Finite Group Databasebeta

← #14540006

Subgroups — #14540006

Order 2176.

362 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C4⋊Q8×D17C211yesmaximal, characteristic, complemented
2#11763344C211yesmaximal, characteristic
3#11763219C211yesmaximal, characteristic, complemented
4#11763171C211yesmaximal, characteristic, complemented
5#11763169C211yesmaximal, characteristic, complemented
6#11763141C212yesmaximal, complemented
7#11763124C212yesmaximal
8#11763103C212yesmaximal, complemented
9#11763098C212yesmaximal, complemented
10#11762864C211yesmaximal, characteristic, complemented
11C17×#501C211yesmaximal, characteristic, complemented
12C2×Q8×D17—21no—
13C17×C4⋊Q8C2211yescharacteristic
14C17×#125C2211yescharacteristic
15#10587630—21no—
16#10587584—21no—
17#10587583C2211yescharacteristic
18#10587581C2211yescharacteristic
19#10587580—21no—
20#10587579—21no—
21#10587579C2211yescharacteristic
22#10587578—21no—
23C4⋊C4×D17—21no—
24C4⋊C4×D17C2212yes—
25#10587574—21no—
26#10587574C2212yes—
27#10587573C2211yescharacteristic
28#10587561C2211yescharacteristic
29C42×D17C2211yescharacteristic
30#10587555C2211yescharacteristic
31C17×#107C2212yes—
32C17×#106C2212yes—
33C17×#97C2211yescharacteristic
34#10587492C2212yes—
35#10587491C2212yes—
36#10587487C2212yes—
37#10587485C2212yes—
38#10587481C2212yes—
39#10587480C2212yes—
40#10587479C2212yes—
41#10587478C2212yes—
42#10587474C2211yescharacteristic
43C2×Q8×C17—21no—
44Q8×D17—44no—
45C2×D68—21no—
46C2×D68C2311yescharacteristic
47C2×C4×D17—22no—
48C2×C4×D17—21no—
49C2×C4×D17D411yescharacteristic
50C2×C4×D17D411yescharacteristic
51C2×C4×D17C2311yescharacteristic
52C2×Dic34—21no—
53C2×Dic34—22no—
54C2×Dic34C2311yescharacteristic
55C2×C8×C17C2312yes—
56C4⋊C4×C17—21no—
57C4⋊C4×C17—21no—
58C4⋊C4×C17—21no—
59C4⋊C4×C17C2311yescharacteristic
60C4⋊C4×C17C2311yescharacteristic
61C4⋊C4×C17C2312yes—
62C42×C17C2311yescharacteristic
63#63227—21no—
64#63227—21no—
65#63227—21no—
66#63227—21no—
67#63227—21no—
68C4⋊Dic17—21no—
69C4⋊Dic17—21no—
70C4⋊Dic17C2312yes—
71#63225—21no—
72#63225—22no—
73#63225—22no—
74C4×Dic17—21no—
75C4×Dic17D411yescharacteristic
76C4×Dic17D411yescharacteristic
77C4×Dic17C2311yescharacteristic
78C2×#3622C2312yes—
79C22×D17—21no—
80C22×D17C2×D411yescharacteristic
81Q8×C17—42no—
82C2×C4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17C2×D411yescharacteristic
86C2×C4×C17C2×D411yescharacteristic
87C2×C4×C17C2411yescharacteristic
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17C2×D411yescharacteristic
91C2×Dic17C2×D411yescharacteristic
92C2×Dic17C2×D411yescharacteristic
93D68—22no—
94D68—42no—
95C4×D17—22no—
96C4×D17—22no—
97C4×D17—22no—
98C4×D17—44no—
99C4×D17—42no—
100Dic34—22no—
101Dic34—44no—
102Dic34—42no—
103C8×C17—22no—
104#3622—22no—
105#3238—171nomaximal, Sylow
106C22×C17C22×D411yescharacteristic
107C2×D17—21no—
108C2×D17—21no—
109C2×D17—41no—
110C2×D17—41no—
111C2×D17—41no—
112C2×D17AGL(1,ℤ/8)12yes—
113C4×C17—21no—
114C4×C17—21no—
115C4×C17—42no—
116C4×C17—41no—
117C4×C17—41no—
118C4×C172−1+412yes—
119Dic17—21no—
