Finite Group Databasebeta

← ΓL(1,𝔽8)×#11763958 #37524671

Subgroups — ΓL(1,𝔽8)×#11763958 #37524671

Order 22932.

168 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 45 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)×#63271C211yesmaximal, characteristic
2Dic13⋊#3862C311yesmaximal, characteristic, complemented
3#21439550C311yesmaximal, characteristic, complemented
4C7×ΓL(1,𝔽8)×Dic13C311yesmaximal, characteristic, complemented
5C7×#11763958C311yesmaximal, characteristic, complemented
6ΓL(1,𝔽8)×#63271C411yescharacteristic, complemented, Hall
7C2×C13⋊#3862C2×C311yescharacteristic
8C2×#14274262C2×C311yescharacteristic
9C2×C7×#63271C2×C311yescharacteristic
10C2×C7×C13×ΓL(1,𝔽8)C2×C311yescharacteristic
11C3×#11763958—71nomaximal
12ΓL(1,𝔽8)×Dic13⋊C3—71nomaximal
13C72×Dic13C3211yescharacteristic, complemented, Hall
14C13⋊#3862C3×C411yescharacteristic, complemented
15#14274262C3×C411yescharacteristic, complemented
16C7×#63271C3×C411yescharacteristic, complemented
17C7×C13×ΓL(1,𝔽8)C3×C411yescharacteristic, complemented
18C4×ΓL(1,𝔽8)2—131nomaximal, Hall
19C2×C3×#63271—71no—
20C2×ΓL(1,𝔽8)×C13⋊C3—71no—
21C2×C72×C13C2×C3211yescharacteristic
22C3×C7×Dic13—71no—
23#11763958—71no—
24#11763958—71no—
25#11763958—71no—
26#11763958—212no—
27#11763958ΓL(1,𝔽8)11yescharacteristic, direct product factor
28#11763957—71no—
29ΓL(1,𝔽8)×Dic13ΓL(1,𝔽8)11yescharacteristic, complemented
30C7×Dic13⋊C3—71no—
31C2×ΓL(1,𝔽8)2—131no—
32C3×#63271—71no—
33ΓL(1,𝔽8)×C13⋊C3—71no—
34C72×C13C32×C411yescharacteristic, complemented, Hall
35C4×#3862—131no—
36C4×#3861—131no—
37C4×C7×ΓL(1,𝔽8)—131no—
38C4×C7×ΓL(1,𝔽8)—391no—
39C2×C3×C7×C13—71no—
40C2×#63271—71no—
41C2×#63271—71no—
42C2×#63271—71no—
43C2×#63271—212no—
44C2×#63271C2×ΓL(1,𝔽8)11yescharacteristic
45C2×#63270—71no—
46C2×C13×ΓL(1,𝔽8)C2×ΓL(1,𝔽8)11yescharacteristic
47C2×C7×C13⋊C3—71no—
48C3×Dic13⋊C3—491noHall
49ΓL(1,𝔽8)2—131noHall
50C7×Dic13—32no—
51C7×Dic13C3×ΓL(1,𝔽8)11yescharacteristic, complemented
52C7×Dic13C3×ΓL(1,𝔽8)11yescharacteristic, complemented
53C2×#3862—131no—
54C2×#3861—131no—
55C2×C7×ΓL(1,𝔽8)—131no—
56C2×C7×ΓL(1,𝔽8)D13⋊C311yescharacteristic
57C3×C7×C13—71no—
58#63271—71no—
59#63271—71no—
60#63271—71no—
61#63271—212no—
62#63271C4×ΓL(1,𝔽8)11yescharacteristic, complemented
63#63270—71no—
64C7×C13⋊C3—71no—
65C13×ΓL(1,𝔽8)C4×ΓL(1,𝔽8)11yescharacteristic, complemented
66C3×C4×ΓL(1,𝔽8)—911no—
67C3×C4×ΓL(1,𝔽8)—911no—
68C2×C3×C13⋊C3—491no—
69C4×C72—391noHall
70C2×C7×C13—32no—
71C2×C7×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic
72C2×C7×C13C2×C3×ΓL(1,𝔽8)11yescharacteristic
73C3×Dic13—71no—
74Dic13⋊C3—71no—
75Dic13⋊C3—491no—
76Dic13⋊C3—491no—
77#3862—131no—
78#3861—131no—
79C7×ΓL(1,𝔽8)—131no—
80C7×ΓL(1,𝔽8)Dic13⋊C311yescharacteristic, complemented
81C2×C3×ΓL(1,𝔽8)—911no—
82C2×C3×ΓL(1,𝔽8)—911no—
83C3×C13⋊C3—491noHall
84C2×C72C3×D13⋊C311yescharacteristic
85C7×C13—32no—
86C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
