Finite Group Databasebeta

← #14535813

Subgroups — #14535813

Order 2176.

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

normal subgroups only

This group has 456 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
1C2×#10587574C212yesmaximal
2#11762739C211yesmaximal, characteristic
3#11762730C212yesmaximal
4#11762729C212yesmaximal
5#11762714C211yesmaximal, characteristic
6#11762704C212yesmaximal
7#11762696C212yesmaximal
8#11762691C212yesmaximal
9C17×#398C211yesmaximal, characteristic
10C22×Dic34C2212yes—
11C22×Dic34C2211yescharacteristic
12C2×C4⋊C4×C17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊Dic17C2212yes—
16C2×C4⋊Dic17C2211yescharacteristic
17C2×#63225C2212yes—
18C2×#63225C2212yes—
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×C4×Dic17C2212yes—
22C2×C4×Dic17C2211yescharacteristic
23#10587574—28no—
24C17×#95C2212yes—
25C17×#95C2211yescharacteristic
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C22×C4×C17C2312yes—
33C22×C4×C17C2311yescharacteristic
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C22×C4×C17C2312yes—
37C22×Dic17C2312yes—
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C2×Dic34—24no—
42C2×Dic34—41no—
43C2×Dic34—41no—
44C2×Dic34—42no—
45C2×Dic34—42no—
46C2×Dic34D414yes—
47C4⋊C4×C17—24no—
48C4⋊C4×C17—41no—
49C4⋊C4×C17—41no—
50C4⋊Dic17—24no—
51C4⋊Dic17—41no—
52#63225—24no—
53#63225—24no—
54#63225—42no—
55#63225Q818yescomplemented
56C4×Dic17—24no—
57C4×Dic17—41no—
58C23×C17C2411yescharacteristic
59C2×C4×C17—22no—
60C2×C4×C17—21no—
61C2×C4×C17—21no—
62C2×C4×C17—22no—
63C2×C4×C17—22no—
64C2×C4×C17—21no—
65C2×C4×C17—22no—
66C2×C4×C17—21no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17C2×D412yes—
71C2×C4×C17C2×Q814yes—
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×Dic17C2×D414yes—
82C2×Dic17C2×Q814yes—
83C2×Dic17C2×Q814yes—
84Dic34—44no—
85Dic34—48no—
86#2734—171nomaximal, Sylow
87C22×C17C22×D411yescharacteristic
88C22×C17C22×Q812yes—
89C22×C17H(5,𝔽2)11yescharacteristic
90C22×C17H(5,𝔽2)11yescharacteristic
91C22×C172−1+412yes—
92C4×C17—22no—
93C4×C17—24no—
94C4×C17—42no—
95C4×C17—41no—
96C4×C17—41no—
97Dic17—24no—
98Dic17—24no—
99Dic17—24no—
100Dic17—42no—
101C2×C4⋊Q8—172no—
102#398—171no—
103#398D1711yescharacteristic
104#395—172no—
105#395—172no—
106#391—171no—
107#389—172no—
108#387—172no—
109#387—172no—
110C2×C17#53411yescharacteristic
111C2×C17#54412yes—
112C2×C17D4×Q812yes—
113C2×C17Q8211yescharacteristic
114C2×C17#56311yescharacteristic
115C22×Q8—171no—
116C22×Q8—172no—
117C4⋊Q8—348no—
118C2×C4⋊C4—172no—
119C2×C4⋊C4—172no—
120C2×C4⋊C4—172no—
121C2×C4⋊C4—172no—
122C2×C4⋊C4—172no—
123C2×C4⋊C4—171no—
124C2×C4⋊C4C2×D1712yes—
125C2×C4⋊C4C2×D1711yescharacteristic
126C2×C4⋊C4C2×D1711yescharacteristic
127C2×C42—172no—
128C2×C42—171no—
129#95—171no—
130#95—172no—
131#95—172no—
132#95—172no—
133#95—172no—
134#95—172no—
135#95C2×D1711yescharacteristic
136#95C2×D1712yes—
137C17#273411yescharacteristic, complemented, Sylow
138C2×Q8—174no—
139C2×Q8—344no—
140C2×Q8—681no—
141C2×Q8—681no—
142C2×Q8—682no—
143C2×Q8—682no—
144C22×C4—172no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—172no—
148C22×C4C22×D1711yescharacteristic
149C22×C4C22×D1711yescharacteristic
150C22×C4C22×D1712yes—
151C22×C4C22×D1711yescharacteristic
152C22×C4C22×D1712yes—
153C4⋊C4—24no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—178no—
157C4⋊C4—344no—
158C4⋊C4—344no—
159C4⋊C4—344no—
160C4⋊C4—682no—
161C4⋊C4—681no—
162C42—344no—
163C42—681no—
164C23C23×D1711yescharacteristic
165Q8—684no—
166Q8—688no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—174no—
179C2×C4—174no—
180C2×C4—174no—
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×C4D4×D1712yes—
191C2×C4Q8×D1714yes—
192C22C2×D4×D1711yescharacteristic
193C22#1058769111yescharacteristic
194C22C2×Q8×D1712yes—
195C22#1058769611yescharacteristic
196C22#1058769712yes—
197C4—24no—
198C4—22no—
199C4—41no—
200C4—41no—
201C4—42no—
202C4—344no—
203C4—344no—
204C4—344no—
205C4—682no—
206C2#1176336711yescharacteristic
207C2#1176343912yes—
208C2#1176347212yes—
209C2#1176353911yescharacteristic
210C2#1176356211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#398C2
hypercenter#398D17
hypocenterC17#2734
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
55. #63225Q8semidirect
137. C17#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
#117627392
#117627302
#117627292
#117627142
#117627042
#117626962
#117626912
C17×#3982
#273417