Finite Group Databasebeta

← #16936666

Subgroups — #16936666

Order 4352.

712 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 137 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×#11763099C212yesmaximal, complemented
2#14529585C211yesmaximal, characteristic, complemented
3C2×D4⋊#3622C212yesmaximal, complemented
4#14526667C211yesmaximal, characteristic, complemented
5C17×#2537C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7#11763099—24no—
8#11763099—24no—
9C2×#10587479—22no—
10C2×C4⋊#3622C2212yes—
11C2×#10587481—22no—
12C2×#10587503—22no—
13C2×C4×#3622C2211yescharacteristic
14D4⋊#3622—24no—
15C2×C17×#121—22no—
16C2×C4×D4×C17C2212yes—
17C17×#394—22no—
18C17×#384C2211yescharacteristic
19C17×#380—22no—
20C22×D4×C17—22no—
21C22×D4×C17D412yes—
22C23×C4×C17—22no—
23C22×Dic34—21no—
24C17×#121—44no—
25C17×#121—44no—
26C4×D4×C17—22no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C4×D4×C17—22no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17D412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊C4×C17—24no—
34C2×C22⋊C4×C17—22no—
35C2×C42×C17C2311yescharacteristic
36C2×C4⋊Dic17—22no—
37C22×#3622—21no—
38C22×#3622—22no—
39#10587555—22no—
40#10587555—21no—
41#10587555D412yescomplemented
42C17×#95—22no—
43#10587503—28no—
44#10587481—44no—
45#10587479—28no—
46C4⋊#3622—22no—
47C4⋊#3622D414yescomplemented
48C4×#3622—22no—
49C24×C17—22no—
50C2×D4×C17—22no—
51C2×D4×C17—22no—
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—48no—
56C2×D4×C17—82no—
57C2×D4×C17SD1614yes—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D412yes—
68C22×Dic17—41no—
69C2×Dic34—22no—
70C2×Dic34—42no—
71C2×Dic34—42no—
72C2×Dic34—41no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—42no—
77C4⋊C4×C17—42no—
78C22⋊C4×C17—48no—
79C22⋊C4×C17—42no—
80C22⋊C4×C17—42no—
81C42×C17C2×D411yescharacteristic
82C42×C17C2×D412yes—
83C42×C17#4111yescharacteristic
84C4⋊Dic17—24no—
85C4⋊Dic17—42no—
86C2×#3622—22no—
87C2×#3622—24no—
88C2×#3622—41no—
89C2×#3622—41no—
90C2×#3622—42no—
91C2×#3622—42no—
92#10235—171nomaximal, Sylow
93C23×C17—24no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17#12011yescharacteristic
101D4×C17—24no—
102D4×C17—24no—
103D4×C17—42no—
104D4×C17—42no—
105D4×C17—88no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—44no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—82no—
114C2×C4×C17—82no—
115C2×C4×C17—82no—
116C2×C4×C17—82no—
117C2×C4×C17#12012yes—
118C2×C4×C17#12111yescharacteristic
119C2×C4×C17#12112yes—
120C2×C4×C17#12711yescharacteristic
121C2×C4×C17C2×SD1612yes—
122C2×C4×C17#13512yes—
123C2×C4×C17AGL(1,ℤ/8)12yes—
124C2×C4×C17#13712yes—
125C2×Dic17—42no—
126C2×Dic17—81no—
127C2×Dic17—81no—
128Dic34—44no—
129Dic34—82no—
130#3622—42no—
131#3622—44no—
132C2×#476—172no—
133#2537D1711yescharacteristic
134#1956—171no—
135C2×D4⋊C8—172no—
136#1296—171no—
137C22×C17—44no—
138C22×C17—44no—
139C22×C17—44no—
140C22×C17—82no—
141C22×C17—82no—
142C22×C17—82no—
143C22×C17—82no—
144C22×C17#39211yescharacteristic
145C22×C17#45012yes—
146C22×C17#45212yes—
147C22×C17#45311yescharacteristic
148C22×C17#45411yescharacteristic
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—82no—
152C4×C17—82no—
