Finite Group Databasebeta

← #14538366

Subgroups — #14538366

Order 2176.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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×#10587590C211yesmaximal, characteristic, complemented
2C22×C8×D17C211yesmaximal, characteristic, complemented
3#11763321C212yesmaximal, complemented
4#11763051C212yesmaximal, complemented
5#11763046C212yesmaximal, complemented
6#11763039C214yesmaximal, complemented
7#11762803C212yesmaximal, complemented
8C17×#427C211yesmaximal, characteristic, complemented
9C2×#63252C2212yes—
10C22×C4×D17C2211yescharacteristic
11C22×C8×C17C2211yescharacteristic
12C17×#117C2212yes—
13C22×#3622C2211yescharacteristic
14C2×Dic68C2211yescharacteristic, complemented
15#10587590—24no—
16C2×D136C2211yescharacteristic, complemented
17C2×#63219C2212yescomplemented
18C2×C8×D17—24no—
19C2×C8×D17C2212yes—
20C2×C8×D17C2212yes—
21#10587583C2214yes—
22#10587581C2214yes—
23#10587572—22no—
24#10587568—22no—
25C4×D68—24no—
26C17×#107C2212yescomplemented
27C17×#106C2212yescomplemented
28#10587481C2214yescomplemented
29#10587480C2214yescomplemented
30C23×D17D411yescharacteristic
31C22×C4×C17C2311yescharacteristic
32C2×C22⋊D17—22no—
33C22×Dic17D411yescharacteristic
34#63252—24no—
35#63252—24no—
36C2×D68—24no—
37C2×D68C2312yes—
38C2×C4×D17—22no—
39C2×C4×D17—22no—
40C2×C4×D17—22no—
41C2×C4×D17D412yes—
42C2×C4×D17C2312yes—
43C2×Dic34C2312yes—
44C2×C8×C17D414yescomplemented
45C2×C8×C17C2311yescharacteristic
46C2×C8×C17C2311yescharacteristic
47C4⋊C4×C17C2314yes—
48C22⋊C4×C17—22no—
49C42×C17—22no—
50#63227—24no—
51#63227—24no—
52C4⋊Dic17—24no—
53C2×#3622—22no—
54C2×#3622C2312yes—
55Dic68—22no—
56D136—22no—
57#63219—24no—
58C8×D17—24no—
59C8×D17—24no—
60C23×C17C2×D411yescharacteristic
61C22×D17—22no—
62C22×D17—24no—
63C22×D17—22no—
64C22×D17C2×D412yes—
65C2×C4×C17—24no—
66C2×C4×C17C2×D412yes—
67C2×C4×C17C2×D411yescharacteristic
68C2×C4×C17#4112yes—
69C2×C4×C17C2411yescharacteristic
70C22⋊D17—44no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17C2×D412yes—
74D68—22no—
75D68—22no—
76D68—48no—
77C4×D17—24no—
78C4×D17—22no—
79C4×D17—22no—
80C4×D17—44no—
81Dic34—22no—
82Dic34—22no—
83C8×C17C2×D412yes—
84C8×C17C2×D412yes—
85#3622—22no—
86#3050—171nomaximal, Sylow
87C22×C17—22no—
88C22×C17#12112yes—
89C22×C17C22×D411yescharacteristic
90C2×D17—28no—
91C2×D17—22no—
92C2×D17—22no—
93C2×D17—42no—
94C2×D17—44no—
95C2×D17#13514yescomplemented
96C4×C17—44no—
97C4×C17#12112yes—
98C4×C17C22×D411yescharacteristic
99C4×C17C2×#4111yescharacteristic
100Dic17—22no—
101Dic17—42no—
102C2×#135—171no—
103C23×C8—171no—
104#525—172no—
105#467—172no—
106#466—172no—
107#465—174no—
108#427D1711yescharacteristic, complemented
109#416—172no—
110C2×C17—22no—
111C2×C17C2×#12111yescharacteristic
