Finite Group Databasebeta

← #14536491

Subgroups — #14536491

Order 2176.

428 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 121 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#11762729C211yesmaximal, characteristic, complemented
2#11762723C212yesmaximal
3#11762721C212yesmaximal, complemented
4#11762703C212yesmaximal, complemented
5#11762702C212yesmaximal, complemented
6#11762697C212yesmaximal, complemented
7#11762674C211yesmaximal, characteristic, complemented
8#11762673C212yesmaximal, complemented
9C17×#394C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C2×C4×Dic17C2212yes—
24C17×#95C2211yescharacteristic
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2312yes—
32C2×D4×C17—41no—
33C2×D4×C17—42no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×Dic34—41no—
43C2×Dic34—42no—
44C4⋊C4×C17—41no—
45C22⋊C4×C17—24no—
46C22⋊C4×C17—42no—
47C22⋊Dic17—24no—
48C22⋊Dic17—24no—
49C22⋊Dic17—42no—
50C22⋊Dic17—42no—
51C4⋊Dic17—41no—
52#63225—42no—
53#63225—42no—
54C4×Dic17—22no—
55C4×Dic17—24no—
56C23×C17—22no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17C2411yescharacteristic
64D4×C17—44no—
65C2×C4×C17—21no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17#4112yes—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17#4114yes—
87Dic34—44no—
88#2785—171nomaximal, Sylow
89C22×C17—42no—
90C22×C17—42no—
91C22×C17—42no—
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17C2×#4111yescharacteristic
97C22×C17C2×#4112yes—
98C22×C17H(5,𝔽2)11yescharacteristic
99C22×C17H(5,𝔽2)12yes—
100C22×C172−1+411yescharacteristic
101C4×C17—22no—
102C4×C17—42no—
103C4×C17—41no—
104C4×C17—41no—
105Dic17—24no—
106Dic17—42no—
107Dic17—42no—
108Dic17—42no—
109#395—171no—
110#394—172no—
111#394D1711yescharacteristic
112#393—172no—
113#388—172no—
114#388—172no—
115#388—172no—
116#382—171no—
117#382—172no—
118C2×C17—82no—
119C2×C17#53912yes—
120C2×C17#53911yescharacteristic
121C2×C17#55511yescharacteristic
122C2×C17#55512yes—
123C2×C17H(3,𝔽4)11yescharacteristic
124C22×Q8—171no—
125C22×D4C2×D1711yescharacteristic
126C2×C4⋊C4—172no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C22⋊C4—172no—
131C2×C22⋊C4—172no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4C2×D1712yes—
135C2×C22⋊C4C2×D1712yes—
136C2×C42—172no—
137C2×C42—171no—
138#95—172no—
139#95—172no—
140#95—172no—
141#95—172no—
142#95—171no—
143#95—172no—
144#95C2×D1711yescharacteristic
145C17#278511yescharacteristic, complemented, Sylow
146C24C22×D1712yes—
147C2×Q8—682no—
148C2×Q8—681no—
149C2×D4—41no—
150C2×D4—42no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—172no—
154C22×C4—172no—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1712yes—
157C22×C4C22×D1711yescharacteristic
158C22×C4C22×D1711yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—682no—
161C4⋊C4—682no—
162C4⋊C4—681no—
163C22⋊C4—24no—
164C22⋊C4—42no—
165C22⋊C4—344no—
166C22⋊C4—344no—
167C22⋊C4—682no—
168C22⋊C4—682no—
169C42—344no—
170C42—342no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23C23×D1711yescharacteristic
179Q8—684no—
180D4—44no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—174no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4#6325412yes—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#6325411yescharacteristic
211C22#1058769111yescharacteristic
212C22#1058769112yes—
213C22#41×D1712yes—
214C22#1058769711yescharacteristic
215C4—22no—
216C4—42no—
217C4—41no—
218C4—41no—
219C4—344no—
220C4—682no—
221C4—682no—
222C4—682no—
223C2—82no—
224C2#1176340212yes—
225C2#1176340411yescharacteristic
226C2#1176352111yescharacteristic
227C2#1176352312yes—
228C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2785
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #11762729C2semidirect
3. #11762721C2semidirect
4. #11762703C2semidirect
5. #11762702C2semidirect
6. #11762697C2semidirect
7. #11762674C2semidirect
8. #11762673C2semidirect
145. C17#2785semidirect

Maximal subgroups

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

subgroupindex
#117627292
#117627232
#117627212
#117627032
#117627022
#117626972
#117626742
#117626732
C17×#3942
#278517