Finite Group Databasebeta

← #89615⋊C3 #13051745

Subgroups — #89615⋊C3 #13051745

Order 1344.

280 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 74 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×D56⋊C3C212yesmaximal, complemented
2#10620710C211yesmaximal, characteristic, complemented
3C4×C4⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
4ΓL(1,𝔽8)×C4⋊C8C211yesmaximal, characteristic, complemented
5#6395⋊C3C212yesmaximal, complemented
6#89615C311yesmaximal, characteristic, complemented, Hall
7C2×C4⋊AGL(1,𝔽7)—21no—
8C2×C4⋊AGL(1,𝔽7)—21no—
9C2×C4⋊AGL(1,𝔽7)C2212yes—
10C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
11C2×C4×AGL(1,𝔽7)—21no—
12C2×C8×ΓL(1,𝔽8)C2212yes—
13C42×ΓL(1,𝔽8)C2211yescharacteristic
14(C2×AGL(1,𝔽7))⋊C4—21no—
15C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
16D56⋊C3—24no—
17C2×D56C2×C312yescomplemented
18#6437C2×C311yescharacteristic, complemented
19C4×D28C2×C311yescharacteristic, complemented
20C7×C4⋊C8C2×C311yescharacteristic, complemented
21#6395C2×C312yescomplemented
22C3×#459—71nomaximal, Hall
23C22×AGL(1,𝔽7)—21no—
24C22×AGL(1,𝔽7)—22no—
25C2×C4×ΓL(1,𝔽8)D411yescharacteristic
26C2×C4×ΓL(1,𝔽8)D411yescharacteristic
27C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
28C2×ΓL(1,𝔽8)⋊C4—21no—
29C4⋊AGL(1,𝔽7)—22no—
30C4⋊AGL(1,𝔽7)—21no—
31C4⋊AGL(1,𝔽7)—22no—
32C4⋊AGL(1,𝔽7)—42no—
33C4⋊AGL(1,𝔽7)—42no—
34C4⋊AGL(1,𝔽7)D412yescomplemented
35C4×AGL(1,𝔽7)—42no—
36C8×ΓL(1,𝔽8)—22no—
37C2×D28—21no—
38C2×D28—21no—
39C2×D28C22×C312yes—
40C2×D28C22×C311yescharacteristic
41C2×C4×D7—21no—
42C2×C7×C8C22×C312yes—
43C42×C7C22×C311yescharacteristic
44(C2×D7)⋊C4—21no—
45C4⋊Dic7C22×C311yescharacteristic
46D56—24no—
47C2×C3×D8—72no—
48C3×#127—71no—
49C3×C4×D4—71no—
50C3×C4⋊C8—71no—
51C3×#102—72no—
52C22×ΓL(1,𝔽8)C2×D411yescharacteristic
53C2×AGL(1,𝔽7)—22no—
54C2×AGL(1,𝔽7)—41no—
55C2×AGL(1,𝔽7)—41no—
56C2×AGL(1,𝔽7)—42no—
57C2×AGL(1,𝔽7)—42no—
58C2×AGL(1,𝔽7)—42no—
59C4×ΓL(1,𝔽8)—21no—
60C4×ΓL(1,𝔽8)D812yescomplemented
61C4×ΓL(1,𝔽8)C2×D411yescharacteristic
62C4×ΓL(1,𝔽8)#4111yescharacteristic
63ΓL(1,𝔽8)⋊C4—41no—
64#459—71noSylow
65C22×D7—21no—
66C22×D7—22no—
67C2×C4×C7C3×D411yescharacteristic
68C2×C4×C7C3×D411yescharacteristic
69C2×C4×C7C23×C311yescharacteristic
70C2×Dic7—21no—
71D28—22no—
72D28—21no—
73D28—22no—
74D28—42no—
75D28—42no—
76D28C3×D412yescomplemented
77C4×D7—42no—
78C7×C8—22no—
79C2×C3×D4—71no—
80C2×C3×D4—72no—
81C2×C3×D4—141no—
82C2×C3×D4—141no—
83C22×C3×C4—141no—
84C3×D8—144no—
85C2×C3×C8—72no—
86C3×C4⋊C4—71no—
87C3×C22⋊C4—141no—
88C3×C42—71no—
89C2×ΓL(1,𝔽8)#12111yescharacteristic
90C2×ΓL(1,𝔽8)C2×D811yescharacteristic
91C2×ΓL(1,𝔽8)AGL(1,ℤ/8)11yescharacteristic
92AGL(1,𝔽7)—42no—
93AGL(1,𝔽7)—82no—
94C2×D8—72no—
95#127—71no—
96C4×D4—71no—
