Finite Group Databasebeta

← C3×#13096375 #72269341

Subgroups — C3×#13096375 #72269341

Order 4704.

434 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C3×#11716033C211yesmaximal, characteristic, complemented
2C3×C7×#1141C211yesmaximal, characteristic, complemented
3C3×#11716036C211yesmaximal, characteristic, complemented
4C3×C7×#1142C211yesmaximal, characteristic, complemented
5C3×#11716100C211yesmaximal, characteristic, complemented
6C3×#11716031C211yesmaximal, characteristic, complemented
7C3×D4×C7⋊D7C211yesmaximal, characteristic, complemented
8#13096375C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C7⋊D7—21no—
10C3×C72×D4C2211yescharacteristic
11C3×C22⋊(C7⋊D7)—21no—
12C3×C4⋊(C7⋊D7)C2211yescharacteristic
13C3×C4×C7⋊D7C2211yescharacteristic
14C3×C7×D28C2211yescharacteristic, complemented
15C3×C7×Dic14C2211yescharacteristic, complemented
16C3×#87641—21no—
17C3×D7×Dic7—21no—
18C3×C7×C7⋊C8C2211yescharacteristic, complemented
19C3×C7×C7⋊C8C2211yescharacteristic, complemented
20D4×C7⋊D7C2×C311yescharacteristic, complemented
21#11716100C2×C311yescharacteristic, complemented
22C7×#1142C2×C311yescharacteristic, complemented
23C7×#1141C2×C311yescharacteristic, complemented
24#11716036C2×C311yescharacteristic, complemented
25#11716033C2×C311yescharacteristic, complemented
26#11716031C2×C311yescharacteristic, complemented
27C3×#6477—71nomaximal
28C3×#6474—71nomaximal
29C22×C3×C72—21no—
30C2×C3×C7⋊D7—21no—
31C2×C3×C7⋊D7—42no—
32C2×C3×C7⋊D7D411yescharacteristic
33C2×C3×C7×D7—21no—
34C3×C4×C72C2311yescharacteristic
35C3×C7⋊Dic7D411yescharacteristic
36C3×C7×Dic7—21no—
37C22×C7⋊D7—21no—
38C72×D4C22×C311yescharacteristic
39C22⋊(C7⋊D7)—21no—
40C4⋊(C7⋊D7)C22×C311yescharacteristic
41C4×C7⋊D7C22×C311yescharacteristic
42C7×D28C22×C311yescharacteristic, complemented
43C7×Dic14C22×C311yescharacteristic, complemented
44#87641—21no—
45D7×Dic7—21no—
46C7×C7⋊C8C22×C311yescharacteristic, complemented
47C7×C7⋊C8C22×C311yescharacteristic, complemented
48C3×#1161—71no—
49C3×#1159—71no—
50C3×D4×D7—71no—
51C3×D4×D7—71no—
52C3×D4×D7—143no—
53C3×C7×SD16—71no—
54C3×C7×D8—71no—
55C3×#1143—71no—
56C3×#1142D711yescharacteristic, complemented
57C3×#1141—71no—
58C3×#1141D711yescharacteristic, complemented
59C3×D56—71no—
60C3×#1132—71no—
61C3×#1131—71no—
62C3×#1131—71no—
63C2×C3×C72—41no—
64C2×C3×C72C2×D411yescharacteristic
65C3×C7⋊D7—21no—
66C3×C7⋊D7—41no—
67C3×C7×D7—41no—
68#6477—71no—
69#6474—71no—
70C22×C72—21no—
71C2×C7⋊D7—21no—
72C2×C7⋊D7—42no—
73C2×C7⋊D7C3×D411yescharacteristic
74C2×C7×D7—21no—
75C4×C72C23×C311yescharacteristic
76C7⋊Dic7C3×D411yescharacteristic
77C7×Dic7—21no—
78C22×C3×D7—141no—
79C22×C3×D7—141no—
80C22×C3×D7—146no—
81C3×C7×Q8—71no—
82C3×C7×D4—23no—
83C3×C7×D4—71no—
84C3×C7×D4C2×D711yescharacteristic
85C3×C7×D4C2×D711yescharacteristic
86C3×C22⋊D7—141no—
87C3×C22⋊D7—141no—
88C3×C22⋊D7—141no—
89C3×C22⋊D7—146no—
90C2×C3×Dic7—141no—
