Finite Group Databasebeta

← #37524849

Subgroups — #37524849

Order 22932.

152 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 24 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#26039565C211yesmaximal, characteristic, complemented
2#26039565C211yesmaximal, characteristic, complemented
3C13×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4#21439671C311yesmaximal, characteristic, complemented
5#64424⋊D13—31nomaximal
6C13×ΓL(1,𝔽8)2C2211yescharacteristic, complemented, Hall
7C13⋊#64418C2×C311yescharacteristic, complemented
8C13⋊#64418C2×C311yescharacteristic, complemented
9C13×ΓL(1,𝔽8)⋊D7—21no—
10#16838460—31no—
11#16838460—31no—
12C13×#64425C2×C311yescharacteristic, complemented
13C13×#64424D311yescharacteristic, complemented
14(C7⋊D7)⋊D13—31noHall
15C13×#3862C22×C311yescharacteristic, complemented
16C13×#3861C2×D311yescharacteristic, complemented
17C7×C13×ΓL(1,𝔽8)—21no—
18(#64418⋊C2)⋊C3—131nomaximal, Hall
19C3×C13×AGL(1,𝔽7)—141no—
20C13×C7⋊D7C3×D311yescharacteristic, complemented
21C7×D91—31no—
22C7×D91—31no—
23D13×AGL(1,𝔽7)—211no—
24D13×AGL(1,𝔽7)—211no—
25ΓL(1,𝔽8)⋊AGL(1,𝔽7)D1311yescharacteristic, complemented
26#64418⋊C3—131no—
27#64418⋊C3—131no—
28C3×C13×ΓL(1,𝔽8)—141no—
29C72×C13C2×C3×D311yescharacteristic, complemented, Hall
30#64423⋊C2—391no—
31#64418⋊C2—131no—
32C3×C13×D7—141no—
33ΓL(1,𝔽8)⋊D13—421no—
34ΓL(1,𝔽8)⋊D13—421no—
35C13×AGL(1,𝔽7)—141no—
36C13×AGL(1,𝔽7)—141no—
37C13×AGL(1,𝔽7)—141no—
38C13×AGL(1,𝔽7)—211no—
39C13×AGL(1,𝔽7)—211no—
40ΓL(1,𝔽8)×D13—211no—
41ΓL(1,𝔽8)×D13—211no—
42C2×C3×D39—491nomaximal, Hall
43ΓL(1,𝔽8)2C2×D1311yescharacteristic, complemented, Hall
44D7×D13—211no—
45D7×D13—211no—
46#64425C3×D1311yescharacteristic, complemented
47#64424D3911yescharacteristic, complemented
48#64423—391no—
49#64423—391no—
50ΓL(1,𝔽8)⋊D7—21no—
51#64418—131no—
52#64418—131no—
53C3×C7×C13—281no—
54C13×ΓL(1,𝔽8)—21no—
55C13×ΓL(1,𝔽8)—141no—
56C13×ΓL(1,𝔽8)—281no—
57C13×ΓL(1,𝔽8)—841no—
58C13×ΓL(1,𝔽8)—841no—
59C2×C32×C13—491no—
60C3×D39—981no—
61C3×D39—981no—
62D72—391noHall
63D91—31no—
64D91—31no—
65C7×D13—211no—
66C7×D13—211no—
67C13×D7—141no—
68C13×D7—211no—
69C13×D7—421no—
70C2×D39—491no—
71C2×C3×D13—1471no—
72#3862C2×C3×D1311yescharacteristic, complemented
73#3861C2×D3911yescharacteristic, complemented
74C7×ΓL(1,𝔽8)—21no—
75C3×AGL(1,𝔽7)—141no—
76C32×C13—1961noHall
77C7⋊D7C3×D3911yescharacteristic, complemented
78C7×D7—391no—
79C7×D7—391no—
80C7×C13—21no—
81C7×C13—31no—
82C7×C13—61no—
83C2×AGL(1,𝔽7)—2731no—
84C2×AGL(1,𝔽7)—2731no—
85C2×C3×C13—491no—
86C2×C3×C13—491no—
87C2×C3×C13—981no—
88D39—981no—
89D39—981no—
90C3×D13—2941no—
91C3×D13—2941no—
92C3×ΓL(1,𝔽8)—141no—
93C2×D13—1471noHall
94C72C2×C3×D3911yescharacteristic, complemented, Sylow
95C3×D7—141no—
96C2×ΓL(1,𝔽8)—2731no—
97C2×ΓL(1,𝔽8)—2731no—
98AGL(1,𝔽7)—141no—
99AGL(1,𝔽7)—141no—
100AGL(1,𝔽7)—141no—
101AGL(1,𝔽7)—211no—
102AGL(1,𝔽7)—5461no—
103AGL(1,𝔽7)—5461no—
104AGL(1,𝔽7)—421no—
105C3×C13—141no—
106C3×C13—1961no—
107C3×C13—1961no—
108C2×C3×D3—6371noHall
109C2×D7—2731no—
110C2×D7—2731no—
111C2×C13—491no—
112D13—211no—
113D13—211no—
114C3×C7—281no—
115ΓL(1,𝔽8)—21no—
116ΓL(1,𝔽8)—281no—
117ΓL(1,𝔽8)—281no—
118ΓL(1,𝔽8)—421no—
119ΓL(1,𝔽8)—841no—
120C2×C32—491no—
121C3×D3—12741no—
122C3×D3—12741no—
123C2×C7—2731no—
124C2×C7—2731no—
125D7—141no—
126D7—391no—
127D7—391no—
128D7—421no—
129D7—421no—
130C13(#64418⋊C2)⋊C311yescharacteristic, complemented, Sylow
131C22×C3—19111no—
132C2×D3—6371no—
133C32—1961noSylow
134C7—21no—
135C7—61no—
136C7—61no—
137C2×C3—491no—
138C2×C3—491no—
139C2×C3—981no—
140C2×C3—38221no—
141C2×C3—38221no—
142D3—12741no—
143D3—12741no—
144C22—19111noSylow
145C3—141no—
146C3—981no—
147C3—981no—
148C2—2731no—
149C2—2731no—
150C2—491no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC13×#3862C22×C3
Fitting, socleC72×C13C2×C3×D3
LCSG ▷ C13×#3862
UCSC1

Complements

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

NH ≅ G/Ntype
1. #26039565C2semidirect
2. #26039565C2semidirect
3. C13×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C2semidirect
4. #21439671C3semidirect
6. C13×ΓL(1,𝔽8)2C22semidirect
7. C13⋊#64418C2×C3semidirect
8. C13⋊#64418C2×C3semidirect
12. C13×#64425C2×C3semidirect
13. C13×#64424D3semidirect
15. C13×#3862C22×C3semidirect
16. C13×#3861C2×D3semidirect
20. C13×C7⋊D7C3×D3semidirect
25. ΓL(1,𝔽8)⋊AGL(1,𝔽7)D13semidirect
29. C72×C13C2×C3×D3semidirect
43. ΓL(1,𝔽8)2C2×D13semidirect
46. #64425C3×D13semidirect
47. #64424D39semidirect
72. #3862C2×C3×D13semidirect
73. #3861C2×D39semidirect
77. C7⋊D7C3×D39semidirect
94. C72C2×C3×D39semidirect
130. C13(#64418⋊C2)⋊C3semidirect

Maximal subgroups

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

subgroupindex
#260395652
C13×ΓL(1,𝔽8)⋊AGL(1,𝔽7)2
#214396713
#64424⋊D133
(#64418⋊C2)⋊C313
C2×C3×D3949