Finite Group Databasebeta

← #89504⋊C3 #13051632

Subgroups — #89504⋊C3 #13051632

Order 1344.

396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C22×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2#6493⋊C3C211yesmaximal, characteristic, complemented
3C2×D56⋊C3C212yesmaximal, complemented
4ΓL(1,𝔽8)×C22⋊C8C211yesmaximal, characteristic, complemented
5#6395⋊C3C212yesmaximal, complemented
6#89504C311yesmaximal, characteristic, complemented, Hall
7C23×AGL(1,𝔽7)—21no—
8C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
9C2×(C22⋊D7)⋊C3—21no—
10C2×C4⋊AGL(1,𝔽7)—21no—
11C2×C4⋊AGL(1,𝔽7)—24no—
12C2×C4⋊AGL(1,𝔽7)C2212yes—
13C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic, complemented
14C2×C8×ΓL(1,𝔽8)C2212yescomplemented
15(C2×AGL(1,𝔽7))⋊C4—21no—
16C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic, complemented
17D56⋊C3—24no—
18C22×D28C2×C311yescharacteristic, complemented
19#6493C2×C311yescharacteristic, complemented
20C2×D56C2×C312yescomplemented
21C7×C22⋊C8C2×C311yescharacteristic, complemented
22#6395C2×C312yescomplemented
23C3×#447—71nomaximal, Hall
24C23×ΓL(1,𝔽8)D411yescharacteristic
25C22×AGL(1,𝔽7)—24no—
26C22×AGL(1,𝔽7)—22no—
27C22×AGL(1,𝔽7)—21no—
28C22×AGL(1,𝔽7)—42no—
29C22×AGL(1,𝔽7)—42no—
30C2×C4×ΓL(1,𝔽8)—22no—
31C2×C4×ΓL(1,𝔽8)D411yescharacteristic
32C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
33(C22⋊D7)⋊C3—42no—
34C2×ΓL(1,𝔽8)⋊C4—21no—
35C4⋊AGL(1,𝔽7)—24no—
36C4⋊AGL(1,𝔽7)—22no—
37C4⋊AGL(1,𝔽7)—22no—
38C4⋊AGL(1,𝔽7)D414yescomplemented
39C8×ΓL(1,𝔽8)—22no—
40C23×D7—21no—
41C22×C4×C7C22×C311yescharacteristic
42C2×C22⋊D7—21no—
43C2×D28—21no—
44C2×D28—24no—
45C2×D28C22×C312yes—
46C2×D28C22×C311yescharacteristic, complemented
47C2×C7×C8C22×C312yescomplemented
48(C2×D7)⋊C4—21no—
49C4⋊Dic7C22×C311yescharacteristic, complemented
50D56—24no—
51C22×C3×D4—71no—
52C2×C3×D8—72no—
53C3×#121—71no—
54C3×#102—72no—
55C3×C22⋊C8—71no—
56C22×ΓL(1,𝔽8)—21no—
57C22×ΓL(1,𝔽8)—21no—
58C22×ΓL(1,𝔽8)D812yescomplemented
59C22×ΓL(1,𝔽8)C2×D411yescharacteristic
60C2×AGL(1,𝔽7)—24no—
61C2×AGL(1,𝔽7)—48no—
62C2×AGL(1,𝔽7)—42no—
63C2×AGL(1,𝔽7)—42no—
64C2×AGL(1,𝔽7)—41no—
65C2×AGL(1,𝔽7)—41no—
66C2×AGL(1,𝔽7)—41no—
67C4×ΓL(1,𝔽8)—21no—
68C4×ΓL(1,𝔽8)C2×D412yes—
69ΓL(1,𝔽8)⋊C4—41no—
70#447—71noSylow
71C23×C7C3×D411yescharacteristic
72C22×D7—24no—
73C22×D7—22no—
74C22×D7—21no—
75C22×D7—42no—
76C22×D7—42no—
77C2×C4×C7—22no—
78C2×C4×C7C3×D411yescharacteristic
79C2×C4×C7C23×C311yescharacteristic
80C22⋊D7—42no—
81C2×Dic7—21no—
82D28—24no—
83D28—22no—
84D28—22no—
85D28C3×D414yescomplemented
86C7×C8—22no—
87C24×C3—141no—
88C2×C3×D4—72no—
89C2×C3×D4—71no—
90C2×C3×D4—141no—
91C2×C3×D4—144no—
92C2×C3×D4—141no—
93C22×C3×C4—71no—
94C3×D8—144no—
95C2×C3×C8—72no—
96C3×C4⋊C4—71no—
97C3×C22⋊C4—141no—
98C2×ΓL(1,𝔽8)—22no—
99C2×ΓL(1,𝔽8)#12011yescharacteristic
100C2×ΓL(1,𝔽8)C2×D811yescharacteristic
101C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
102AGL(1,𝔽7)—44no—
103AGL(1,𝔽7)—81no—
104C22×D4—71no—
105C2×D8—72no—
106#121—71no—
107#102—72no—
108C22⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
109C22×C7—21no—
110C22×C7—21no—