91C3×D28—71no—
92C3×D28—71no—
93C3×D28—143no—
94C3×D28—141no—
95C3×D28C2×D711yescharacteristic, complemented
96C3×C4×D7—71no—
97C3×C4×D7—71no—
98C3×C4×D7—143no—
99C3×C4×D7—141no—
100C3×Dic14C2×D711yescharacteristic, complemented
101C3×C7×C8—71no—
102C3×C7×C8—71no—
103C3×C7⋊C8C2×D711yescharacteristic, complemented
104C3×C7⋊C8C2×D711yescharacteristic, complemented
105C3×C72AGL(1,ℤ/8)11yescharacteristic, complemented, Hall
106#1161—71no—
107#1159—71no—
108D4×D7—71no—
109D4×D7—71no—
110D4×D7—143no—
111C7×SD16—71no—
112C7×D8—71no—
113#1143—71no—
114#1142C3×D711yescharacteristic, complemented
115#1141—71no—
116#1141C3×D711yescharacteristic, complemented
117D56—71no—
118#1132—71no—
119#1131—71no—
120#1131—71no—
121C2×C72—41no—
122C2×C72C2×C3×D411yescharacteristic
123C7⋊D7—21no—
124C7⋊D7—41no—
125C7×D7—41no—
126C3×AGL(1,ℤ/8)—491noHall
127C22×C3×C7—26no—
128C22×C3×C7—21no—
129C22×C3×C7—21no—
130C22×C3×C7—141no—
131C2×C3×D7—21no—
132C2×C3×D7—71no—
133C2×C3×D7—71no—
134C2×C3×D7—143no—
135C2×C3×D7—146no—
136C2×C3×D7—141no—
137C2×C3×D7—141no—
138C2×C3×D7—2812no—
139C2×C3×D7—282no—
140C2×C3×D7—282no—
141C3×C4×C7—23no—
142C3×C4×C7—141no—
143C3×C4×C7C22×D711yescharacteristic
144C3×C4×C7C22×D711yescharacteristic
145C3×Dic7—21no—
146C3×Dic7—71no—
147C3×Dic7—71no—
148C3×Dic7—143no—
149C22×D7—141no—
150C22×D7—141no—
151C22×D7—146no—
152C7×Q8—71no—
153C7×D4—23no—
154C7×D4—71no—
155C7×D4C2×C3×D711yescharacteristic
156C7×D4C2×C3×D711yescharacteristic
157C22⋊D7—141no—
158C22⋊D7—141no—
159C22⋊D7—141no—
160C22⋊D7—146no—
161C2×Dic7—141no—
162D28—71no—
163D28—71no—
164D28—143no—
165D28—141no—
166D28C2×C3×D711yescharacteristic, complemented
167C4×D7—71no—
168C4×D7—71no—
169C4×D7—143no—
170C4×D7—141no—
171Dic14C2×C3×D711yescharacteristic, complemented
172C7×C8—71no—
173C7×C8—71no—
174C7⋊C8C2×C3×D711yescharacteristic, complemented
175C7⋊C8C2×C3×D711yescharacteristic, complemented
176C72C3×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
177C3×#41—491no—
178C2×C3×D4—491no—
179C3×SD16—71no—
180C3×SD16—491no—
181C3×D8—71no—
182C3×D8—491no—
183C3×M16—491no—
184C2×C3×C7—23no—
185C2×C3×C7—46no—
186C2×C3×C7—41no—
187C2×C3×C7—41no—
188C2×C3×C7—281no—
189C2×C3×C7D4×D711yescharacteristic
190C2×C3×C7D4×D711yescharacteristic
191C3×D7—41no—
192C3×D7—146no—
193C3×D7—141no—
194C3×D7—141no—
195C3×D7—286no—
196C3×D7—281no—
197C3×D7—281no—
198AGL(1,ℤ/8)—491noSylow
199C22×C7—26no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—141no—
203C2×D7—21no—
204C2×D7—71no—
205C2×D7—71no—
206C2×D7—143no—
207C2×D7—146no—
208C2×D7—141no—
209C2×D7—141no—
210C2×D7—2812no—
211C2×D7—282no—
212C2×D7—282no—
213C4×C7—23no—
214C4×C7—141no—
215C4×C7C22×C3×D711yescharacteristic
216C4×C7C22×C3×D711yescharacteristic
217Dic7—21no—
218Dic7—71no—
219Dic7—71no—
220Dic7—143no—
221C23×C3—981no—
222C3×Q8—71no—
223C3×D4—71no—
224C3×D4—491no—
225C3×D4—981no—
226C3×D4—981no—
227C3×D4D7211yescharacteristic
