Finite Group Databasebeta

← #14533247

Subgroups — #14533247

Order 2176.

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

normal subgroups only

This group has 568 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#11762723C214yesmaximal, complemented
3#11762712C211yesmaximal, characteristic, complemented
4#11762712C211yesmaximal, characteristic, complemented
5#11762697C214yesmaximal, complemented
6#11762678C211yesmaximal, characteristic, complemented
7#11762674C211yesmaximal, characteristic, complemented
8C4×C22⋊Dic17C211yesmaximal, characteristic, complemented
9C2×C17×#121C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C22×D4×C17C2212yes—
12C23×C4×C17C2211yescharacteristic
13C17×#121—24no—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2214yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2214yes—
20#10587612—24no—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2214yes—
24C2×C4×Dic17C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2214yes—
29C24×C17C2312yes—
30C24×C17C2311yescharacteristic
31C2×D4×C17—24no—
32C2×D4×C17—22no—
33C2×D4×C17—22no—
34C2×D4×C17—22no—
35C2×D4×C17—44no—
36C22×C4×C17—22no—
37C22×C4×C17—22no—
38C22×C4×C17D414yes—
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C4⋊C4×C17—22no—
47C22⋊C4×C17—24no—
48C22⋊Dic17—24no—
49C22⋊Dic17—24no—
50C22⋊Dic17—28no—
51C22⋊Dic17—44no—
52C4⋊Dic17—22no—
53C4⋊Dic17—22no—
54#63225—44no—
55C4×Dic17—24no—
56C4×Dic17—22no—
57C4×Dic17—22no—
58C23×C17—22no—
59C23×C17—21no—
60C23×C17—21no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—24no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—24no—
67C23×C17C2×D412yes—
68C23×C17C2411yescharacteristic
69D4×C17—48no—
70D4×C17—44no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—28no—
74C2×C4×C17—22no—
75C2×C4×C17—24no—
76C2×C4×C17C2×D412yes—
77C2×C4×C17C2×D412yes—
78C2×C4×C17#4112yes—
79C2×C4×C17#4112yes—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—24no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—24no—
89C2×Dic17#4114yes—
90#2500—171nomaximal, Sylow
91C22×C17—22no—
92C22×C17—22no—
93C22×C17—22no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17—42no—
99C22×C17—44no—
100C22×C17—44no—
101C22×C17C22×D411yescharacteristic
102C22×C17C2×#4111yescharacteristic
103C22×C17C2×#4111yescharacteristic
104C22×C17C2×#4112yes—
105C22×C17H(5,𝔽2)12yes—
106C4×C17—22no—
107C4×C17—42no—
108C4×C17#12314yes—
109Dic17—24no—
110Dic17—42no—
111Dic17—42no—
112Dic17—42no—
113C2×#121D1711yescharacteristic
114C2×#117—171no—
115#394—174no—
116#390—171no—
117#390—171no—
118#388—174no—
119#384—171no—
120#382—171no—
121C4×C22⋊C4—171no—
122C2×C17—42no—
123C2×C17—82no—
124C2×C17C2×#12311yescharacteristic
125C2×C17#53211yescharacteristic
126C2×C17#53511yescharacteristic
127C2×C17#55312yes—
128C2×C17#55512yes—
129C22×D4C2×D1711yescharacteristic
130C22×D4C2×D1712yes—
131C23×C4C2×D1711yescharacteristic
132#121—24no—
133#117—344no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4—171no—
136C2×C4⋊C4—174no—
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—174no—
141C2×C22⋊C4—174no—
142C2×C22⋊C4C2×D1712yes—
143C2×C42—172no—
144C2×C42—171no—
145C2×C42—171no—
146#95—172no—
147#95—174no—
148C17#250011yescharacteristic, complemented, Sylow
149C24C22×D1712yes—
150C24C22×D1711yescharacteristic
151C2×D4—24no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—44no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—172no—
161C22×C4—172no—
162C22×C4C22⋊D1714yes—
163C22×C4C22×D1712yes—
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1711yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—342no—
168C4⋊C4—342no—
169C4⋊C4—684no—
170C22⋊C4—24no—
171C22⋊C4—344no—
172C22⋊C4—348no—
173C22⋊C4—344no—
174C22⋊C4—684no—
175C42—342no—
176C42—342no—
177C42—344no—
178C23—21no—
179C23—21no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—24no—
185C23—24no—
186C23—22no—
187C23C2×C22⋊D1712yes—
188C23C23×D1711yescharacteristic
189D4—48no—
190D4—44no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—174no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—344no—
202C2×C4—342no—
203C2×C4—344no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4#6325412yes—
207C2×C4#6325412yes—
208C2×C4C2×C22⋊D1712yes—
209C2×C4C2×C22⋊D1712yes—
210C22—22no—
211C22—22no—
212C22—22no—
213C22—42no—
214C22—42no—
215C22—44no—
216C22—44no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22C2×#6325411yescharacteristic
221C22C2×#6325411yescharacteristic
222C22#1058769112yes—
223C22#41×D1712yes—
224C22C22×C22⋊D1711yescharacteristic
225C4—22no—
226C4—42no—
227C4—344no—
228C4—682no—
229C4—682no—
230C4—682no—
231C4#1058762114yes—
232C2—42no—
233C2—82no—
234C2C2×#1058762111yescharacteristic
235C2#1176335511yescharacteristic
236C2#1176337611yescharacteristic
237C2#1176350612yes—
238C2#1176352312yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
Fitting, prime radicalC2×C17×#121C2
hypercenterC2×#121D17
hypocenterC17#2500
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#10587612C2semidirect
2. #11762723C2semidirect
3. #11762712C2semidirect
4. #11762712C2semidirect
5. #11762697C2semidirect
6. #11762678C2semidirect
7. #11762674C2semidirect
8. C4×C22⋊Dic17C2semidirect
148. C17#2500semidirect

Maximal subgroups

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

subgroupindex
C2×#105876122
#117627232
#117627122
#117626972
#117626782
#117626742
C4×C22⋊Dic172
C2×C17×#1212
#250017