Finite Group Databasebeta

← #72218399

Subgroups — #72218399

Order 3360.

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

normal subgroups only

This group has 304 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×D5×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic
2(C22⋊(C7⋊AGL(1,𝔽5)))⋊C3C211yesmaximal, characteristic
3ΓL(1,𝔽8)×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
4(ΓL(1,𝔽8)⋊C4)⋊AGL(1,𝔽5)C212yesmaximal, complemented
5AGL(1,𝔽5)×ΓL(1,𝔽8)⋊C4C212yesmaximal, complemented
6#11765326C311yesmaximal, characteristic, complemented, Hall
7C22×D5×ΓL(1,𝔽8)C2211yescharacteristic
8C2×ΓL(1,𝔽8)⋊AGL(1,𝔽5)C2212yes—
9C2×AGL(1,𝔽5)×ΓL(1,𝔽8)C2212yes—
10C2×ΓL(1,𝔽8)⋊Dic5C411yescharacteristic, complemented
11C2×C5×ΓL(1,𝔽8)⋊C4C411yescharacteristic, complemented
12D5×ΓL(1,𝔽8)⋊C4C411yescharacteristic, complemented
13D5×ΓL(1,𝔽8)⋊C4C411yescharacteristic, complemented
14D5×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
15D5×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
16#6440⋊C3—51nomaximal, Hall
17C2×D5×Dic7C2×C311yescharacteristic
18C22⋊(C7⋊AGL(1,𝔽5))C2×C311yescharacteristic
19C7×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
20Dic7⋊AGL(1,𝔽5)C2×C312yescomplemented
21AGL(1,𝔽5)×Dic7C2×C312yescomplemented
22C3×#4130—71nomaximal, Hall
23C22×C5×ΓL(1,𝔽8)C2×C411yescharacteristic
24C2×D5×ΓL(1,𝔽8)—22no—
25C2×D5×ΓL(1,𝔽8)C2×C411yescharacteristic
26C2×D5×ΓL(1,𝔽8)C2311yescharacteristic
27ΓL(1,𝔽8)⋊AGL(1,𝔽5)—22no—
28AGL(1,𝔽5)×ΓL(1,𝔽8)—22no—
29ΓL(1,𝔽8)⋊Dic5C2×C411yescharacteristic
30ΓL(1,𝔽8)⋊Dic5C2×C411yescharacteristic
31C5×ΓL(1,𝔽8)⋊C4C2×C411yescharacteristic
32C5×ΓL(1,𝔽8)⋊C4C2×C411yescharacteristic
33C22×ΓL(1,𝔽8)⋊C4—51no—
34C22⋊C4×ΓL(1,𝔽8)—51no—
35(C22⋊Dic7)⋊C3—51no—
36(ΓL(1,𝔽8)⋊C4)⋊C4—52no—
37C4×ΓL(1,𝔽8)⋊C4—52no—
38C22×C7×D5C22×C311yescharacteristic
39C2×C7⋊AGL(1,𝔽5)C22×C312yes—
40C2×C7×AGL(1,𝔽5)C22×C312yes—
41C2×Dic35C3×C411yescharacteristic, complemented
42C2×C5×Dic7C3×C411yescharacteristic, complemented
43D5×Dic7C3×C411yescharacteristic, complemented
44D5×Dic7C3×C411yescharacteristic, complemented
45D5×Dic7C22×C311yescharacteristic
46D5×Dic7C22×C311yescharacteristic
47C2×C3×C4×D5—71no—
48C3×C22⋊AGL(1,𝔽5)—71no—
49C3×C22⋊AGL(1,𝔽5)—71no—
50C3×C4⋊AGL(1,𝔽5)—72no—
51C3×C4×AGL(1,𝔽5)—72no—
52#6440—51noHall
53C2×C5×ΓL(1,𝔽8)—21no—
54C2×C5×ΓL(1,𝔽8)C22×C411yescharacteristic
55D5×ΓL(1,𝔽8)—21no—
56D5×ΓL(1,𝔽8)#4112yes—
57C23×ΓL(1,𝔽8)—51no—
58C2×C4×ΓL(1,𝔽8)—52no—
59C2×ΓL(1,𝔽8)⋊C4—51no—
60C2×ΓL(1,𝔽8)⋊C4—51no—
61C2×ΓL(1,𝔽8)⋊C4—51no—
62C2×ΓL(1,𝔽8)⋊C4—51no—
