Finite Group Databasebeta

← #31004485

Subgroups — #31004485

Order 17152.

3798 conjugacy classes of subgroups; only the 323 normal subgroups are shown.

normal subgroups only

This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 323 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
1C24×Dic134C211yesmaximal, characteristic
2C2×#16863551C216yesmaximal, complemented
3#22560143C216yesmaximal, complemented
4#22560119C211yesmaximal, characteristic
5C67×#2848C211yesmaximal, characteristic
6C23×Dic134—224no—
7C23×Dic134—23no—
8C23×Dic134C2216yes—
9C24×Dic67—21no—
10C22×#11761618C2214yescomplemented
11#16863551—224no—
12C2×#14523966C22112yescomplemented
13C2×C22⋊(C67⋊C8)C2216yes—
14C24×C4×C67C2211yescharacteristic
15C2×C22⋊Q8×C67C2213yes—
16#385×C67—23no—
17#380×C67C2213yes—
18#379×C67—21no—
19C22×Dic134—2128no—
20C22×Dic134—224no—
21C22×Dic134—248no—
22C22×Dic134—248no—
23C22×Dic134—212no—
24C22×Dic134—26no—
25C22×Dic134D4124yes—
26C22×Dic134C2314yes—
27C23×Dic67—224no—
28C23×Dic67—26no—
29C2×#11761618—224no—
30C22×C67⋊C8C2314yes—
31#14523966—224no—
32C22⋊(C67⋊C8)—212no—
33C25×C67D411yescharacteristic
34C22×Q8×C67C2311yescharacteristic
35C23×C4×C67—212no—
36C23×C4×C67D413yes—
37C23×C4×C67C2313yes—
38C22⋊Q8×C67—212no—
39C2×C4⋊C4×C67—23no—
40C2×C4⋊C4×C67C2313yes—
41C2×C22⋊C4×C67—23no—
42C2×C22⋊C4×C67—23no—
43#95×C67—23no—
44C2×Dic134—2192no—
45C2×Dic134—248no—
46C2×Dic134—296no—
47C2×Dic134—224no—
48C2×Dic134—212no—
49C2×Dic134—296no—
50C2×Dic134C2×D4124yes—
51C22×Dic67—264no—
52C22×Dic67—248no—
53C22×Dic67—224no—
54C22×Dic67—24no—
55#11761618—416no—
56C2×C67⋊C8—212no—
57C24×C67—26no—
58C24×C67—26no—
59C24×C67Q1614yescomplemented
60C24×C67C2×D413yes—
61C2×Q8×C67—26no—
62C22×C4×C67—232no—
63C22×C4×C67—26no—
64C22×C4×C67—212no—
65C22×C4×C67—212no—
66C22×C4×C67—21no—
67C22×C4×C67—23no—
68C22×C4×C67C2×D4112yes—
69C22×C4×C67C2×D413yes—
70C22×C4×C67C2411yescharacteristic
71C4⋊C4×C67—26no—
72C4⋊C4×C67—46no—
73C22⋊C4×C67—46no—
74C22⋊C4×C67—46no—
75C23×C67—28no—
76C23×C67—224no—
77C23×C67—26no—
78C23×C67—26no—
79C23×C67—224no—
80C23×C67#12016yes—
81C23×C67C2×Q1616yes—
82C23×C67#13716yes—
83C23×C67C22×D411yescharacteristic
84Q8×C67—44no—
85C2×C4×C67—224no—
86C2×C4×C67—26no—
87C2×C4×C67—224no—
88C2×C4×C67—43no—
89C2×C4×C67—43no—
90C2×C4×C67—46no—
91C2×C4×C67#12016yes—
92C2×C4×C67C22×D416yes—
93C2×Dic67—296no—
94C2×Dic67—224no—
95Dic134—296no—
96Dic134—224no—
97Dic134#120116yes—
98C67⋊C8—44no—
99C22×C67—232no—
100C22×C67—26no—
101C22×C67—212no—
102C22×C67—26no—
103C22×C67—212no—
104C22×C67#451112yescomplemented
105C22×C67C2×#12013yes—
106C22×C67C22×Q1611yescharacteristic
107C22×C67C2×#13713yes—
108C4×C67—26no—
109C4×C67—81no—
110C4×C67—83no—
111C4×C67C2×#12014yes—
112Dic67—216no—
113#32882—671nomaximal, Sylow
114C2×C67—212no—
115C2×C67#284611yescharacteristic
116C2×C67C2×#45113yes—
117C2×C67#300413yes—
