Finite Group Databasebeta

← #13057485

Subgroups — #13057485

Order 1344.

516 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 186 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
1#10621519C211yesmaximal, characteristic, complemented
2#10621518C211yesmaximal, characteristic, complemented
3ΓL(1,𝔽8)×#131C211yesmaximal, characteristic, complemented
4#6472⋊C3C213yesmaximal, complemented
5#6471⋊C3C213yesmaximal, complemented
6#6467⋊C3C213yesmaximal, complemented
7C2×#1132⋊C3C213yesmaximal, complemented
8#90442C311yesmaximal, characteristic, complemented, Hall
9#41×ΓL(1,𝔽8)C2211yescharacteristic
10#66659—21no—
11Q8×AGL(1,𝔽7)—21no—
12#1159⋊C3—23no—
13D4×AGL(1,𝔽7)—23no—
14#1157⋊C3C2213yes—
15#1157⋊C3C2213yescomplemented
16C2×C4⋊AGL(1,𝔽7)C2213yes—
17C2×ΓL(1,𝔽8)⋊Q8C2213yescomplemented
18M16×ΓL(1,𝔽8)C2213yescomplemented
19C2×C8×ΓL(1,𝔽8)C2213yescomplemented
20Dic28⋊C3C2213yescomplemented
21D56⋊C3C2213yescomplemented
22#1132⋊C3C2213yescomplemented
23#1132⋊C3C2216yescomplemented
24#1132⋊C3C2211yescharacteristic, complemented
25#6554C2×C311yescharacteristic, complemented
26#6553C2×C311yescharacteristic, complemented
27C7×#131C2×C311yescharacteristic, complemented
28#6472C2×C313yescomplemented
29#6471C2×C313yescomplemented
30#6467C2×C313yescomplemented
31C2×#1132C2×C313yescomplemented
32C3×#577—71nomaximal, Hall
33C22×AGL(1,𝔽7)—23no—
34Q8×ΓL(1,𝔽8)D411yescharacteristic
35D4×ΓL(1,𝔽8)D413yes—
36C2×C4×ΓL(1,𝔽8)C2313yes—
37(C22⋊D7)⋊C3—23no—
38(C22⋊D7)⋊C3—23no—
39C2×ΓL(1,𝔽8)⋊C4—23no—
40C4⋊AGL(1,𝔽7)—23no—
41C4⋊AGL(1,𝔽7)C2313yes—
42C4⋊AGL(1,𝔽7)C2311yescharacteristic
43C4×AGL(1,𝔽7)—23no—
44C4×AGL(1,𝔽7)—23no—
45ΓL(1,𝔽8)⋊Q8—23no—
46ΓL(1,𝔽8)⋊Q8C2313yes—
47ΓL(1,𝔽8)⋊Q8C2311yescharacteristic
48C8×ΓL(1,𝔽8)C2311yescharacteristic
49C8×ΓL(1,𝔽8)C2313yes—
50C7×#41C22×C311yescharacteristic
51#1161—21no—
52Q8×D7—21no—
53#1159—23no—
54D4×D7—23no—
55#1157C22×C313yes—
56#1157C22×C313yescomplemented
57C2×D28C22×C313yes—
58C2×Dic14C22×C313yescomplemented
59C7×M16C22×C313yescomplemented
60C2×C7×C8C22×C313yescomplemented
61Dic28C22×C313yescomplemented
62D56C22×C313yescomplemented
63#1132C22×C313yescomplemented
64#1132C22×C316yescomplemented
65#1132C22×C311yescharacteristic, complemented
66C3×2−1+4—71no—
67C3×H(5,𝔽2)—71no—
68C3×#137—73no—
69C3×AGL(1,ℤ/8)—73no—
70C3×#135—73no—
71C2×C3×SD16—73no—
72C3×#131—71no—
73C22×ΓL(1,𝔽8)C2×D413yes—
74C2×AGL(1,𝔽7)—21no—
75C2×AGL(1,𝔽7)—23no—
76C2×AGL(1,𝔽7)—83no—
77C4×ΓL(1,𝔽8)C2×D413yes—
78C4×ΓL(1,𝔽8)C2411yescharacteristic
79ΓL(1,𝔽8)⋊C4—23no—
80ΓL(1,𝔽8)⋊C4—21no—
81#577—71noSylow
82C22×D7—23no—
83C7×Q8C3×D411yescharacteristic
84C7×D4C3×D413yes—
85C2×C4×C7C23×C313yes—
86C22⋊D7—23no—
87C22⋊D7—23no—
88C2×Dic7—23no—
89D28—23no—
90D28C23×C313yes—
91D28C23×C311yescharacteristic
92C4×D7—23no—
93C4×D7—23no—
94Dic14—23no—
95Dic14C23×C313yes—
96Dic14C23×C311yescharacteristic
97C7×C8C23×C311yescharacteristic
98C7×C8C23×C313yes—
99C3×#41—73no—
100C3×#41—73no—
101C3×#41—71no—
102C3×#41—143no—
103C3×#41—141no—
104C2×C3×Q8—73no—
105C2×C3×Q8—141no—
106C2×C3×D4—73no—
107C2×C3×D4—143no—
108C3×Q16—73no—
109C3×SD16—73no—
110C3×SD16—76no—
111C3×SD16—71no—
112C3×D8—73no—
113C3×M16—73no—
114C2×C3×C8—73no—
115C2×ΓL(1,𝔽8)—23no—
116C2×ΓL(1,𝔽8)C22×D411yescharacteristic
117AGL(1,𝔽7)—41no—
118AGL(1,𝔽7)—43no—
1192−1+4—71no—
120H(5,𝔽2)—71no—
