Finite Group Databasebeta

← #14539920

Subgroups — #14539920

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#11763359C211yesmaximal, characteristic
2#11763212C211yesmaximal, characteristic, complemented
3#11763205C211yesmaximal, characteristic, complemented
4#11763157C211yesmaximal, characteristic, complemented
5#11763157C211yesmaximal, characteristic, complemented
6#11763133C214yesmaximal, complemented
7#11763106C214yesmaximal, complemented
8#11762862C211yesmaximal, characteristic, complemented
9C17×#533C211yesmaximal, characteristic, complemented
10C2×#63252—21no—
11C17×C4⋊Q8C2211yescharacteristic
12C17×C4⋊Q8C2211yescharacteristic
13C17×#125C2212yes—
14C17×C22⋊Q8—21no—
15C17×C22⋊Q8—21no—
16C4×Q8×C17—21no—
17C4×Q8×C17—21no—
18C17×#117C2212yes—
19C2×C42×C17C2211yescharacteristic
20#10587616—21no—
21#10587616C2212yes—
22#10587561—21no—
23#10587560C2211yescharacteristic
24C4×D68—21no—
25C4×D68C2212yes—
26#10587555C2211yescharacteristic
27C22⋊#3622C2212yes—
28#10587480C2214yes—
29#10587480C2214yes—
30#10587479C2214yes—
31#10587478C2214yes—
32C4⋊#3622C2212yes—
33C4×#3622C2211yescharacteristic
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17—21no—
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×Dic34—21no—
47C4⋊C4×C17—22no—
48C4⋊C4×C17—22no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17—21no—
51C4⋊C4×C17C2312yes—
52C4⋊C4×C17C2312yes—
53C22⋊C4×C17—22no—
54C42×C17—22no—
55C42×C17D411yescharacteristic
56C42×C17D411yescharacteristic
57C42×C17C2311yescharacteristic
58C42×C17C2311yescharacteristic
59#63227—22no—
60#63227—22no—
61C4⋊Dic17—21no—
62C4⋊Dic17C2312yes—
63C2×#3622C2312yes—
64C2×#3622C2312yes—
65C23×C17C2×D411yescharacteristic
66C22×D17—22no—
67Q8×C17—42no—
68Q8×C17—42no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2411yescharacteristic
80C22⋊D17—44no—
81C2×Dic17—22no—
82D68—24no—
83D68—42no—
84D68—42no—
85C4×D17—44no—
86Dic34—42no—
87#3622—22no—
88#3622—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—42no—
101C4×C17—42no—
102C4×C17#13512yes—
103C4×C17AGL(1,ℤ/8)12yes—
104C4×C172−1+412yes—
105Dic17—42no—
106#533D1711yescharacteristic, complemented
107#532—171no—
108#500—171no—
109#498—171no—
110#486—171no—
111#486—171no—
112#482—174no—
113#478—174no—
114#431—171no—
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⋊Q8C2×D1711yescharacteristic
123C4⋊Q8C2×D1711yescharacteristic
124#127—171no—
125#125C2×D1712yes—
126#124—341no—
127C22⋊Q8—21no—
128C22⋊Q8—21no—
129#121—172no—
130#121—341no—
131C4×Q8—21no—
132C4×Q8—21no—
133C4×D4—172no—
134C4×D4—341no—
135#117C2×D1712yes—
136C2×C42C2×D1711yescharacteristic
137#107—174no—
138#106—174no—
139C4⋊C8—172no—
140#102—174no—
141#102—174no—
142C22⋊C8—172no—
143C4×C8—171no—
144C4×C8—171no—
145C17#322811yescharacteristic, complemented, Sylow
146#41—684no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—341no—
150C2×D4—172no—
151C2×D4—342no—
152C2×D4—341no—
153C2×D4—341no—
154C22×C4—342no—
155C22×C4C22⋊D1711yescharacteristic
156C22×C4C22⋊D1711yescharacteristic
157C22×C4C22×D1711yescharacteristic
158C2×C8—172no—
159C2×C8—172no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—172no—
165C4⋊C4—341no—
166C4⋊C4C22×D1712yes—
167C4⋊C4C22×D1712yes—
168C22⋊C4—22no—
169C22⋊C4—342no—
170C22⋊C4—342no—
171C42—22no—
172C42C22⋊D1711yescharacteristic
173C42C22⋊D1711yescharacteristic
174C42C22×D1711yescharacteristic
175C42C22×D1711yescharacteristic
176C23—342no—
177C23C2×C22⋊D1711yescharacteristic
178Q8—42no—
179Q8—42no—
180Q8—682no—
181D4—344no—
182D4—684no—
183D4—682no—
184D4—682no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—342no—
191C2×C4—684no—
192C2×C4C2×C22⋊D1711yescharacteristic
193C2×C4C2×C22⋊D1711yescharacteristic
194C2×C4C2×C22⋊D1711yescharacteristic
195C2×C4C2×C22⋊D1711yescharacteristic
196C2×C4C2×C22⋊D1711yescharacteristic
197C2×C4C23×D1711yescharacteristic
198C8—342no—
199C8—342no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—682no—
204C22—682no—
205C22—682no—
206C22C22×C22⋊D1711yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—682no—
213C4#1058763512yes—
214C4#1058763612yes—
215C4#1058769412yes—
216C2—41no—
217C2—1362no—
218C2#1176338311yescharacteristic
219C2C2×#1058763611yescharacteristic
220C2C2×#1058763511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radical#11763359C2
hypercenter#533D17
hypocenterC17#3228
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. #11763212C2semidirect
3. #11763205C2semidirect
4. #11763157C2semidirect
5. #11763157C2semidirect
6. #11763133C2semidirect
7. #11763106C2semidirect
8. #11762862C2semidirect
9. C17×#533C2semidirect
106. #533D17semidirect
145. C17#3228semidirect

Maximal subgroups

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

subgroupindex
#117633592
#117632122
#117632052
#117631572
#117631332
#117631062
#117628622
C17×#5332
#322817