Finite Group Databasebeta

← #14536822

Subgroups — #14536822

Order 2176.

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

normal subgroups only

This group has 444 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
1C2×#10587564C212yesmaximal
2#11762751C212yesmaximal, complemented
3#11762750C212yesmaximal, complemented
4#11762737C212yesmaximal
5#11762734C211yesmaximal, characteristic
6#11762730C212yesmaximal, complemented
7#11762729C212yesmaximal, complemented
8#11762670C211yesmaximal, characteristic
9C17×#397C211yesmaximal, characteristic
10C23×Dic17C2211yescharacteristic
11C22×Dic34C2212yes—
12C2×C4⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊C4×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22#10587564—28no—
23C17×#95C2212yes—
24C17×#95C2211yescharacteristic
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C24×C17C2311yescharacteristic
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×Dic17—41no—
38C22×Dic17—41no—
39C22×Dic17D414yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×Dic34—24no—
45C2×Dic34—42no—
46C2×Dic34—42no—
47C4⋊C4×C17—41no—
48C22⋊C4×C17—24no—
49C22⋊C4×C17—41no—
50C22⋊Dic17—24no—
51C22⋊Dic17—41no—
52C4⋊Dic17—24no—
53C4⋊Dic17—41no—
54#63225—24no—
55#63225—24no—
56#63225—42no—
57C23×C17—21no—
58C23×C17—21no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17C2×D412yes—
62C23×C17C2411yescharacteristic
63C2×C4×C17—22no—
64C2×C4×C17—22no—
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—22no—
72C2×C4×C17—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—22no—
83C2×Dic17—22no—
84C2×Dic17—44no—
85C2×Dic17C2×D414yes—
86Dic34—48no—
87#2817—171nomaximal, Sylow
88C22×C17—22no—
89C22×C17—24no—
90C22×C17—41no—
91C22×C17—42no—
92C22×C17—41no—
93C22×C17C22×D411yescharacteristic
94C22×C17H(5,𝔽2)11yescharacteristic
95C22×C17H(5,𝔽2)11yescharacteristic
96C22×C172−1+412yes—
97C22×C172−1+412yes—
98C4×C17—42no—
99C4×C17—42no—
100C4×C17—41no—
101C4×C17—41no—
102Dic17—24no—
103Dic17—42no—
104Dic17—42no—
105Dic17—42no—
106C2×C22⋊Q8—172no—
107#400—172no—
108#400—172no—
109#397—171no—
110#397—172no—
111#397D1711yescharacteristic
112#395—172no—
113#395—172no—
114#381—171no—
115C2×C17—42no—
116C2×C17#53411yescharacteristic
117C2×C17#53612yes—
118C2×C17#56311yescharacteristic
119C2×C17#56312yes—
120C2×C17#56311yescharacteristic
121C22×Q8—172no—
122C23×C4—171no—
123C22⋊Q8—348no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—172no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4—171no—
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C22⋊C4—172no—
131C2×C22⋊C4—171no—
132C2×C22⋊C4C2×D1712yes—
133C2×C22⋊C4C2×D1711yescharacteristic
134#95—172no—
135#95—172no—
136#95—172no—
137#95—172no—
138#95—172no—
139#95—172no—
140#95—171no—
141#95C2×D1712yes—
142#95C2×D1711yescharacteristic
143C17#281711yescharacteristic, complemented, Sylow
144C24C22×D1711yescharacteristic
145C2×Q8—344no—
146C2×Q8—682no—
147C2×Q8—682no—
148C22×C4—172no—
149C22×C4—174no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—172no—
153C22×C4—681no—
154C22×C4—681no—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1711yescharacteristic
157C22×C4C22×D1712yes—
158C22×C4C22×D1712yes—
159C4⋊C4—41no—
160C4⋊C4—344no—
161C4⋊C4—344no—
162C4⋊C4—344no—
163C4⋊C4—682no—
164C4⋊C4—681no—
165C22⋊C4—24no—
166C22⋊C4—41no—
167C22⋊C4—344no—
168C22⋊C4—681no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23D4×D1712yes—
174C23C23×D1711yescharacteristic
175Q8—688no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—174no—
187C2×C4—342no—
188C2×C4—342no—
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—684no—
199C22—22no—
200C22—24no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22C2×D4×D1711yescharacteristic
205C22#1058769111yescharacteristic
206C22#1058769611yescharacteristic
207C22#1058769712yes—
208C22#1058769712yes—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—42no—
213C4—344no—
214C4—682no—
215C4—682no—
216C4—682no—
217C2—42no—
218C2#1176336711yescharacteristic
219C2#1176338112yes—
220C2#1176355911yescharacteristic
221C2#1176356112yes—
222C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#397C2
prime radicalC24×C17C23
hypercenter#397D17
hypocenterC17#2817
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #11762751C2semidirect
3. #11762750C2semidirect
6. #11762730C2semidirect
7. #11762729C2semidirect
143. C17#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#105875642
#117627512
#117627502
#117627372
#117627342
#117627302
#117627292
#117626702
C17×#3972
#281717