Finite Group Databasebeta

← #14534027

Subgroups — #14534027

Order 2176.

476 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 129 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#11762721C212yesmaximal, complemented
3#11762714C211yesmaximal, characteristic, complemented
4#11762702C212yesmaximal, complemented
5#11762689C212yesmaximal
6#11762676C212yesmaximal, complemented
7#11762670C212yesmaximal, complemented
8C22⋊C4×Dic17C212yesmaximal, complemented
9C17×#394C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C23×Dic17C2212yes—
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4×Dic17C2212yes—
21C2×C4×Dic17C2211yescharacteristic
22C17×#95C2211yescharacteristic
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31C24×C17C2312yes—
32C2×D4×C17—24no—
33C2×D4×C17—41no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17—24no—
39C22×Dic17—24no—
40C22×Dic17—42no—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C2×Dic34—22no—
46C2×Dic34—41no—
47C2×Dic34—41no—
48C4⋊C4×C17—41no—
49C22⋊C4×C17—24no—
50C22⋊C4×C17—24no—
51C22⋊Dic17—24no—
52C22⋊Dic17—42no—
53C22⋊Dic17—42no—
54C4⋊Dic17—41no—
55C4×Dic17—24no—
56C4×Dic17—41no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17#4114yes—
64C23×C17C2411yescharacteristic
65D4×C17—44no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17#4112yes—
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—
88C2×Dic17#4114yes—
89Dic34—44no—
90#2572—171nomaximal, Sylow
91C22×C17—24no—
92C22×C17—24no—
93C22×C17—24no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17C2×#4111yescharacteristic
99C22×C17C2×#4112yes—
100C22×C17C2×#4112yes—
101C22×C17H(5,𝔽2)11yescharacteristic
102C22×C17H(5,𝔽2)11yescharacteristic
103C4×C17—22no—
104C4×C17—42no—
105C4×C17—41no—
106C4×C17—41no—
107Dic17—24no—
108Dic17—42no—
109Dic17—42no—
110Dic17—42no—
111#400—171no—
112#394D1711yescharacteristic
113#393—172no—
114#391—171no—
115#388—172no—
116#386—172no—
117#383—172no—
118#381—172no—
119C4×C22⋊C4—172no—
120C2×C17—44no—
121C2×C17#52912yes—
122C2×C17#53811yescharacteristic
123C2×C17#55112yes—
124C2×C17#55111yescharacteristic
125C2×C17#55511yescharacteristic
126C22×Q8—171no—
127C22×D4C2×D1711yescharacteristic
128C23×C4—172no—
129C2×C4⋊C4—171no—
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—172no—
137C2×C42—171no—
138#95—172no—
139#95—172no—
140#95—171no—
141#95—172no—
142#95—172no—
143#95—172no—
144#95—172no—
145#95—172no—
146#95C2×D1711yescharacteristic
147C17#257211yescharacteristic, complemented, Sylow
148C24C22×D1712yes—
149C2×Q8—342no—
150C2×Q8—681no—
151C2×Q8—681no—
152C2×D4—24no—
153C2×D4—41no—
154C22×C4—172no—
155C22×C4—172no—
156C22×C4—172no—
157C22×C4—172no—
158C22×C4—344no—
159C22×C4—344no—
160C22×C4—682no—
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1712yes—
163C22×C4C22×D1711yescharacteristic
164C22×C4C22×D1711yescharacteristic
165C4⋊C4—41no—
166C4⋊C4—681no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—344no—
170C22⋊C4—682no—
171C22⋊C4—682no—
172C42—344no—
173C42—681no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23#6325414yes—
181C23C23×D1711yescharacteristic
182Q8—684no—
183D4—44no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—174no—
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—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—688no—
206C2×C4#6325412yes—
207C22—24no—
208C22—24no—
209C22—24no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×#6325412yes—
215C22C2×#6325411yescharacteristic
216C22#1058769111yescharacteristic
217C22#41×D1712yes—
218C22#1058769611yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—344no—
224C4—682no—
225C4—682no—
226C4—682no—
227C2—44no—
228C2#1176334012yes—
229C2#1176339611yescharacteristic
230C2#1176348212yes—
231C2#1176349011yescharacteristic
232C2#1176352111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2572
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #11762748C2semidirect
2. #11762721C2semidirect
3. #11762714C2semidirect
4. #11762702C2semidirect
6. #11762676C2semidirect
7. #11762670C2semidirect
8. C22⋊C4×Dic17C2semidirect
147. C17#2572semidirect

Maximal subgroups

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

subgroupindex
#117627482
#117627212
#117627142
#117627022
#117626892
#117626762
#117626702
C22⋊C4×Dic172
C17×#3942
#257217