Finite Group Databasebeta

← #17955943

Subgroups — #17955943

Order 4352.

558 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 124 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#14544701C211yesmaximal, characteristic, complemented
2#14544499C211yesmaximal, characteristic, complemented
3C2×Q8⋊#3624C212yesmaximal, complemented
4C2×D4⋊#3624C212yesmaximal
5#124×Dic17C211yesmaximal, characteristic
6C2×C4⋊#3624C2212yes—
7C2×C4×#3624C2212yes—
8#11763675—22no—
9#11763675—22no—
10Q8⋊#3624—24no—
11D4⋊#3624—24no—
12C2×Q8×Dic17C2211yescharacteristic
13C2×D4×Dic17C2211yescharacteristic
14#11762716C411yescharacteristic
15#11762712C411yescharacteristic
16#11762702—22no—
17#11762675C411yescharacteristic
18C22⋊C4×Dic17—22no—
19C42×Dic17C2211yescharacteristic
20C2×C17×#124C411yescharacteristic
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17C2×C411yescharacteristic
23C23×Dic17—21no—
24C22×#3624—22no—
25C22×#3624—22no—
26C17×#124—22no—
27C17×#124—22no—
28C2×C22⋊C4×C17—22no—
29C2×C42×C17C2×C411yescharacteristic
30C2×C22⋊Dic17—22no—
31C2×C22⋊Dic17—21no—
32Q8×Dic17—22no—
33Q8×Dic17—22no—
34D4×Dic17—22no—
35D4×Dic17—22no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4×Dic17—21no—
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17D412yes—
42C2×C4×Dic17D412yes—
43C2×C4×Dic17C2311yescharacteristic
44C4⋊#3624—22no—
45C4⋊#3624—22no—
46C4×#3624—22no—
47C4×#3624D414yes—
48#10587502—22no—
49#10587502—22no—
50C24×C17—21no—
51C2×Q8×C17—21no—
52C2×Q8×C17—22no—
53C2×Q8×C17—22no—
54C2×Q8×C17M1612yes—
55C2×D4×C17—22no—
56C2×D4×C17—21no—
57C2×D4×C17—22no—
58C2×D4×C17M1612yes—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17—44no—
67C22×Dic17C22⋊C412yes—
68C22×Dic17C2×D411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C2×#3624—24no—
71C2×#3624—44no—
72C2×#3624—42no—
73C2×#3624—44no—
74C22⋊C4×C17—28no—
75C42×C17—21no—
76C42×C17—21no—
77C22⋊Dic17—44no—
78C22⋊Dic17—42no—
79C4⋊Dic17—22no—
80C4⋊Dic17—22no—
81C4⋊Dic17—42no—
82C4×Dic17—22no—
83C4×Dic17—24no—
84C4×Dic17—42no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C4×Dic17C2×D412yes—
88C4×Dic17#4112yes—
89#9645—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—42no—
94C23×C17—42no—
95C23×C17C2×C22⋊C411yescharacteristic
96Q8×C17—24no—
97Q8×C17—42no—
98D4×C17—24no—
99D4×C17—42no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—22no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17C4×D412yes—
107C2×C4×C17C2×M1611yescharacteristic
108C2×C4×C17C2×M1611yescharacteristic
109C2×C4×C17#13112yes—
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—82no—
119C2×Dic17—82no—
120C2×Dic17#12112yes—
121C2×Dic17#12411yescharacteristic
122C2×Dic17#12711yescharacteristic
123C2×Dic17#13514yes—
124#3624—44no—
125#3624—82no—
126C4×#124—171no—
127#1831—171no—
128#1744—171no—
129C2×Q8⋊C8—172no—
130C2×D4⋊C8—172no—
131C22×C17—42no—
132C22×C17—44no—
133C22×C17—81no—
134C22×C17—81no—
135C22×C17—82no—
136C22×C17#39011yescharacteristic
137C22×C17#40711yescharacteristic
138C22×C17#40811yescharacteristic
139C22×C17#41612yes—
140C22×C17#42112yes—
141C4×C17—42no—
142C4×C17—42no—
143C4×C17#43512yes—
144C4×C17#43512yes—
145Dic17—22no—
146Dic17—42no—
147Dic17—81no—
148Dic17—81no—
149Dic17#48812yes—
150Dic17#49512yes—
