Finite Group Databasebeta

← #14539868

Subgroups — #14539868

Order 2176.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#11763422C211yesmaximal, characteristic, complemented
2#11763384C211yesmaximal, characteristic
3#11763203C211yesmaximal, characteristic, complemented
4#11763188C211yesmaximal, characteristic, complemented
5#11763166C211yesmaximal, characteristic, complemented
6#11763034C212yesmaximal, complemented
7#11763018C212yesmaximal, complemented
8#11762974C212yesmaximal
9#11762966C212yesmaximal, complemented
10#11762871C211yesmaximal, characteristic, complemented
11C17×#491C211yesmaximal, characteristic, complemented
12C2×#63256C2211yescharacteristic
13C2×Q8×D17C2211yescharacteristic
14C17×C4⋊Q8C2211yescharacteristic
15C17×#125C2211yescharacteristic
16#10587633C2211yescharacteristic
17#10587630C2211yescharacteristic
18C2×#63231C2212yes—
19C2×#63231C2212yes—
20C2×Dic68C2212yes—
21C2×#63219C2212yes—
22#10587584—22no—
23#10587583C2212yes—
24#10587582—22no—
25#10587580—22no—
26#10587576—22no—
27#10587573C2212yes—
28#10587561C2211yescharacteristic
29#10587558C2211yescharacteristic
30#10587555C2211yescharacteristic
31C17×#103C2212yes—
32C17×#103C2212yes—
33C17×#97C2211yescharacteristic
34#10587491C2212yes—
35#10587485C2212yes—
36#10587481C2212yes—
37#10587480C2212yes—
38C4×#3622C2211yescharacteristic
39C2×Q8×C17C2311yescharacteristic
40C2×Q8×C17C2311yescharacteristic
41#63256—42no—
42Q8×D17—42no—
43C2×D68—21no—
44C2×D68C2311yescharacteristic
45C2×C4×D17—21no—
46C2×C4×D17—21no—
47C2×C4×D17C2311yescharacteristic
48C2×Dic34—21no—
49C2×Dic34C2311yescharacteristic
50C2×Dic34C2312yes—
51C2×C8×C17C2312yes—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17C2312yes—
55C42×C17C2311yescharacteristic
56#63231—42no—
57#63231—42no—
58#63227—22no—
59#63227—22no—
60#63227—22no—
61#63227D412yes—
62C4⋊Dic17—22no—
63C4⋊Dic17—22no—
64#63225—22no—
65#63225—22no—
66#63225D412yes—
67C4×Dic17—22no—
68C4×Dic17C2311yescharacteristic
69C2×#3622C2312yes—
70Dic68—42no—
71#63219—42no—
72C22×D17—21no—
73C22×D17C2×D411yescharacteristic
74Q8×C17—22no—
75Q8×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17C2×D412yes—
80C2×C4×C17C2411yescharacteristic
81C2×Dic17—21no—
82C2×Dic17—22no—
83C2×Dic17C2×D411yescharacteristic
84C2×Dic17C2×D412yes—
85D68—22no—
86D68—42no—
87C4×D17—22no—
88C4×D17—42no—
89C4×D17—42no—
90Dic34—22no—
91Dic34—22no—
92Dic34—22no—
93Dic34—42no—
94C8×C17—22no—
95#3622—22no—
96#3223—171nomaximal, Sylow
97C22×C17C22×D411yescharacteristic
98C2×D17—21no—
99C2×D17—22no—
100C2×D17—41no—
101C2×D17—42no—
102C4×C17—22no—
103C4×C17—41no—
104C4×C17—41no—
105C4×C17—42no—
106C4×C17H(5,𝔽2)12yes—
107Dic17—21no—
108Dic17—22no—
109Dic17—41no—
110Dic17—42no—
111#541—171no—
112#536—171no—
113#497—171no—
114#494—171no—
115#491D1711yescharacteristic, complemented
116#488—171no—
117#464—172no—
118#462—172no—
119#452—172no—
120#451—172no—
121#433—171no—
122C2×C17#53511yescharacteristic
123C2×C17#57812yes—
124D17—41no—
125D17—81no—
126C2×#41—171no—
127C22×Q8—171no—
128C2×Q16—172no—
129C2×Q16—172no—
130C2×Q16—172no—
131C2×SD16—172no—
132C4⋊Q8—171no—
133C4⋊Q8—171no—
134C4⋊Q8C2×D1711yescharacteristic
135#126—342no—
136#125—342no—
137#125C2×D1711yescharacteristic
138#124—171no—
139#124—171no—
140#123—342no—
141C22⋊Q8—172no—
142C22⋊Q8—342no—
143C4×Q8—172no—
144#117—171no—
145C4⋊C8—172no—
146#103—172no—
147#103C2×D1712yes—
148#103C2×D1712yes—
149#102—172no—
150C22⋊C8—172no—
151#97C2×D1711yescharacteristic
152C4×C8—171no—
153C17#322311yescharacteristic, complemented, Sylow
154#41—682no—
155C2×Q8—171no—
156C2×Q8—172no—
157C2×Q8—341no—
158C2×Q8—682no—
159C2×Q8C22×D1711yescharacteristic
160C2×Q8C22×D1711yescharacteristic
161C2×D4—171no—
162C2×D4—341no—
163C22×C4—171no—
164C22×C4—341no—
165C22×C4—341no—
166Q16—682no—
167Q16—682no—
168Q16—682no—
169SD16—682no—
170C2×C8—172no—
171C2×C8C22×D1712yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—172no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4C22×D1712yes—
180C22⋊C4—172no—
181C22⋊C4—342no—
182C22⋊C4—342no—
183C22⋊C4—342no—
184C42—171no—
185C42—342no—
186C42C22×D1711yescharacteristic
187C23—171no—
188C23—341no—
189Q8—22no—
190Q8—22no—
191Q8—342no—
192Q8—342no—
193Q8—342no—
194Q8—682no—
195D4—342no—
196D4—682no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—172no—
201C2×C4—171no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—341no—
205C2×C4—682no—
206C2×C4—682no—
207C2×C4D4×D1712yes—
208C2×C4C23×D1711yescharacteristic
209C8—22no—
210C8—342no—
211C22—342no—
212C22—341no—
213C22—682no—
214C22—681no—
215C22C2×D4×D1711yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—342no—
221C4—341no—
222C4—681no—
223C4—682no—
224C4#1058769612yes—
225C2—681no—
226C2—1361no—
227C2#1176337711yescharacteristic
228C2#1176362512yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#491C2
socleC22×C17C22×D4
prime radicalC2×#63256C22
hypercenter#491D17
hypocenterC17#3223
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #11763422C2semidirect
3. #11763203C2semidirect
4. #11763188C2semidirect
5. #11763166C2semidirect
6. #11763034C2semidirect
7. #11763018C2semidirect
9. #11762966C2semidirect
10. #11762871C2semidirect
11. C17×#491C2semidirect
115. #491D17semidirect
153. C17#3223semidirect

Maximal subgroups

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

subgroupindex
#117634222
#117633842
#117632032
#117631882
#117631662
#117630342
#117630182
#117629742
#117629662
#117628712
C17×#4912
#322317