Finite Group Databasebeta

← #31004488

Subgroups — #31004488

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×#16863554C216yesmaximal, complemented
3#22568496C211yesmaximal, characteristic
4#22560145C216yesmaximal, complemented
5C67×#1779C211yesmaximal, characteristic
6C23×Dic134—224no—
7C23×Dic134—23no—
8C23×Dic134C2216yes—
9C24×Dic67—21no—
10C22×Dic268C2214yescomplemented
11C2×#14524022C2213yes—
12#16863554—224no—
13C2×#14523968C22112yescomplemented
14#16863269—23no—
15#16863256C2213yes—
16#16863252—21no—
17C24×C4×C67C2211yescharacteristic
18C2×C22⋊C8×C67C2216yes—
19C22×Dic134—2128no—
20C22×Dic134—224no—
21C22×Dic134—248no—
22C22×Dic134—248no—
23C22×Dic134—212no—
24C22×Dic134—26no—
25C22×Dic134D4124yes—
26C22×Dic134C2311yescharacteristic
27C22×Dic134C2314yes—
28C23×Dic67—224no—
29C23×Dic67—26no—
30C2×Dic268—224no—
31#14524022—212no—
32C2×C4⋊Dic67C2313yes—
33C2×Dic67⋊C4—23no—
34C2×C22⋊Dic67—23no—
35C2×C22⋊Dic67—23no—
36#14523968—224no—
37C67⋊#95—23no—
38C25×C67D411yescharacteristic
39C23×C4×C67—212no—
40C23×C4×C67D413yes—
41C23×C4×C67C2313yes—
42C22×C8×C67C2314yes—
43C22⋊C8×C67—212no—
44C2×Dic134—2192no—
45C2×Dic134—248no—
46C2×Dic134—296no—
47C2×Dic134—224no—
48C2×Dic134—212no—
49C2×Dic134—296no—
50C2×Dic134—26no—
51C2×Dic134C2×D4124yes—
52C22×Dic67—264no—
53C22×Dic67—248no—
54C22×Dic67—224no—
55C22×Dic67—24no—
56C22×Dic67—23no—
57C22×Dic67—21no—
58Dic268—416no—
59C4⋊Dic67—26no—
60Dic67⋊C4—46no—
61C22⋊Dic67—46no—
62C22⋊Dic67—46no—
63C24×C67—26no—
64C24×C67—26no—
65C24×C67Q1614yescomplemented
66C24×C67C2×D413yes—
67C22×C4×C67—232no—
68C22×C4×C67—26no—
69C22×C4×C67—212no—
70C22×C4×C67—212no—
71C22×C4×C67C2×D4112yes—
72C22×C4×C67C2×D413yes—
73C22×C4×C67C2411yescharacteristic
74C2×C8×C67—212no—
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
84C2×C4×C67—224no—
85C2×C4×C67—26no—
86C2×C4×C67—224no—
87C2×C4×C67#12016yes—
88C2×C4×C67C22×D416yes—
89C2×Dic67—296no—
90C2×Dic67—224no—
91C2×Dic67—43no—
92C2×Dic67—43no—
93C2×Dic67—46no—
94Dic134—296no—
95Dic134—224no—
96Dic134—44no—
97Dic134#120116yes—
98C8×C67—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×C67C2×#12014yes—
110Dic67—216no—
111Dic67—81no—
112Dic67—83no—
113#32882—671nomaximal, Sylow
114C2×C67—212no—
115C2×C67#284611yescharacteristic
116C2×C67C2×#45113yes—
117C2×C67#300413yes—
118C24×Q8—671no—
119C2×#451—676no—
120#2848—671no—
121#1787—676no—
122#1779D6711yescharacteristic
123C67#3288211yescharacteristic, complemented, Sylow
124C23×Q8—676no—
125C23×Q8—13424no—
126C23×Q8—1343no—
127C24×C4—1341no—
128C24×C4C2×D6711yescharacteristic
129C22×Q16—674no—
130C2×C22⋊Q8—673no—