63C2×ΓL(1,𝔽8)⋊C4—51no—
64C2×ΓL(1,𝔽8)⋊C4—52no—
65C2×ΓL(1,𝔽8)⋊C4AGL(1,𝔽5)11yescharacteristic, complemented
66#4130—71noHall
67C22×C5×C7C2×C3×C411yescharacteristic
68C2×C7×D5—22no—
69C2×C7×D5C2×C3×C411yescharacteristic
70C2×C7×D5C23×C311yescharacteristic
71C7⋊AGL(1,𝔽5)—22no—
72C7×AGL(1,𝔽5)—22no—
73Dic35C2×C3×C411yescharacteristic
74Dic35C2×C3×C411yescharacteristic
75C5×Dic7C2×C3×C411yescharacteristic
76C5×Dic7C2×C3×C411yescharacteristic
77C22×C3×D5—71no—
78C2×C3×AGL(1,𝔽5)—72no—
79C2×C3×AGL(1,𝔽5)—72no—
80C2×C3×C4×C5—71no—
81C2×C3×Dic5—71no—
82C3×C4×D5—71no—
83C3×C4×D5—71no—
84C3×C4×D5—71no—
85C3×C4×D5—71no—
86C22×Dic7—51no—
87C7×C22⋊C4—51no—
88C22⋊Dic7—51no—
89Dic7⋊C4—52no—
90C4×Dic7—52no—
91C5×ΓL(1,𝔽8)#11711yescharacteristic, complemented, Hall
92C3×#117—351noHall
93C22×ΓL(1,𝔽8)—51no—
94C22×ΓL(1,𝔽8)—51no—
95C22×ΓL(1,𝔽8)—102no—
96C22×ΓL(1,𝔽8)C2×AGL(1,𝔽5)11yescharacteristic
97C4×ΓL(1,𝔽8)—102no—
98ΓL(1,𝔽8)⋊C4—51no—
99ΓL(1,𝔽8)⋊C4—51no—
100ΓL(1,𝔽8)⋊C4—102no—
101ΓL(1,𝔽8)⋊C4C2×AGL(1,𝔽5)11yescharacteristic
102ΓL(1,𝔽8)⋊C4C2×AGL(1,𝔽5)11yescharacteristic
103C2×C4×D5—71no—
104C22⋊AGL(1,𝔽5)—71no—
105C22⋊AGL(1,𝔽5)AGL(1,𝔽7)11yescharacteristic
106C4⋊AGL(1,𝔽5)—72no—
107C4×AGL(1,𝔽5)—72no—
108C2×C5×C7—21no—
109C2×C5×C7C22×C3×C411yescharacteristic
110C7×D5—21no—
111C7×D5C3×#4112yes—
112C22×C3×C5—71no—
113C2×C3×D5—71no—
114C2×C3×D5—71no—
115C2×C3×D5—142no—
116C3×AGL(1,𝔽5)—142no—
117C3×AGL(1,𝔽5)—142no—
118C3×C4×C5—71no—
119C3×C4×C5—71no—
120C3×Dic5—71no—
121C3×Dic5—71no—
122C23×C7—51no—
123C2×C4×C7—52no—
124C2×Dic7—51no—
125C2×Dic7—51no—
126C2×Dic7—51no—
127C2×Dic7—51no—
128C2×Dic7—51no—
129C2×Dic7—52no—
130C2×Dic7C3×AGL(1,𝔽5)11yescharacteristic, complemented
131C22×C3×C4—351no—
132C3×C4⋊C4—352no—
133C3×C22⋊C4—351no—
134C3×C22⋊C4—351no—
135C3×C42—352no—
136C2×ΓL(1,𝔽8)—21no—
137C2×ΓL(1,𝔽8)—52no—
138C2×ΓL(1,𝔽8)—101no—
139C2×ΓL(1,𝔽8)C22×AGL(1,𝔽5)11yescharacteristic
140C22×D5C2×AGL(1,𝔽7)11yescharacteristic
141C2×AGL(1,𝔽5)—72no—
142C2×AGL(1,𝔽5)C2×AGL(1,𝔽7)12yes—
143C2×C4×C5—71no—
144C2×Dic5—71no—
145C4×D5—71no—
146C4×D5—71no—
147C4×D5—71no—
148C4×D5—71no—
149C5×C7C3×#11711yescharacteristic, complemented, Hall
150#117—351noSylow
151C2×C3×C5—71no—
152C2×C3×C5—141no—
153C3×D5—72no—
154C3×D5—141no—
155C22×C7—51no—
156C22×C7—51no—
157C22×C7—102no—
158C22×C7C2×C3×AGL(1,𝔽5)11yescharacteristic
159C4×C7—102no—
160Dic7—51no—
161Dic7—51no—
162Dic7—102no—
163Dic7C2×C3×AGL(1,𝔽5)11yescharacteristic
164Dic7C2×C3×AGL(1,𝔽5)11yescharacteristic
165C23×C3—351no—
166C2×C3×C4—71no—
167C2×C3×C4—352no—
