Finite Group Databasebeta

← #14539710

Subgroups — #14539710

Order 2176.

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

normal subgroups only

This group has 428 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#11763360C211yesmaximal, characteristic
2C4⋊Q8×D17C211yesmaximal, characteristic, complemented
3#11763183C211yesmaximal, characteristic, complemented
4#11763182C211yesmaximal, characteristic, complemented
5#11763161C211yesmaximal, characteristic, complemented
6#11763013C214yesmaximal, complemented
7#11762949C214yesmaximal, complemented
8#11762860C211yesmaximal, characteristic, complemented
9C17×#487C211yesmaximal, characteristic, complemented
10C2×#63256—21no—
11C2×Q8×D17C2212yes—
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14#10587633—21no—
15#10587630C2212yes—
16C2×#63230C2214yes—
17C2×#63219C2214yes—
18#10587581—21no—
19#10587581C2212yes—
20#10587579—21no—
21#10587579C2212yes—
22C4⋊C4×D17—21no—
23C4⋊C4×D17—21no—
24#10587574—21no—
25#10587574—21no—
26#10587560C2211yescharacteristic
27C42×D17C2211yescharacteristic
28#10587555C2211yescharacteristic
29C17×#103C2214yes—
30C4×C8×C17C2211yescharacteristic
31#10587491C2212yes—
32#10587485C2212yes—
33#10587481C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17C2312yes—
37#63256—44no—
38Q8×D17—44no—
39C2×D68—22no—
40C2×D68—21no—
41C2×D68—21no—
42C2×D68C2312yes—
43C2×C4×D17—22no—
44C2×C4×D17—22no—
45C2×C4×D17D411yescharacteristic
46C2×C4×D17D411yescharacteristic
47C2×C4×D17C2311yescharacteristic
48C2×Dic34—22no—
49C2×Dic34C2312yes—
50C2×C8×C17C2312yes—
51C4⋊C4×C17—21no—
52C4⋊C4×C17—21no—
53C4⋊C4×C17—21no—
54C4⋊C4×C17C2312yes—
55C42×C17C2311yescharacteristic
56#63230—44no—
57#63227—22no—
58#63227—22no—
59C4⋊Dic17—21no—
60C4⋊Dic17—21no—
61#63225—22no—
62#63225—22no—
63C4×Dic17D411yescharacteristic
64C4×Dic17D411yescharacteristic
65C4×Dic17C2311yescharacteristic
66C2×#3622C2312yes—
67#63219—44no—
68C22×D17—22no—
69C22×D17C2×D411yescharacteristic
70Q8×C17—24no—
71Q8×C17—42no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2×D411yescharacteristic
76C2×C4×C17C2411yescharacteristic
77C2×Dic17—22no—
78C2×Dic17C2×D411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80C2×Dic17C2×D411yescharacteristic
81D68—24no—
82D68—44no—
83D68—42no—
84D68—42no—
85C4×D17—22no—
86C4×D17—22no—
87C4×D17—44no—
88C4×D17—44no—
89C4×D17SD1614yescomplemented
90Dic34—24no—
91Dic34—44no—
92C8×C17—22no—
93#3622—22no—
94#3206—171nomaximal, Sylow
95C22×C17C22×D411yescharacteristic
96C2×D17—21no—
97C2×D17—21no—
98C2×D17—42no—
99C2×D17—42no—
100C2×D17—42no—
101C2×D17C2×SD1612yes—
102C4×C17—42no—
103C4×C17—42no—
104C4×C17C2×SD1612yes—
105C4×C17#13712yes—
106C4×C17H(5,𝔽2)12yes—
107Dic17—21no—
108Dic17—21no—
109Dic17—42no—
110Dic17C2×SD1612yes—
111#532—171no—
112C2×C4⋊Q8—171no—
113#492—171no—
114#492—171no—
115#487—171no—
116#487D1711yescharacteristic, complemented
117#461—174no—
118#448—174no—
119#431—171no—
120C2×C17#53511yescharacteristic
121C2×C17C22×SD1611yescharacteristic
122C2×C17C2×#13711yescharacteristic
123D17—22no—
124D17—82no—
125C2×#41—341no—
126C22×Q8—172no—
127C2×SD16—174no—
128C2×SD16—174no—
129C4⋊Q8—172no—
130C4⋊Q8—171no—
131C4⋊Q8—341no—
132C4⋊Q8—341no—
133C4⋊Q8C2×D1711yescharacteristic
134C4⋊Q8C2×D1711yescharacteristic
135#127—171no—
136#124—341no—
137#121—172no—
138#121—341no—
139C4×D4—172no—
140C4×D4—341no—
141C2×C4⋊C4—341no—
142C2×C4⋊C4—341no—
143C2×C42—171no—
144C4⋊C8—172no—
145#103—174no—
146#103C2×D1714yes—
147C22⋊C8—172no—
148C4×C8—171no—
149C4×C8C2×D1711yescharacteristic
150C17#320611yescharacteristic, complemented, Sylow
151#41—684no—
152C2×Q8—21no—
153C2×Q8—172no—
154C2×Q8—342no—
155C2×Q8—684no—
156C2×Q8C22×D1712yes—
157C2×D4—172no—
158C2×D4—342no—
159C2×D4—341no—
160C2×D4—341no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—342no—
165C22×C4—342no—
166SD16—684no—
167SD16—684no—
168C2×C8—172no—
169C2×C8C22×D1712yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—341no—
176C4⋊C4—341no—
177C4⋊C4C22×D1712yes—
178C22⋊C4—342no—
179C22⋊C4—342no—
180C42—171no—
181C42—171no—
182C42—171no—
183C42C22×D1711yescharacteristic
184C23—171no—
185C23—342no—
186Q8—24no—
187Q8—42no—
188Q8—344no—
189Q8—684no—
190D4—344no—
191D4—684no—
192D4—682no—
193D4—682no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—171no—
197C2×C4—174no—
198C2×C4—171no—
199C2×C4—171no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—684no—
204C2×C4—684no—
205C2×C4D4×D1711yescharacteristic
206C2×C4D4×D1711yescharacteristic
207C2×C4C23×D1711yescharacteristic
208C8—22no—
209C8—342no—
210C22—172no—
211C22—341no—
212C22—341no—
213C22—682no—
214C22—682no—
215C22—682no—
216C22C2×D4×D1711yescharacteristic
217C4—42no—
218C4—42no—
219C4—172no—
220C4—341no—
221C4—341no—
222C4—682no—
223C4SD16×D1712yes—
224C4#1058760412yes—
225C4#1058769612yes—
226C2—342no—
227C2—1362no—
228C2#1176337711yescharacteristic
229C2C2×SD16×D1711yescharacteristic
230C2C2×#1058760411yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#487C2
socleC22×C17C22×D4
prime radical#11763360C2
hypercenter#487D17
hypocenterC17#3206
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D17C2semidirect
3. #11763183C2semidirect
4. #11763182C2semidirect
5. #11763161C2semidirect
6. #11763013C2semidirect
7. #11762949C2semidirect
8. #11762860C2semidirect
9. C17×#487C2semidirect
89. C4×D17SD16semidirect
116. #487D17semidirect
150. C17#3206semidirect

Maximal subgroups

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

subgroupindex
#117633602
C4⋊Q8×D172
#117631832
#117631822
#117631612
#117630132
#117629492
#117628602
C17×#4872
#320617