Finite Group Databasebeta

← #14540010

Subgroups — #14540010

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#11763338C211yesmaximal, characteristic
3#11763218C211yesmaximal, characteristic, complemented
4#11763215C211yesmaximal, characteristic, complemented
5#11763168C211yesmaximal, characteristic, complemented
6#11763143C212yesmaximal, complemented
7#11763126C212yesmaximal
8#11763092C212yesmaximal, complemented
9#11763087C212yesmaximal, complemented
10#11762864C211yesmaximal, characteristic, complemented
11C17×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D17—21no—
13C17×C4⋊Q8C2211yescharacteristic
14C17×#124C2211yescharacteristic
15#10587632C2211yescharacteristic
16Q8×Dic17C2211yescharacteristic
17#10587630—21no—
18#10587624C2211yescharacteristic
19D4×Dic17C2211yescharacteristic
20C4⋊C4×D17—21no—
21C4⋊C4×D17C2212yes—
22#10587574—21no—
23#10587574C2212yes—
24#10587569—21no—
25#10587567—21no—
26#10587565—21no—
27#10587563—21no—
28C42×D17C2211yescharacteristic
29#10587556C2211yescharacteristic
30#10587555C2211yescharacteristic
31C17×#103C2212yes—
32C17×#102C2212yes—
33C17×#97C2211yescharacteristic
34#10587506C2212yes—
35#10587503C2212yes—
36#10587491C2212yes—
37#10587489C2212yes—
38#10587488C2212yes—
39#10587485C2212yes—
40#10587479C2212yes—
41#10587478C2212yes—
42#10587474C2211yescharacteristic
43C2×Q8×C17—21no—
44C2×Q8×C17C2311yescharacteristic
45C2×D4×C17C2311yescharacteristic
46C2×C22⋊D17—21no—
47C22×Dic17—21no—
48Q8×D17—44no—
49C2×C4×D17—22no—
50C2×C4×D17D411yescharacteristic
51C2×C4×D17D411yescharacteristic
52C2×C4×D17C2311yescharacteristic
53C2×Dic34—21no—
54C2×Dic34—22no—
55C2×C8×C17C2312yes—
56C4⋊C4×C17—21no—
57C4⋊C4×C17C2312yes—
58C22⋊C4×C17—21no—
59C22⋊C4×C17—21no—
60C42×C17C2311yescharacteristic
61C22⋊Dic17—21no—
62C22⋊Dic17—21no—
63#63227—21no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17C2311yescharacteristic
67C4⋊Dic17C2311yescharacteristic
68C4⋊Dic17C2312yes—
69#63225—21no—
70#63225—21no—
71#63225—21no—
72#63225—22no—
73#63225—22no—
74C4×Dic17—21no—
75C4×Dic17D411yescharacteristic
76C4×Dic17D411yescharacteristic
77C4×Dic17C2311yescharacteristic
78C2×#3622C2312yes—
79C23×C17—21no—
80C22×D17C2×D411yescharacteristic
81Q8×C17—22no—
82Q8×C17—42no—
83D4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17C2×D411yescharacteristic
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2411yescharacteristic
89C22⋊D17—42no—
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—22no—
93C2×Dic17—42no—
94C2×Dic17C2×D411yescharacteristic
95C2×Dic17C2×D411yescharacteristic
96C2×Dic17C2×D411yescharacteristic
97C4×D17—22no—
98C4×D17—22no—
99C4×D17—22no—
100C4×D17—44no—
101Dic34—42no—
102Dic34—44no—
103C8×C17—22no—
104#3622—22no—
105#3239—171nomaximal, Sylow
106C22×C17—41no—
107C22×C17—41no—
108C22×C17—41no—
109C22×C17C22×D411yescharacteristic
110C2×D17—21no—
111C2×D17—21no—
112C2×D17#13712yes—
113C4×C17—21no—
114C4×C17—21no—
115C4×C17—41no—
116C4×C17—42no—
117C4×C172−1+412yes—
118Dic17—21no—
119Dic17—21no—
120Dic17—42no—