151C2×#124#59511yescharacteristic
152C2×C4×Q8—171no—
153C2×C4×D4—171no—
154C2×C4⋊C8—172no—
155C2×C4×C8—172no—
156#391—171no—
157#390—171no—
158#388—342no—
159#382—171no—
160C4×C22⋊C4—342no—
161C43—171no—
162#336—342no—
163#336—342no—
164Q8⋊C8—344no—
165D4⋊C8—344no—
166C2×C17—82no—
167C2×C17#148212yes—
168C2×C17#148812yes—
169C2×C17#195511yescharacteristic
170C2×C17#195811yescharacteristic
171C2×C17#196311yescharacteristic
172C22×Q8C2×#59511yescharacteristic
173C22×D4C2×#59511yescharacteristic
174C23×C4—341no—
175C22×C8—342no—
176C22×C8—342no—
177#124—22no—
178#124—22no—
179C4×Q8—342no—
180C4×Q8—342no—
181C4×D4—342no—
182C4×D4—342no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—171no—
185C2×C4⋊C4—341no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—341no—
188C2×C22⋊C4—342no—
189C2×C42—171no—
190C2×C42—172no—
191C2×C42—171no—
192C2×C42—172no—
193C2×C42—341no—
194C2×C42C2×#59511yescharacteristic
195C4⋊C8—342no—
196C4⋊C8—342no—
197C4×C8—174no—
198C4×C8—342no—
199#95—342no—
200#95—342no—
201C17#964511yescharacteristic, complemented, Sylow
202C24—21no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8#6324712yes—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—21no—
210C2×D4#6324712yes—
211C22×C4—21no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—172no—
215C22×C4—341no—
216C22×C4—341no—
217C22×C4—681no—
218C22×C4—681no—
219C22×C4—684no—
220C22×C4C22⋊#59512yes—
221C22×C4C22×#59511yescharacteristic
222C2×C8—344no—
223C2×C8—684no—
224C2×C8—682no—
225C2×C8—684no—
226C4⋊C4—342no—
227C4⋊C4—342no—
228C4⋊C4—682no—
229C22⋊C4—28no—
230C22⋊C4—682no—
231C22⋊C4—684no—
232C42—21no—
233C42—21no—
234C42—172no—
235C42—172no—
236C42—344no—
237C42—342no—
238C42—682no—
239C42—681no—
240C42—681no—
241C23—22no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23C2×C22⋊#59511yescharacteristic
247Q8—24no—
248Q8—42no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—171no—
258C2×C4—171no—
259C2×C4—174no—
260C2×C4—172no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—342no—
265C2×C4—681no—
266C2×C4—681no—
267C2×C4—682no—
268C2×C4—682no—
269C2×C4—1362no—
270C2×C4—1362no—
271C2×C4#1058767812yes—
272C2×C4D4×#59512yes—
273C2×C4C2×#6324711yescharacteristic
274C2×C4C2×#6324711yescharacteristic
275C8—684no—
276C8—1362no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—81no—
281C22—82no—
282C22#1176376111yescharacteristic
283C22#1176377911yescharacteristic
284C22#1176378311yescharacteristic
285C22#1176379912yes—
286C22#1176380612yes—
287C4—42no—
288C4—42no—
289C4—172no—
290C4—172no—
291C4—342no—
292C4—682no—
293C4—1361no—
294C4—1361no—
295C4#1176384012yes—
296C4#1176384012yes—
297C2—82no—
298C2#1454409612yes—
299C2#1454411512yes—
300C2#1454490811yescharacteristic
301C2#1454491311yescharacteristic
302C2#1454492111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#124C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenterC2×#124#595
hypocenterC17#9645
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #14544701C2semidirect
2. #14544499C2semidirect
3. C2×Q8⋊#3624C2semidirect
201. C17#9645semidirect

Maximal subgroups

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

subgroupindex
#145447012
#145444992
C2×Q8⋊#36242
C2×D4⋊#36242
#124×Dic172
#964517