Finite Group Databasebeta

← #33083358

Subgroups — #33083358

Order 18688.

2036 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989354C211yesmaximal, characteristic
2#22623134C211yesmaximal, characteristic
3C2×#17988343C2112yesmaximal, complemented
4C73×#3428C211yesmaximal, characteristic
5#17989354—24no—
6#17989354—24no—
7C2×#14556598C418yescomplemented
8C22×#11924012C2216yes—
9#17989070—24no—
10#17989065—21no—
11#17989062—23no—
12#41×Dic73C418yescomplemented
13C2×#14556550C2216yes—
14#17988343C22116yes—
15C22×#41×C73C2211yescharacteristic
16#520×C73—24no—
17#519×C73—21no—
18#518×C73—23no—
19C4×#41×C73C418yescomplemented
20C2×#117×C73C2216yes—
21#14556598—232no—
22C2×#11924012—212no—
23C2×#11924012—212no—
24C2×#11924012C2×C4112yescomplemented
25C2×#11924012C2314yes—
26C22×#10596831C2×C4112yescomplemented
27Q8×Dic73—28no—
28D4×Dic73—224no—
29#14556550—26no—
30#14556550—26no—
31#14556550C2×C4112yescomplemented
32#14556550C2314yes—
33C2×C4⋊Dic73—26no—
34C2×C22⋊Dic73—26no—
35C2×C4×Dic73C2×C4112yescomplemented
36C2×#41×C73C2×C414yes—
37C2×#41×C73C2×C414yes—
38C2×#41×C73C2×C414yes—
39C2×#41×C73D414yes—
40C2×#41×C73D414yes—
41C2×#41×C73Q814yes—
42C22×Q8×C73D411yescharacteristic
43C22×D4×C73D413yes—
44C23×C4×C73C2313yes—
45C4×Q8×C73—28no—
46C4×D4×C73—224no—
47#117×C73—26no—
48#117×C73—26no—
49#117×C73C2×C4112yescomplemented
50#117×C73C2314yes—
51C2×C4⋊C4×C73—26no—
52C2×C22⋊C4×C73—26no—
53C2×C42×C73C2×C4112yescomplemented
54C22×Dic73—212no—
55#11924012—224no—
56#11924012—24no—
57#11924012—24no—
58C2×#10596831—224no—
59C2×#10596831C22×C418yes—
60C4⋊Dic73—212no—
61C4⋊Dic73—24no—
62C22⋊Dic73—212no—
63C22⋊Dic73—24no—
64C4×Dic73—212no—
65C4×Dic73C22×C418yes—
66C24×C73C2×D413yes—
67#41×C73C42116yes—
68#41×C73C22⋊C4116yes—
69#41×C73C4⋊C4116yes—
70#41×C73C4⋊C4116yes—
71C2×Q8×C73C22⋊C414yes—
72C2×Q8×C73C22⋊C414yes—
73C2×Q8×C73C4⋊C414yes—
74C2×D4×C73C22⋊C4112yes—
75C2×D4×C73C22⋊C4112yes—
76C2×D4×C73C4⋊C4112yes—
77C22×C4×C73—212no—
78C22×C4×C73C22×C416yes—
79C22×C4×C73C22×C416yes—
80C22×C4×C73C22×C416yes—
81C22×C4×C73C2×D413yes—
82C22×C4×C73C2×D416yes—
83C22×C4×C73C2×D416yes—
84C22×C4×C73C2×Q816yes—
85C22×C4×C73C2411yescharacteristic
86C4⋊C4×C73—212no—
87C4⋊C4×C73—24no—
88C22⋊C4×C73—212no—
89C22⋊C4×C73—24no—
90C42×C73—212no—
91C42×C73C22×C418yes—
92C23×C73—83no—
93C23×C73C2×C22⋊C416yes—
94C23×C73C2×C22⋊C416yes—
95C23×C73C2×C4⋊C416yes—
96C23×C73C22×D411yescharacteristic
97Q8×C73#95116yes—
98D4×C73#95148yes—