121Dic17—41no—
122Dic17—41no—
123Dic17AGL(1,ℤ/8)12yes—
124C2×C4⋊Q8—171no—
125#529—171no—
126#501—171no—
127#501—171no—
128#489—171no—
129#489D1711yescharacteristic, complemented
130#484—172no—
131#481—172no—
132#475—172no—
133#474—172no—
134#432—171no—
135C2×C17—81no—
136C2×C17#53611yescharacteristic
137C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C17C2×#13711yescharacteristic
139D17—22no—
140C22×Q8—341no—
141C4⋊Q8—172no—
142C4⋊Q8—171no—
143C4⋊Q8—341no—
144C4⋊Q8—341no—
145C4⋊Q8C2×D1711yescharacteristic
146#126—341no—
147#125—171no—
148#124C2×D1711yescharacteristic
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#107—172no—
161#106—172no—
162#106—172no—
163C4⋊C8—172no—
164#103—172no—
165#103C2×D1712yes—
166#102—172no—
167#102C2×D1712yes—
168C22⋊C8—172no—
169#97—171no—
170#97C2×D1711yescharacteristic
171C17#323911yescharacteristic, complemented, Sylow
172C2×Q8—21no—
173C2×Q8—342no—
174C2×Q8—341no—
175C2×Q8—684no—
176C2×Q8C22×D1711yescharacteristic
177C2×D4—341no—
178C2×D4C22×D1711yescharacteristic
179C22×C4—171no—
180C22×C4—171no—
181C22×C4—171no—
182C22×C4—342no—
183C22×C4—341no—
184C2×C8—172no—
185C2×C8C22×D1712yes—
186C4⋊C4—21no—
187C4⋊C4—172no—
188C4⋊C4—171no—
189C4⋊C4—171no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—341no—
193C4⋊C4—341no—
194C4⋊C4—341no—
195C4⋊C4—341no—
196C4⋊C4—341no—
197C4⋊C4C22×D1712yes—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—341no—
201C22⋊C4—341no—
202C22⋊C4—341no—
203C42—171no—
204C42—171no—
205C42—171no—
206C42—341no—
207C42C22×D1711yescharacteristic
208C23—21no—
209C23—171no—
210Q8—22no—
211Q8—42no—
212Q8—682no—
213Q8—684no—
214D4—22no—
215D4—682no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—171no—
219C2×C4—171no—
220C2×C4—171no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—341no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—684no—
228C2×C4—682no—
229C2×C4D4×D1711yescharacteristic
230C2×C4D4×D1711yescharacteristic
231C2×C4C23×D1711yescharacteristic
232C8—22no—
233C8—342no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—172no—
238C22—341no—
239C22—341no—
240C22C2×D4×D1711yescharacteristic
241C4—21no—
242C4—21no—
243C4—42no—
244C4—41no—
245C4—172no—
246C4—341no—
247C4—341no—
248C4—682no—
249C4—681no—
250C4—681no—
251C4#1058769712yes—
252C2—81no—
253C2—342no—
254C2#1176337911yescharacteristic
255C2AGL(1,ℤ/8)×D1711yescharacteristic
256C2#137×D1711yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#489C2
socleC22×C17C22×D4
prime radical#10587624C22
hypercenter#489D17
hypocenterC17#3239
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D17C2semidirect
3. #11763218C2semidirect
4. #11763215C2semidirect
5. #11763168C2semidirect
6. #11763143C2semidirect
8. #11763092C2semidirect
9. #11763087C2semidirect
10. #11762864C2semidirect
11. C17×#489C2semidirect
129. #489D17semidirect
171. C17#3239semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D172
#117633382
#117632182
#117632152
#117631682
#117631432
#117631262
#117630922
#117630872
#117628642
C17×#4892
#323917