121#137—73no—
122AGL(1,ℤ/8)—73no—
123#135—73no—
124C2×SD16—73no—
125#131AGL(1,𝔽7)11yescharacteristic, complemented
126C22×C7C2×C3×D413yes—
127C2×D7—21no—
128C2×D7—23no—
129C2×D7—83no—
130C4×C7C2×C3×D413yes—
131C4×C7C24×C311yescharacteristic
132Dic7—23no—
133Dic7—21no—
134C23×C3—143no—
135C3×Q8—73no—
136C3×Q8—71no—
137C3×Q8—71no—
138C3×Q8—143no—
139C3×D4—71no—
140C3×D4—73no—
141C3×D4—73no—
142C3×D4—143no—
143C3×D4—143no—
144C3×D4—143no—
145C2×C3×C4—73no—
146C2×C3×C4—143no—
147C2×C3×C4—143no—
148C2×C3×C4—143no—
149C3×C8—71no—
150C3×C8—73no—
151ΓL(1,𝔽8)#57711yescharacteristic, complemented, Hall
152#41—73no—
153#41—73no—
154#41—141no—
155#41—143no—
156#41C2×AGL(1,𝔽7)11yescharacteristic
157C2×Q8—73no—
158C2×Q8—141no—
159C2×D4—73no—
160C2×D4—143no—
161Q16—73no—
162SD16—73no—
163SD16—76no—
164SD16—71no—
165D8—73no—
166M16C2×AGL(1,𝔽7)13yescomplemented
167C2×C8C2×AGL(1,𝔽7)13yescomplemented
168C2×C7—23no—
169C2×C7C22×C3×D411yescharacteristic
170D7—43no—
171D7—41no—
172C22×C3—73no—
173C22×C3—141no—
174C22×C3—143no—
175C22×C3—563no—
176C3×C4—73no—
177C3×C4—71no—
178C3×C4—141no—
179C3×C4—143no—
180C23—143no—
181Q8—73no—
182Q8—71no—
183Q8—143no—
184Q8C4⋊AGL(1,𝔽7)11yescharacteristic
185D4—71no—
186D4—73no—
187D4—143no—
188D4—143no—
189D4—143no—
190D4C4⋊AGL(1,𝔽7)13yes—
191C2×C4—143no—
192C2×C4—143no—
193C2×C4—143no—
194C2×C4C22×AGL(1,𝔽7)13yes—
195C8C22×AGL(1,𝔽7)11yescharacteristic
196C8C22×AGL(1,𝔽7)13yes—
197C7C3×#57711yescharacteristic, complemented, Sylow
198C2×C3—71no—
199C2×C3—143no—
200C2×C3—283no—
201C2×C3—281no—
202C22—141no—
203C22—143no—
204C22—563no—
205C22C2×C4⋊AGL(1,𝔽7)13yes—
206C4—141no—
207C4—143no—
208C4C2×C4⋊AGL(1,𝔽7)13yes—
209C4C23×AGL(1,𝔽7)11yescharacteristic
210C3—71noSylow
211C2—23no—
212C2—283no—
213C2—281no—
214C2C22×C4⋊AGL(1,𝔽7)11yescharacteristic
subgroupG/H
centerC2C22×C4⋊AGL(1,𝔽7)
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×AGL(1,𝔽7)
FittingC7×#131C2×C3
socleC2×C7C22×C3×D4
hypercenter#131AGL(1,𝔽7)
hypocenterC7C3×#577
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ #41 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #10621519C2semidirect
2. #10621518C2semidirect
3. ΓL(1,𝔽8)×#131C2semidirect
4. #6472⋊C3C2semidirect
5. #6471⋊C3C2semidirect
6. #6467⋊C3C2semidirect
7. C2×#1132⋊C3C2semidirect
8. #90442C3semidirect
15. #1157⋊C3C22semidirect
17. C2×ΓL(1,𝔽8)⋊Q8C22semidirect
18. M16×ΓL(1,𝔽8)C22semidirect
19. C2×C8×ΓL(1,𝔽8)C22semidirect
20. Dic28⋊C3C22semidirect
21. D56⋊C3C22semidirect
22. #1132⋊C3C22semidirect
23. #1132⋊C3C22semidirect
24. #1132⋊C3C22semidirect
25. #6554C2×C3semidirect
26. #6553C2×C3semidirect
27. C7×#131C2×C3semidirect
28. #6472C2×C3semidirect
29. #6471C2×C3semidirect
30. #6467C2×C3semidirect
31. C2×#1132C2×C3semidirect
56. #1157C22×C3semidirect
58. C2×Dic14C22×C3semidirect
59. C7×M16C22×C3semidirect
60. C2×C7×C8C22×C3semidirect
61. Dic28C22×C3semidirect
62. D56C22×C3semidirect
63. #1132C22×C3semidirect
64. #1132C22×C3semidirect
65. #1132C22×C3semidirect
125. #131AGL(1,𝔽7)semidirect
151. ΓL(1,𝔽8)#577semidirect
166. M16C2×AGL(1,𝔽7)semidirect
167. C2×C8C2×AGL(1,𝔽7)semidirect
197. C7C3×#577semidirect

Maximal subgroups

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

subgroupindex
#106215192
#106215182
ΓL(1,𝔽8)×#1312
#6472⋊C32
#6471⋊C32
#6467⋊C32
C2×#1132⋊C32
#904423
C3×#5777