87C7×C13C3×C4×ΓL(1,𝔽8)11yescharacteristic, complemented
88C3×C4×C7—2731no—
89C3×C4×C7—911no—
90C4×ΓL(1,𝔽8)—131no—
91C4×ΓL(1,𝔽8)—2732no—
92C4×ΓL(1,𝔽8)—391no—
93C4×ΓL(1,𝔽8)—911no—
94C4×ΓL(1,𝔽8)—911no—
95C4×ΓL(1,𝔽8)—911no—
96C4×ΓL(1,𝔽8)—911no—
97C2×C3×C13—71no—
98C2×C13⋊C3—71no—
99C2×C13⋊C3—491no—
100C2×C13⋊C3—491no—
101C3×ΓL(1,𝔽8)—911no—
102C3×ΓL(1,𝔽8)—911no—
103Dic13ΓL(1,𝔽8)211yescharacteristic, complemented, Hall
104C72C3×Dic13⋊C311yescharacteristic, complemented, Sylow
105C2×C3×C7—71no—
106C2×C3×C7—911no—
107C2×ΓL(1,𝔽8)—131no—
108C2×ΓL(1,𝔽8)—2732no—
109C2×ΓL(1,𝔽8)—911no—
110C2×ΓL(1,𝔽8)—911no—
111C2×ΓL(1,𝔽8)—911no—
112C2×ΓL(1,𝔽8)—911no—
113C2×ΓL(1,𝔽8)#1058771711yescharacteristic
114C3×C13—71no—
115C13⋊C3—71no—
116C13⋊C3—491no—
117C13⋊C3—491no—
118C32×C4—6371noHall
119C4×C7—391no—
120C4×C7—391no—
121C4×C7—1172no—
122C2×C13C2×ΓL(1,𝔽8)211yescharacteristic
123C3×C7—71no—
124C3×C7—911no—
125ΓL(1,𝔽8)—131no—
126ΓL(1,𝔽8)—2732no—
127ΓL(1,𝔽8)—911no—
128ΓL(1,𝔽8)—911no—
129ΓL(1,𝔽8)—911no—
130ΓL(1,𝔽8)—911no—
131ΓL(1,𝔽8)#1176395811yescharacteristic, direct product factor
132C2×C32—6371no—
133C2×C7—32no—
134C2×C7ΓL(1,𝔽8)×D13⋊C311yescharacteristic
135C2×C7C3×#1058771711yescharacteristic
136C13C4×ΓL(1,𝔽8)211yescharacteristic, complemented, Sylow
137C3×C4—2731no—
138C3×C4—911no—
139C3×C4—6371no—
140C3×C4—6371no—
141C32—6371noSylow
142C7—32no—
143C7ΓL(1,𝔽8)×Dic13⋊C311yescharacteristic, complemented
144C7C3×#1176395811yescharacteristic, complemented
145C2×C3—71no—
146C2×C3—911no—
147C2×C3—6371no—
148C2×C3—6371no—
149C4—391noSylow
150C3—71no—
151C3—911no—
152C3—6371no—
153C3—6371no—
154C2ΓL(1,𝔽8)×#1058771711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×#10587717
commutator, hypocenterC72×C13C32×C4
Fitting, socle, cosocleC2×C72×C13C2×C32
prime radicalC2×ΓL(1,𝔽8)×#63271C2
LCSG ▷ C72×C13
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. Dic13⋊#3862C3semidirect
3. #21439550C3semidirect
4. C7×ΓL(1,𝔽8)×Dic13C3semidirect
5. C7×#11763958C3semidirect
6. ΓL(1,𝔽8)×#63271C4semidirect
13. C72×Dic13C32semidirect
14. C13⋊#3862C3×C4semidirect
15. #14274262C3×C4semidirect
16. C7×#63271C3×C4semidirect
17. C7×C13×ΓL(1,𝔽8)C3×C4semidirect
27. #11763958ΓL(1,𝔽8)direct (N × H)
29. ΓL(1,𝔽8)×Dic13ΓL(1,𝔽8)semidirect
34. C72×C13C32×C4semidirect
51. C7×Dic13C3×ΓL(1,𝔽8)semidirect
52. C7×Dic13C3×ΓL(1,𝔽8)semidirect
62. #63271C4×ΓL(1,𝔽8)semidirect
65. C13×ΓL(1,𝔽8)C4×ΓL(1,𝔽8)semidirect
80. C7×ΓL(1,𝔽8)Dic13⋊C3semidirect
86. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
87. C7×C13C3×C4×ΓL(1,𝔽8)semidirect
103. Dic13ΓL(1,𝔽8)2semidirect
104. C72C3×Dic13⋊C3semidirect
131. ΓL(1,𝔽8)#11763958direct (N × H)
136. C13C4×ΓL(1,𝔽8)2semidirect
143. C7ΓL(1,𝔽8)×Dic13⋊C3semidirect
144. C7C3×#11763958semidirect

Maximal subgroups

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

subgroupindex
C2×ΓL(1,𝔽8)×#632712
Dic13⋊#38623
#214395503
C7×ΓL(1,𝔽8)×Dic133
C7×#117639583
C3×#117639587
ΓL(1,𝔽8)×Dic13⋊C37
C4×ΓL(1,𝔽8)213