Finite Group Databasebeta

← #14534649

Subgroups — #14534649

Order 2176.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#10587622C211yesmaximal, characteristic
2C2×#10587573C211yesmaximal, characteristic, complemented
3#11762745C212yesmaximal, complemented
4#11762730C211yesmaximal, characteristic, complemented
5#11762729C211yesmaximal, characteristic, complemented
6#11762723C211yesmaximal, characteristic
7#11762721C212yesmaximal, complemented
8#11762703C212yesmaximal, complemented
9#11762702C212yesmaximal, complemented
10#11762676C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21C2×C22⋊Dic17C2212yes—
22#10587622—24no—
23C2×C4⋊Dic17C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587573—24no—
31C17×#95C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C24×C17C2312yes—
38C2×D4×C17—22no—
39C2×D4×C17—42no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—22no—
52C2×Dic34—22no—
53C2×Dic34—22no—
54C2×Dic34—42no—
55C4⋊C4×C17—22no—
56C22⋊C4×C17—24no—
57C22⋊C4×C17—42no—
58C22⋊Dic17—24no—
59C22⋊Dic17—24no—
60C22⋊Dic17—24no—
61C22⋊Dic17—42no—
62C4⋊Dic17—42no—
63#63225—22no—
64#63225—22no—
65#63225—42no—
66C4×Dic17—22no—
67C4×Dic17—22no—
68C4×Dic17D414yes—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17C2411yescharacteristic
77D4×C17—44no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17C2×D412yes—
87C2×C4×C17#4112yes—
88C2×Dic17—21no—
89C2×Dic17—21no—
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103C2×Dic17#4112yes—
104Dic34—28no—
105Dic34—44no—
106#2632—171nomaximal, Sylow
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17—42no—
112C22×C17—42no—
113C22×C17—42no—
114C22×C17C22×D411yescharacteristic
115C22×C17C2×#4111yescharacteristic
116C22×C17C2×#4111yescharacteristic
117C22×C17H(5,𝔽2)12yes—
118C22×C172−1+412yes—
119C4×C17—22no—
120C4×C17—22no—
121C4×C17—41no—
122C4×C17—42no—
123Dic17—22no—
124Dic17—22no—
125Dic17—22no—
126Dic17—42no—
127Dic17—41no—
128Dic17—42no—
129C2×#124—171no—
130C2×C4×Q8—171no—
131#399—172no—
132#395—171no—
133#395—171no—
134#394—171no—
135#394D1711yescharacteristic
136#393—172no—
137#388—172no—
138#388—172no—
139#383—171no—
140C2×C17—82no—
141C2×C17#52511yescharacteristic
142C2×C17#53511yescharacteristic
143C2×C17#53611yescharacteristic
144C2×C17#54212yes—
145C2×C17#54212yes—
146C22×Q8—171no—
147C22×Q8—171no—
148C22×D4C2×D1711yescharacteristic
149#124—344no—
150C4×Q8—344no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4—171no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4C2×D1711yescharacteristic
156C2×C22⋊C4—172no—
157C2×C22⋊C4—172no—
158C2×C22⋊C4—172no—
159C2×C22⋊C4—172no—
160C2×C22⋊C4C2×D1712yes—
161C2×C22⋊C4C2×D1712yes—
162C2×C42—171no—
163C2×C42—171no—
164C2×C42—171no—
165#95—172no—
166#95—172no—
167#95—171no—
168#95—172no—
169#95—172no—
170#95C2×D1711yescharacteristic
171C17#263211yescharacteristic, complemented, Sylow
172C24C22×D1712yes—
173C2×Q8—342no—
174C2×Q8—342no—
175C2×Q8—342no—
176C2×Q8—342no—
177C2×Q8—682no—
178C2×D4—22no—
179C2×D4—42no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—171no—
183C22×C4—172no—
184C22×C4—172no—
185C22×C4—171no—
186C22×C4C22×D1712yes—
187C22×C4C22×D1711yescharacteristic
188C22×C4C22×D1711yescharacteristic
189C22×C4C22×D1711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—342no—
192C4⋊C4—342no—
193C4⋊C4—682no—
194C4⋊C4—682no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—344no—
198C22⋊C4—344no—
199C22⋊C4—344no—
200C22⋊C4—682no—
201C42—174no—
202C42—342no—
203C42—342no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23C23×D1711yescharacteristic
212Q8—348no—
213Q8—684no—
214D4—44no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—172no—
224C2×C4—172no—
225C2×C4—172no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—341no—
229C2×C4—341no—
230C2×C4—342no—
231C2×C4—341no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4D4×D1712yes—
240C2×C4#6325412yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D1711yescharacteristic
249C22C2×#6325411yescharacteristic
250C22#1058769112yes—
251C22#41×D1711yescharacteristic
252C22#1058769712yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—342no—
258C4—342no—
259C4—342no—
260C4—682no—
261C4—681no—
262C4—682no—
263C2—82no—
264C2#1176331911yescharacteristic
265C2#1176337411yescharacteristic
266C2#1176337911yescharacteristic
267C2#1176342612yes—
268C2#1176342712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2632
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#10587573C2semidirect
3. #11762745C2semidirect
4. #11762730C2semidirect
5. #11762729C2semidirect
7. #11762721C2semidirect
8. #11762703C2semidirect
9. #11762702C2semidirect
10. #11762676C2semidirect
171. C17#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#105876222
C2×#105875732
#117627452
#117627302
#117627292
#117627232
#117627212
#117627032
#117627022
#117626762
C17×#3942
#263217