Finite Group Databasebeta

← #14539936

Subgroups — #14539936

Order 2176.

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

normal subgroups only

This group has 360 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#11763364C211yesmaximal, characteristic
2C4⋊Q8×D17C211yesmaximal, characteristic
3#11763212C211yesmaximal, characteristic, complemented
4#11763161C211yesmaximal, characteristic, complemented
5#11763160C211yesmaximal, characteristic, complemented
6#11763146C214yesmaximal
7#11763103C214yesmaximal, complemented
8#11762860C211yesmaximal, characteristic
9C17×#500C211yesmaximal, characteristic, complemented
10C2×Q8×D17—21no—
11C17×C4⋊Q8C2211yescharacteristic
12C17×C4⋊Q8C2211yescharacteristic
13#10587632—21no—
14Q8×Dic17—21no—
15#10587630—21no—
16#10587583C2212yes—
17#10587578—21no—
18C4⋊C4×D17—21no—
19C4⋊C4×D17C2212yes—
20#10587576—21no—
21#10587574—21no—
22#10587574C2212yes—
23#10587573C2212yes—
24C42×D17C2211yescharacteristic
25#10587555C2211yescharacteristic
26#10587555C2211yescharacteristic
27C17×#107C2214yes—
28C4×C8×C17C2211yescharacteristic
29#10587491C2212yes—
30#10587487C2214yes—
31#10587485C2212yes—
32#10587481C2214yes—
33#10587478C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17—21no—
37Q8×D17—44no—
38C2×C4×D17—22no—
39C2×C4×D17D411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17C2311yescharacteristic
42C2×Dic34—21no—
43C2×Dic34—22no—
44C2×Dic34C2312yes—
45C2×C8×C17C2312yes—
46C4⋊C4×C17—21no—
47C4⋊C4×C17—21no—
48C4⋊C4×C17C2312yes—
49C4⋊C4×C17C2312yes—
50C42×C17C2311yescharacteristic
51#63227—22no—
52C4⋊Dic17—21no—
53C4⋊Dic17—21no—
54C4⋊Dic17—21no—
55C4⋊Dic17—22no—
56C4⋊Dic17C2312yes—
57#63225—22no—
58#63225—22no—
59#63225—22no—
60C4×Dic17—22no—
61C4×Dic17D411yescharacteristic
62C4×Dic17D411yescharacteristic
63C4×Dic17C2311yescharacteristic
64C2×#3622C2312yes—
65C22×D17C2×D411yescharacteristic
66Q8×C17—42no—
67Q8×C17—42no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2411yescharacteristic
73C2×Dic17—22no—
74C2×Dic17—22no—
75C2×Dic17C2×D411yescharacteristic
76C2×Dic17C2×D411yescharacteristic
77C2×Dic17C2×D411yescharacteristic
78C4×D17—22no—
79C4×D17—22no—
80C4×D17—44no—
81C4×D17Q1614yes—
82Dic34—24no—
83Dic34—44no—
84Dic34—42no—
85C8×C17—22no—
86#3622—22no—
87#3230—171nomaximal, Sylow
88C22×C17C22×D411yescharacteristic
89C2×D17—21no—
90C2×D17—21no—
91C2×D17C2×Q1612yes—
92C4×C17—42no—
93C4×C17—42no—
94C4×C17C2×Q1612yes—
95C4×C17AGL(1,ℤ/8)12yes—
96C4×C172−1+412yes—
97Dic17—21no—
98Dic17—21no—
99Dic17—42no—
100Dic17—42no—
101Dic17C2×Q1612yes—
102#533—171no—
103C2×C4⋊Q8—171no—
104#500—171no—
105#500D1711yescharacteristic, complemented
106#487—171no—
107#487—171no—
108#484—174no—
109#477—174no—
110#431—171no—
111C2×C17#53611yescharacteristic
112C2×C17C22×Q1611yescharacteristic
113C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
114D17—22no—
115C22×Q8—341no—
116C4⋊Q8—172no—
117C4⋊Q8—171no—
118C4⋊Q8—171no—
119C4⋊Q8—341no—
120C4⋊Q8—341no—
121C4⋊Q8C2×D1711yescharacteristic
122C4⋊Q8C2×D1711yescharacteristic
123#125—341no—
124C22⋊Q8—172no—
125C22⋊Q8—341no—
126C4×Q8—172no—
127C4×Q8—341no—
128#117—341no—
129C2×C4⋊C4—172no—
130C2×C4⋊C4—341no—
131C2×C42—171no—
132#107—174no—
133#107C2×D1714yes—
134C4⋊C8—172no—
135#103—174no—
136#103—174no—
137C22⋊C8—172no—
138C4×C8—171no—
139C4×C8C2×D1711yescharacteristic
140C17#323011yescharacteristic, complemented, Sylow
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—172no—
144C2×Q8—342no—
145C2×Q8—341no—
146C2×Q8—684no—
147C22×C4—171no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—342no—
151C2×C8—172no—
152C2×C8C22×D1712yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—172no—
156C4⋊C4—342no—
157C4⋊C4—342no—
158C4⋊C4—342no—
159C4⋊C4—342no—
160C4⋊C4—341no—
161C4⋊C4—341no—
162C4⋊C4—341no—
163C4⋊C4C22×D1712yes—
164C4⋊C4C22×D1712yes—
165C22⋊C4—342no—
166C42—171no—
167C42—171no—
168C42—171no—
169C42—342no—
170C42C22×D1711yescharacteristic
171C23—171no—
172Q8—42no—
173Q8—42no—
174Q8—344no—
175Q8—684no—
176Q8—682no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—171no—
180C2×C4—174no—
181C2×C4—171no—
182C2×C4—171no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—684no—
188C2×C4D4×D1711yescharacteristic
189C2×C4D4×D1711yescharacteristic
190C2×C4C23×D1711yescharacteristic
191C8—22no—
192C8—342no—
193C22—172no—
194C22—341no—
195C22—341no—
196C22C2×D4×D1711yescharacteristic
197C4—42no—
198C4—42no—
199C4—172no—
200C4—341no—
201C4—341no—
202C4—682no—
203C4—682no—
204C4#1058759712yes—
205C4Q16×D1712yes—
206C4#1058769412yes—
207C2—342no—
208C2#1176338411yescharacteristic
209C2C2×Q16×D1711yescharacteristic
210C2C2×#1058759711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#500C2
socleC22×C17C22×D4
prime radicalC22×D17C2×D4
hypercenter#500D17
hypocenterC17#3230
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
3. #11763212C2semidirect
4. #11763161C2semidirect
5. #11763160C2semidirect
7. #11763103C2semidirect
9. C17×#500C2semidirect
105. #500D17semidirect
140. C17#3230semidirect

Maximal subgroups

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

subgroupindex
#117633642
C4⋊Q8×D172
#117632122
#117631612
#117631602
#117631462
#117631032
#117628602
C17×#5002
#323017