Finite Group Databasebeta

← ΓL(1,𝔽8)⋊C4×#63270 #37524653

Subgroups — ΓL(1,𝔽8)⋊C4×#63270 #37524653

Order 22932.

152 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 46 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×ΓL(1,𝔽8)×#63270C211yesmaximal, characteristic
2Dic7×#63270C311yesmaximal, characteristic, complemented
3#21439541C311yesmaximal, characteristic, complemented
4#21439534C311yesmaximal, characteristic, complemented
5C7×C13×ΓL(1,𝔽8)⋊C4C311yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)×#63270C411yescharacteristic, complemented, Hall
7C2×C13⋊#3862C2×C311yescharacteristic
8C2×#14274263C2×C311yescharacteristic
9C2×C7×#63270C2×C311yescharacteristic
10C2×C7×C13×ΓL(1,𝔽8)C2×C311yescharacteristic
11C3×C4×#63270—71nomaximal
12C13⋊C3×ΓL(1,𝔽8)⋊C4—71nomaximal
13C7×C13×Dic7C3211yescharacteristic, complemented, Hall
14C13⋊#3862C3×C411yescharacteristic, complemented
15#14274263C3×C411yescharacteristic, complemented
16C7×#63270C3×C411yescharacteristic, complemented
17C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
18ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4—131nomaximal, Hall
19C2×C3×#63270—71no—
20C2×ΓL(1,𝔽8)×C13⋊C3—71no—
21C2×C72×C13C2×C3211yescharacteristic
22C3×C4×C7×C13—71no—
23C4×#63270—71no—
24C4×#63270—71no—
25C4×#63270—211no—
26#11763962—71no—
27#11763961—71no—
28C13×ΓL(1,𝔽8)⋊C4ΓL(1,𝔽8)11yescharacteristic, complemented
29Dic7×C13⋊C3—71no—
30C2×ΓL(1,𝔽8)2—131no—
31C3×#63270—71no—
32ΓL(1,𝔽8)×C13⋊C3—71no—
33C72×C13C32×C411yescharacteristic, complemented, Hall
34#10596873—131no—
35#10596872—131no—
36ΓL(1,𝔽8)×Dic7—131no—
37C7×ΓL(1,𝔽8)⋊C4C13⋊C311yescharacteristic, complemented
38C2×C3×C7×C13—71no—
39C2×#63271—71no—
40C2×#63270—71no—
41C2×#63270—71no—
42C2×#63270—71no—
43C2×#63270—421no—
44C2×#63270AGL(1,𝔽7)11yescharacteristic
45C2×C13×ΓL(1,𝔽8)C2×ΓL(1,𝔽8)11yescharacteristic
46C2×C7×C13⋊C3—71no—
47C3×C4×C13⋊C3—491noHall
48ΓL(1,𝔽8)2—131noHall
49C4×C7×C13—211no—
50C13×Dic7C3×ΓL(1,𝔽8)11yescharacteristic, complemented
51C2×#3862—131no—
52C2×#3861—131no—
53C2×C7×ΓL(1,𝔽8)—131no—
54C2×C7×ΓL(1,𝔽8)C2×C13⋊C311yescharacteristic
55C3×C7×C13—71no—
56#63271—71no—
57#63270—71no—
58#63270—71no—
59#63270—71no—
60#63270—421no—
61#63270ΓL(1,𝔽8)⋊C411yescharacteristic, direct product factor
62C7×C13⋊C3—71no—
63C13×ΓL(1,𝔽8)C4×ΓL(1,𝔽8)11yescharacteristic, complemented
64C3×C4×ΓL(1,𝔽8)—911no—
65C3×ΓL(1,𝔽8)⋊C4—911no—
66C2×C3×C13⋊C3—491no—
67C7×Dic7C3×C13⋊C311yescharacteristic, complemented, Hall
68C2×C7×C13—61no—
69C2×C7×C13C3×AGL(1,𝔽7)11yescharacteristic
70C2×C7×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic
71C3×C4×C13—71no—
72C4×C13⋊C3—491no—
73C4×C13⋊C3—491no—
74C4×C13⋊C3—1471no—
75#3862—131no—
76#3861—131no—
77C7×ΓL(1,𝔽8)—131no—
78C7×ΓL(1,𝔽8)C4×C13⋊C311yescharacteristic, complemented
79C2×C3×ΓL(1,𝔽8)—911no—
80C2×C3×ΓL(1,𝔽8)—911no—
81C3×C13⋊C3—491noHall
82C2×C72C2×C3×C13⋊C311yescharacteristic
83C7×C13—61no—
84C7×C13C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
