Finite Group Databasebeta

← C3×#6380⋊C3 #72246155

Subgroups — C3×#6380⋊C3 #72246155

Order 2016.

312 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 86 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
1C2×C3×(C22⋊D7)⋊C3C211yesmaximal, characteristic
2C3×C22⋊C4×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C3×(C22⋊Dic7)⋊C3C211yesmaximal, characteristic, complemented
4C3×#6380C311yesmaximal, characteristic, complemented
5#6380⋊C3C311yesmaximal, characteristic, direct product factor
6#6380⋊C3C312yesmaximal, direct product factor
7C23×C3×ΓL(1,𝔽8)C2211yescharacteristic
8C22×C3×AGL(1,𝔽7)C411yescharacteristic, complemented
9C2×C3×C4×ΓL(1,𝔽8)—21no—
10C3×(C22⋊D7)⋊C3—21no—
11C3×(C22⋊D7)⋊C3—21no—
12C2×C3×ΓL(1,𝔽8)⋊C4—21no—
13C2×C3×ΓL(1,𝔽8)⋊C4C411yescharacteristic, complemented
14C2×C3×C22⋊D7C2×C311yescharacteristic
15C2×(C22⋊D7)⋊C3C2×C311yescharacteristic
16C2×(C22⋊D7)⋊C3C2×C312yes—
17C3×C7×C22⋊C4C2×C311yescharacteristic, complemented
18C3×C22⋊Dic7C2×C311yescharacteristic, complemented
19C22⋊C4×ΓL(1,𝔽8)C2×C311yescharacteristic, complemented
20C22⋊C4×ΓL(1,𝔽8)C2×C312yescomplemented
21(C22⋊Dic7)⋊C3C2×C311yescharacteristic, complemented
22(C22⋊Dic7)⋊C3C2×C312yescomplemented
23C32×#99—71nomaximal, Hall
24C22×C3×ΓL(1,𝔽8)—41no—
25C22×C3×ΓL(1,𝔽8)C2×C411yescharacteristic
26C22×C3×ΓL(1,𝔽8)D411yescharacteristic
27C22×C3×ΓL(1,𝔽8)D411yescharacteristic
28C2×C3×AGL(1,𝔽7)—21no—
29C2×C3×AGL(1,𝔽7)—41no—
30C3×C4×ΓL(1,𝔽8)—41no—
31C3×ΓL(1,𝔽8)⋊C4—21no—
32C3×ΓL(1,𝔽8)⋊C4—41no—
33#6380C3211yescharacteristic, complemented, Hall
34C23×C3×C7C22×C311yescharacteristic
35C22×C3×D7C3×C411yescharacteristic, complemented
36C23×ΓL(1,𝔽8)C22×C311yescharacteristic
37C23×ΓL(1,𝔽8)C22×C312yes—
38C22×AGL(1,𝔽7)C3×C411yescharacteristic, complemented
39C22×AGL(1,𝔽7)C3×C412yescomplemented
40C2×C3×C4×C7—21no—
41C3×C22⋊D7—21no—
42C3×C22⋊D7—21no—
43C2×C3×Dic7—21no—
44C2×C3×Dic7C3×C411yescharacteristic, complemented
45C2×C4×ΓL(1,𝔽8)—21no—
46C2×C4×ΓL(1,𝔽8)—22no—
47(C22⋊D7)⋊C3—21no—
48(C22⋊D7)⋊C3—21no—
49(C22⋊D7)⋊C3—22no—
50(C22⋊D7)⋊C3—22no—
51C2×ΓL(1,𝔽8)⋊C4—21no—
52C2×ΓL(1,𝔽8)⋊C4—22no—
53C2×ΓL(1,𝔽8)⋊C4C3×C411yescharacteristic, complemented
54C2×ΓL(1,𝔽8)⋊C4C3×C412yescomplemented
55C2×C32×D4—71no—
56C32×C22⋊C4—71no—
57C32×C22⋊C4—141no—
58C2×C3×ΓL(1,𝔽8)—21no—
59C2×C3×ΓL(1,𝔽8)—21no—
60C2×C3×ΓL(1,𝔽8)—21no—
61C2×C3×ΓL(1,𝔽8)C22⋊C411yescharacteristic
62C3×AGL(1,𝔽7)—41no—
63C2×C22⋊D7C2×C3211yescharacteristic
64C7×C22⋊C4C2×C3211yescharacteristic, complemented
65C22⋊Dic7C2×C3211yescharacteristic, complemented
66C3×#99—71no—
67C3×#99—72no—
68C3×#99—71no—
69C22×C3×C7—41no—
70C22×C3×C7C2×C3×C411yescharacteristic
71C22×C3×C7C3×D411yescharacteristic
72C22×C3×C7C3×D411yescharacteristic
73C2×C3×D7—21no—
