Finite Group Databasebeta

← #14539860

Subgroups — #14539860

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#11763420C211yesmaximal, characteristic, complemented
2#11763384C211yesmaximal, characteristic
3#11763201C211yesmaximal, characteristic, complemented
4#11763177C211yesmaximal, characteristic, complemented
5#11763164C211yesmaximal, characteristic, complemented
6#11763033C212yesmaximal, complemented
7#11763017C212yesmaximal, complemented
8#11762970C212yesmaximal
9#11762963C212yesmaximal, complemented
10#11762868C211yesmaximal, characteristic, complemented
11C17×#494C211yesmaximal, characteristic, complemented
12C2×#63256C2211yescharacteristic
13C2×Q8×D17C2211yescharacteristic
14C2×Q16×C17C2212yes—
15C2×Q16×C17C2212yes—
16C17×C4⋊Q8C2211yescharacteristic
17C17×C4⋊Q8C2211yescharacteristic
18#10587633C2211yescharacteristic
19#10587632C2212yes—
20Q8×Dic17C2212yes—
21#10587630C2211yescharacteristic
22C2×#63231C2212yes—
23C2×#63230C2212yes—
24#10587584—22no—
25#10587582—22no—
26#10587580—22no—
27#10587575—22no—
28#10587561C2211yescharacteristic
29#10587558C2211yescharacteristic
30#10587556C2211yescharacteristic
31C4×C8×C17C2211yescharacteristic
32#10587506C2212yes—
33#10587506C2212yes—
34#10587492C2212yes—
35#10587491C2212yes—
36#10587487C2212yes—
37#10587485C2212yes—
38#10587474C2211yescharacteristic
39C2×Q8×C17C2311yescharacteristic
40C2×Q8×C17C2311yescharacteristic
41C2×Q8×C17C2312yes—
42#63256—42no—
43Q8×D17—42no—
44C2×D68—21no—
45C2×D68C2311yescharacteristic
46C2×C4×D17—21no—
47C2×C4×D17—21no—
48C2×C4×D17C2311yescharacteristic
49C2×Dic34—21no—
50C2×Dic34C2311yescharacteristic
51Q16×C17—42no—
52Q16×C17—42no—
53C2×C8×C17C2312yes—
54C4⋊C4×C17—22no—
55C4⋊C4×C17—22no—
56C42×C17C2311yescharacteristic
57#63231—42no—
58#63230—42no—
59#63227—22no—
60#63227—22no—
61#63227—22no—
62#63227D412yes—
63C4⋊Dic17—22no—
64C4⋊Dic17C2312yes—
65#63225—22no—
66#63225—22no—
67#63225—22no—
68#63225D412yes—
69C4×Dic17—22no—
70C4×Dic17C2311yescharacteristic
71C2×#3622C2312yes—
72C22×D17—21no—
73C22×D17C2×D411yescharacteristic
74Q8×C17—22no—
75Q8×C17—22no—
76Q8×C17—22no—
77Q8×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2411yescharacteristic
83C2×Dic17—21no—
84C2×Dic17—22no—
85C2×Dic17C2×D411yescharacteristic
86C2×Dic17C2×D412yes—
87D68—22no—
88D68—42no—
89C4×D17—22no—
90C4×D17—42no—
91C4×D17—42no—
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#494D1711yescharacteristic, complemented
115#491—171no—
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×Q16C2×D1712yes—
130C2×Q16C2×D1712yes—
131C2×SD16—172no—
132C4⋊Q8—171no—
133C4⋊Q8C2×D1711yescharacteristic
134C4⋊Q8C2×D1711yescharacteristic
135#126—342no—
136#125—171no—
137#125—342no—
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#103—172no—
148#103—172no—
149#102—172no—
150C22⋊C8—172no—
151#97—171no—
152C4×C8C2×D1711yescharacteristic
153C17#322311yescharacteristic, complemented, Sylow
154#41—682no—
155C2×Q8—171no—
156C2×Q8—341no—
157C2×Q8—682no—
158C2×Q8C22×D1711yescharacteristic
159C2×Q8C22×D1711yescharacteristic
160C2×Q8C22×D1712yes—
161C2×D4—171no—
162C2×D4—341no—
163C22×C4—171no—
164C22×C4—341no—
165C22×C4—341no—
166Q16—42no—
167Q16—42no—
168Q16—682no—
169SD16—682no—
170C2×C8—172no—
171C2×C8C22×D1712yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—172no—
175C4⋊C4—172no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4—342no—
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—22no—
192Q8—22no—
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—41no—
218C4—41no—
219C4—42no—
220C4—342no—
221C4—341no—
222C4—682no—
223C4—681no—
224C4#1058769112yes—
225C2—681no—
226C2—1361no—
227C2#1176337211yescharacteristic
228C2#1176362212yes—
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#494C2
socleC22×C17C22×D4
prime radicalC2×#63256C22
hypercenter#494D17
hypocenterC17#3223
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #11763420C2semidirect
3. #11763201C2semidirect
4. #11763177C2semidirect
5. #11763164C2semidirect
6. #11763033C2semidirect
7. #11763017C2semidirect
9. #11762963C2semidirect
10. #11762868C2semidirect
11. C17×#494C2semidirect
114. #494D17semidirect
153. C17#3223semidirect

Maximal subgroups

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

subgroupindex
#117634202
#117633842
#117632012
#117631772
#117631642
#117630332
#117630172
#117629702
#117629632
#117628682
C17×#4942
#322317