153C4×C17#44812yes—
154C4×C17#44912yes—
155C4×C17#46512yes—
156C4×C17#46812yes—
157Dic17—82no—
158C2×C4⋊Q8—171no—
159C2×#121—22no—
160C2×C4×D4C2×D1712yes—
161#476—344no—
162#476—344no—
163C2×#106—342no—
164C2×C4⋊C8—172no—
165C2×#103—342no—
166C2×#102—342no—
167C2×C4×C8—171no—
168#394—22no—
169#384C2×D1711yescharacteristic
170#380—22no—
171D4⋊C8—344no—
172C2×C17—84no—
173C2×C17#162911yescharacteristic
174C2×C17#163411yescharacteristic
175C2×C17#163512yes—
176C2×C17#200012yes—
177C2×C17#200611yescharacteristic
178C22×Q8—341no—
179C22×D4—22no—
180C22×D4C22⋊D1712yes—
181C23×C4—22no—
182C22×C8—341no—
183C22×C8—342no—
184C4⋊Q8—172no—
185C4⋊Q8—342no—
186C4⋊Q8—341no—
187#121—44no—
188#121—44no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—342no—
195C2×C4⋊C4C22⋊D1712yes—
196C2×C4⋊C4C22⋊D1712yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C42C22×D1711yescharacteristic
200#106—348no—
201C4⋊C8—174no—
202C4⋊C8—342no—
203#103—684no—
204#102—348no—
205C4×C8—342no—
206#95—22no—
207C17#1023511yescharacteristic, complemented, Sylow
208C24—22no—
209C2×Q8—342no—
210C2×Q8—682no—
211C2×Q8—682no—
212C2×Q8—681no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—48no—
219C2×D4—82no—
220C2×D4#6322914yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—681no—
230C22×C4C2×C22⋊D1711yescharacteristic
231C22×C4C2×C22⋊D1712yes—
232C2×C8—342no—
233C2×C8—344no—
234C2×C8—681no—
235C2×C8—682no—
236C2×C8—681no—
237C2×C8—682no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—344no—
244C4⋊C4—682no—
245C22⋊C4—48no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42D4×D1711yescharacteristic
249C42D4×D1712yes—
250C42#6325411yescharacteristic
251C23—24no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23#1058763911yescharacteristic
259Q8—684no—
260Q8—1362no—
261D4—24no—
262D4—24no—
263D4—42no—
264D4—42no—
265D4—88no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—682no—
278C2×C4—1361no—
279C2×C4—1361no—
280C2×C4#1058761812yes—
281C2×C4C2×#6322912yes—
282C2×C4#1058762312yes—
283C2×C4#1058762411yescharacteristic
284C2×C4#1058762512yes—
285C2×C4#1058762611yescharacteristic
286C2×C4#1058763612yes—
287C2×C4#1058763712yes—
288C8—682no—
289C8—684no—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22#1176271711yescharacteristic
298C22#1176296012yes—
299C22#1176297112yes—
300C22#1176297711yescharacteristic
301C22#1176298211yescharacteristic
302C4—21no—
303C4—21no—
304C4—82no—
305C4—82no—
306C4—1362no—
307C4#1176294712yes—
308C4#1176295212yes—
309C4#1176303612yes—
310C4#1176305312yes—
311C2—84no—
312C2#1452790011yescharacteristic
313C2#1452792911yescharacteristic
314C2#1452793512yes—
315C2#1452978712yes—
316C2#1452982611yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
Fitting, prime radicalC17×#2537C2
socleC23×C17#120
hypercenter#2537D17
hypocenterC17#10235
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#11763099C2semidirect
2. #14529585C2semidirect
3. C2×D4⋊#3622C2semidirect
4. #14526667C2semidirect
41. #10587555D4semidirect
47. C4⋊#3622D4semidirect
207. C17#10235semidirect

Maximal subgroups

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

subgroupindex
C2×#117630992
#145295852
C2×D4⋊#36222
#145266672
C17×#25372
#1023517