Finite Group Databasebeta

← #14528701

Subgroups — #14528701

Order 2176.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#63252C211yesmaximal, characteristic
2C2×#10587477C214yesmaximal, complemented
3#11762672C211yesmaximal, characteristic, complemented
4C17×#381C211yesmaximal, characteristic, complemented
5C23×C4×C17C2211yescharacteristic
6C22×C22⋊D17C2212yes—
7C2×#63252—22no—
8C2×#63252—22no—
9C2×#63252—28no—
10C22×D68C411yescharacteristic, complemented
11C22×C4×D17C412yescomplemented
12C22×Dic34C411yescharacteristic, complemented
13C2×C22⋊C4×C17C2212yes—
14C2×C22⋊Dic17C2212yes—
15C17×#95—22no—
16#10587502—22no—
17#10587477—28no—
18C24×C17C2311yescharacteristic
19C23×D17C2×C412yes—
20C22×C4×C17—24no—
21C22×C4×C17—22no—
22C22×C4×C17C2×C411yescharacteristic
23C22×C4×C17C2×C411yescharacteristic
24C22×C4×C17D412yes—
25C22×C4×C17D412yes—
26C2×C22⋊D17—22no—
27C2×C22⋊D17—24no—
28C2×C22⋊D17—22no—
29C2×C22⋊D17D418yes—
30C22×Dic17—22no—
31C22×Dic17C2×C412yes—
32#63252—216no—
33#63252—216no—
34C2×D68—24no—
35C2×D68—22no—
36C2×C4×D17—24no—
37C2×C4×D17—24no—
38C2×C4×D17—24no—
39C2×Dic34—22no—
40C2×Dic34—24no—
41C22⋊C4×C17—24no—
42C22⋊Dic17—24no—
43C23×C17—42no—
44C23×C17C22×C411yescharacteristic
45C23×C17C2×D411yescharacteristic
46C23×C17C2×D414yes—
47C23×C17C2×D411yescharacteristic
48C22×D17—22no—
49C22×D17—22no—
50C22×D17—44no—
51C22×D17C22⋊C414yes—
52C2×C4×C17—22no—
53C2×C4×C17—22no—
54C2×C4×C17—24no—
55C2×C4×C17—24no—
56C2×C4×C17—44no—
57C2×C4×C17—42no—
58C2×C4×C17C22⋊C412yes—
59C2×C4×C17C22⋊C412yes—
60C22⋊D17—28no—
61C22⋊D17—28no—
62C2×Dic17—22no—
63C2×Dic17—22no—
64C2×Dic17—44no—
65C2×Dic17—42no—
66C2×Dic17C22⋊C414yes—
67D68—28no—
68C4×D17—216no—
69Dic34—28no—
70#1782—171nomaximal, Sylow
71C22×C17—22no—
72C22×C17—22no—
73C22×C17—22no—
74C22×C17—44no—
75C22×C17C2×C22⋊C412yes—
76C22×C17C2×C22⋊C411yescharacteristic
77C22×C17#12012yes—
78C22×C17#12012yes—
79C2×D17—24no—
80C2×D17—44no—
81C2×D17—44no—
82C2×D17—44no—
83C4×C17—22no—
84C4×C17—22no—
85C4×C17—82no—
86Dic17—24no—
87Dic17—82no—
88C22×#41—171no—
89C2×#99—174no—
90#381—171no—
91#381D1711yescharacteristic, complemented
92C2×C17—22no—
93C2×C17—22no—
94C2×C17—22no—
95C2×C17#37911yescharacteristic
96C2×C17#41012yes—
97D17—44no—
98C2×#41—348no—
99C2×#41—342no—
100C2×#41—342no—
101C22×Q8—171no—
102C22×D4—172no—
103C22×D4—171no—
104C23×C4—172no—
105C23×C4C2×D1711yescharacteristic
106C2×C22⋊C4—172no—
107C2×C22⋊C4C2×D1712yes—
108#99—348no—
109#95—22no—
110#95—342no—
111C17#178211yescharacteristic, complemented, Sylow
112C24—172no—
113C24C22×D1711yescharacteristic
114#41—3416no—
115#41—3416no—
116C2×Q8—344no—
117C2×Q8—342no—
118C2×D4—178no—
119C2×D4—344no—
120C2×D4—342no—
121C2×D4—344no—
122C2×D4—342no—
123C2×D4—342no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—172no—
127C22×C4—344no—
128C22×C4—344no—
129C22×C4—344no—
130C22×C4—342no—
131C22×C4C4×D1711yescharacteristic
132C22×C4C4×D1711yescharacteristic
133C22×C4D6812yes—
134C22×C4C22⋊D1712yes—
135C22⋊C4—24no—
136C22⋊C4—344no—
137C23—42no—
138C23—174no—
139C23—342no—
140C23—342no—
141C23—684no—
142C23C2×C4×D1711yescharacteristic
143C23C2×D6811yescharacteristic
144C23D4×D1714yes—
145C23C2×C22⋊D1711yescharacteristic
146Q8—348no—
147D4—348no—
148D4—348no—
149D4—348no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—44no—
155C2×C4—42no—
156C2×C4—174no—
157C2×C4—3416no—
158C2×C4—342no—
159C2×C4—342no—
160C2×C4—684no—
161C2×C4—682no—
162C2×C4#6322712yes—
163C2×C4#6322712yes—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—344no—
169C22—684no—
170C22—684no—
171C22—684no—
172C22C22⋊C4×D1712yes—
173C22#1058756812yes—
174C22C2×#6322711yescharacteristic
175C22#1058762312yes—
176C4—22no—
177C4—22no—
178C4—82no—
179C4—344no—
180C4—1362no—
181C2—22no—
182C2—22no—
183C2—22no—
184C2—684no—
185C2#1176266411yescharacteristic
186C2#1176278212yes—
subgroupG/H
centerC22C2×#63227
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC17×#381C2
socleC22×C17C2×C22⋊C4
prime radicalC22×#63252C2
cosocleC24×C17C23
hypercenter#381D17
hypocenterC17#1782
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#10587477C2semidirect
3. #11762672C2semidirect
4. C17×#381C2semidirect
10. C22×D68C4semidirect
11. C22×C4×D17C4semidirect
12. C22×Dic34C4semidirect
91. #381D17semidirect
111. C17#1782semidirect

Maximal subgroups

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

subgroupindex
C22×#632522
C2×#105874772
#117626722
C17×#3812
#178217