118C24×Q8—671no—
119C2×#451—676no—
120#2848D6711yescharacteristic
121#1787—676no—
122#1779—671no—
123C67#3288211yescharacteristic, complemented, Sylow
124C23×Q8—676no—
125C23×Q8—13424no—
126C23×Q8—1343no—
127C24×C4—1341no—
128C24×C4C2×D6711yescharacteristic
129C22×Q16—674no—
130C2×C22⋊Q8C2×D6713yes—
131#451—13424no—
132C2×#103—6712no—
133C2×C22⋊C8—676no—
134#385—23no—
135#380C2×D6713yes—
136#379—21no—
137C25C22⋊D6711yescharacteristic
138C22×Q8—6724no—
139C22×Q8—674no—
140C22×Q8—13448no—
141C22×Q8—134128no—
142C22×Q8—13424no—
143C22×Q8—13448no—
144C22×Q8—13412no—
145C22×Q8—1346no—
146C22×Q8C22×D6711yescharacteristic
147C23×C4—212no—
148C23×C4—13424no—
149C23×C4—1346no—
150C23×C4C22⋊D6713yes—
151C23×C4C22×D6713yes—
152C2×Q16—13424no—
153C22×C8—674no—
154C22⋊Q8—212no—
155C2×C4⋊C4—23no—
156C2×C4⋊C4C22×D6713yes—
157C2×C22⋊C4—23no—
158C2×C22⋊C4—23no—
159#103—13424no—
160C22⋊C8—13412no—
161#95—23no—
162C24—26no—
163C24—26no—
164C24#1176161814yescomplemented
165C24C2×C22⋊D6713yes—
166C2×Q8—26no—
167C2×Q8—6724no—
168C2×Q8—13496no—
169C2×Q8—134192no—
170C2×Q8—13448no—
171C2×Q8—13496no—
172C2×Q8—13424no—
173C2×Q8—13412no—
174C22×C4—232no—
175C22×C4—26no—
176C22×C4—212no—
177C22×C4—212no—
178C22×C4—23no—
179C22×C4—21no—
180C22×C4—13464no—
181C22×C4—13448no—
182C22×C4—13424no—
183C22×C4—1344no—
184C22×C4D4×D67112yes—
185C22×C4C2×C22⋊D6713yes—
186C22×C4C23×D6711yescharacteristic
187Q16—26816no—
188C2×C8—13412no—
189C4⋊C4—26no—
190C4⋊C4—46no—
191C22⋊C4—46no—
192C22⋊C4—46no—
193C23—28no—
194C23—224no—
195C23—224no—
196C23—26no—
197C23—26no—
198C23#1452401116yes—
199C23C2×#1176161816yes—
200C23#1452407316yes—
201C23C22×C22⋊D6711yescharacteristic
202Q8—44no—
203Q8—6716no—
204Q8—13496no—
205Q8—13424no—
206C2×C4—224no—
207C2×C4—26no—
208C2×C4—224no—
209C2×C4—43no—
210C2×C4—43no—
211C2×C4—46no—
212C2×C4—13496no—
213C2×C4—13424no—
214C2×C4#1452401116yes—
215C2×C4C2×D4×D6716yes—
216C8—2684no—
217C22—232no—
218C22—26no—
219C22—212no—
220C22—26no—
221C22—212no—
222C22#16863551112yescomplemented
223C22C2×#1452401113yes—
224C22C22×#1176161811yescharacteristic
225C22C2×#1452407313yes—
226C4—26no—
227C4—81no—
228C4—83no—
229C4—13416no—
230C4#120×D6714yes—
231C2—212no—
232C2#2256848511yescharacteristic
233C2C2×#1686355113yes—
234C2#2256944213yes—
subgroupG/H
centerC23C22×C22⋊D67
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC67×#2848C2
socleC23×C67C22×D4
prime radicalC25×C67D4
hypercenter#2848D67
hypocenterC67#32882
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
2. C2×#16863551C2semidirect
3. #22560143C2semidirect
10. C22×#11761618C22semidirect
12. C2×#14523966C22semidirect
59. C24×C67Q16semidirect
104. C22×C67#451semidirect
123. C67#32882semidirect
164. C24#11761618semidirect
222. C22#16863551semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1342
C2×#168635512
#225601432
#225601192
C67×#28482
#3288267