Finite Group Databasebeta

← #14539923

Subgroups — #14539923

Order 2176.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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#11763364C211yesmaximal, characteristic, complemented
2#11763360C211yesmaximal, characteristic
3#11763205C211yesmaximal, characteristic, complemented
4#11763157C211yesmaximal, characteristic, complemented
5#11763156C211yesmaximal, characteristic, complemented
6#11763136C214yesmaximal, complemented
7#11763112C214yesmaximal, complemented
8#11762860C211yesmaximal, characteristic, complemented
9C17×#500C211yesmaximal, characteristic, complemented
10C2×#63256—21no—
11C17×C4⋊Q8C2211yescharacteristic
12C17×C4⋊Q8C2211yescharacteristic
13#10587633—21no—
14#10587632—21no—
15Q8×Dic17—21no—
16#10587583—21no—
17#10587581—21no—
18#10587581C2212yes—
19#10587579—21no—
20#10587579C2212yes—
21#10587578C2212yes—
22#10587576C2212yes—
23#10587573—21no—
24#10587560C2211yescharacteristic
25C42×D17C2211yescharacteristic
26#10587555C2211yescharacteristic
27C17×#107C2214yes—
28C4×C8×C17C2211yescharacteristic
29#10587492C2214yes—
30#10587491C2212yes—
31#10587485C2212yes—
32#10587480C2214yes—
33#10587479C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17—21no—
37#63256—44no—
38C2×D68—22no—
39C2×D68—21no—
40C2×D68—21no—
41C2×D68C2312yes—
42C2×C4×D17—22no—
43C2×C4×D17D411yescharacteristic
44C2×C4×D17D411yescharacteristic
45C2×C4×D17C2311yescharacteristic
46C2×Dic34—21no—
47C2×C8×C17C2312yes—
48C4⋊C4×C17—21no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17C2312yes—
51C4⋊C4×C17C2312yes—
52C42×C17C2311yescharacteristic
53#63227—22no—
54#63227—22no—
55#63227—22no—
56C4⋊Dic17—21no—
57C4⋊Dic17—22no—
58C4⋊Dic17C2312yes—
59#63225—22no—
60C4×Dic17—22no—
61C4×Dic17D411yescharacteristic
62C4×Dic17D411yescharacteristic
63C4×Dic17C2311yescharacteristic
64C2×#3622C2312yes—
65C22×D17—22no—
66C22×D17C2×D411yescharacteristic
67Q8×C17—42no—
68Q8×C17—42no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2411yescharacteristic
74C2×Dic17—22no—
75C2×Dic17C2×D411yescharacteristic
76C2×Dic17C2×D411yescharacteristic
77C2×Dic17C2×D411yescharacteristic
78D68—24no—
79D68—44no—
80D68—42no—
81D68—42no—
82C4×D17—22no—
83C4×D17—22no—
84C4×D17—22no—
85C4×D17—44no—
86Dic34—42no—
87C8×C17—22no—
88#3622—22no—
89#3228—171nomaximal, Sylow
90C22×C17C22×D411yescharacteristic
91C2×D17—21no—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—42no—
95C2×D17—42no—
96C2×D17—42no—
97C4×C17—42no—
98C4×C17—42no—
99C4×C17#13512yes—
100C4×C17AGL(1,ℤ/8)12yes—
101C4×C172−1+412yes—
102Dic17—21no—
103Dic17—21no—
104Dic17—21no—
105Dic17—42no—
106#533—171no—
107#532—171no—
108#500D1711yescharacteristic, complemented
109#498—171no—
110#486—171no—
111#486—171no—
112#482—174no—
113#478—174no—
114#431—171no—
115C2×C17#53611yescharacteristic
116C2×C17C2×#13511yescharacteristic
117C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
118D17—41no—
119D17—82no—
120C2×#41—341no—
121C4⋊Q8—171no—
122C4⋊Q8C2×D1711yescharacteristic
123C4⋊Q8C2×D1711yescharacteristic
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×C42—171no—
137#107C2×D1714yes—
138#106—174no—
139C4⋊C8—172no—
140#102—174no—
141#102—174no—
142C22⋊C8—172no—
143C4×C8—171no—
144C4×C8C2×D1711yescharacteristic
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—171no—
155C22×C4—171no—
156C22×C4—171no—
157C22×C4—342no—
158C2×C8—172no—
159C2×C8C22×D1712yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—172no—
163C4⋊C4—342no—
164C4⋊C4—342no—
165C4⋊C4—341no—
166C4⋊C4C22×D1712yes—
167C4⋊C4C22×D1712yes—
168C22⋊C4—342no—
169C22⋊C4—342no—
170C22⋊C4—342no—
171C42—171no—
172C42—171no—
173C42—171no—
174C42—342no—
175C42C22×D1711yescharacteristic
176C23—171no—
177C23—342no—
178Q8—42no—
179Q8—42no—
180Q8—682no—
181D4—344no—
182D4—684no—
183D4—682no—
184D4—682no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—171no—
188C2×C4—171no—
189C2×C4—171no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—684no—
195C2×C4D4×D1711yescharacteristic
196C2×C4D4×D1711yescharacteristic
197C2×C4C23×D1711yescharacteristic
198C8—22no—
199C8—342no—
200C22—341no—
201C22—341no—
202C22—341no—
203C22—682no—
204C22—682no—
205C22—682no—
206C22C2×D4×D1711yescharacteristic
207C4—42no—
208C4—42no—
209C4—341no—
210C4—341no—
211C4—341no—
212C4—682no—
213C4#1058759712yes—
214C4#1058760512yes—
215C4#1058769412yes—
216C2—681no—
217C2—1362no—
218C2#1176338411yescharacteristic
219C2C2×#1058760511yescharacteristic
220C2C2×#1058759711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#500C2
socleC22×C17C22×D4
prime radical#11763360C2
hypercenter#500D17
hypocenterC17#3228
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #11763364C2semidirect
3. #11763205C2semidirect
4. #11763157C2semidirect
5. #11763156C2semidirect
6. #11763136C2semidirect
7. #11763112C2semidirect
8. #11762860C2semidirect
9. C17×#500C2semidirect
108. #500D17semidirect
145. C17#3228semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117633602
#117632052
#117631572
#117631562
#117631362
#117631122
#117628602
C17×#5002
#322817