228C2×C3×C4—491no—
229C2×C3×C4—981no—
230C3×C8—71no—
231C3×C8—71no—
232C3×C7—23no—
233C3×C7#647411yescharacteristic, complemented
234C3×C7#647711yescharacteristic, complemented
235#41—491no—
236C2×D4—491no—
237SD16—71no—
238SD16—491no—
239D8—71no—
240D8—491no—
241M16—491no—
242C2×C7—23no—
243C2×C7—46no—
244C2×C7—41no—
245C2×C7—41no—
246C2×C7—281no—
247C2×C7C3×D4×D711yescharacteristic
248C2×C7C3×D4×D711yescharacteristic
249D7—41no—
250D7—146no—
251D7—141no—
252D7—141no—
253D7—286no—
254D7—281no—
255D7—281no—
256C22×C3—21no—
257C22×C3—141no—
258C22×C3—491no—
259C22×C3—981no—
260C22×C3—1962no—
261C3×C4—141no—
262C3×C4—491no—
263C3×C4C2×D7211yescharacteristic
264C23—981no—
265Q8—71no—
266D4—71no—
267D4—491no—
268D4—981no—
269D4—981no—
270D4C3×D7211yescharacteristic
271C2×C4—491no—
272C2×C4—981no—
273C8—71no—
274C8—71no—
275C7—23no—
276C7C3×#647411yescharacteristic, complemented
277C7C3×#647711yescharacteristic, complemented
278C2×C3—41no—
279C2×C3—281no—
280C2×C3—981no—
281C2×C3—1961no—
282C2×C3#1171611511yescharacteristic
283C22—21no—
284C22—141no—
285C22—491no—
286C22—981no—
287C22—1962no—
288C4—141no—
289C4—491no—
290C4C2×C3×D7211yescharacteristic
291C3#1309637511yescharacteristic, direct product factor, Sylow
292C2—41no—
293C2—281no—
294C2—981no—
295C2—1961no—
296C2C3×#1171611511yescharacteristic
subgroupG/H
centerC2×C3#11716115
commutator, cosocleC4×C72C23×C3
FrattiniC4C2×C3×D72
FittingC3×C72×D4C22
socleC2×C3×C72C2×D4
hypercenterC3×D4D72
hypocenterC72C3×AGL(1,ℤ/8)
LCSG ▷ C4×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#11716033C2semidirect
2. C3×C7×#1141C2semidirect
3. C3×#11716036C2semidirect
4. C3×C7×#1142C2semidirect
5. C3×#11716100C2semidirect
6. C3×#11716031C2semidirect
7. C3×D4×C7⋊D7C2semidirect
8. #13096375C3direct (N × H)
14. C3×C7×D28C22semidirect
15. C3×C7×Dic14C22semidirect
18. C3×C7×C7⋊C8C22semidirect
19. C3×C7×C7⋊C8C22semidirect
20. D4×C7⋊D7C2×C3semidirect
21. #11716100C2×C3semidirect
22. C7×#1142C2×C3semidirect
23. C7×#1141C2×C3semidirect
24. #11716036C2×C3semidirect
25. #11716033C2×C3semidirect
26. #11716031C2×C3semidirect
42. C7×D28C22×C3semidirect
43. C7×Dic14C22×C3semidirect
46. C7×C7⋊C8C22×C3semidirect
47. C7×C7⋊C8C22×C3semidirect
56. C3×#1142D7semidirect
58. C3×#1141D7semidirect
95. C3×D28C2×D7semidirect
100. C3×Dic14C2×D7semidirect
103. C3×C7⋊C8C2×D7semidirect
104. C3×C7⋊C8C2×D7semidirect
105. C3×C72AGL(1,ℤ/8)semidirect
114. #1142C3×D7semidirect
116. #1141C3×D7semidirect
166. D28C2×C3×D7semidirect
171. Dic14C2×C3×D7semidirect
174. C7⋊C8C2×C3×D7semidirect
175. C7⋊C8C2×C3×D7semidirect
176. C72C3×AGL(1,ℤ/8)semidirect
233. C3×C7#6474semidirect
234. C3×C7#6477semidirect
276. C7C3×#6474semidirect
277. C7C3×#6477semidirect
291. C3#13096375direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#117160332
C3×C7×#11412
C3×#117160362
C3×C7×#11422
C3×#117161002
C3×#117160312
C3×D4×C7⋊D72
#130963753
C3×#64777
C3×#64747