Finite Group Databasebeta

← #14533516

Subgroups — #14533516

Order 2176.

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

normal subgroups only

This group has 420 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#11762737C212yesmaximal
2#11762734C211yesmaximal, characteristic
3#11762721C211yesmaximal, characteristic
4#11762713C212yesmaximal, complemented
5#11762695C212yesmaximal, complemented
6#11762690C212yesmaximal, complemented
7#11762676C212yesmaximal, complemented
8C4×C22⋊Dic17C212yesmaximal
9C17×#385C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2212yes—
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×#63225C2212yes—
19C2×C4×Dic17C2212yes—
20C2×C4×Dic17C2212yes—
21C17×#95C2212yes—
22#10587502C2211yescharacteristic
23#10587502C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31C24×C17C2311yescharacteristic
32C22×C4×C17—24no—
33C22×C4×C17—41no—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×Dic34—24no—
43C2×Dic34—41no—
44C4⋊C4×C17—24no—
45C22⋊C4×C17—22no—
46C22⋊C4×C17—41no—
47C22⋊Dic17—24no—
48C22⋊Dic17—24no—
49C4⋊Dic17—41no—
50#63225—24no—
51C4×Dic17—24no—
52C4×Dic17—42no—
53C23×C17—22no—
54C23×C17—22no—
55C23×C17—21no—
56C23×C17—21no—
57C23×C17#4112yes—
58C23×C17C2411yescharacteristic
59C2×C4×C17—22no—
60C2×C4×C17—22no—
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—21no—
68C2×C4×C17—44no—
69C2×C4×C17#4114yes—
70C2×Dic17—22no—
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#4114yes—
82Dic34—44no—
83#2523—171nomaximal, Sylow
84C22×C17—24no—
85C22×C17—22no—
86C22×C17—41no—
87C22×C17—41no—
88C22×C17—42no—
89C22×C17C2×#4111yescharacteristic
90C22×C17C2×#4112yes—
91C22×C17C2×#4112yes—
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C172−1+411yescharacteristic
94C4×C17—24no—
95C4×C17—42no—
96C4×C17—41no—
97C4×C17—41no—
98Dic17—24no—
99Dic17—42no—
100Dic17—42no—
101Dic17—42no—
102#397—171no—
103#397—172no—
104#393—171no—
105#391—172no—
106#387—172no—
107#387—172no—
108#385D1711yescharacteristic
109#383—172no—
110C4×C22⋊C4—172no—
111C2×C17—42no—
112C2×C17#52912yes—
113C2×C17#53911yescharacteristic
114C2×C17#55111yescharacteristic
115C2×C17#55111yescharacteristic
116C2×C17#55612yes—
117C22×Q8—171no—
118C23×C4C2×D1711yescharacteristic
119C2×C4⋊C4—172no—
120C2×C4⋊C4—171no—
121C2×C4⋊C4C2×D1712yes—
122C2×C22⋊C4—172no—
123C2×C22⋊C4—172no—
124C2×C22⋊C4C2×D1711yescharacteristic
125C2×C22⋊C4C2×D1711yescharacteristic
126C2×C42—172no—
127C2×C42—172no—
128#95—172no—
129#95—172no—
130#95—171no—
131#95—172no—
132#95—172no—
133#95—172no—
134#95—172no—
135#95—171no—
136#95—172no—
137#95C2×D1712yes—
138C17#252311yescharacteristic, complemented, Sylow
139C24C22×D1711yescharacteristic
140C2×Q8—344no—
141C2×Q8—681no—
142C22×C4—24no—
143C22×C4—41no—
144C22×C4—172no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4C22×D1711yescharacteristic
149C22×C4C22×D1712yes—
150C22×C4C22×D1711yescharacteristic
151C22×C4C22×D1712yes—
152C4⋊C4—24no—
153C4⋊C4—344no—
154C4⋊C4—681no—
155C22⋊C4—22no—
156C22⋊C4—41no—
157C22⋊C4—344no—
158C22⋊C4—344no—
159C42—344no—
160C42—682no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23—22no—
165C23#6325212yes—
166C23C23×D1711yescharacteristic
167Q8—684no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—174no—
179C2×C4—342no—
180C2×C4—342no—
181C2×C4—342no—
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#6325414yes—
191C22—24no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22C2×#6325211yescharacteristic
197C22C2×#6325412yes—
198C22#1058769111yescharacteristic
199C22#1058769411yescharacteristic
200C22#41×D1712yes—
201C4—24no—
202C4—41no—
203C4—41no—
204C4—42no—
205C4—344no—
206C4—682no—
207C4—682no—
208C4—682no—
209C2—42no—
210C2#1176333912yes—
211C2#1176340611yescharacteristic
212C2#1176348711yescharacteristic
213C2#1176349011yescharacteristic
214C2#1176353012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2523
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #11762713C2semidirect
5. #11762695C2semidirect
6. #11762690C2semidirect
7. #11762676C2semidirect
138. C17#2523semidirect

Maximal subgroups

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

subgroupindex
#117627372
#117627342
#117627212
#117627132
#117626952
#117626902
#117626762
C4×C22⋊Dic172
C17×#3852
#252317