C8—174no—
184C4⋊C8—342no—
185C4⋊C8—344no—
186C22⋊C8—342no—
187C22⋊C8—342no—
188C4×C8—342no—
189C4×C8—342no—
190C4×C8—344no—
191#95—342no—
192#95—342no—
193C17#895811yescharacteristic, complemented, Sylow
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—21no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8C2×#362412yes—
200C2×Q8C2×#362412yes—
201C2×Q8#6324712yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—171no—
205C22×C4—171no—
206C22×C4—172no—
207C22×C4—341no—
208C22×C4—341no—
209C22×C4C22⋊#59512yes—
210C22×C4C22×#59511yescharacteristic
211C2×C8—344no—
212C2×C8—682no—
213C4⋊C4—28no—
214C4⋊C4—344no—
215C4⋊C4—342no—
216C4⋊C4—342no—
217C4⋊C4—684no—
218C4⋊C4—682no—
219C42—21no—
220C42—172no—
221C42—172no—
222C42—342no—
223C42—342no—
224C42—344no—
225C42—342no—
226C42—682no—
227C42—681no—
228C42—681no—
229C42C2×#362412yes—
230C23C2×C22⋊#59511yescharacteristic
231Q8—24no—
232Q8—24no—
233Q8—24no—
234Q8—42no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—171no—
244C2×C4—171no—
245C2×C4—172no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—681no—
250C2×C4—681no—
251C2×C4—681no—
252C2×C4—682no—
253C2×C4—682no—
254C2×C4—681no—
255C2×C4C22⋊#362414yes—
256C2×C4#1058767812yes—
257C2×C4D4×#59512yes—
258C2×C4C22×#362411yescharacteristic
259C2×C4C2×#6324711yescharacteristic
260C8—684no—
261C8—1362no—
262C22—21no—
263C22—21no—
264C22#1176375711yescharacteristic
265C22C2×C22⋊#362411yescharacteristic
266C22#1176378211yescharacteristic
267C4—24no—
268C4—24no—
269C4—42no—
270C4—172no—
271C4—342no—
272C4—341no—
273C4—682no—
274C4—682no—
275C4—1361no—
276C4D4×#362412yes—
277C4#1176384012yes—
278C2—21no—
279C2—21no—
280C2#1454475711yescharacteristic
281C2#1454478611yescharacteristic
282C2#1454479011yescharacteristic
subgroupG/H
centerC22#11763757
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#8958
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
24. C17×C4⋊Q8C8semidirect
165. C4⋊Q8#3624semidirect
193. C17#8958semidirect

Maximal subgroups

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

subgroupindex
#145456772
#145440862
#145440272
#145436122
C4⋊Q8×Dic172
#895817