Finite Group Databasebeta

← #14535690

Subgroups — #14535690

Order 2176.

448 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587565C212yesmaximal
2#11762741C212yesmaximal, complemented
3#11762739C212yesmaximal, complemented
4#11762734C211yesmaximal, characteristic
5#11762734C211yesmaximal, characteristic
6#11762708C212yesmaximal, complemented
7#11762694C212yesmaximal, complemented
8#11762669C212yesmaximal
9C17×#385C211yesmaximal, characteristic
10C23×C4×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2212yes—
12C2×C22⋊C4×C17C2211yescharacteristic
13C2×C22⋊C4×C17C2211yescharacteristic
14C2×C22⋊Dic17C2212yes—
15C2×C22⋊Dic17C2212yes—
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×C4×Dic17C2212yes—
24#10587565—28no—
25C17×#95C2212yes—
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C24×C17C2311yescharacteristic
33C22×C4×C17—24no—
34C22×C4×C17—41no—
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2312yes—
38C22×C4×C17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C4⋊C4×C17—24no—
44C22⋊C4×C17—41no—
45C22⋊C4×C17D414yes—
46C22⋊Dic17—24no—
47C22⋊Dic17—24no—
48C4⋊Dic17—24no—
49C4⋊Dic17—24no—
50C4⋊Dic17—42no—
51C4⋊Dic17—41no—
52C4⋊Dic17—41no—
53#63225—24no—
54#63225—24no—
55C4×Dic17—24no—
56C23×C17—22no—
57C23×C17—21no—
58C23×C17—21no—
59C23×C17—22no—
60C23×C17C2×D412yes—
61C23×C17C2411yescharacteristic
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—22no—
70C2×C4×C17—44no—
71C2×C4×C17C2×D414yes—
72C2×C4×C17#4114yes—
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—22no—
85#2722—171nomaximal, Sylow
86C22×C17—22no—
87C22×C17—24no—
88C22×C17—42no—
89C22×C17—41no—
90C22×C17—41no—
91C22×C17C22×D411yescharacteristic
92C22×C17C2×#4112yes—
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C172−1+411yescharacteristic
95C22×C172−1+412yes—
96C4×C17—24no—
97C4×C17—24no—
98C4×C17—41no—
99C4×C17—41no—
100Dic17—42no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104C2×#126—172no—
105#398—172no—
106#398—172no—
107#397—171no—
108#397—171no—
109#389—172no—
110#387—172no—
111#385D1711yescharacteristic
112#380—172no—
113C2×C17—42no—
114C2×C17#53711yescharacteristic
115C2×C17#54112yes—
116C2×C17#54712yes—
117C2×C17#55111yescharacteristic
118C2×C17#56311yescharacteristic
119C23×C4C2×D1711yescharacteristic
120#126—348no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—171no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4C2×D1712yes—
129C2×C22⋊C4—172no—
130C2×C22⋊C4—172no—
131C2×C22⋊C4C2×D1711yescharacteristic
132C2×C22⋊C4C2×D1711yescharacteristic
133C2×C42—172no—
134#95—172no—
135#95—172no—
136#95—171no—
137#95—172no—
138#95—172no—
139#95—171no—
140#95C2×D1712yes—
141C17#272211yescharacteristic, complemented, Sylow
142C24C22×D1711yescharacteristic
143C22×C4—24no—
144C22×C4—41no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4—172no—
149C22×C4C22×D1711yescharacteristic
150C22×C4C22×D1711yescharacteristic
151C22×C4C22×D1712yes—
152C22×C4C22×D1712yes—
153C4⋊C4—24no—
154C4⋊C4—344no—
155C4⋊C4—344no—
156C4⋊C4—344no—
157C4⋊C4—344no—
158C4⋊C4—682no—
159C4⋊C4—681no—
160C4⋊C4—681no—
161C22⋊C4—41no—
162C22⋊C4—344no—
163C22⋊C4—344no—
164C22⋊C4D6814yes—
165C42—344no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23—22no—
170C23C2×D6812yes—
171C23C23×D1711yescharacteristic
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—44no—
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—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4C2×D6814yes—
194C2×C4#6325414yes—
195C22—24no—
196C22—22no—
197C22—41no—
198C22—41no—
199C22—42no—
200C22C22×D6811yescharacteristic
201C22C2×#6325412yes—
202C22#1058769111yescharacteristic
203C22#1058769411yescharacteristic
204C22#1058769712yes—
205C4—24no—
206C4—24no—
207C4—41no—
208C4—41no—
209C4—682no—
210C4—682no—
211C4—682no—
212C4—682no—
213C2—42no—
214C2#1176339011yescharacteristic
215C2#1176341912yes—
216C2#1176346312yes—
217C2#1176349011yescharacteristic
218C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2722
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #11762741C2semidirect
3. #11762739C2semidirect
6. #11762708C2semidirect
7. #11762694C2semidirect
141. C17#2722semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
#117627412
#117627392
#117627342
#117627082
#117626942
#117626692
C17×#3852
#272217