Finite Group Databasebeta

← #37524892

Subgroups — #37524892

Order 22932.

152 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 35 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#26039554C211yesmaximal, characteristic
2#21439658C311yesmaximal, characteristic, complemented
3#21439629C311yesmaximal, characteristic, complemented
4#21439627C311yesmaximal, characteristic, complemented
5C7×#11763984C311yesmaximal, characteristic, complemented
6#3861⋊(C13⋊C3)C411yescharacteristic, complemented, Hall
7D13⋊#3862C2×C311yescharacteristic
8D13×#3861C2×C311yescharacteristic
9C7×#10587717C2×C311yescharacteristic
10C7×#10587716C2×C311yescharacteristic
11ΓL(1,𝔽8)×AGL(1,𝔽13)—71nomaximal
12ΓL(1,𝔽8)⋊AGL(1,𝔽13)—71nomaximal
13C7×C7⋊#261C3211yescharacteristic, complemented, Hall
14C13⋊#3862C3×C411yescharacteristic, complemented
15C13×#3861C3×C411yescharacteristic, complemented
16C7×#63270C3×C411yescharacteristic, complemented
17C7×#63271C3×C411yescharacteristic, complemented
18ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4—131nomaximal, Hall
19ΓL(1,𝔽8)×D13⋊C3—71no—
20ΓL(1,𝔽8)×D13⋊C3—71no—
21C72×D13C2×C3211yescharacteristic
22#11763985—71no—
23#11763984ΓL(1,𝔽8)11yescharacteristic, complemented
24ΓL(1,𝔽8)⋊#261—71no—
25#11763982—71no—
26#11763981—71no—
27ΓL(1,𝔽8)×#261—71no—
28#11763979—71no—
29C7×AGL(1,𝔽13)—71no—
30C2×ΓL(1,𝔽8)2—131no—
31ΓL(1,𝔽8)×C13⋊C3—71no—
32ΓL(1,𝔽8)×C13⋊C3—71no—
33C72×C13C32×C411yescharacteristic, complemented, Hall
34#10596873—391no—
35#10596872—131no—
36ΓL(1,𝔽8)×Dic7—131no—
37C7×ΓL(1,𝔽8)⋊C4—131no—
38#10587717—71no—
39#10587717C2×ΓL(1,𝔽8)11yescharacteristic
40#10587716—71no—
41#10587716AGL(1,𝔽7)11yescharacteristic
42ΓL(1,𝔽8)×D13—71no—
43ΓL(1,𝔽8)×D13—71no—
44ΓL(1,𝔽8)×D13—421no—
45C7×D13⋊C3—71no—
46C7×D13⋊C3—71no—
47C3×AGL(1,𝔽13)—491noHall
48ΓL(1,𝔽8)2—131noHall
49C7⋊#261C3×ΓL(1,𝔽8)11yescharacteristic, complemented
50C7×#261—71no—
51C2×#3862—131no—
52C2×#3861—391no—
53C2×C7×ΓL(1,𝔽8)—131no—
54C2×C7×ΓL(1,𝔽8)—131no—
55#63271—71no—
56#63271C4×ΓL(1,𝔽8)11yescharacteristic, complemented
57#63270—71no—
58#63270ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
59C7×C13⋊C3—71no—
60C7×C13⋊C3—71no—
61C13×ΓL(1,𝔽8)—71no—
62C13×ΓL(1,𝔽8)—71no—
63C13×ΓL(1,𝔽8)—421no—
64C3×C4×ΓL(1,𝔽8)—911no—
65C3×ΓL(1,𝔽8)⋊C4—911no—
66C3×D13⋊C3—491no—
67C7×Dic7—391noHall
68C7×D13—61no—
69C7×D13C3×AGL(1,𝔽7)11yescharacteristic
70C7×D13C2×C3×ΓL(1,𝔽8)11yescharacteristic
71C3×#261—491no—
72AGL(1,𝔽13)—71no—
73AGL(1,𝔽13)—491no—
74AGL(1,𝔽13)—1471no—
75#3862—131no—
76#3861AGL(1,𝔽13)11yescharacteristic, complemented
77C7×ΓL(1,𝔽8)—131no—
78C7×ΓL(1,𝔽8)—131no—
79C2×C3×ΓL(1,𝔽8)—911no—
80C2×C3×ΓL(1,𝔽8)—911no—
81C3×C13⋊C3—491noHall
82C2×C72—391no—
83C7×C13—61no—
84C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
85C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
86C3×C4×C7—911no—
87C3×Dic7—911no—
88C4×ΓL(1,𝔽8)—911no—
89C4×ΓL(1,𝔽8)—911no—
90C4×ΓL(1,𝔽8)—911no—
91ΓL(1,𝔽8)⋊C4—131no—
92ΓL(1,𝔽8)⋊C4—911no—
93ΓL(1,𝔽8)⋊C4—911no—
94C3×D13—491no—
95D13⋊C3—71no—
96D13⋊C3—71no—
97D13⋊C3—491no—
98C3×ΓL(1,𝔽8)—911no—
99C3×ΓL(1,𝔽8)—911no—
100#261—211noHall
101C72C3×AGL(1,𝔽13)11yescharacteristic, complemented, Sylow
102C2×C3×C7—911no—
103C2×C3×C7—911no—
104C2×ΓL(1,𝔽8)—131no—
105C2×ΓL(1,𝔽8)—131no—
106C2×ΓL(1,𝔽8)—2731no—
107C2×ΓL(1,𝔽8)—2731no—
108C2×ΓL(1,𝔽8)—5461no—
109C2×ΓL(1,𝔽8)—911no—
110C2×ΓL(1,𝔽8)—911no—
111C3×C13—491no—
112C13⋊C3—71no—
113C13⋊C3—71no—
114C13⋊C3—491no—
115C32×C4—6371noHall
116C4×C7—8191no—
117Dic7—131no—
118D13ΓL(1,𝔽8)×AGL(1,𝔽7)11yescharacteristic
119C3×C7—911no—
120C3×C7—911no—
121ΓL(1,𝔽8)—71no—
122ΓL(1,𝔽8)—71no—
123ΓL(1,𝔽8)—131no—
124ΓL(1,𝔽8)—131no—
125ΓL(1,𝔽8)—421no—
126ΓL(1,𝔽8)—911no—
127ΓL(1,𝔽8)—911no—
128C2×C32—6371no—
129C2×C7—391no—
130C2×C7—391no—
131C2×C7—781no—
132C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
133C3×C4—911no—
134C3×C4—19111no—
135C3×C4—19111no—
136C3×C4—6371no—
137C32—6371noSylow
138C7—61no—
139C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)11yescharacteristic, complemented
140C7ΓL(1,𝔽8)×AGL(1,𝔽13)11yescharacteristic, complemented
141C2×C3—911no—
142C2×C3—911no—
143C2×C3—19111no—
144C2×C3—6371no—
145C4—8191noSylow
146C3—491no—
147C3—911no—
148C3—911no—
149C3—6371no—
150C2—391no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC72×C13C32×C4
prime radical#26039554C2
cosocleC72×D13C2×C32
LCSG ▷ C72×C13
UCSC1

