Finite Group Databasebeta

← #14535552

Subgroups — #14535552

Order 2176.

556 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 137 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#11762734C211yesmaximal, characteristic, complemented
3#11762726C212yesmaximal, complemented
4#11762722C212yesmaximal, complemented
5#11762718C211yesmaximal, characteristic
6#11762714C212yesmaximal, complemented
7#11762701C212yesmaximal, complemented
8#11762688C212yesmaximal, complemented
9C17×#385C211yesmaximal, characteristic, complemented
10C23×C4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×D68C2211yescharacteristic
13C22×Dic34C2212yes—
14C2×C4⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2211yescharacteristic
16C2×C22⋊C4×C17C2211yescharacteristic
17C2×C22⋊Dic17C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4×Dic17C2212yes—
25#10587571—28no—
26C17×#95C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C24×C17C2311yescharacteristic
33C23×D17C2312yes—
34C22×C4×C17—42no—
35C22×C4×C17—41no—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C2×C22⋊D17—24no—
41C2×C22⋊D17—24no—
42C2×C22⋊D17—42no—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C2×D68—41no—
47C2×D68—42no—
48C2×Dic34—24no—
49C2×Dic34—24no—
50C2×Dic34—42no—
51C4⋊C4×C17—42no—
52C22⋊C4×C17—41no—
53C22⋊C4×C17D414yescomplemented
54C22⋊Dic17—24no—
55#63227—24no—
56#63227—24no—
57#63227—24no—
58#63227—42no—
59#63227—42no—
60C4⋊Dic17—41no—
61C4×Dic17—24no—
62C23×C17—22no—
63C23×C17—21no—
64C23×C17—22no—
65C23×C17—21no—
66C23×C17C2×D412yes—
67C23×C17C2411yescharacteristic
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—44no—
85C2×C4×C17C2×D414yes—
86C22⋊D17—48no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17#4114yes—
96D68—44no—
97Dic34—48no—
98#2712—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—24no—
101C22×C17—42no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17C22×D411yescharacteristic
105C22×C17C2×#4112yes—
106C22×C17H(5,𝔽2)11yescharacteristic
107C22×C17H(5,𝔽2)12yes—
108C22×C172−1+411yescharacteristic
109C2×D17—42no—
110C2×D17—42no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17—42no—
114C2×D17—42no—
115C2×D17—42no—
116C4×C17—24no—
117C4×C17—41no—
118C4×C17—41no—
119C4×C17—42no—
120Dic17—24no—
121Dic17—42no—
122Dic17—42no—
123C2×#124—172no—
124#397—171no—
125#394—172no—
126#393—172no—
127#392—171no—
128#391—172no—
129#388—172no—
130#386—172no—
131#385D1711yescharacteristic, complemented
132C2×C17—42no—
133C2×C17#53711yescharacteristic
134C2×C17#54212yes—
135C2×C17#54612yes—
136C2×C17#55911yescharacteristic
137C2×C17H(3,𝔽4)11yescharacteristic
138D17—82no—
139C22×Q8—172no—
140C22×D4—172no—
141C22×D4—171no—
142C23×C4C2×D1711yescharacteristic
143#124—348no—
144C2×C4⋊C4—171no—
145C2×C4⋊C4C2×D1712yes—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—172no—
150C2×C22⋊C4—172no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4C2×D1711yescharacteristic
153C2×C22⋊C4C2×D1711yescharacteristic
154C2×C42—172no—
155#95—172no—
156#95—172no—
157#95—171no—
158#95—172no—
159#95—171no—
160#95C2×D1712yes—
161C17#271211yescharacteristic, complemented, Sylow
162C24—172no—
163C24C22×D1711yescharacteristic
164C2×Q8—344no—
165C2×Q8—344no—
166C2×Q8—682no—
167C2×D4—344no—
168C2×D4—344no—
169C2×D4—681no—
170C2×D4—682no—
171C2×D4—682no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—172no—
175C22×C4—172no—
176C22×C4—172no—
177C22×C4C22×D1711yescharacteristic
178C22×C4C22×D1712yes—
179C22×C4C22×D1711yescharacteristic
180C22×C4C22×D1712yes—
181C4⋊C4—42no—
182C4⋊C4—681no—
183C22⋊C4—41no—
184C22⋊C4—344no—
185C22⋊C4—344no—
186C22⋊C4—344no—
187C22⋊C4—344no—
188C22⋊C4—682no—
189C22⋊C4—682no—
190C22⋊C4D6814yescomplemented
191C42—344no—
192C23—21no—
193C23—22no—
194C23—21no—
195C23—22no—
196C23—342no—
197C23—342no—
198C23—342no—
199C23—342no—
200C23—342no—
201C23—342no—
202C23—342no—
203C23C2×D6812yes—
204C23C23×D1711yescharacteristic
205Q8—688no—
206D4—684no—
207D4—688no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—44no—
218C2×C4—174no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4C2×D6814yes—
228C22—22no—
229C22—24no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—682no—
234C22—682no—
235C22—682no—
236C22—682no—
237C22—682no—
238C22—682no—
239C22—682no—
240C22C22×D6811yescharacteristic
241C22#1058769111yescharacteristic
242C22#1058769112yes—
243C22#1058769411yescharacteristic
244C22#41×D1712yes—
245C4—24no—
246C4—41no—
247C4—42no—
248C4—41no—
249C4—344no—
250C4—682no—
251C4—682no—
252C2—42no—
253C2—1362no—
254C2#1176339011yescharacteristic
255C2#1176342912yes—
256C2#1176345412yes—
257C2#1176354311yescharacteristic
258C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radical#11762718C2
hypercenter#385D17
hypocenterC17#2712
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#10587571C2semidirect
2. #11762734C2semidirect
3. #11762726C2semidirect
4. #11762722C2semidirect
6. #11762714C2semidirect
7. #11762701C2semidirect
8. #11762688C2semidirect
9. C17×#385C2semidirect
53. C22⋊C4×C17D4semidirect
131. #385D17semidirect
161. C17#2712semidirect
190. C22⋊C4D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875712
#117627342
#117627262
#117627222
#117627182
#117627142
#117627012
#117626882
C17×#3852
#271217