Finite Group Databasebeta

← #14536462

Subgroups — #14536462

Order 2176.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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#11762749C212yesmaximal, complemented
2#11762748C211yesmaximal, characteristic, complemented
3#11762737C212yesmaximal, complemented
4#11762697C212yesmaximal, complemented
5#11762689C212yesmaximal
6#11762684C212yesmaximal, complemented
7#11762684C212yesmaximal, complemented
8#11762674C211yesmaximal, characteristic, complemented
9C17×#394C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23C17×#95C2211yescharacteristic
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2212yes—
30#10587502C2212yes—
31C24×C17C2312yes—
32C2×D4×C17—24no—
33C2×D4×C17—41no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—42no—
39C22×Dic17—42no—
40C22×Dic17—42no—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C4⋊C4×C17—41no—
46C22⋊C4×C17—24no—
47C22⋊C4×C17—24no—
48C22⋊Dic17—24no—
49C22⋊Dic17—42no—
50C22⋊Dic17—42no—
51C4⋊Dic17—41no—
52C4⋊Dic17—41no—
53C4⋊Dic17—42no—
54#63225—42no—
55C4×Dic17—22no—
56C23×C17—22no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17#4114yes—
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—22no—
87C2×Dic17—48no—
88#2782—171nomaximal, Sylow
89C22×C17—24no—
90C22×C17—24no—
91C22×C17—24no—
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17C2×#4112yes—
97C22×C17C2×#4111yescharacteristic
98C22×C17H(5,𝔽2)11yescharacteristic
99C22×C17H(5,𝔽2)12yes—
100C22×C172−1+411yescharacteristic
101C4×C17—22no—
102C4×C17—41no—
103C4×C17—41no—
104C4×C17—42no—
105Dic17—42no—
106Dic17—42no—
107Dic17—42no—
108Dic17—42no—
109#400—171no—
110#400—172no—
111#397—172no—
112#394D1711yescharacteristic
113#388—172no—
114#386—172no—
115#385—172no—
116#385—172no—
117#382—171no—
118C2×C17—44no—
119C2×C17#53912yes—
120C2×C17#53911yescharacteristic
121C2×C17#55111yescharacteristic
122C2×C17#55312yes—
123C2×C17H(3,𝔽4)11yescharacteristic
124C22×D4C2×D1711yescharacteristic
125C23×C4—172no—
126C2×C4⋊C4—171no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4C2×D1711yescharacteristic
131C2×C22⋊C4—172no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4—172no—
134C2×C22⋊C4C2×D1712yes—
135C2×C22⋊C4C2×D1712yes—
136C2×C42—171no—
137#95—172no—
138#95—172no—
139#95—172no—
140#95—172no—
141#95—172no—
142#95—172no—
143#95—171no—
144#95C2×D1711yescharacteristic
145C17#278211yescharacteristic, complemented, Sylow
146C24C22×D1712yes—
147C2×D4—24no—
148C2×D4—41no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—682no—
154C22×C4—682no—
155C22×C4—682no—
156C22×C4C22×D1711yescharacteristic
157C22×C4C22×D1711yescharacteristic
158C22×C4C22×D1712yes—
159C22×C4C22×D1711yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—682no—
162C4⋊C4—681no—
163C4⋊C4—681no—
164C4⋊C4—682no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—344no—
168C22⋊C4—682no—
169C22⋊C4—682no—
170C42—342no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23#6325414yes—
178C23C23×D1711yescharacteristic
179D4—44no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—342no—
190C2×C4—342no—
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—688no—
202C2×C4#6325412yes—
203C22—24no—
204C22—24no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#6325412yes—
211C22C2×#6325411yescharacteristic
212C22#1058769111yescharacteristic
213C22#1058769612yes—
214C22#1058769711yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—682no—
220C4—682no—
221C4—682no—
222C4—682no—
223C2—44no—
224C2#1176340312yes—
225C2#1176340411yescharacteristic
226C2#1176349011yescharacteristic
227C2#1176351112yes—
228C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2782
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #11762749C2semidirect
2. #11762748C2semidirect
3. #11762737C2semidirect
4. #11762697C2semidirect
6. #11762684C2semidirect
7. #11762684C2semidirect
8. #11762674C2semidirect
145. C17#2782semidirect

Maximal subgroups

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

subgroupindex
#117627492
#117627482
#117627372
#117626972
#117626892
#117626842
#117626742
C17×#3942
#278217