74C2×C3×D7—41no—
75C22×ΓL(1,𝔽8)—41no—
76C22×ΓL(1,𝔽8)—42no—
77C22×ΓL(1,𝔽8)C2×C3×C411yescharacteristic
78C22×ΓL(1,𝔽8)C2×C3×C412yes—
79C22×ΓL(1,𝔽8)C3×D411yescharacteristic
80C22×ΓL(1,𝔽8)C3×D411yescharacteristic
81C22×ΓL(1,𝔽8)C3×D412yes—
82C22×ΓL(1,𝔽8)C3×D412yes—
83C2×AGL(1,𝔽7)—21no—
84C2×AGL(1,𝔽7)—22no—
85C2×AGL(1,𝔽7)—41no—
86C2×AGL(1,𝔽7)—42no—
87C3×C4×C7—41no—
88C3×Dic7—21no—
89C3×Dic7—41no—
90C4×ΓL(1,𝔽8)—41no—
91C4×ΓL(1,𝔽8)—42no—
92ΓL(1,𝔽8)⋊C4—21no—
93ΓL(1,𝔽8)⋊C4—22no—
94ΓL(1,𝔽8)⋊C4—41no—
95ΓL(1,𝔽8)⋊C4—42no—
96C23×C32—71no—
97C23×C32—141no—
98C32×D4—141no—
99C32×D4—141no—
100C2×C32×C4—71no—
101C2×C32×C4—141no—
102C2×C32×C4—141no—
103C3×ΓL(1,𝔽8)#9911yescharacteristic, complemented, Hall
104C23×C7C22×C3211yescharacteristic
105C22×D7C32×C411yescharacteristic, complemented
106C2×C4×C7—21no—
107C22⋊D7—21no—
108C22⋊D7—21no—
109C2×Dic7—21no—
110C2×Dic7C32×C411yescharacteristic, complemented
111C2×C3×D4—71no—
112C2×C3×D4—72no—
113C2×C3×D4—71no—
114C3×C22⋊C4—71no—
115C3×C22⋊C4—72no—
116C3×C22⋊C4—71no—
117C3×C22⋊C4—71no—
118C3×C22⋊C4—142no—
119C3×C22⋊C4AGL(1,𝔽7)11yescharacteristic, complemented
120C2×C3×C7—21no—
121C2×C3×C7—21no—
122C2×C3×C7—21no—
123C2×C3×C7C3×C22⋊C411yescharacteristic
124C3×D7—41no—
125C2×ΓL(1,𝔽8)—21no—
126C2×ΓL(1,𝔽8)—21no—
127C2×ΓL(1,𝔽8)—22no—
128C2×ΓL(1,𝔽8)—21no—
129C2×ΓL(1,𝔽8)—22no—
130C2×ΓL(1,𝔽8)—22no—
131C2×ΓL(1,𝔽8)C3×C22⋊C411yescharacteristic
132C2×ΓL(1,𝔽8)C3×C22⋊C412yes—
133AGL(1,𝔽7)—41no—
134AGL(1,𝔽7)—42no—
135C22×C32—141no—
136C22×C32—141no—
137C22×C32—141no—
138C22×C32—141no—
139C22×C32—281no—
140C22×C32—281no—
141C32×C4—141no—
142C32×C4—281no—
143C32×C4—281no—
144#99—71noSylow
145C22×C7—41no—
146C22×C7C2×C32×C411yescharacteristic
147C22×C7C32×D411yescharacteristic
148C22×C7C32×D411yescharacteristic
149C2×D7—21no—
150C2×D7—41no—
151C4×C7—41no—
152Dic7—21no—
153Dic7—41no—
154C23×C3—71no—
155C23×C3—72no—
156C23×C3—71no—
157C23×C3—71no—
158C23×C3—142no—
159C23×C3C2×AGL(1,𝔽7)11yescharacteristic
160C3×D4—141no—
161C3×D4—141no—
162C3×D4—142no—
163C3×D4—142no—
164C3×D4—141no—
165C3×D4—141no—
166C2×C3×C4—21no—
167C2×C3×C4—71no—
168C2×C3×C4—72no—
169C2×C3×C4—71no—
170C2×C3×C4—141no—
171C2×C3×C4—142no—
172C2×C3×C4—141no—
173C2×C3×C4—142no—
174C2×C3×C4—141no—
175C3×C7C3×#9911yescharacteristic, complemented
176ΓL(1,𝔽8)C3×#9911yescharacteristic, complemented
177ΓL(1,𝔽8)C3×#9912yescomplemented
178C2×C32—141no—
179C2×C32—141no—
180C2×C32—281no—
181C2×C32—281no—
182C2×C32—281no—
183C2×D4—71no—
184C22⋊C4—71no—
185C22⋊C4C3×AGL(1,𝔽7)11yescharacteristic, complemented
186C2×C7—21no—
187C2×C7—21no—
188C2×C7—21no—
189C2×C7C32×C22⋊C411yescharacteristic
190D7—41no—
191C22×C3—41no—
192C22×C3—71no—
193C22×C3—71no—
194C22×C3—71no—
195C22×C3—141no—
