Finite Group Databasebeta

← #14536207

Subgroups — #14536207

Order 2176.

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

normal subgroups only

This group has 400 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#11762750C212yesmaximal, complemented
2#11762743C212yesmaximal, complemented
3#11762734C211yesmaximal, characteristic
4#11762721C211yesmaximal, characteristic
5#11762713C212yesmaximal, complemented
6#11762703C212yesmaximal
7#11762682C212yesmaximal
8#11762676C212yesmaximal, complemented
9C17×#381C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C22⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C4⋊Dic17C2211yescharacteristic
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×C4×Dic17C2212yes—
21C17×#95C2212yes—
22C17×#95C2212yes—
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2211yescharacteristic
29#10587502C2211yescharacteristic
30#10587502C2212yes—
31C24×C17C2311yescharacteristic
32C22×C4×C17—22no—
33C22×C4×C17—41no—
34C22×C4×C17—41no—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C2×Dic34—24no—
44C2×Dic34—41no—
45C22⋊C4×C17—41no—
46C22⋊C4×C17—41no—
47C22⋊Dic17—24no—
48C22⋊Dic17—24no—
49C4⋊Dic17—41no—
50#63225—42no—
51#63225—42no—
52#63225—42no—
53C4×Dic17—24no—
54C23×C17—22no—
55C23×C17—21no—
56C23×C17—21no—
57C23×C17—22no—
58C23×C17#4112yes—
59C23×C17C2411yescharacteristic
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—21no—
64C2×C4×C17—22no—
65C2×C4×C17—21no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—44no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17#4114yes—
83Dic34—44no—
84#2764—171nomaximal, Sylow
85C22×C17—22no—
86C22×C17—24no—
87C22×C17—42no—
88C22×C17—41no—
89C22×C17—41no—
90C22×C17C2×#4111yescharacteristic
91C22×C17C2×#4112yes—
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C172−1+412yes—
95C4×C17—42no—
96C4×C17—42no—
97C4×C17—41no—
98C4×C17—41no—
99Dic17—24no—
100Dic17—42no—
101Dic17—42no—
102Dic17—42no—
103#400—172no—
104#398—172no—
105#397—171no—
106#393—171no—
107#391—172no—
108#388—172no—
109#385—172no—
110#383—172no—
111#381D1711yescharacteristic
112C2×C17—42no—
113C2×C17#53811yescharacteristic
114C2×C17#54212yes—
115C2×C17#55111yescharacteristic
116C2×C17#55212yes—
117C2×C17#56311yescharacteristic
118C22×Q8—171no—
119C23×C4C2×D1711yescharacteristic
120C2×C4⋊C4—172no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—171no—
124C2×C22⋊C4—172no—
125C2×C22⋊C4—172no—
126C2×C22⋊C4C2×D1711yescharacteristic
127C2×C22⋊C4C2×D1711yescharacteristic
128C2×C42—172no—
129#95—171no—
130#95—172no—
131#95—172no—
132#95—172no—
133#95—172no—
134#95—172no—
135#95—172no—
136#95—171no—
137#95C2×D1712yes—
138#95C2×D1712yes—
139C17#276411yescharacteristic, complemented, Sylow
140C24C22×D1711yescharacteristic
141C2×Q8—344no—
142C2×Q8—681no—
143C22×C4—22no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4C22×D1711yescharacteristic
151C22×C4C22×D1711yescharacteristic
152C22×C4C22×D1712yes—
153C22×C4C22×D1712yes—
154C4⋊C4—682no—
155C4⋊C4—682no—
156C4⋊C4—682no—
157C4⋊C4—681no—
158C22⋊C4—41no—
159C22⋊C4—41no—
160C22⋊C4—344no—
161C22⋊C4—344no—
162C42—344no—
163C23—21no—
164C23—21no—
165C23—22no—
166C23—22no—
167C23#6325212yes—
168C23C23×D1711yescharacteristic
169Q8—684no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—174no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C22—24no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—42no—
198C22C2×#6325211yescharacteristic
199C22#1058769111yescharacteristic
200C22#1058769111yescharacteristic
201C22#41×D1712yes—
202C22#1058769712yes—
203C4—41no—
204C4—41no—
205C4—42no—
206C4—42no—
207C4—344no—
208C4—682no—
209C4—682no—
210C4—682no—
211C2—42no—
212C2#1176339711yescharacteristic
213C2#1176342712yes—
214C2#1176348711yescharacteristic
215C2#1176349212yes—
216C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC24×C17C23
hypercenter#381D17
hypocenterC17#2764
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. #11762750C2semidirect
2. #11762743C2semidirect
5. #11762713C2semidirect
8. #11762676C2semidirect
139. C17#2764semidirect

Maximal subgroups

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

subgroupindex
#117627502
#117627432
#117627342
#117627212
#117627132
#117627032
#117626822
#117626762
C17×#3812
#276417