Finite Group Databasebeta

← #14533230

Subgroups — #14533230

Order 2176.

546 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587612C211yesmaximal, characteristic, complemented
2#11762748C211yesmaximal, characteristic, complemented
3#11762745C211yesmaximal, characteristic, complemented
4#11762739C211yesmaximal, characteristic, complemented
5#11762734C211yesmaximal, characteristic, complemented
6#11762697C212yesmaximal, complemented
7#11762690C212yesmaximal, complemented
8#11762689C212yesmaximal
9#11762667C212yesmaximal, complemented
10C2×#10587502C211yesmaximal, characteristic, complemented
11C2×C17×#123C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C23×C4×C17C2211yescharacteristic
14C23×Dic17C2212yes—
15C17×#123—24no—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2211yescharacteristic
18C2×C22⋊C4×C17C2211yescharacteristic
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊C4×C17C2211yescharacteristic
21C2×C22⋊Dic17C2212yes—
22C2×C22⋊Dic17C2212yes—
23C2×C22⋊Dic17C2212yes—
24#10587612—24no—
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29C2×C4×Dic17C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2211yescharacteristic
36#10587502C2211yescharacteristic
37C24×C17C2311yescharacteristic
38C24×C17C2311yescharacteristic
39C2×D4×C17—22no—
40C2×D4×C17—24no—
41C22×C4×C17—24no—
42C22×C4×C17D414yes—
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C22×Dic17—24no—
49C22×Dic17—24no—
50C22×Dic17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C22×Dic17C2312yes—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—22no—
57C22⋊C4×C17—22no—
58C22⋊C4×C17—22no—
59C22⋊C4×C17—22no—
60C22⋊Dic17—24no—
61C22⋊Dic17—24no—
62C22⋊Dic17—42no—
63C4⋊Dic17—22no—
64C4⋊Dic17—22no—
65#63225—24no—
66#63225—42no—
67C4×Dic17—24no—
68C23×C17—22no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—21no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—21no—
75C23×C17—21no—
76C23×C17C2×D412yes—
77C23×C17#4114yes—
78C23×C17C2411yescharacteristic
79D4×C17—44no—
80C2×C4×C17—28no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17C2×D412yes—
91C2×C4×C17C2×D412yes—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—48no—
104C2×Dic17#4114yes—
105#2498—171nomaximal, Sylow
106C22×C17—22no—
107C22×C17—22no—
108C22×C17—24no—
109C22×C17—24no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17—41no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17#12314yes—
116C22×C17C22×D411yescharacteristic
117C22×C17C2×#4112yes—
118C22×C17C2×#4112yes—
119C22×C17H(5,𝔽2)11yescharacteristic
120C22×C172−1+411yescharacteristic
121C4×C17—22no—
122C4×C17—22no—
123C4×C17—41no—
124C4×C17—41no—
125C4×C17—41no—
126Dic17—24no—
127Dic17—42no—
128Dic17—42no—
129Dic17—42no—
130C2×#123D1711yescharacteristic
131C2×#117—171no—
132#400—171no—
133#399—171no—
134#398—171no—
135#397—171no—
136#388—172no—
137#387—172no—
138#386—172no—
139#380—172no—
140C2×#95—171no—
141C2×C17—24no—
142C2×C17—81no—
143C2×C17C2×#12311yescharacteristic
144C2×C17#53211yescharacteristic
145C2×C17#53711yescharacteristic
146C2×C17#55112yes—
147C2×C17#55212yes—
148C22×D4C2×D1711yescharacteristic
149C23×C4—172no—
150C23×C4C2×D1711yescharacteristic
151#123—24no—
152#117—344no—
153C2×C4⋊C4—171no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4—172no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4C2×D1711yescharacteristic
158C2×C4⋊C4C2×D1711yescharacteristic
159C2×C22⋊C4—172no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4C2×D1711yescharacteristic
163C2×C22⋊C4C2×D1711yescharacteristic
164C2×C22⋊C4C2×D1711yescharacteristic
165C2×C42—172no—
166#95—172no—
167#95—171no—
168#95—171no—
169#95—172no—
170#95—172no—
171#95—172no—
172#95—172no—
173C17#249811yescharacteristic, complemented, Sylow
174C24C22×D1711yescharacteristic
175C24C22×D1711yescharacteristic
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—24no—
179C22×C4—172no—
180C22×C4—172no—
181C22×C4—172no—
182C22×C4—172no—
183C22×C4—344no—
184C22×C4—344no—
185C22×C4—682no—
186C22×C4C22⋊D1714yes—
187C22×C4C22×D1711yescharacteristic
188C22×C4C22×D1711yescharacteristic
189C22×C4C22×D1711yescharacteristic
190C22×C4C22×D1711yescharacteristic
191C22×C4C22×D1711yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—344no—
195C4⋊C4—342no—
196C4⋊C4—342no—
197C4⋊C4—682no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—344no—
202C22⋊C4—344no—
203C22⋊C4—682no—
204C42—344no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23#6325414yes—
214C23C2×C22⋊D1712yes—
215C23C23×D1711yescharacteristic
216D4—44no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—28no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—174no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—342no—
237C2×C4—342no—
238C2×C4—342no—
239C2×C4—688no—
240C2×C4C2×C22⋊D1712yes—
241C2×C4C2×C22⋊D1712yes—
242C22—24no—
243C22—24no—
244C22—22no—
245C22—22no—
246C22—42no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22#1058762114yes—
252C22C2×#6325412yes—
253C22#41×D1712yes—
254C22#1058769611yescharacteristic
255C22#1058769711yescharacteristic
256C22C22×C22⋊D1711yescharacteristic
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—344no—
263C4—682no—
264C4—682no—
265C4—682no—
266C2—24no—
267C2—81no—
268C2C2×#1058762111yescharacteristic
269C2#1176335511yescharacteristic
270C2#1176338711yescharacteristic
271C2#1176348212yes—
272C2#1176349612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#123C2
prime radicalC22×D4×C17C22
hypercenterC2×#123D17
hypocenterC17#2498
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#10587612C2semidirect
2. #11762748C2semidirect
3. #11762745C2semidirect
4. #11762739C2semidirect
5. #11762734C2semidirect
6. #11762697C2semidirect
7. #11762690C2semidirect
9. #11762667C2semidirect
10. C2×#10587502C2semidirect
173. C17#2498semidirect

Maximal subgroups

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

subgroupindex
C2×#105876122
#117627482
#117627452
#117627392
#117627342
#117626972
#117626902
#117626892
#117626672
C2×#105875022
C2×C17×#1232
#249817