131#451—13424no—
132C2×#103—6712no—
133C2×C22⋊C8C2×D6716yes—
134#385—1343no—
135#380—673no—
136#379—1341no—
137C25D26811yescharacteristic
138C22×Q8—6724no—
139C22×Q8—674no—
140C22×Q8—671no—
141C22×Q8—13448no—
142C22×Q8—134128no—
143C22×Q8—13424no—
144C22×Q8—13448no—
145C22×Q8—13412no—
146C22×Q8—1346no—
147C23×C4—212no—
148C23×C4—13424no—
149C23×C4—1346no—
150C23×C4D26813yes—
151C23×C4C22×D6713yes—
152C2×Q16—13424no—
153C22×C8C22×D6714yes—
154C22⋊Q8—13412no—
155C2×C4⋊C4—673no—
156C2×C4⋊C4—1343no—
157C2×C22⋊C4—1343no—
158C2×C22⋊C4—1343no—
159#103—13424no—
160C22⋊C8—212no—
161#95—1343no—
162C24—26no—
163C24—26no—
164C24Dic26814yescomplemented
165C24C2×D26813yes—
166C2×Q8—6724no—
167C2×Q8—13496no—
168C2×Q8—134192no—
169C2×Q8—13448no—
170C2×Q8—13496no—
171C2×Q8—1346no—
172C2×Q8—13424no—
173C2×Q8—13412no—
174C22×C4—232no—
175C22×C4—26no—
176C22×C4—212no—
177C22×C4—212no—
178C22×C4—1343no—
179C22×C4—1341no—
180C22×C4—13464no—
181C22×C4—13448no—
182C22×C4—13424no—
183C22×C4—1344no—
184C22×C4D4×D67112yes—
185C22×C4C2×D26813yes—
186C22×C4C23×D6711yescharacteristic
187Q16—26816no—
188C2×C8—212no—
189C4⋊C4—2686no—
190C4⋊C4—1346no—
191C22⋊C4—2686no—
192C22⋊C4—2686no—
193C23—28no—
194C23—224no—
195C23—224no—
196C23—26no—
197C23—26no—
198C23#1452401316yes—
199C23C2×Dic26816yes—
200C23#1452407616yes—
201C23C22×D26811yescharacteristic
202Q8—2684no—
203Q8—6716no—
204Q8—13496no—
205Q8—13424no—
206C2×C4—224no—
207C2×C4—26no—
208C2×C4—224no—
209C2×C4—2683no—
210C2×C4—2683no—
211C2×C4—2686no—
212C2×C4—13496no—
213C2×C4—13424no—
214C2×C4#1452401316yes—
215C2×C4C2×D4×D6716yes—
216C8—44no—
217C22—232no—
218C22—26no—
219C22—212no—
220C22—26no—
221C22—212no—
222C22#16863554112yescomplemented
223C22C2×#1452401313yes—
224C22C22×Dic26811yescharacteristic
225C22C2×#1452407613yes—
226C4—26no—
227C4—5361no—
228C4—5363no—
229C4—13416no—
230C4#120×D6714yes—
231C2—212no—
232C2#2256848711yescharacteristic
233C2C2×#1686355413yes—
234C2#2256944613yes—
subgroupG/H
centerC23C22×D268
commutator, cosocleC22×C4×C67C24
FrattiniC22×C4C23×D67
FittingC67×#1779C2
socleC23×C67C22×D4
prime radicalC25×C67D4
hypercenter#1779D67
hypocenterC67#32882
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C2×#16863554C2semidirect
4. #22560145C2semidirect
10. C22×Dic268C22semidirect
13. C2×#14523968C22semidirect
65. C24×C67Q16semidirect
104. C22×C67#451semidirect
123. C67#32882semidirect
164. C24Dic268semidirect
222. C22#16863554semidirect

Maximal subgroups

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

subgroupindex
C24×Dic1342
C2×#168635542
#225684962
#225601452
C67×#17792
#3288267