112C2×C17C2×#13512yes—
113D17—24no—
114D17—82no—
115C2×#41—172no—
116C23×C4—171no—
117#135—344no—
118C2×Q16—171no—
119C2×SD16—172no—
120C2×D8—171no—
121C22×C8—171no—
122C22×C8—172no—
123C22×C8—172no—
124C22×C8—344no—
125C22×C8C2×D1711yescharacteristic
126#123—342no—
127C22⋊Q8—174no—
128#121—174no—
129#120—342no—
130C4×D4—344no—
131#117C2×D1712yes—
132#107C2×D1712yescomplemented
133#106C2×D1712yescomplemented
134#103—174no—
135#102—174no—
136C17#305011yescharacteristic, complemented, Sylow
137C24—171no—
138#41—344no—
139#41—344no—
140C2×Q8—172no—
141C2×D4—172no—
142C2×D4—344no—
143C2×D4—342no—
144C22×C4—171no—
145C22×C4—172no—
146C22×C4—172no—
147C22×C4—342no—
148C22×C4—342no—
149C22×C4—342no—
150C22×C4C22×D1711yescharacteristic
151Q16—342no—
152SD16—344no—
153D8—342no—
154C2×C8—172no—
155C2×C8—342no—
156C2×C8—344no—
157C2×C8—344no—
158C2×C8D6814yescomplemented
159C2×C8C22×D1711yescharacteristic
160C2×C8C22×D1711yescharacteristic
161C4⋊C4—344no—
162C4⋊C4C22×D1714yes—
163C22⋊C4—22no—
164C22⋊C4—344no—
165C22⋊C4—344no—
166C42—22no—
167C23—172no—
168C23—344no—
169C23—342no—
170C23—342no—
171C23D4×D1711yescharacteristic
172Q8—342no—
173Q8—342no—
174D4—342no—
175D4—342no—
176D4—688no—
177D4—684no—
178C2×C4—24no—
179C2×C4—172no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—344no—
184C2×C4—342no—
185C2×C4—684no—
186C2×C4C2×D6812yes—
187C2×C4D4×D1711yescharacteristic
188C2×C4#6325612yes—
189C2×C4C23×D1711yescharacteristic
190C8—342no—
191C8C2×D6812yes—
192C8C2×D6812yes—
193C22—22no—
194C22—174no—
195C22—348no—
196C22—342no—
197C22—342no—
198C22—682no—
199C22—684no—
200C22#1058758112yes—
201C22C2×D4×D1711yescharacteristic
202C4—44no—
203C4—342no—
204C4—682no—
205C4#1058758112yes—
206C4C22×D6811yescharacteristic
207C4C2×#6325611yescharacteristic
208C2—22no—
209C2—344no—
210C2—1362no—
211C2C2×#1058758111yescharacteristic
212C2#135×D1712yes—
subgroupG/H
centerC2×C4D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#427C2
socleC22×C17C22×D4
hypercenter#427D17
hypocenterC17#3050
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C2×#10587590C2semidirect
2. C22×C8×D17C2semidirect
3. #11763321C2semidirect
4. #11763051C2semidirect
5. #11763046C2semidirect
6. #11763039C2semidirect
7. #11762803C2semidirect
8. C17×#427C2semidirect
14. C2×Dic68C22semidirect
16. C2×D136C22semidirect
17. C2×#63219C22semidirect
26. C17×#107C22semidirect
27. C17×#106C22semidirect
28. #10587481C22semidirect
29. #10587480C22semidirect
44. C2×C8×C17D4semidirect
95. C2×D17#135semidirect
108. #427D17semidirect
132. #107C2×D17semidirect
133. #106C2×D17semidirect
136. C17#3050semidirect
158. C2×C8D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875902
C22×C8×D172
#117633212
#117630512
#117630462
#117630392
#117628032
C17×#4272
#305017