111C22×C7C3×D812yescomplemented
112C22×C7C2×C3×D411yescharacteristic
113C2×D7—24no—
114C2×D7—48no—
115C2×D7—42no—
116C2×D7—42no—
117C2×D7—41no—
118C2×D7—41no—
119C2×D7—41no—
120C4×C7—21no—
121C4×C7C2×C3×D412yes—
122Dic7—41no—
123C23×C3—71no—
124C23×C3—142no—
125C23×C3—141no—
126C23×C3—144no—
127C23×C3—282no—
128C23×C3—282no—
129C3×D4—74no—
130C3×D4—142no—
131C3×D4—142no—
132C3×D4—144no—
133C3×D4—282no—
134C2×C3×C4—71no—
135C2×C3×C4—71no—
136C2×C3×C4—142no—
137C2×C3×C4—141no—
138C3×C8—142no—
139ΓL(1,𝔽8)#44711yescharacteristic, complemented, Hall
140C24—141no—
141C2×D4—72no—
142C2×D4—71no—
143C2×D4—144no—
144C2×D4—141no—
145C2×D4—141no—
146C22×C4C2×AGL(1,𝔽7)11yescharacteristic
147D8—144no—
148C2×C8C2×AGL(1,𝔽7)12yescomplemented
149C4⋊C4—71no—
150C22⋊C4—141no—
151C2×C7—22no—
152C2×C7C3×#12011yescharacteristic
153C2×C7C2×C3×D811yescharacteristic
154C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
155D7—44no—
156D7—81no—
157C22×C3—71no—
158C22×C3—72no—
159C22×C3—141no—
160C22×C3—141no—
161C22×C3—144no—
162C22×C3—282no—
163C22×C3—282no—
164C22×C3—281no—
165C22×C3—281no—
166C22×C3—281no—
167C22×C3—288no—
168C3×C4—72no—
169C3×C4—141no—
170C3×C4—281no—
171C23—144no—
172C23—142no—
173C23—141no—
174C23—282no—
175C23—282no—
176C23C4⋊AGL(1,𝔽7)11yescharacteristic
177D4—74no—
178D4—144no—
179D4—142no—
180D4—142no—
181D4—282no—
182C2×C4—22no—
183C2×C4—141no—
184C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
185C2×C4C22×AGL(1,𝔽7)11yescharacteristic
186C8—22no—
187C7C3×#44711yescharacteristic, complemented, Sylow
188C2×C3—71no—
189C2×C3—71no—
190C2×C3—71no—
191C2×C3—142no—
192C2×C3—284no—
193C2×C3—561no—
194C22—21no—
195C22—21no—
196C22—144no—
197C22—288no—
198C22—282no—
199C22—282no—
200C22—281no—
201C22—281no—
202C22—281no—
203C22D56⋊C312yescomplemented
204C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
205C4—21no—
206C4—281no—
207C4D4×AGL(1,𝔽7)12yes—
208C3—71noSylow
209C2—22no—
210C2—284no—
211C2—561no—
212C2#6445⋊C311yescharacteristic
213C2C2×D56⋊C311yescharacteristic
214C2#6471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C22⋊C8C2×C3
socleC22×C7C2×C3×D4
hypercenterC22⋊C8AGL(1,𝔽7)
hypocenterC7C3×#447
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊AGL(1,𝔽7)C2semidirect
2. #6493⋊C3C2semidirect
3. C2×D56⋊C3C2semidirect
4. ΓL(1,𝔽8)×C22⋊C8C2semidirect
5. #6395⋊C3C2semidirect
6. #89504C3semidirect
13. C2×C4⋊AGL(1,𝔽7)C22semidirect
14. C2×C8×ΓL(1,𝔽8)C22semidirect
16. C4⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
18. C22×D28C2×C3semidirect
19. #6493C2×C3semidirect
20. C2×D56C2×C3semidirect
21. C7×C22⋊C8C2×C3semidirect
22. #6395C2×C3semidirect
38. C4⋊AGL(1,𝔽7)D4semidirect
46. C2×D28C22×C3semidirect
47. C2×C7×C8C22×C3semidirect
49. C4⋊Dic7C22×C3semidirect
58. C22×ΓL(1,𝔽8)D8semidirect
85. D28C3×D4semidirect
108. C22⋊C8AGL(1,𝔽7)semidirect
111. C22×C7C3×D8semidirect
139. ΓL(1,𝔽8)#447semidirect
148. C2×C8C2×AGL(1,𝔽7)semidirect
187. C7C3×#447semidirect
203. C22D56⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊AGL(1,𝔽7)2
#6493⋊C32
C2×D56⋊C32
ΓL(1,𝔽8)×C22⋊C82
#6395⋊C32
#895043
C3×#4477