Finite Group Databasebeta

← #14536746

Subgroups — #14536746

Order 2176.

392 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 149 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#11762742C212yesmaximal
2#11762741C212yesmaximal
3#11762729C212yesmaximal
4#11762715C211yesmaximal, characteristic
5#11762715C211yesmaximal, characteristic
6#11762713C212yesmaximal
7#11762708C212yesmaximal
8#11762695C212yesmaximal
9C17×#389C211yesmaximal, characteristic
10C22×Dic34C2211yescharacteristic
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C42×C17C2211yescharacteristic
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×C4⋊Dic17C2212yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×C4×Dic17C2212yes—
23C17×#95C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2212yes—
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31C22×C4×C17C2312yes—
32C22×C4×C17C2311yescharacteristic
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2312yes—
35C22×C4×C17C2311yescharacteristic
36C22×Dic17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C2×Dic34—22no—
41C2×Dic34—41no—
42C2×Dic34—41no—
43C2×Dic34—41no—
44C2×Dic34—41no—
45C2×Dic34Q814yescomplemented
46C4⋊C4×C17—24no—
47C4⋊C4×C17—41no—
48C4⋊C4×C17Q814yescomplemented
49C42×C17—41no—
50C4⋊Dic17—42no—
51C4⋊Dic17—42no—
52C4⋊Dic17—41no—
53C4⋊Dic17—41no—
54#63225—42no—
55#63225—42no—
56C4×Dic17—42no—
57C23×C17C2411yescharacteristic
58C2×C4×C17—22no—
59C2×C4×C17—21no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—21no—
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×C17C2×Q812yes—
70C2×C4×C17C2×Q814yes—
71C2×C4×C17C2×Q812yes—
72C2×C4×C17#4112yes—
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×Dic17C2×Q814yes—
85Dic34—44no—
86Dic34—44no—
87#2806—171nomaximal, Sylow
88C22×C17C22×Q811yescharacteristic
89C22×C17C22×Q811yescharacteristic
90C22×C17C2×#4111yescharacteristic
91C22×C17H(5,𝔽2)11yescharacteristic
92C22×C17H(5,𝔽2)11yescharacteristic
93C22×C17H(5,𝔽2)11yescharacteristic
94C22×C172−1+411yescharacteristic
95C4×C17—22no—
96C4×C17—22no—
97C4×C17—22no—
98C4×C17—24no—
99C4×C17—42no—
100Dic17—24no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104#398—172no—
105#398—172no—
106#395—172no—
107#391—171no—
108#391—171no—
109#391—172no—
110#389—172no—
111#389D1711yescharacteristic
112#387—172no—
113C2×C17#53811yescharacteristic
114C2×C17#54311yescharacteristic
115C2×C17#54411yescharacteristic
116C2×C17#55411yescharacteristic
117C2×C17#55711yescharacteristic
118C2×C17Q8211yescharacteristic
119C2×C17#56211yescharacteristic
120C22×Q8—171no—
121C22×Q8—171no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—171no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4—171no—
127C2×C4⋊C4—172no—
128C2×C4⋊C4C2×D1712yes—
129C2×C4⋊C4C2×D1711yescharacteristic
130C2×C4⋊C4C2×D1711yescharacteristic
131C2×C42—172no—
132C2×C42C2×D1711yescharacteristic
133#95—172no—
134#95—172no—
135#95—172no—
136#95—172no—
137#95—172no—
138#95—172no—
139#95—172no—
140#95C2×D1712yes—
141C17#280611yescharacteristic, complemented, Sylow
142C2×Q8—174no—
143C2×Q8—342no—
144C2×Q8—681no—
145C2×Q8—681no—
146C2×Q8—681no—
147C2×Q8—681no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4C22×D1712yes—
153C22×C4C22×D1711yescharacteristic
154C22×C4C22×D1712yes—
155C22×C4C22×D1711yescharacteristic
156C22×C4C22×D1711yescharacteristic
157C4⋊C4—24no—
158C4⋊C4—41no—
159C4⋊C4—682no—
160C4⋊C4—682no—
161C4⋊C4—682no—
162C4⋊C4—681no—
163C4⋊C4—681no—
164C4⋊C4—682no—
165C4⋊C4Dic3414yescomplemented
166C42—41no—
167C42—682no—
168C23C23×D1711yescharacteristic
169Q8—684no—
170Q8—684no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—174no—
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×C4—342no—
194C2×C4C2×Dic3414yes—
195C2×C4C2×Dic3412yes—
196C2×C4#6325412yes—
197C2×C4Q8×D1712yes—
198C22C22×Dic3411yescharacteristic
199C22C2×#6325411yescharacteristic
200C22#1058769111yescharacteristic
201C22C2×Q8×D1711yescharacteristic
202C22#1058769411yescharacteristic
203C22#1058769611yescharacteristic
204C22#1058769611yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—344no—
211C4—682no—
212C4—682no—
213C4—682no—
214C2#1176339611yescharacteristic
215C2#1176343411yescharacteristic
216C2#1176344011yescharacteristic
217C2#1176351611yescharacteristic
218C2#1176353411yescharacteristic
219C2Q8×Dic3411yescharacteristic
220C2#1176355711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#389C2
hypercenter#389D17
hypocenterC17#2806
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
45. C2×Dic34Q8semidirect
48. C4⋊C4×C17Q8semidirect
141. C17#2806semidirect
165. C4⋊C4Dic34semidirect

Maximal subgroups

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

subgroupindex
#117627422
#117627412
#117627292
#117627152
#117627132
#117627082
#117626952
C17×#3892
#280617