Finite Group Databasebeta

← #14534526

Subgroups — #14534526

Order 2176.

632 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 165 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×#10587615C212yesmaximal, complemented
2C2×#10587616C211yesmaximal, characteristic
3#11762739C212yesmaximal, complemented
4#11762731C214yesmaximal, complemented
5#11762726C212yesmaximal, complemented
6#11762688C212yesmaximal, complemented
7#11762669C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C17C211yesmaximal, characteristic, complemented
9C23×C4×C17C2211yescharacteristic
10C23×C4×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C22×D68C2211yescharacteristic
14C2×C4⋊C4×C17—24no—
15C2×C4⋊C4×C17C2212yes—
16C2×C4⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2212yes—
18#10587616—24no—
19#10587615—28no—
20C2×#63227C2214yes—
21C2×#63227C2212yes—
22C2×#63227C2214yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2212yes—
25C2×#63225C2214yes—
26#10587502C2214yes—
27#10587502C2212yes—
28C24×C17C2311yescharacteristic
29C23×D17C2312yes—
30C22×C4×C17—22no—
31C22×C4×C17—22no—
32C22×C4×C17—22no—
33C22×C4×C17—42no—
34C22×C4×C17—42no—
35C22×C4×C17D414yes—
36C22×C4×C17D414yescomplemented
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C2×C22⋊D17—24no—
42C2×C22⋊D17—24no—
43C2×C22⋊D17—24no—
44C22×Dic17C2312yes—
45C22×Dic17C2314yes—
46C2×D68—22no—
47C2×D68—42no—
48C4⋊C4×C17—24no—
49C4⋊C4×C17—24no—
50C22⋊Dic17—24no—
51#63227—28no—
52#63227—24no—
53#63227—44no—
54C4⋊Dic17—22no—
55C4⋊Dic17—42no—
56#63225—28no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17C2×D412yes—
60C23×C17C2×D412yes—
61C23×C17#4112yes—
62C23×C17C2411yescharacteristic
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—24no—
67C22×D17—24no—
68C2×C4×C17—28no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17—44no—
76C2×C4×C17—44no—
77C2×C4×C17C2×D414yes—
78C2×C4×C17C2×D412yes—
79C2×C4×C17C2×D412yes—
80C22⋊D17—48no—
81C2×Dic17—24no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—24no—
86C2×Dic17—24no—
87D68—44no—
88#2618—171nomaximal, Sylow
89C22×C17—22no—
90C22×C17—22no—
91C22×C17—24no—
92C22×C17—24no—
93C22×C17#12114yes—
94C22×C17C22×D411yescharacteristic
95C22×C17C22×D411yescharacteristic
96C22×C17C2×#4111yescharacteristic
97C22×C17H(5,𝔽2)12yes—
98C22×C172−1+412yes—
99C2×D17—42no—
100C2×D17—42no—
101C2×D17—42no—
102C2×D17—44no—
103C2×D17—44no—
104C4×C17—24no—
105C4×C17—22no—
106C4×C17—22no—
107C4×C17—42no—
108Dic17—42no—
109Dic17—44no—
110C2×#123—172no—
111C2×#121—171no—
112C22×C4⋊C4D1711yescharacteristic, complemented
113#398—172no—
114#396—174no—
115#394—172no—
116#386—172no—
117#380—171no—
118C2×C17—24no—
119C2×C17C2×#12111yescharacteristic
120C2×C17#53411yescharacteristic
121C2×C17#53611yescharacteristic
122C2×C17#53712yes—
123C2×C17#53912yes—
124D17—82no—
125C22×D4—172no—
126C22×D4—171no—
127C23×C4C2×D1711yescharacteristic
128C23×C4C2×D1711yescharacteristic
129C23×C4C2×D1711yescharacteristic
130#123—348no—
131#121—344no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—171no—
134C2×C4⋊C4—174no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4C2×D1712yes—
137C2×C4⋊C4C2×D1712yes—
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—174no—
141C2×C22⋊C4—174no—
142#95—172no—
143#95—174no—
144C17#261811yescharacteristic, complemented, Sylow
145C24—172no—
146C24C22×D1711yescharacteristic
147C2×D4—344no—
148C2×D4—344no—
149C2×D4—344no—
150C2×D4—342no—
151C2×D4—682no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—174no—
158C22×C4—172no—
159C22×C4D6814yescomplemented
160C22×C4C22⋊D1714yes—
161C22×C4C22×D1712yes—
162C22×C4C22×D1712yes—
163C22×C4C22×D1711yescharacteristic
164C22×C4C22×D1711yescharacteristic
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—348no—
168C4⋊C4—342no—
169C4⋊C4—682no—
170C22⋊C4—344no—
171C22⋊C4—348no—
172C22⋊C4—344no—
173C22⋊C4—684no—
174C23—22no—
175C23—22no—
176C23—342no—
177C23—342no—
178C23—344no—
179C23—344no—
180C23—342no—
181C23C2×D6812yes—
182C23#6325212yes—
183C23C2×C22⋊D1712yes—
184C23C23×D1711yescharacteristic
185D4—684no—
186D4—688no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—28no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—44no—
196C2×C4—344no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—344no—
200C2×C4—344no—
201C2×C4—342no—
202C2×C4C2×D6814yes—
203C2×C4C2×C22⋊D1712yes—
204C2×C4C2×C22⋊D1712yes—
205C22—22no—
206C22—22no—
207C22—24no—
208C22—24no—
209C22—682no—
210C22—682no—
211C22—682no—
212C22—684no—
213C22—684no—
214C22#1058761614yes—
215C22C22×D6811yescharacteristic
216C22C2×#6325211yescharacteristic
217C22#1058769112yes—
218C22#1058769412yes—
219C22C22×C22⋊D1711yescharacteristic
220C4—24no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—682no—
225C4—684no—
226C2—24no—
227C2—1362no—
228C2C2×#1058761611yescharacteristic
229C2#1176336811yescharacteristic
230C2#1176338311yescharacteristic
231C2#1176339012yes—
232C2#1176340612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C4⋊C4×C17C2
prime radicalC2×#10587616C2
hypercenterC22×C4⋊C4D17
hypocenterC17#2618
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
3. #11762739C2semidirect
4. #11762731C2semidirect
5. #11762726C2semidirect
6. #11762688C2semidirect
7. #11762669C2semidirect
8. C22×C4⋊C4×C17C2semidirect
36. C22×C4×C17D4semidirect
112. C22×C4⋊C4D17semidirect
144. C17#2618semidirect
159. C22×C4D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876162
#117627392
#117627312
#117627262
#117626882
#117626692
C22×C4⋊C4×C172
#261817