Finite Group Databasebeta

← #14535807

Subgroups — #14535807

Order 2176.

592 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 129 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×#10587571C212yesmaximal, complemented
2#11762726C211yesmaximal, characteristic, complemented
3#11762718C212yesmaximal, complemented
4#11762717C212yesmaximal, complemented
5#11762714C211yesmaximal, characteristic, complemented
6#11762709C212yesmaximal, complemented
7#11762703C212yesmaximal, complemented
8#11762698C212yesmaximal, complemented
9C17×#394C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63225C2212yes—
25C2×C4×Dic17C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27#10587571—28no—
28C17×#95C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C24×C17C2312yes—
33C23×D17C2312yes—
34C2×D4×C17—41no—
35C2×D4×C17—42no—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C2×C22⋊D17—24no—
41C2×C22⋊D17—24no—
42C2×C22⋊D17—42no—
43C2×C22⋊D17—42no—
44C2×C22⋊D17—42no—
45C2×C22⋊D17—42no—
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C22×Dic17C2312yes—
49C2×D68—42no—
50C2×D68—41no—
51C2×Dic34—41no—
52C2×Dic34—41no—
53C2×Dic34D414yescomplemented
54C4⋊C4×C17—41no—
55C22⋊C4×C17—24no—
56C22⋊C4×C17—42no—
57C22⋊Dic17—24no—
58C22⋊Dic17—24no—
59#63227—24no—
60#63227—24no—
61#63227—24no—
62#63227—42no—
63#63225—42no—
64C4×Dic17—24no—
65C4×Dic17—41no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17C2411yescharacteristic
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81D4×C17—44no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17C2×D412yes—
92C22⋊D17—48no—
93C22⋊D17—48no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17C2×D414yes—
102C2×Dic17#4114yes—
103D68—44no—
104Dic34—44no—
105#2733—171nomaximal, Sylow
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17—42no—
112C22×C17—42no—
113C22×C17C22×D411yescharacteristic
114C22×C17C2×#4112yes—
115C22×C17H(5,𝔽2)11yescharacteristic
116C22×C17H(5,𝔽2)12yes—
117C22×C17H(5,𝔽2)11yescharacteristic
118C2×D17—42no—
119C2×D17—42no—
120C2×D17—42no—
121C2×D17—42no—
122C2×D17—42no—
123C2×D17—42no—
124C2×D17—42no—
125C4×C17—22no—
126C4×C17—42no—
127C4×C17—41no—
128C4×C17—41no—
129Dic17—24no—
130Dic17—24no—
131Dic17—42no—
132C2×#124—172no—
133#394—171no—
134#394D1711yescharacteristic, complemented
135#392—172no—
136#392—172no—
137#391—171no—
138#390—172no—
139#388—172no—
140#388—172no—
141C2×C17—82no—
142C2×C17#53411yescharacteristic
143C2×C17#53812yes—
144C2×C17#55012yes—
145C2×C17#55511yescharacteristic
146C2×C17H(3,𝔽4)11yescharacteristic
147D17—82no—
148C22×Q8—171no—
149C22×D4—172no—
150C22×D4—172no—
151C22×D4—171no—
152C22×D4C2×D1711yescharacteristic
153#124—348no—
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⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4C2×D1712yes—
163C2×C22⋊C4C2×D1712yes—
164C2×C42—172no—
165C2×C42—171no—
166#95—171no—
167#95—172no—
168#95—172no—
169#95C2×D1711yescharacteristic
170C17#273311yescharacteristic, complemented, Sylow
171C24—172no—
172C24C22×D1712yes—
173C2×Q8—174no—
174C2×Q8—681no—
175C2×Q8—681no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—344no—
179C2×D4—344no—
180C2×D4—682no—
181C2×D4—681no—
182C2×D4—682no—
183C2×D4—682no—
184C2×D4—682no—
185C2×D4—682no—
186C22×C4—172no—
187C22×C4—172no—
188C22×C4—172no—
189C22×C4C22×D1711yescharacteristic
190C22×C4C22×D1711yescharacteristic
191C22×C4C22×D1712yes—
192C22×C4C22×D1711yescharacteristic
193C4⋊C4—41no—
194C4⋊C4—682no—
195C22⋊C4—24no—
196C22⋊C4—42no—
197C22⋊C4—344no—
198C22⋊C4—344no—
199C22⋊C4—344no—
200C22⋊C4—344no—
201C22⋊C4—344no—
202C22⋊C4—682no—
203C42—344no—
204C42—681no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—342no—
213C23—342no—
214C23—342no—
215C23—342no—
216C23—342no—
217C23—342no—
218C23—342no—
219C23C23×D1711yescharacteristic
220Q8—684no—
221D4—44no—
222D4—684no—
223D4—688no—
224D4—688no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—174no—
235C2×C4—174no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—342no—
240C2×C4—342no—
241C2×C4—342no—
242C2×C4—342no—
243C2×C4D4×D1712yes—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—682no—
252C22—682no—
253C22—682no—
254C22—682no—
255C22—682no—
256C22—682no—
257C22—682no—
258C22C2×D4×D1711yescharacteristic
259C22#1058769112yes—
260C22#1058769111yescharacteristic
261C22#41×D1712yes—
262C22#1058769611yescharacteristic
263C4—22no—
264C4—42no—
265C4—41no—
266C4—41no—
267C4—344no—
268C4—344no—
269C4—682no—
270C2—82no—
271C2—1362no—
272C2#1176336711yescharacteristic
273C2#1176339512yes—
274C2#1176347712yes—
275C2#1176352111yescharacteristic
276C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
hypercenter#394D17
hypocenterC17#2733
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587571C2semidirect
2. #11762726C2semidirect
3. #11762718C2semidirect
4. #11762717C2semidirect
5. #11762714C2semidirect
6. #11762709C2semidirect
7. #11762703C2semidirect
8. #11762698C2semidirect
9. C17×#394C2semidirect
53. C2×Dic34D4semidirect
134. #394D17semidirect
170. C17#2733semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
#117627262
#117627182
#117627172
#117627142
#117627092
#117627032
#117626982
C17×#3942
#273317