99C2×C4×C73—28no—
100C2×C4×C73—812no—
101C2×C4×C73C2×C42112yes—
102C2×C4×C73C2×C22⋊C4112yes—
103C2×C4×C73C2×C22⋊C416yes—
104C2×C4×C73C2×C22⋊C416yes—
105C2×C4×C73C2×C4⋊C4112yes—
106C2×C4×C73C2×C4⋊C416yes—
107C2×C4×C73C2×C4⋊C4112yes—
108C2×C4×C73C23×C411yescharacteristic
109C2×C4×C73C23×C411yescharacteristic
110C2×C4×C73C23×C411yescharacteristic
111C2×C4×C73C22×D411yescharacteristic
112C2×C4×C73C22×D411yescharacteristic
113C2×C4×C73C22×Q811yescharacteristic
114C2×Dic73—28no—
115C2×Dic73—812no—
116#10596831—28no—
117C22×C73—46no—
118C22×C73—46no—
119C22×C73—41no—
120C22×C73—46no—
121C22×C73C2×#95112yes—
122C22×C73C22×C22⋊C411yescharacteristic
123C22×C73C22×C22⋊C411yescharacteristic
124C22×C73C22×C4⋊C411yescharacteristic
125C4×C73—48no—
126C4×C73C2×#95112yes—
127C4×C73C22×C4211yescharacteristic
128C4×C73C22×C22⋊C411yescharacteristic
129C4×C73C22×C4⋊C411yescharacteristic
130C4×C73C22×C4⋊C411yescharacteristic
131Dic73—48no—
132#20338—731nomaximal, Sylow
133C2×C73—21no—
134C2×C73—21no—
135C2×C73—21no—
136C2×C73—212no—
137C2×C73C22×#9511yescharacteristic
138C2×#568—731no—
139#3428—731no—
140#3428D7311yescharacteristic
141C2×H(3,ℤ/4)—7312no—
142C73#2033811yescharacteristic, complemented, Sylow
143C22×#41C2×D7311yescharacteristic
144#568—1464no—
145#568—1464no—
146C2×#131—738no—
147C22×M16—736no—
148#520—24no—
149#520—1464no—
150#519—21no—
151#519—1461no—
152#518—23no—
153#518—1463no—
154C4×#41—738no—
155C4×#41Dic7318yescomplemented
156C2×#117—736no—
157C2×#117C2×D7316yes—
158H(3,ℤ/4)—7316no—
159C2×#41Dic14614yes—
160C2×#41C4×D7314yes—
161C2×#41C4×D7314yes—
162C2×#41D29214yes—
163C2×#41C2×Dic7314yes—
164C2×#41C22⋊D7314yes—
165C22×Q8C22⋊D7311yescharacteristic
166C22×D4C22⋊D7313yes—
167C23×C4C22×D7313yes—
168#131—14632no—
169C2×M16—7312no—
170C2×M16—734no—
171C2×M16—14612no—
172C2×M16—14612no—
173C22×C8—7312no—
174C4×Q8—28no—
175C4×Q8—1468no—
176C4×D4—224no—
177C4×D4—14624no—
178#117—26no—
179#117—26no—
180#117—7312no—
181#117—734no—
182#117—1466no—
183#117—1466no—
184#117C2×Dic73112yescomplemented
185#117C22×D7314yes—
186C2×C4⋊C4—26no—
187C2×C4⋊C4—1466no—
188C2×C22⋊C4—26no—
189C2×C22⋊C4—1466no—
190C2×C42—7312no—
191C2×C42C2×Dic73112yescomplemented
192C24C2×C22⋊D7313yes—
193#41C4×Dic73116yes—
194#41#11924005116yes—
195#41#11924007116yes—
196#41C4⋊Dic73116yes—
197C2×Q8#1192400514yes—
198C2×Q8C22⋊Dic7314yes—
199C2×Q8#1192400714yes—
200C2×D4#11924005112yes—
201C2×D4C22⋊Dic73112yes—
202C2×D4#11924007112yes—
203C22×C4—212no—