97C4⋊C8AGL(1,𝔽7)11yescharacteristic, complemented
98#102—72no—
99C22×C7C2×C3×D411yescharacteristic
100C2×D7—22no—
101C2×D7—41no—
102C2×D7—41no—
103C2×D7—42no—
104C2×D7—42no—
105C2×D7—42no—
106C4×C7—21no—
107C4×C7C3×D812yescomplemented
108C4×C7C2×C3×D411yescharacteristic
109C4×C7C3×#4111yescharacteristic
110Dic7—41no—
111C23×C3—141no—
112C23×C3—142no—
113C3×D4—72no—
114C3×D4—141no—
115C3×D4—142no—
116C3×D4—142no—
117C3×D4—282no—
118C3×D4—282no—
119C2×C3×C4—71no—
120C2×C3×C4—71no—
121C2×C3×C4—71no—
122C2×C3×C4—141no—
123C2×C3×C4—282no—
124C3×C8—142no—
125ΓL(1,𝔽8)#45911yescharacteristic, complemented, Hall
126C2×D4—71no—
127C2×D4—72no—
128C2×D4—141no—
129C2×D4—141no—
130C22×C4—141no—
131D8—144no—
132C2×C8C2×AGL(1,𝔽7)12yes—
133C4⋊C4—71no—
134C22⋊C4—141no—
135C42C2×AGL(1,𝔽7)11yescharacteristic
136C2×C7C3×#12111yescharacteristic
137C2×C7C2×C3×D811yescharacteristic
138C2×C7C3×AGL(1,ℤ/8)11yescharacteristic
139D7—42no—
140D7—82no—
141C22×C3—71no—
142C22×C3—142no—
143C22×C3—281no—
144C22×C3—281no—
145C22×C3—282no—
146C22×C3—282no—
147C22×C3—282no—
148C3×C4—71no—
149C3×C4—71no—
150C3×C4—72no—
151C3×C4—141no—
152C3×C4—281no—
153C23—141no—
154C23—142no—
155D4—72no—
156D4—142no—
157D4—141no—
158D4—142no—
159D4—282no—
160D4—282no—
161C2×C4—141no—
162C2×C4—282no—
163C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
164C2×C4C4⋊AGL(1,𝔽7)11yescharacteristic
165C2×C4C22×AGL(1,𝔽7)11yescharacteristic
166C8—22no—
167C7C3×#45911yescharacteristic, complemented, Sylow
168C2×C3—71no—
169C2×C3—71no—
170C2×C3—71no—
171C2×C3—282no—
172C2×C3—562no—
173C22—142no—
174C22—281no—
175C22—281no—
176C22—282no—
177C22—282no—
178C22—282no—
179C22C2×C4⋊AGL(1,𝔽7)11yescharacteristic
180C4—21no—
181C4—281no—
182C4D56⋊C312yescomplemented
183C4D4×AGL(1,𝔽7)11yescharacteristic
184C4#6665911yescharacteristic
185C3—71noSylow
186C2—282no—
187C2—562no—
188C2#6458⋊C311yescharacteristic
189C2C2×D56⋊C311yescharacteristic
190C2#6471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×C4⋊C8C2×C3
socleC22×C7C2×C3×D4
hypercenterC4⋊C8AGL(1,𝔽7)
hypocenterC7C3×#459
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×D56⋊C3C2semidirect
2. #10620710C2semidirect
3. C4×C4⋊AGL(1,𝔽7)C2semidirect
4. ΓL(1,𝔽8)×C4⋊C8C2semidirect
5. #6395⋊C3C2semidirect
6. #89615C3semidirect
17. C2×D56C2×C3semidirect
18. #6437C2×C3semidirect
19. C4×D28C2×C3semidirect
20. C7×C4⋊C8C2×C3semidirect
21. #6395C2×C3semidirect
34. C4⋊AGL(1,𝔽7)D4semidirect
60. C4×ΓL(1,𝔽8)D8semidirect
76. D28C3×D4semidirect
97. C4⋊C8AGL(1,𝔽7)semidirect
107. C4×C7C3×D8semidirect
125. ΓL(1,𝔽8)#459semidirect
167. C7C3×#459semidirect
182. C4D56⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D56⋊C32
#106207102
C4×C4⋊AGL(1,𝔽7)2
ΓL(1,𝔽8)×C4⋊C82
#6395⋊C32
#896153
C3×#4597