85C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
86C3×C4×C7—71no—
87C3×Dic7—911no—
88C4×ΓL(1,𝔽8)—2731no—
89C4×ΓL(1,𝔽8)—911no—
90C4×ΓL(1,𝔽8)—911no—
91ΓL(1,𝔽8)⋊C4—911no—
92ΓL(1,𝔽8)⋊C4—911no—
93ΓL(1,𝔽8)⋊C4#6327011yescharacteristic, direct product factor
94C2×C3×C13—71no—
95C2×C13⋊C3—71no—
96C2×C13⋊C3—491no—
97C2×C13⋊C3—491no—
98C3×ΓL(1,𝔽8)—911no—
99C3×ΓL(1,𝔽8)—911no—
100C4×C13—211noHall
101C72C3×C4×C13⋊C311yescharacteristic, complemented, Sylow
102C2×C3×C7—71no—
103C2×C3×C7—911no—
104C2×ΓL(1,𝔽8)—131no—
105C2×ΓL(1,𝔽8)—5461no—
106C2×ΓL(1,𝔽8)—911no—
107C2×ΓL(1,𝔽8)—911no—
108C2×ΓL(1,𝔽8)—911no—
109C2×ΓL(1,𝔽8)—911no—
110C2×ΓL(1,𝔽8)C2×#6327011yescharacteristic
111C3×C13—71no—
112C13⋊C3—71no—
113C13⋊C3—491no—
114C13⋊C3—491no—
115C32×C4—6371noHall
116C4×C7—211no—
117Dic7C3×#6327011yescharacteristic, complemented
118C2×C13ΓL(1,𝔽8)×AGL(1,𝔽7)11yescharacteristic
119C3×C7—71no—
120C3×C7—911no—
121ΓL(1,𝔽8)—131no—
122ΓL(1,𝔽8)—5461no—
123ΓL(1,𝔽8)—911no—
124ΓL(1,𝔽8)—911no—
125ΓL(1,𝔽8)—911no—
126ΓL(1,𝔽8)—911no—
127ΓL(1,𝔽8)C4×#6327011yescharacteristic, complemented
128C2×C32—6371no—
129C2×C7—61no—
130C2×C7C13⋊C3×AGL(1,𝔽7)11yescharacteristic
131C2×C7C2×C3×#6327011yescharacteristic
132C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented, Sylow
133C3×C4—71no—
134C3×C4—19111no—
135C3×C4—6371no—
136C3×C4—6371no—
137C32—6371noSylow
138C7—61no—
139C7C13⋊C3×ΓL(1,𝔽8)⋊C411yescharacteristic, complemented
140C7C3×C4×#6327011yescharacteristic, complemented
141C2×C3—71no—
142C2×C3—911no—
143C2×C3—6371no—
144C2×C3—6371no—
145C4—211noSylow
146C3—71no—
147C3—911no—
148C3—6371no—
149C3—6371no—
150C2AGL(1,𝔽7)×#6327011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽7)×#63270
commutator, hypocenterC72×C13C32×C4
Fitting, socle, cosocleC2×C72×C13C2×C32
prime radicalC2×ΓL(1,𝔽8)×#63270C2
LCSG ▷ C72×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. Dic7×#63270C3semidirect
3. #21439541C3semidirect
4. #21439534C3semidirect
5. C7×C13×ΓL(1,𝔽8)⋊C4C3semidirect
6. ΓL(1,𝔽8)×#63270C4semidirect
13. C7×C13×Dic7C32semidirect
14. C13⋊#3862C3×C4semidirect
15. #14274263C3×C4semidirect
16. C7×#63270C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
28. C13×ΓL(1,𝔽8)⋊C4ΓL(1,𝔽8)semidirect
33. C72×C13C32×C4semidirect
37. C7×ΓL(1,𝔽8)⋊C4C13⋊C3semidirect
50. C13×Dic7C3×ΓL(1,𝔽8)semidirect
61. #63270ΓL(1,𝔽8)⋊C4direct (N × H)
63. C13×ΓL(1,𝔽8)C4×ΓL(1,𝔽8)semidirect
67. C7×Dic7C3×C13⋊C3semidirect
78. C7×ΓL(1,𝔽8)C4×C13⋊C3semidirect
84. C7×C13C3×ΓL(1,𝔽8)⋊C4semidirect
85. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
93. ΓL(1,𝔽8)⋊C4#63270direct (N × H)
101. C72C3×C4×C13⋊C3semidirect
117. Dic7C3×#63270semidirect
127. ΓL(1,𝔽8)C4×#63270semidirect
132. C13ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C4semidirect
139. C7C13⋊C3×ΓL(1,𝔽8)⋊C4semidirect
140. C7C3×C4×#63270semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×#632702
Dic7×#632703
#214395413
#214395343
C7×C13×ΓL(1,𝔽8)⋊C43
C3×C4×#632707
C13⋊C3×ΓL(1,𝔽8)⋊C47
ΓL(1,𝔽8)×ΓL(1,𝔽8)⋊C413