168C2×C3×C4—351no—
169C2×C3×C4—351no—
170C2×C3×C4—351no—
171C2×C3×C4—351no—
172C2×C3×C4—351no—
173C2×C3×C4—352no—
174ΓL(1,𝔽8)#413011yescharacteristic, complemented, Hall
175C22×C5C4×AGL(1,𝔽7)11yescharacteristic
176C2×D5—22no—
177C2×D5C4×AGL(1,𝔽7)11yescharacteristic
178C2×D5C22×AGL(1,𝔽7)11yescharacteristic
179AGL(1,𝔽5)—22no—
180AGL(1,𝔽5)—142no—
181C4×C5—71no—
182C4×C5—71no—
183Dic5—71no—
184Dic5—71no—
185C22×C4—351no—
186C4⋊C4—352no—
187C22⋊C4—51no—
188C22⋊C4—351no—
189C42—352no—
190C3×C5—71noHall
191C2×C7—21no—
192C2×C7—52no—
193C2×C7—101no—
194C2×C7C22×C3×AGL(1,𝔽5)11yescharacteristic
195C22×C3—71no—
196C22×C3—351no—
197C22×C3—351no—
198C22×C3—702no—
199C3×C4—71no—
200C3×C4—71no—
201C3×C4—351no—
202C3×C4—351no—
203C3×C4—702no—
204C3×C4—702no—
205C2×C5—21no—
206C2×C5C2×C4×AGL(1,𝔽7)11yescharacteristic
207D5—21no—
208D5#1159⋊C312yes—
209C23—51no—
210C2×C4—52no—
211C2×C4—71no—
212C2×C4—351no—
213C2×C4—351no—
214C2×C4—351no—
215C2×C4—351no—
216C2×C4—351no—
217C2×C4—352no—
218C7C3×#413011yescharacteristic, complemented, Sylow
219C2×C3—71no—
220C2×C3—141no—
221C2×C3—352no—
222C2×C3—701no—
223C5#6440⋊C311yescharacteristic, complemented, Sylow
224C22—51no—
225C22—51no—
226C22—102no—
227C22AGL(1,𝔽5)×AGL(1,𝔽7)11yescharacteristic
228C4—71no—
229C4—71no—
230C4—102no—
231C4—351no—
232C4—351no—
233C4—702no—
234C3—71noSylow
235C2—21no—
236C2—52no—
237C2—101no—
238C2C2×AGL(1,𝔽5)×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽5)×AGL(1,𝔽7)
commutator, socleC2×C5×C7C22×C3×C4
FittingC22×C5×C7C2×C3×C4
prime radicalC22×D5×ΓL(1,𝔽8)C22
cosocleC2×C7×D5C23×C3
hypercenterC22AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C3×#117
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
4. (ΓL(1,𝔽8)⋊C4)⋊AGL(1,𝔽5)C2semidirect
5. AGL(1,𝔽5)×ΓL(1,𝔽8)⋊C4C2semidirect
6. #11765326C3semidirect
10. C2×ΓL(1,𝔽8)⋊Dic5C4semidirect
11. C2×C5×ΓL(1,𝔽8)⋊C4C4semidirect
12. D5×ΓL(1,𝔽8)⋊C4C4semidirect
13. D5×ΓL(1,𝔽8)⋊C4C4semidirect
20. Dic7⋊AGL(1,𝔽5)C2×C3semidirect
21. AGL(1,𝔽5)×Dic7C2×C3semidirect
41. C2×Dic35C3×C4semidirect
42. C2×C5×Dic7C3×C4semidirect
43. D5×Dic7C3×C4semidirect
44. D5×Dic7C3×C4semidirect
65. C2×ΓL(1,𝔽8)⋊C4AGL(1,𝔽5)semidirect
91. C5×ΓL(1,𝔽8)#117semidirect
130. C2×Dic7C3×AGL(1,𝔽5)semidirect
149. C5×C7C3×#117semidirect
174. ΓL(1,𝔽8)#4130semidirect
218. C7C3×#4130semidirect
223. C5#6440⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D5×ΓL(1,𝔽8)⋊C42
(C22⋊(C7⋊AGL(1,𝔽5)))⋊C32
ΓL(1,𝔽8)×C22⋊AGL(1,𝔽5)2
(ΓL(1,𝔽8)⋊C4)⋊AGL(1,𝔽5)2
AGL(1,𝔽5)×ΓL(1,𝔽8)⋊C42
#117653263
#6440⋊C35
C3×#41307