Finite Group Databasebeta

← ΓL(1,𝔽8)×Dic7×C13⋊C3 #37524677

Subgroups — ΓL(1,𝔽8)×Dic7×C13⋊C3 #37524677

Order 22932.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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×C7×ΓL(1,𝔽8)×C13⋊C3C211yesmaximal, characteristic
2Dic7×#63271C311yesmaximal, characteristic, complemented
3Dic7×#63270C311yesmaximal, characteristic, complemented
4C13×ΓL(1,𝔽8)×Dic7C311yesmaximal, characteristic, complemented
5C7×Dic7×C13⋊C3C311yesmaximal, characteristic, complemented
6C7×ΓL(1,𝔽8)×C13⋊C3C411yescharacteristic, complemented, Hall
7C2×C7×#63271C2×C311yescharacteristic
8C2×C7×#63270C2×C311yescharacteristic
9C2×C7×C13×ΓL(1,𝔽8)C2×C311yescharacteristic
10C2×C72×C13⋊C3C2×C311yescharacteristic
11C3×Dic7×C13⋊C3—71nomaximal
12C4×ΓL(1,𝔽8)×C13⋊C3—71nomaximal
13C7×C13×Dic7C3211yescharacteristic, complemented, Hall
14C7×#63270C3×C411yescharacteristic, complemented
15C7×#63271C3×C411yescharacteristic, complemented
16C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
17C72×C13⋊C3C3×C411yescharacteristic, complemented
18C3×ΓL(1,𝔽8)×Dic7—131nomaximal, Hall
19C2×C3×C7×C13⋊C3—71no—
20C2×ΓL(1,𝔽8)×C13⋊C3D711yescharacteristic
21C2×C72×C13C2×C3211yescharacteristic
22C3×C13×Dic7—71no—
23C4×#63271—71no—
24C4×#63270—71no—
25C4×C13×ΓL(1,𝔽8)—71no—
26C4×C7×C13⋊C3—71no—
27Dic7×C13⋊C3—71no—
28Dic7×C13⋊C3—71no—
29Dic7×C13⋊C3ΓL(1,𝔽8)11yescharacteristic, direct product factor
30C2×C3×C7×ΓL(1,𝔽8)—131no—
31C3×C7×C13⋊C3—71no—
32ΓL(1,𝔽8)×C13⋊C3Dic711yescharacteristic, direct product factor
33C72×C13C32×C411yescharacteristic, complemented, Hall
34C3×C7×Dic7—131no—
35ΓL(1,𝔽8)×Dic7—131no—
36ΓL(1,𝔽8)×Dic7—131no—
37ΓL(1,𝔽8)×Dic7C13⋊C311yescharacteristic, direct product factor
38C2×C3×C7×C13—71no—
39C2×#63271C3×D711yescharacteristic
40C2×#63270C3×D711yescharacteristic
41C2×C13×ΓL(1,𝔽8)C3×D711yescharacteristic
42C2×C7×C13⋊C3—61no—
43C2×C7×C13⋊C3—71no—
44C2×C7×C13⋊C3—71no—
45C2×C7×C13⋊C3C2×ΓL(1,𝔽8)11yescharacteristic
46C2×C7×C13⋊C3C3×D711yescharacteristic
47C3×C4×C13⋊C3—491noHall
48C3×C7×ΓL(1,𝔽8)—131noHall
49C4×C7×C13—71no—
50C13×Dic7C3×ΓL(1,𝔽8)11yescharacteristic, complemented
51C2×C3×C72—131no—
52C2×C7×ΓL(1,𝔽8)—131no—
53C2×C7×ΓL(1,𝔽8)—131no—
54C2×C7×ΓL(1,𝔽8)C2×C13⋊C311yescharacteristic
55C3×C7×C13—71no—
56#63271C3×Dic711yescharacteristic, complemented
57#63270C3×Dic711yescharacteristic, complemented
58C7×C13⋊C3—61no—
59C7×C13⋊C3—71no—
60C7×C13⋊C3—71no—
61C7×C13⋊C3C4×ΓL(1,𝔽8)11yescharacteristic, complemented
62C7×C13⋊C3C3×Dic711yescharacteristic, complemented
63C13×ΓL(1,𝔽8)C3×Dic711yescharacteristic, complemented
64C32×Dic7—911no—
65C3×C4×ΓL(1,𝔽8)—911no—
66C2×C3×C13⋊C3—71no—
67C7×Dic7C3×C13⋊C311yescharacteristic, complemented, Hall
68C2×C7×C13—61no—
69C2×C7×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic
70C2×C7×C13C32×D711yescharacteristic
71C3×C4×C13—491no—
72C4×C13⋊C3—71no—
73C4×C13⋊C3—491no—
74C4×C13⋊C3—491no—
75C3×C72—131no—
76C7×ΓL(1,𝔽8)—131no—
77C7×ΓL(1,𝔽8)—131no—
78C7×ΓL(1,𝔽8)C4×C13⋊C311yescharacteristic, complemented
79C2×C32×C7—911no—
80C2×C3×ΓL(1,𝔽8)—131no—
81C3×C13⋊C3—71noHall
82C2×C72C2×C3×C13⋊C311yescharacteristic
83C7×C13—61no—