Complements

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

NH ≅ G/Ntype
2. #21439658C3semidirect
3. #21439629C3semidirect
4. #21439627C3semidirect
5. C7×#11763984C3semidirect
6. #3861⋊(C13⋊C3)C4semidirect
13. C7×C7⋊#261C32semidirect
14. C13⋊#3862C3×C4semidirect
15. C13×#3861C3×C4semidirect
16. C7×#63270C3×C4semidirect
17. C7×#63271C3×C4semidirect
23. #11763984ΓL(1,𝔽8)semidirect
33. C72×C13C32×C4semidirect
49. C7⋊#261C3×ΓL(1,𝔽8)semidirect
56. #63271C4×ΓL(1,𝔽8)semidirect
58. #63270ΓL(1,𝔽8)⋊C4semidirect
76. #3861AGL(1,𝔽13)semidirect
84. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
85. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
101. C72C3×AGL(1,𝔽13)semidirect
132. C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4semidirect
139. C7ΓL(1,𝔽8)⋊AGL(1,𝔽13)semidirect
140. C7ΓL(1,𝔽8)×AGL(1,𝔽13)semidirect

Maximal subgroups

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

subgroupindex
#260395542
#214396583
#214396293
#214396273
C7×#117639843
ΓL(1,𝔽8)×AGL(1,𝔽13)7
ΓL(1,𝔽8)⋊AGL(1,𝔽13)7
ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C413