120Dic17—21no—
121Dic17—41no—
122Dic17—42no—
123Dic17#13712yes—
124C2×C4⋊Q8—171no—
125#529—171no—
126#501—171no—
127#501D1711yescharacteristic, complemented
128#489—171no—
129#489—171no—
130#483—172no—
131#480—172no—
132#477—172no—
133#476—172no—
134#432—171no—
135C2×C17#53611yescharacteristic
136C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C17C2×#13711yescharacteristic
138D17—22no—
139D17—81no—
140C22×Q8—341no—
141C4⋊Q8—172no—
142C4⋊Q8—171no—
143C4⋊Q8—341no—
144C4⋊Q8—341no—
145C4⋊Q8C2×D1711yescharacteristic
146#126—341no—
147#125C2×D1711yescharacteristic
148#124—171no—
149#123—341no—
150C22⋊Q8—171no—
151#121—171no—
152C4×Q8—171no—
153C4×D4—171no—
154C4×D4—341no—
155#117—341no—
156C2×C4⋊C4—172no—
157C2×C4⋊C4—341no—
158C2×C42—171no—
159#107—172no—
160#107C2×D1712yes—
161#106—172no—
162#106C2×D1712yes—
163C4⋊C8—172no—
164#103—172no—
165#103—172no—
166#102—172no—
167#102—172no—
168C22⋊C8—172no—
169#97—171no—
170#97C2×D1711yescharacteristic
171C17#323811yescharacteristic, complemented, Sylow
172C2×Q8—21no—
173C2×Q8—171no—
174C2×Q8—342no—
175C2×Q8—341no—
176C2×Q8—684no—
177C2×D4—171no—
178C2×D4—341no—
179C22×C4—171no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—341no—
183C22×C4—342no—
184C2×C8—172no—
185C2×C8C22×D1712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—172no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—341no—
193C4⋊C4—341no—
194C4⋊C4—341no—
195C4⋊C4C22×D1712yes—
196C4⋊C4C22×D1711yescharacteristic
197C4⋊C4C22×D1711yescharacteristic
198C22⋊C4—341no—
199C22⋊C4—341no—
200C22⋊C4—341no—
201C22⋊C4—341no—
202C22⋊C4—341no—
203C42—171no—
204C42—171no—
205C42—171no—
206C42—341no—
207C42C22×D1711yescharacteristic
208C23—171no—
209C23—341no—
210Q8—42no—
211Q8—342no—
212Q8—682no—
213Q8—684no—
214D4—342no—
215D4—682no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—171no—
220C2×C4—171no—
221C2×C4—171no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—682no—
228C2×C4—684no—
229C2×C4D4×D1711yescharacteristic
230C2×C4D4×D1711yescharacteristic
231C2×C4C23×D1711yescharacteristic
232C8—22no—
233C8—342no—
234C22—172no—
235C22—341no—
236C22—341no—
237C22—681no—
238C22—681no—
239C22—681no—
240C22C2×D4×D1711yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—172no—
247C4—341no—
248C4—341no—
249C4—682no—
250C4—681no—
251C4#1058769412yes—
252C2—342no—
253C2—1361no—
254C2#1176338411yescharacteristic
255C2AGL(1,ℤ/8)×D1711yescharacteristic
256C2#137×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#501C2
socleC22×C17C22×D4
prime radical#10587581C22
hypercenter#501D17
hypocenterC17#3238
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D17C2semidirect
3. #11763219C2semidirect
4. #11763171C2semidirect
5. #11763169C2semidirect
6. #11763141C2semidirect
8. #11763103C2semidirect
9. #11763098C2semidirect
10. #11762864C2semidirect
11. C17×#501C2semidirect
127. #501D17semidirect
171. C17#3238semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D172
#117633442
#117632192
#117631712
#117631692
#117631412
#117631242
#117631032
#117630982
#117628642
C17×#5012
#323817