Finite Group Databasebeta

← #14539736

Subgroups — #14539736

Order 2176.

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

normal subgroups only

This group has 394 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
1#11763359C211yesmaximal, characteristic
2#11763188C211yesmaximal, characteristic, complemented
3#11763183C211yesmaximal, characteristic, complemented
4#11763161C211yesmaximal, characteristic, complemented
5#11763157C211yesmaximal, characteristic, complemented
6#11763031C214yesmaximal, complemented
7#11762969C214yesmaximal, complemented
8#11762862C211yesmaximal, characteristic, complemented
9C17×#533C211yesmaximal, characteristic, complemented
10C2×#63252C2212yes—
11C17×C4⋊Q8C2211yescharacteristic
12C17×C4⋊Q8C2211yescharacteristic
13C17×#125—21no—
14C17×C22⋊Q8—21no—
15C17×C22⋊Q8C2212yes—
16C4×Q8×C17—21no—
17C4×Q8×C17C2212yes—
18C17×#117—21no—
19C2×C42×C17C2211yescharacteristic
20C2×#63231C2214yes—
21C2×#63230C2214yes—
22#10587616—21no—
23#10587616—21no—
24#10587561C2212yes—
25#10587560C2211yescharacteristic
26C4×D68—21no—
27C4×D68—21no—
28#10587555C2211yescharacteristic
29C22⋊#3622C2212yes—
30#10587481C2214yes—
31#10587480C2214yes—
32C4⋊#3622C2212yes—
33C4×#3622C2211yescharacteristic
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17C2312yes—
37C22×C4×C17D411yescharacteristic
38C22×C4×C17D411yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C2×C22⋊D17—22no—
41#63252—44no—
42C2×D68—21no—
43C2×D68—21no—
44C2×D68C2312yes—
45C2×C4×D17—22no—
46C2×Dic34C2312yes—
47C4⋊C4×C17—22no—
48C4⋊C4×C17—21no—
49C4⋊C4×C17—22no—
50C4⋊C4×C17—21no—
51C4⋊C4×C17—21no—
52C4⋊C4×C17C2312yes—
53C22⋊C4×C17—22no—
54C42×C17—22no—
55C42×C17D411yescharacteristic
56C42×C17D411yescharacteristic
57C42×C17C2311yescharacteristic
58C42×C17C2311yescharacteristic
59#63231—44no—
60#63230—44no—
61#63227—22no—
62#63227—22no—
63C4⋊Dic17—21no—
64C4⋊Dic17—21no—
65C2×#3622C2312yes—
66C2×#3622C2312yes—
67C23×C17C2×D411yescharacteristic
68C22×D17—22no—
69Q8×C17—24no—
70Q8×C17—42no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2411yescharacteristic
82C22⋊D17—44no—
83C2×Dic17—22no—
84D68—24no—
85D68—42no—
86D68—42no—
87C4×D17—44no—
88Dic34—24no—
89#3622—22no—
90#3622—22no—
91#3210—171nomaximal, Sylow
92C22×C17—21no—
93C22×C17—21no—
94C22×C17—21no—
95C22×C17C22×D411yescharacteristic
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—42no—
103C4×C17—42no—
104C4×C17#13512yes—
105C4×C17#13712yes—
106C4×C17H(5,𝔽2)12yes—
107Dic17—42no—
108#533D1711yescharacteristic, complemented
109#532—171no—
110#494—171no—
111#492—171no—
112#487—171no—
113#486—171no—
114#464—174no—
115#452—174no—
116#431—171no—
117C2×C17—41no—
118C2×C17#53511yescharacteristic
119C2×C17C2×#13511yescharacteristic
120C2×C17C2×#13711yescharacteristic
121D17—82no—
122C2×#41—172no—
123C2×Q16—174no—
124C2×SD16—174no—
125C4⋊Q8—171no—
126C4⋊Q8C2×D1711yescharacteristic
127C4⋊Q8C2×D1711yescharacteristic
128#127—171no—
129#125—21no—
130#124—172no—
131C22⋊Q8—21no—
132C22⋊Q8C2×D1712yes—
133#121—341no—
134#121—341no—
135C4×Q8—21no—
136C4×Q8C2×D1712yes—
137C4×D4—341no—
138C4×D4—341no—
139#117—21no—
140C2×C42C2×D1711yescharacteristic
141C4⋊C8—172no—
142#103—174no—
143#102—174no—
144C22⋊C8—172no—
145C4×C8—171no—
146C4×C8—171no—
147C17#321011yescharacteristic, complemented, Sylow
148#41—684no—
149C2×Q8—21no—
150C2×Q8—172no—
151C2×Q8C22×D1712yes—
152C2×D4—172no—
153C2×D4—342no—
154C2×D4—341no—
155C2×D4—341no—
156C22×C4—342no—
157C22×C4C22⋊D1711yescharacteristic
158C22×C4C22⋊D1711yescharacteristic
159C22×C4C22×D1711yescharacteristic
160Q16—684no—
161SD16—684no—
162C2×C8—172no—
163C2×C8—172no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—341no—
170C4⋊C4—341no—
171C4⋊C4C22×D1712yes—
172C22⋊C4—22no—
173C22⋊C4—342no—
174C22⋊C4—342no—
175C42—22no—
176C42C22⋊D1711yescharacteristic
177C42C22⋊D1711yescharacteristic
178C42C22×D1711yescharacteristic
179C42C22×D1711yescharacteristic
180C23—342no—
181C23C2×C22⋊D1711yescharacteristic
182Q8—24no—
183Q8—42no—
184Q8—344no—
185D4—344no—
186D4—684no—
187D4—682no—
188D4—682no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—342no—
195C2×C4—684no—
196C2×C4C2×C22⋊D1711yescharacteristic
197C2×C4C2×C22⋊D1711yescharacteristic
198C2×C4C2×C22⋊D1711yescharacteristic
199C2×C4C2×C22⋊D1711yescharacteristic
200C2×C4C2×C22⋊D1711yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—342no—
203C8—342no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—682no—
208C22—682no—
209C22—682no—
210C22C22×C22⋊D1711yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—682no—
217C4#1058762812yes—
218C4#1058763612yes—
219C4#1058769612yes—
220C2—41no—
221C2—1362no—
222C2#1176337311yescharacteristic
223C2C2×#1058763611yescharacteristic
224C2C2×#1058762811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radical#11763359C2
hypercenter#533D17
hypocenterC17#3210
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #11763188C2semidirect
3. #11763183C2semidirect
4. #11763161C2semidirect
5. #11763157C2semidirect
6. #11763031C2semidirect
7. #11762969C2semidirect
8. #11762862C2semidirect
9. C17×#533C2semidirect
108. #533D17semidirect
147. C17#3210semidirect

Maximal subgroups

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

subgroupindex
#117633592
#117631882
#117631832
#117631612
#117631572
#117630312
#117629692
#117628622
C17×#5332
#321017