Finite Group Databasebeta

← #14539743

Subgroups — #14539743

Order 2176.

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

normal subgroups only

This group has 394 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#11763188C211yesmaximal, characteristic, complemented
4#11763182C211yesmaximal, characteristic, complemented
5#11763157C211yesmaximal, characteristic, complemented
6#11763034C214yesmaximal, complemented
7#11762974C214yesmaximal, complemented
8#11762860C211yesmaximal, characteristic, complemented
9C17×#487C211yesmaximal, characteristic, complemented
10C2×#63256C2212yes—
11C17×C4⋊Q8C2211yescharacteristic
12C17×C4⋊Q8C2211yescharacteristic
13#10587633C2212yes—
14#10587632—21no—
15Q8×Dic17—21no—
16C2×#63231C2214yes—
17C2×#63219C2214yes—
18#10587583C2212yes—
19#10587581—21no—
20#10587581—21no—
21#10587579—21no—
22#10587579—21no—
23#10587578—21no—
24#10587576—21no—
25#10587573C2212yes—
26#10587560C2211yescharacteristic
27C42×D17C2211yescharacteristic
28#10587555C2211yescharacteristic
29C17×#103C2214yes—
30C4×C8×C17C2211yescharacteristic
31#10587491C2212yes—
32#10587485C2212yes—
33#10587480C2214yes—
34C4×#3622C2211yescharacteristic
35C2×Q8×C17—21no—
36C2×Q8×C17C2312yes—
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×Dic34C2312yes—
47C2×C8×C17C2312yes—
48C4⋊C4×C17—21no—
49C4⋊C4×C17—21no—
50C4⋊C4×C17—21no—
51C4⋊C4×C17C2312yes—
52C42×C17C2311yescharacteristic
53#63231—44no—
54#63227—22no—
55#63227—22no—
56#63227—22no—
57C4⋊Dic17—21no—
58C4⋊Dic17—21no—
59C4⋊Dic17—22no—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17D411yescharacteristic
63C4×Dic17D411yescharacteristic
64C4×Dic17C2311yescharacteristic
65C2×#3622C2312yes—
66#63219—44no—
67C22×D17—22no—
68C22×D17C2×D411yescharacteristic
69Q8×C17—24no—
70Q8×C17—42no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2×D411yescharacteristic
75C2×C4×C17C2411yescharacteristic
76C2×Dic17—22no—
77C2×Dic17C2×D411yescharacteristic
78C2×Dic17C2×D411yescharacteristic
79C2×Dic17C2×D411yescharacteristic
80D68—24no—
81D68—44no—
82D68—42no—
83D68—42no—
84C4×D17—22no—
85C4×D17—22no—
86C4×D17—22no—
87C4×D17—44no—
88Dic34—24no—
89C8×C17—22no—
90#3622—22no—
91#3210—171nomaximal, Sylow
92C22×C17C22×D411yescharacteristic
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—21no—
96C2×D17—42no—
97C2×D17—42no—
98C2×D17—42no—
99C4×C17—42no—
100C4×C17—42no—
101C4×C17#13512yes—
102C4×C17#13712yes—
103C4×C17H(5,𝔽2)12yes—
104Dic17—21no—
105Dic17—21no—
106Dic17—21no—
107Dic17—42no—
108#533—171no—
109#532—171no—
110#494—171no—
111#492—171no—
112#487D1711yescharacteristic, complemented
113#486—171no—
114#464—174no—
115#452—174no—
116#431—171no—
117C2×C17#53511yescharacteristic
118C2×C17C2×#13511yescharacteristic
119C2×C17C2×#13711yescharacteristic
120D17—41no—
121D17—82no—
122C2×#41—172no—
123C2×Q16—174no—
124C2×SD16—174no—
125C4⋊Q8—171no—
126C4⋊Q8C2×D1711yescharacteristic
127C4⋊Q8C2×D1711yescharacteristic
128#127—171no—
129#125—341no—
130#124—172no—
131C22⋊Q8—172no—
132C22⋊Q8—341no—
133#121—341no—
134#121—341no—
135C4×Q8—172no—
136C4×Q8—341no—
137C4×D4—341no—
138C4×D4—341no—
139#117—341no—
140C2×C42—171no—
141C4⋊C8—172no—
142#103C2×D1714yes—
143#102—174no—
144C22⋊C8—172no—
145C4×C8—171no—
146C4×C8C2×D1711yescharacteristic
147C17#321011yescharacteristic, complemented, Sylow
148#41—684no—
149C2×Q8—21no—
150C2×Q8—172no—
151C2×Q8C22×D1712yes—
152C2×D4—172no—
153C2×D4—342no—
154C2×D4—341no—
155C2×D4—341no—
156C22×C4—171no—
157C22×C4—171no—
158C22×C4—171no—
159C22×C4—342no—
160Q16—684no—
161SD16—684no—
162C2×C8—172no—
163C2×C8C22×D1712yes—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—342no—
168C4⋊C4—342no—
169C4⋊C4—341no—
170C4⋊C4—341no—
171C4⋊C4C22×D1712yes—
172C22⋊C4—342no—
173C22⋊C4—342no—
174C22⋊C4—342no—
175C42—171no—
176C42—171no—
177C42—171no—
178C42—342no—
179C42C22×D1711yescharacteristic
180C23—171no—
181C23—342no—
182Q8—24no—
183Q8—42no—
184Q8—344no—
185D4—344no—
186D4—684no—
187D4—682no—
188D4—682no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—171no—
192C2×C4—171no—
193C2×C4—171no—
194C2×C4—342no—
195C2×C4—342no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—684no—
199C2×C4D4×D1711yescharacteristic
200C2×C4D4×D1711yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—22no—
203C8—342no—
204C22—341no—
205C22—341no—
206C22—341no—
207C22—682no—
208C22—682no—
209C22—682no—
210C22C2×D4×D1711yescharacteristic
211C4—42no—
212C4—42no—
213C4—341no—
214C4—341no—
215C4—341no—
216C4—682no—
217C4#1058760212yes—
218C4#1058760412yes—
219C4#1058769612yes—
220C2—681no—
221C2—1362no—
222C2#1176337711yescharacteristic
223C2C2×#1058760211yescharacteristic
224C2C2×#1058760411yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#487C2
socleC22×C17C22×D4
prime radical#11763360C2
hypercenter#487D17
hypocenterC17#3210
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
1. #11763364C2semidirect
3. #11763188C2semidirect
4. #11763182C2semidirect
5. #11763157C2semidirect
6. #11763034C2semidirect
7. #11762974C2semidirect
8. #11762860C2semidirect
9. C17×#487C2semidirect
112. #487D17semidirect
147. C17#3210semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117633602
#117631882
#117631822
#117631572
#117630342
#117629742
#117628602
C17×#4872
#321017