Finite Group Databasebeta

← #14539922

Subgroups — #14539922

Order 2176.

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

normal subgroups only

This group has 370 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#11763366C211yesmaximal, characteristic, complemented
2#11763359C211yesmaximal, characteristic
3#11763214C211yesmaximal, characteristic, complemented
4#11763207C211yesmaximal, characteristic, complemented
5#11763156C211yesmaximal, characteristic, complemented
6#11763156C211yesmaximal, characteristic, complemented
7#11763134C214yesmaximal, complemented
8#11763110C214yesmaximal, complemented
9C17×#431C211yesmaximal, characteristic, complemented
10C2×#63252—21no—
11C2×C42×C17C2211yescharacteristic
12#10587616—21no—
13#10587616C2212yes—
14#10587612C2212yes—
15#10587610—21no—
16#10587610—21no—
17#10587561—21no—
18#10587560C2211yescharacteristic
19C4×D68—21no—
20C4×D68C2212yes—
21#10587556C2212yes—
22#10587555C2211yescharacteristic
23#10587555C2211yescharacteristic
24#10587555C2211yescharacteristic
25C4×Dic34—21no—
26C4×Dic34—21no—
27C17×C4⋊C8C2212yes—
28C17×C22⋊C8C2212yes—
29C4×C8×C17C2211yescharacteristic
30C4×C8×C17C2211yescharacteristic
31#10587492C2214yes—
32#10587492C2214yes—
33#10587489C2214yes—
34#10587488C2214yes—
35C22×C4×C17D411yescharacteristic
36C22×C4×C17D411yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C2×C22⋊D17—22no—
39#63252—44no—
40C2×D68—21no—
41C2×D68—21no—
42C2×D68C2312yes—
43C2×C4×D17—22no—
44C2×Dic34—21no—
45C2×Dic34—21no—
46C2×Dic34—21no—
47C2×C8×C17C2312yes—
48C2×C8×C17C2312yes—
49C42×C17D411yescharacteristic
50C42×C17D411yescharacteristic
51C42×C17C2311yescharacteristic
52C42×C17C2311yescharacteristic
53C22⋊Dic17—22no—
54#63227—22no—
55#63227—22no—
56C4⋊Dic17—21no—
57C4⋊Dic17—21no—
58C4⋊Dic17—21no—
59C4⋊Dic17C2312yes—
60C4⋊Dic17C2312yes—
61C4⋊Dic17C2312yes—
62#63225—22no—
63#63225—22no—
64C4×Dic17—22no—
65C23×C17C2×D411yescharacteristic
66C22×D17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2411yescharacteristic
76C22⋊D17—44no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80D68—24no—
81D68—42no—
82D68—42no—
83C4×D17—44no—
84Dic34—42no—
85Dic34—42no—
86Dic34—42no—
87C8×C17—22no—
88C8×C17—22no—
89#3228—171nomaximal, Sylow
90C22×C17—21no—
91C22×C17—21no—
92C22×C17—21no—
93C22×C17C22×D411yescharacteristic
94C2×D17—42no—
95C2×D17—42no—
96C2×D17—42no—
97C4×C17—21no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17#13512yes—
101C4×C17AGL(1,ℤ/8)12yes—
102C4×C172−1+412yes—
103Dic17—42no—
104Dic17—42no—
105Dic17—42no—
106#533—171no—
107#532—171no—
108#500—171no—
109#498—171no—
110#486—171no—
111#486—171no—
112#482—174no—
113#478—174no—
114#431D1711yescharacteristic, complemented
115C2×C17—41no—
116C2×C17#53611yescharacteristic
117C2×C17C2×#13511yescharacteristic
118C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
119D17—82no—
120C2×#41—341no—
121C4⋊Q8—171no—
122C4⋊Q8—171no—
123C4⋊Q8—171no—
124#127—171no—
125#125—172no—
126#124—341no—
127C22⋊Q8—341no—
128C22⋊Q8—341no—
129#121—172no—
130#121—341no—
131C4×Q8—341no—
132C4×Q8—341no—
133C4×D4—172no—
134C4×D4—341no—
135#117—172no—
136C2×C42C2×D1711yescharacteristic
137#107—174no—
138#106—174no—
139C4⋊C8C2×D1712yes—
140#102—174no—
141#102—174no—
142C22⋊C8C2×D1712yes—
143C4×C8C2×D1711yescharacteristic
144C4×C8C2×D1711yescharacteristic
145C17#322811yescharacteristic, complemented, Sylow
146#41—684no—
147C2×Q8—341no—
148C2×Q8—341no—
149C2×Q8—341no—
150C2×D4—172no—
151C2×D4—342no—
152C2×D4—341no—
153C2×D4—341no—
154C22×C4—342no—
155C22×C4D6811yescharacteristic
156C22×C4D6811yescharacteristic
157C22×C4C22×D1711yescharacteristic
158C2×C8C22×D1712yes—
159C2×C8C22×D1712yes—
160C4⋊C4—172no—
161C4⋊C4—172no—
162C4⋊C4—172no—
163C4⋊C4—341no—
164C4⋊C4—342no—
165C4⋊C4—342no—
166C4⋊C4—341no—
167C4⋊C4—341no—
168C22⋊C4—342no—
169C22⋊C4—342no—
170C22⋊C4—342no—
171C42—342no—
172C42D6811yescharacteristic
173C42D6811yescharacteristic
174C42C22×D1711yescharacteristic
175C42C22×D1711yescharacteristic
176C23—342no—
177C23C2×D6811yescharacteristic
178Q8—682no—
179Q8—682no—
180Q8—682no—
181D4—344no—
182D4—684no—
183D4—682no—
184D4—682no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—684no—
192C2×C4C2×D6811yescharacteristic
193C2×C4C2×D6811yescharacteristic
194C2×C4C2×D6811yescharacteristic
195C2×C4C2×D6811yescharacteristic
196C2×C4C2×D6811yescharacteristic
197C2×C4C23×D1711yescharacteristic
198C8—22no—
199C8—22no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—682no—
204C22—682no—
205C22—682no—
206C22C22×D6811yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—682no—
211C4—682no—
212C4—682no—
213C4#1058759012yes—
214C4#1058759412yes—
215C4#1058769712yes—
216C2—41no—
217C2—1362no—
218C2#1176338511yescharacteristic
219C2C2×#1058759011yescharacteristic
220C2C2×#1058759411yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#431C2
socleC22×C17C22×D4
prime radical#11763359C2
hypercenter#431D17
hypocenterC17#3228
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
3. #11763214C2semidirect
4. #11763207C2semidirect
5. #11763156C2semidirect
6. #11763156C2semidirect
7. #11763134C2semidirect
8. #11763110C2semidirect
9. C17×#431C2semidirect
114. #431D17semidirect
145. C17#3228semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117633592
#117632142
#117632072
#117631562
#117631342
#117631102
C17×#4312
#322817