Finite Group Databasebeta

← #14533688

Subgroups — #14533688

Order 2176.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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×#10587564C212yesmaximal, complemented
2C2×#10587573C211yesmaximal, characteristic, complemented
3C2×D4×Dic17C211yesmaximal, characteristic
4#11762741C211yesmaximal, characteristic, complemented
5#11762739C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic
7#11762703C212yesmaximal, complemented
8#11762702C212yesmaximal, complemented
9#11762673C211yesmaximal, characteristic, complemented
10#11762667C212yesmaximal, complemented
11C17×#394C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C23×Dic17C2212yes—
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21D4×Dic17—24no—
22C2×C4⋊Dic17C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30C2×C4×Dic17C2211yescharacteristic
31#10587573—24no—
32#10587564—28no—
33C17×#95C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2211yescharacteristic
38C24×C17C2312yes—
39C2×D4×C17—24no—
40C2×D4×C17—22no—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×Dic17—24no—
46C22×Dic17—24no—
47C22×Dic17—24no—
48C22×Dic17C2312yes—
49C22×Dic17C2312yes—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2311yescharacteristic
54C2×Dic34—22no—
55C2×Dic34—22no—
56C2×Dic34D414yescomplemented
57C4⋊C4×C17—22no—
58C22⋊C4×C17—24no—
59C22⋊C4×C17—24no—
60C22⋊Dic17—24no—
61C22⋊Dic17—24no—
62C22⋊Dic17—24no—
63C4⋊Dic17—22no—
64C4⋊Dic17—24no—
65#63225—24no—
66#63225—22no—
67#63225—24no—
68#63225—22no—
69C4×Dic17—22no—
70C4×Dic17—22no—
71C4×Dic17D414yes—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17—22no—
78C23×C17#4114yes—
79C23×C17C2411yescharacteristic
80D4×C17—28no—
81C2×C4×C17—21no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17C2×D412yes—
90C2×C4×C17C2×D412yes—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—21no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—48no—
103C2×Dic17C2×D412yes—
104C2×Dic17C2×D412yes—
105C2×Dic17C2×D412yes—
106C2×Dic17C2×D412yes—
107C2×Dic17#4112yes—
108Dic34—28no—
109#2536—171nomaximal, Sylow
110C22×C17—24no—
111C22×C17—24no—
112C22×C17—24no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17C22×D411yescharacteristic
118C22×C17C22×D411yescharacteristic
119C22×C17C2×#4112yes—
120C22×C17C2×#4111yescharacteristic
121C22×C172−1+412yes—
122C4×C17—22no—
123C4×C17—22no—
124C4×C17—41no—
125C4×C17—42no—
126Dic17—22no—
127Dic17—22no—
128Dic17—42no—
129Dic17—41no—
130Dic17—42no—
131Dic17#12114yes—
132C2×C22⋊Q8—172no—
133C2×C4×Q8—171no—
134C2×C4×D4—171no—
135#398—171no—
136#398—171no—
137#394—171no—
138#394D1711yescharacteristic
139#388—172no—
140#388—172no—
141#382—171no—
142#380—172no—
143C2×C17—44no—
144C2×C17C2×#12111yescharacteristic
145C2×C17#52511yescharacteristic
146C2×C17#53611yescharacteristic
147C2×C17#54812yes—
148C2×C17#55212yes—
149C22×Q8—171no—
150C22×D4C2×D1711yescharacteristic
151C23×C4—172no—
152C22⋊Q8—348no—
153C4×Q8—344no—
154C4×D4—344no—
155C2×C4⋊C4—171no—
156C2×C4⋊C4—171no—
157C2×C4⋊C4—171no—
158C2×C4⋊C4—172no—
159C2×C4⋊C4—172no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4C2×D1711yescharacteristic
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4C2×D1712yes—
166C2×C22⋊C4C2×D1712yes—
167C2×C42—171no—
168C2×C42—171no—
169C2×C42—171no—
170#95—172no—
171#95—172no—
172#95—172no—
173#95—171no—
174#95C2×D1711yescharacteristic
175C17#253611yescharacteristic, complemented, Sylow
176C24C22×D1712yes—
177C2×Q8—174no—
178C2×Q8—342no—
179C2×Q8—342no—
180C2×D4—24no—
181C2×D4—22no—
182C22×C4—171no—
183C22×C4—171no—
184C22×C4—171no—
185C22×C4—172no—
186C22×C4—171no—
187C22×C4—172no—
188C22×C4—344no—
189C22×C4—344no—
190C22×C4—344no—
191C22×C4C22×D1712yes—
192C22×C4C22×D1711yescharacteristic
193C22×C4C22×D1711yescharacteristic
194C22×C4C22×D1711yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—342no—
197C4⋊C4—344no—
198C4⋊C4—344no—
199C4⋊C4—342no—
200C4⋊C4—342no—
201C4⋊C4—344no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—344no—
205C22⋊C4—344no—
206C22⋊C4—344no—
207C42—174no—
208C42—342no—
209C42—342no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23#6325414yes—
217C23C23×D1711yescharacteristic
218Q8—348no—
219D4—28no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—172no—
229C2×C4—172no—
230C2×C4—172no—
231C2×C4—172no—
232C2×C4—172no—
233C2×C4—341no—
234C2×C4—341no—
235C2×C4—342no—
236C2×C4—341no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4—342no—
244C2×C4—688no—
245C2×C4D4×D1712yes—
246C2×C4D4×D1712yes—
247C22—24no—
248C22—24no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×D4×D1711yescharacteristic
255C22C2×D4×D1711yescharacteristic
256C22C2×#6325412yes—
257C22#41×D1711yescharacteristic
258C22#1058769712yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—41no—
263C4—174no—
264C4—342no—
265C4—342no—
266C4—682no—
267C4—682no—
268C4—681no—
269C2—44no—
270C2#121×D1711yescharacteristic
271C2#1176331511yescharacteristic
272C2#1176337911yescharacteristic
273C2#1176346512yes—
274C2#1176349612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2536
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587564C2semidirect
2. C2×#10587573C2semidirect
4. #11762741C2semidirect
5. #11762739C2semidirect
7. #11762703C2semidirect
8. #11762702C2semidirect
9. #11762673C2semidirect
10. #11762667C2semidirect
56. C2×Dic34D4semidirect
175. C17#2536semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
C2×#105875732
C2×D4×Dic172
#117627412
#117627392
#117627232
#117627032
#117627022
#117626732
#117626672
C17×#3942
#253617