Finite Group Databasebeta

← #14536907

Subgroups — #14536907

Order 2176.

508 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#11762748C211yesmaximal, characteristic, complemented
2#11762736C212yesmaximal, complemented
3#11762725C212yesmaximal, complemented
4#11762725C212yesmaximal, complemented
5#11762718C212yesmaximal, complemented
6#11762717C212yesmaximal, complemented
7#11762710C211yesmaximal, characteristic, complemented
8#11762702C212yesmaximal, complemented
9C17×#394C211yesmaximal, characteristic, complemented
10C22×D4×C17C2211yescharacteristic
11C22×C22⋊D17C2212yes—
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2212yes—
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4×Dic17C2211yescharacteristic
26C17×#95C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31C24×C17C2312yes—
32C23×D17C2312yes—
33C2×D4×C17—41no—
34C2×D4×C17—42no—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C2×C22⋊D17—42no—
40C2×C22⋊D17—42no—
41C2×C22⋊D17—42no—
42C2×C22⋊D17—42no—
43C2×C22⋊D17—42no—
44C2×C22⋊D17—42no—
45C22×Dic17C2312yes—
46C22×Dic17C2312yes—
47C22×Dic17C2312yes—
48C2×D68—22no—
49C2×D68—41no—
50C2×D68—41no—
51C4⋊C4×C17—41no—
52C22⋊C4×C17—24no—
53C22⋊C4×C17—42no—
54C22⋊Dic17—42no—
55C22⋊Dic17—42no—
56#63227—42no—
57#63227—42no—
58#63227—42no—
59#63227—42no—
60C4⋊Dic17—42no—
61C4⋊Dic17—41no—
62C4×Dic17—41no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17C2411yescharacteristic
71C22×D17—22no—
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—22no—
75C22×D17—22no—
76C22×D17—22no—
77C22×D17—22no—
78D4×C17—44no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—21no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17#4112yes—
89C22⋊D17—48no—
90C22⋊D17—48no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100D68—44no—
101#2828—171nomaximal, Sylow
102C22×C17—42no—
103C22×C17—42no—
104C22×C17—42no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17C2×#4111yescharacteristic
110C22×C17H(5,𝔽2)11yescharacteristic
111C22×C17H(5,𝔽2)12yes—
112C22×C17H(5,𝔽2)11yescharacteristic
113C22×C17H(5,𝔽2)12yes—
114C2×D17—42no—
115C2×D17—42no—
116C2×D17—42no—
117C2×D17—42no—
118C2×D17—42no—
119C2×D17—42no—
120C2×D17—42no—
121C4×C17—22no—
122C4×C17—42no—
123C4×C17—41no—
124C4×C17—41no—
125Dic17—42no—
126Dic17—42no—
127Dic17—42no—
128#400—171no—
129#397—172no—
130#394—172no—
131#394—172no—
132#394D1711yescharacteristic, complemented
133#392—172no—
134#392—172no—
135#390—171no—
136#388—172no—
137C2×C17—82no—
138C2×C17#53811yescharacteristic
139C2×C17#54012yes—
140C2×C17#56011yescharacteristic
141C2×C17#56012yes—
142C2×C17H(3,𝔽4)11yescharacteristic
143D17—82no—
144C22×D4—172no—
145C22×D4—172no—
146C22×D4—171no—
147C22×D4C2×D1711yescharacteristic
148C2×C4⋊C4—172no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4C2×D1711yescharacteristic
151C2×C22⋊C4—172no—
152C2×C22⋊C4—172no—
153C2×C22⋊C4—172no—
154C2×C22⋊C4—172no—
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C22⋊C4C2×D1712yes—
158C2×C22⋊C4C2×D1712yes—
159C2×C42—171no—
160#95—172no—
161#95—172no—
162#95—172no—
163#95—171no—
164#95C2×D1711yescharacteristic
165C17#282811yescharacteristic, complemented, Sylow
166C24—172no—
167C24C22×D1712yes—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—342no—
171C2×D4—681no—
172C2×D4—681no—
173C2×D4—682no—
174C2×D4—682no—
175C2×D4—682no—
176C2×D4—682no—
177C2×D4—682no—
178C2×D4—682no—
179C22×C4—172no—
180C22×C4—172no—
181C22×C4—172no—
182C22×C4C22×D1711yescharacteristic
183C22×C4C22×D1712yes—
184C22×C4C22×D1711yescharacteristic
185C22×C4C22×D1711yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—681no—
188C4⋊C4—682no—
189C22⋊C4—24no—
190C22⋊C4—42no—
191C22⋊C4—682no—
192C22⋊C4—682no—
193C22⋊C4—682no—
194C22⋊C4—682no—
195C22⋊C4—682no—
196C22⋊C4—682no—
197C42—681no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—342no—
206C23—342no—
207C23—342no—
208C23—342no—
209C23—342no—
210C23—342no—
211C23—342no—
212C23C23×D1711yescharacteristic
213D4—44no—
214D4—684no—
215D4—688no—
216D4—688no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4#6325412yes—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—682no—
244C22—682no—
245C22—682no—
246C22—682no—
247C22—682no—
248C22—682no—
249C22—682no—
250C22C2×#6325411yescharacteristic
251C22#1058769112yes—
252C22#1058769111yescharacteristic
253C22#1058769611yescharacteristic
254C22#1058769612yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—682no—
260C4—682no—
261C4—682no—
262C2—82no—
263C2—1362no—
264C2#1176339611yescharacteristic
265C2#1176341212yes—
266C2#1176354811yescharacteristic
267C2#1176354912yes—
268C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
hypercenter#394D17
hypocenterC17#2828
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #11762748C2semidirect
2. #11762736C2semidirect
3. #11762725C2semidirect
4. #11762725C2semidirect
5. #11762718C2semidirect
6. #11762717C2semidirect
7. #11762710C2semidirect
8. #11762702C2semidirect
9. C17×#394C2semidirect
132. #394D17semidirect
165. C17#2828semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627362
#117627252
#117627182
#117627172
#117627102
#117627022
C17×#3942
#282817