204C22×C4—14612no—
205C22×C4C2×C4×D7316yes—
206C22×C4C2×C4×D7316yes—
207C22×C4C22×Dic7316yes—
208C22×C4C2×C22⋊D7313yes—
209C22×C4C2×C22⋊D7316yes—
210C22×C4C2×D29216yes—
211C22×C4C2×Dic14616yes—
212C22×C4C23×D7311yescharacteristic
213M16—14624no—
214M16—1464no—
215M16—1464no—
216C2×C8—738no—
217C2×C8—14624no—
218C4⋊C4—212no—
219C4⋊C4—24no—
220C4⋊C4—14612no—
221C4⋊C4—1464no—
222C22⋊C4—212no—
223C22⋊C4—24no—
224C22⋊C4—14612no—
225C22⋊C4—1464no—
226C42—212no—
227C42—738no—
228C42—14612no—
229C42C22×Dic7318yes—
230C23—83no—
231C23C2×#1192400516yes—
232C23C2×C22⋊Dic7316yes—
233C23C2×#1192400716yes—
234C23C22×C22⋊D7311yescharacteristic
235Q8#14556498116yes—
236D4#14556498148yes—
237C2×C4—28no—
238C2×C4—812no—
239C2×C4—58412no—
240C2×C4—1468no—
241C2×C4C2×C4×Dic73112yes—
242C2×C4C2×#1192400516yes—
243C2×C4C2×#11924005112yes—
244C2×C4C2×C22⋊Dic7316yes—
245C2×C4C2×#1192400716yes—
246C2×C4C2×#11924007112yes—
247C2×C4C2×C4⋊Dic73112yes—
248C2×C4C22×C4×D7311yescharacteristic
249C2×C4C22×C4×D7311yescharacteristic
250C2×C4C23×Dic7311yescharacteristic
251C2×C4C22×C22⋊D7311yescharacteristic
252C2×C4C22×D29211yescharacteristic
253C2×C4C22×Dic14611yescharacteristic
254C8—1468no—
255C22—46no—
256C22—46no—
257C22—41no—
258C22—46no—
259C22C2×#14556498112yes—
260C22C22×#1192400511yescharacteristic
261C22C22×C22⋊Dic7311yescharacteristic
262C22C22×#1192400711yescharacteristic
263C4—48no—
264C4—2928no—
265C4C2×#14556498112yes—
266C4C22×C4×Dic7311yescharacteristic
267C4C22×#1192400511yescharacteristic
268C4C22×#1192400711yescharacteristic
269C4C22×C4⋊Dic7311yescharacteristic
270C2—21no—
271C2—21no—
272C2—21no—
273C2—212no—
274C2C22×#1455649811yescharacteristic
subgroupG/H
centerC4C22×C4×Dic73
commutatorC4×C73C22×C42
FrattiniC22×C4C23×D73
FittingC73×#3428C2
socleC2×C73C22×#95
prime radicalC22×#41×C73C22
cosocleC22×C4×C73C24
hypercenter#3428D73
hypocenterC73#20338
LCSG ▷ C4×C73 ▷ C73
UCSC1 ◁ C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
3. C2×#17988343C2semidirect
7. C2×#14556598C4semidirect
12. #41×Dic73C4semidirect
19. C4×#41×C73C4semidirect
24. C2×#11924012C2×C4semidirect
26. C22×#10596831C2×C4semidirect
31. #14556550C2×C4semidirect
35. C2×C4×Dic73C2×C4semidirect
49. #117×C73C2×C4semidirect
53. C2×C42×C73C2×C4semidirect
142. C73#20338semidirect
155. C4×#41Dic73semidirect
184. #117C2×Dic73semidirect
191. C2×C42C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179893542
#226231342
C2×#179883432
C73×#34282
#2033873