84C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
85C7×C13C32×Dic711yescharacteristic, complemented
86C3×C4×C7—911no—
87C3×Dic7—71no—
88C3×Dic7—131no—
89C3×Dic7—911no—
90C3×Dic7—911no—
91C4×ΓL(1,𝔽8)—71no—
92C4×ΓL(1,𝔽8)—911no—
93C4×ΓL(1,𝔽8)—911no—
94C2×C3×C13—71no—
95C2×C13⋊C3—71no—
96C2×C13⋊C3—71no—
97C2×C13⋊C3D7×ΓL(1,𝔽8)11yescharacteristic
98C32×C7—911no—
99C3×ΓL(1,𝔽8)—131no—
100C4×C13—71noHall
101C72C3×C4×C13⋊C311yescharacteristic, complemented, Sylow
102C2×C3×C7—71no—
103C2×C3×C7—131no—
104C2×C3×C7—131no—
105C2×C3×C7—781no—
106C2×C3×C7—911no—
107C2×C3×C7—911no—
108C2×ΓL(1,𝔽8)—131no—
109C2×ΓL(1,𝔽8)—131no—
110C2×ΓL(1,𝔽8)D7×C13⋊C311yescharacteristic
111C3×C13—71no—
112C13⋊C3—71no—
113C13⋊C3—71no—
114C13⋊C3ΓL(1,𝔽8)×Dic711yescharacteristic, direct product factor
115C32×C4—6371noHall
116C4×C7—71no—
117Dic7ΓL(1,𝔽8)×C13⋊C311yescharacteristic, direct product factor
118C2×C13C3×D7×ΓL(1,𝔽8)11yescharacteristic
119C3×C7—71no—
120C3×C7—131no—
121C3×C7—131no—
122C3×C7—781no—
123C3×C7—911no—
124C3×C7—911no—
125ΓL(1,𝔽8)—131no—
126ΓL(1,𝔽8)—131no—
127ΓL(1,𝔽8)Dic7×C13⋊C311yescharacteristic, direct product factor
128C2×C32—911no—
129C2×C7—61no—
130C2×C7C2×ΓL(1,𝔽8)×C13⋊C311yescharacteristic
131C2×C7C3×D7×C13⋊C311yescharacteristic
132C13C3×ΓL(1,𝔽8)×Dic711yescharacteristic, complemented, Sylow
133C3×C4—491no—
134C3×C4—911no—
135C3×C4—6371no—
136C3×C4—6371no—
137C32—911noSylow
138C7—61no—
139C7C4×ΓL(1,𝔽8)×C13⋊C311yescharacteristic, complemented
140C7C3×Dic7×C13⋊C311yescharacteristic, complemented
141C2×C3—71no—
142C2×C3—131no—
143C2×C3—911no—
144C2×C3—911no—
145C4—71noSylow
146C3—71no—
147C3—131no—
148C3—911no—
149C3—911no—
150C2D7×ΓL(1,𝔽8)×C13⋊C311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D7×ΓL(1,𝔽8)×C13⋊C3
commutator, hypocenterC72×C13C32×C4
Fitting, socle, cosocleC2×C72×C13C2×C32
prime radicalC2×C7×ΓL(1,𝔽8)×C13⋊C3C2
LCSG ▷ C72×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. Dic7×#63271C3semidirect
3. Dic7×#63270C3semidirect
4. C13×ΓL(1,𝔽8)×Dic7C3semidirect
5. C7×Dic7×C13⋊C3C3semidirect
6. C7×ΓL(1,𝔽8)×C13⋊C3C4semidirect
13. C7×C13×Dic7C32semidirect
14. C7×#63270C3×C4semidirect
15. C7×#63271C3×C4semidirect
16. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
17. C72×C13⋊C3C3×C4semidirect
29. Dic7×C13⋊C3ΓL(1,𝔽8)direct (N × H)
32. ΓL(1,𝔽8)×C13⋊C3Dic7direct (N × H)
33. C72×C13C32×C4semidirect
37. ΓL(1,𝔽8)×Dic7C13⋊C3direct (N × H)
50. C13×Dic7C3×ΓL(1,𝔽8)semidirect
56. #63271C3×Dic7semidirect
57. #63270C3×Dic7semidirect
61. C7×C13⋊C3C4×ΓL(1,𝔽8)semidirect
62. C7×C13⋊C3C3×Dic7semidirect
63. C13×ΓL(1,𝔽8)C3×Dic7semidirect
67. C7×Dic7C3×C13⋊C3semidirect
78. C7×ΓL(1,𝔽8)C4×C13⋊C3semidirect
84. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
85. C7×C13C32×Dic7semidirect
101. C72C3×C4×C13⋊C3semidirect
114. C13⋊C3ΓL(1,𝔽8)×Dic7direct (N × H)
117. Dic7ΓL(1,𝔽8)×C13⋊C3direct (N × H)
127. ΓL(1,𝔽8)Dic7×C13⋊C3direct (N × H)
132. C13C3×ΓL(1,𝔽8)×Dic7semidirect
139. C7C4×ΓL(1,𝔽8)×C13⋊C3semidirect
140. C7C3×Dic7×C13⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C7×ΓL(1,𝔽8)×C13⋊C32
Dic7×#632713
Dic7×#632703
C13×ΓL(1,𝔽8)×Dic73
C7×Dic7×C13⋊C33
C3×Dic7×C13⋊C37
C4×ΓL(1,𝔽8)×C13⋊C37
C3×ΓL(1,𝔽8)×Dic713