196C22×C3—142no—
197C22×C3—141no—
198C22×C3—142no—
199C22×C3—142no—
200C22×C3—142no—
201C22×C3—281no—
202C22×C3—282no—
203C22×C3—281no—
204C22×C3—281no—
205C22×C3—282no—
206C22×C3C4×AGL(1,𝔽7)11yescharacteristic
207C22×C3C4⋊AGL(1,𝔽7)11yescharacteristic
208C22×C3(C22⋊D7)⋊C311yescharacteristic
209C3×C4—41no—
210C3×C4—141no—
211C3×C4—142no—
212C3×C4—141no—
213C3×C4—281no—
214C3×C4—282no—
215C3×C4—281no—
216C3×C4—281no—
217C3×C4—282no—
218C32—141noSylow
219C23—71no—
220C23C2×C3×AGL(1,𝔽7)11yescharacteristic
221D4—141no—
222D4—141no—
223C2×C4—21no—
224C2×C4—71no—
225C2×C4—141no—
226C7C32×#9911yescharacteristic, complemented, Sylow
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—71no—
231C2×C3—141no—
232C2×C3—141no—
233C2×C3—141no—
234C2×C3—142no—
235C2×C3—142no—
236C2×C3—282no—
237C2×C3—282no—
238C2×C3—281no—
239C2×C3—282no—
240C2×C3—281no—
241C2×C3(C2×AGL(1,𝔽7))⋊C411yescharacteristic
242C22—41no—
243C22—141no—
244C22—281no—
245C22C3×C4×AGL(1,𝔽7)11yescharacteristic
246C22C3×C4⋊AGL(1,𝔽7)11yescharacteristic
247C22C3×(C22⋊D7)⋊C311yescharacteristic
248C4—41no—
249C4—141no—
250C4—281no—
251C3—71no—
252C3—142no—
253C3#6380⋊C311yescharacteristic, direct product factor
254C2—21no—
255C2—21no—
256C2—21no—
257C2—281no—
258C2C3×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
subgroupG/H
centerC2×C3(C2×AGL(1,𝔽7))⋊C4
commutatorC22×C7C2×C32×C4
FrattiniC23C2×C3×AGL(1,𝔽7)
FittingC3×C7×C22⋊C4C2×C3
socleC2×C3×C7C3×C22⋊C4
prime radicalC2×C3×(C22⋊D7)⋊C3C2
cosocleC23×C7C22×C32
hypercenterC3×C22⋊C4AGL(1,𝔽7)
hypocenterC7C32×#99
LCSG ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×C22⋊C4×ΓL(1,𝔽8)C2semidirect
3. C3×(C22⋊Dic7)⋊C3C2semidirect
4. C3×#6380C3semidirect
5. #6380⋊C3C3direct (N × H)
6. #6380⋊C3C3direct (N × H)
8. C22×C3×AGL(1,𝔽7)C4semidirect
13. C2×C3×ΓL(1,𝔽8)⋊C4C4semidirect
17. C3×C7×C22⋊C4C2×C3semidirect
18. C3×C22⋊Dic7C2×C3semidirect
19. C22⋊C4×ΓL(1,𝔽8)C2×C3semidirect
20. C22⋊C4×ΓL(1,𝔽8)C2×C3semidirect
21. (C22⋊Dic7)⋊C3C2×C3semidirect
22. (C22⋊Dic7)⋊C3C2×C3semidirect
33. #6380C32semidirect
35. C22×C3×D7C3×C4semidirect
38. C22×AGL(1,𝔽7)C3×C4semidirect
39. C22×AGL(1,𝔽7)C3×C4semidirect
44. C2×C3×Dic7C3×C4semidirect
53. C2×ΓL(1,𝔽8)⋊C4C3×C4semidirect
54. C2×ΓL(1,𝔽8)⋊C4C3×C4semidirect
64. C7×C22⋊C4C2×C32semidirect
65. C22⋊Dic7C2×C32semidirect
103. C3×ΓL(1,𝔽8)#99semidirect
105. C22×D7C32×C4semidirect
110. C2×Dic7C32×C4semidirect
119. C3×C22⋊C4AGL(1,𝔽7)semidirect
175. C3×C7C3×#99semidirect
176. ΓL(1,𝔽8)C3×#99semidirect
177. ΓL(1,𝔽8)C3×#99semidirect
185. C22⋊C4C3×AGL(1,𝔽7)semidirect
226. C7C32×#99semidirect
253. C3#6380⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×(C22⋊D7)⋊C32
C3×C22⋊C4×ΓL(1,𝔽8)2
C3×(C22⋊Dic7)⋊C32
C3×#63803
#6380⋊C33
C32×#997