Finite Group Databasebeta

← #14533228

Subgroups — #14533228

Order 2176.

546 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 157 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×#10587615C211yesmaximal, characteristic
2C2×#10587612C211yesmaximal, characteristic, complemented
3#11762749C211yesmaximal, characteristic, complemented
4#11762746C211yesmaximal, characteristic, complemented
5#11762741C211yesmaximal, characteristic, complemented
6#11762736C211yesmaximal, characteristic, complemented
7#11762701C212yesmaximal, complemented
8#11762694C212yesmaximal, complemented
9#11762688C212yesmaximal
10#11762669C212yesmaximal, complemented
11C2×C17×#95C211yesmaximal, characteristic, complemented
12C23×C4×C17C2212yes—
13C23×C4×C17C2211yescharacteristic
14C22×C22⋊D17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17#10587615—24no—
18C2×#63227C2212yes—
19C2×#63227C2211yescharacteristic
20C2×#63227C2211yescharacteristic
21C2×#63227C2212yes—
22#10587612—24no—
23C2×C4⋊Dic17C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×C4⋊Dic17C2212yes—
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×C4×Dic17C2212yes—
30C17×#95C2212yes—
31C17×#95C2211yescharacteristic
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2311yescharacteristic
38C23×D17C2311yescharacteristic
39C22×C4×C17—24no—
40C22×C4×C17—24no—
41C22×C4×C17—24no—
42C22×C4×C17—42no—
43C22×C4×C17D414yes—
44C22×C4×C17C2312yes—
45C22×C4×C17C2311yescharacteristic
46C22×C4×C17C2311yescharacteristic
47C22×C4×C17C2312yes—
48C2×C22⋊D17—22no—
49C2×C22⋊D17—24no—
50C22×Dic17C2311yescharacteristic
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C22⋊Dic17—24no—
56C22⋊Dic17—22no—
57#63227—22no—
58#63227—22no—
59#63227—24no—
60#63227—42no—
61C4⋊Dic17—22no—
62C4⋊Dic17—22no—
63C4⋊Dic17—24no—
64C4⋊Dic17—42no—
65#63225—22no—
66#63225—22no—
67C4×Dic17—24no—
68C23×C17—22no—
69C23×C17—21no—
70C23×C17—21no—
71C23×C17C2×D412yes—
72C23×C17#4114yes—
73C23×C17C2411yescharacteristic
74C22×D17—21no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—21no—
78C22×D17—21no—
79C2×C4×C17—28no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—48no—
88C2×C4×C17C2×D412yes—
89C2×C4×C17C2×D412yes—
90C2×C4×C17#4114yes—
91C22⋊D17—44no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—21no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—21no—
105#2498—171nomaximal, Sylow
106C22×C17—22no—
107C22×C17—22no—
108C22×C17—24no—
109C22×C17—24no—
110C22×C17#12314yes—
111C22×C17C22×D411yescharacteristic
112C22×C17C2×#4112yes—
113C22×C17C2×#4112yes—
114C22×C17H(5,𝔽2)11yescharacteristic
115C22×C172−1+411yescharacteristic
116C2×D17—41no—
117C2×D17—41no—
118C2×D17—41no—
119C2×D17—42no—
120C2×D17—42no—
121C4×C17—22no—
122C4×C17—22no—
123C4×C17—24no—
124C4×C17—42no—
125Dic17—41no—
126Dic17—42no—
127Dic17—41no—
128Dic17—42no—
129Dic17—41no—
130C2×#123—171no—
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×#95D1711yescharacteristic, complemented
141C2×C17—24no—
142C2×C17C2×#12311yescharacteristic
143C2×C17#53211yescharacteristic
144C2×C17#53711yescharacteristic
145C2×C17#55112yes—
146C2×C17#55212yes—
147D17—81no—
148C22×D4—171no—
149C23×C4C2×D1712yes—
150C23×C4C2×D1711yescharacteristic
151#123—344no—
152#117—344no—
153C2×C4⋊C4—171no—
154C2×C4⋊C4—171no—
155C2×C4⋊C4—171no—
156C2×C4⋊C4—171no—
157C2×C4⋊C4—172no—
158C2×C4⋊C4—172no—
159C2×C22⋊C4—172no—
160C2×C22⋊C4—171no—
161C2×C22⋊C4—171no—
162C2×C22⋊C4—171no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C42—172no—
166#95—172no—
167#95—172no—
168#95—172no—
169#95—172no—
170#95C2×D1712yes—
171#95C2×D1711yescharacteristic
172#95C2×D1711yescharacteristic
173C17#249811yescharacteristic, complemented, Sylow
174C24—171no—
175C24C22×D1711yescharacteristic
176C2×D4—342no—
177C2×D4—344no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—171no—
183C22×C4—172no—
184C22×C4—172no—
185C22×C4—171no—
186C22×C4—171no—
187C22×C4D6814yes—
188C22×C4C22×D1712yes—
189C22×C4C22×D1711yescharacteristic
190C22×C4C22×D1711yescharacteristic
191C22×C4C22×D1712yes—
192C4⋊C4—342no—
193C4⋊C4—342no—
194C4⋊C4—342no—
195C4⋊C4—342no—
196C4⋊C4—344no—
197C4⋊C4—682no—
198C22⋊C4—344no—
199C22⋊C4—342no—
200C22⋊C4—342no—
201C22⋊C4—342no—
202C22⋊C4—344no—
203C22⋊C4—682no—
204C42—344no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—341no—
209C23—342no—
210C23—342no—
211C23—341no—
212C23—341no—
213C23C2×D6812yes—
214C23#6325414yes—
215C23C23×D1711yescharacteristic
216D4—684no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—48no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—341no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—341no—
237C2×C4—342no—
238C2×C4—341no—
239C2×C4C2×D6812yes—
240C2×C4C2×D6812yes—
241C2×C4#6325214yes—
242C22—24no—
243C22—24no—
244C22—22no—
245C22—22no—
246C22—682no—
247C22—682no—
248C22—681no—
249C22—681no—
250C22—681no—
251C22#1058757214yes—
252C22C22×D6811yescharacteristic
253C22C2×#6325212yes—
254C22C2×#6325412yes—
255C22#1058769111yescharacteristic
256C22#1058769411yescharacteristic
257C4—24no—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—681no—
262C4—682no—
263C4—681no—
264C4—682no—
265C4—681no—
266C2—24no—
267C2—1361no—
268C2C2×#1058757211yescharacteristic
269C2#1176335911yescharacteristic
270C2#1176339011yescharacteristic
271C2#1176348512yes—
272C2#1176349412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#95C2
prime radicalC22×C22⋊D17C22
hypercenterC2×#95D17
hypocenterC17#2498
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. C2×#10587612C2semidirect
3. #11762749C2semidirect
4. #11762746C2semidirect
5. #11762741C2semidirect
6. #11762736C2semidirect
7. #11762701C2semidirect
8. #11762694C2semidirect
10. #11762669C2semidirect
11. C2×C17×#95C2semidirect
140. C2×#95D17semidirect
173. C17#2498semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876122
#117627492
#117627462
#117627412
#117627362
#117627012
#117626942
#117626882
#117626692
C2×C17×#952
#249817