Finite Group Databasebeta

← #89898⋊C3 #13052026

Subgroups — #89898⋊C3 #13052026

Order 1344.

380 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C4×AGL(1,𝔽7)C211yesmaximal, characteristic
2C2×M16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C2×#1136⋊C3C211yesmaximal, characteristic, complemented
4(C2×AGL(1,𝔽7))⋊C8C214yesmaximal, complemented
5#89898C311yesmaximal, characteristic, complemented, Hall
6C23×AGL(1,𝔽7)C411yescharacteristic
7C22×C4×ΓL(1,𝔽8)C2211yescharacteristic
8C22×ΓL(1,𝔽8)⋊C4C411yescharacteristic
9C2×C4×AGL(1,𝔽7)—22no—
10C2×C4×AGL(1,𝔽7)—22no—
11C2×C4×AGL(1,𝔽7)C412yes—
12C2×C4×AGL(1,𝔽7)C2212yes—
13M16×ΓL(1,𝔽8)—22no—
14C2×C8×ΓL(1,𝔽8)C2212yescomplemented
15#1136⋊C3—22no—
16C2×ΓL(1,𝔽8)⋊C8C2212yescomplemented
17C22×C4×D7C2×C311yescharacteristic
18C2×C7×M16C2×C311yescharacteristic, complemented
19C2×#1136C2×C311yescharacteristic, complemented
20(C2×D7)⋊C8C2×C314yescomplemented
21C3×#407—71nomaximal, Hall
22C23×ΓL(1,𝔽8)C2×C411yescharacteristic
23C22×AGL(1,𝔽7)—22no—
24C22×AGL(1,𝔽7)—24no—
25C22×AGL(1,𝔽7)C2×C412yes—
26C2×C4×ΓL(1,𝔽8)C2×C411yescharacteristic
27C2×C4×ΓL(1,𝔽8)D412yes—
28C2×C4×ΓL(1,𝔽8)D412yes—
29C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
30C2×ΓL(1,𝔽8)⋊C4—22no—
31C2×ΓL(1,𝔽8)⋊C4C2×C412yes—
32C4×AGL(1,𝔽7)—24no—
33C4×AGL(1,𝔽7)—22no—
34C4×AGL(1,𝔽7)—22no—
35C8×ΓL(1,𝔽8)—22no—
36ΓL(1,𝔽8)⋊C8—22no—
37C23×D7C3×C411yescharacteristic
38C22×C4×C7C22×C311yescharacteristic
39C22×Dic7C3×C411yescharacteristic
40C2×C4×D7—22no—
41C2×C4×D7—22no—
42C2×C4×D7C3×C412yes—
43C2×C4×D7C22×C312yes—
44C7×M16—22no—
45C2×C7×C8C22×C312yescomplemented
46#1136—22no—
47C2×C7⋊C8C22×C312yescomplemented
48C23×C3×C4—71no—
49C2×C3×M16—71no—
50C2×C3×M16—71no—
51C3×C22⋊C8—74no—
52C22×ΓL(1,𝔽8)—22no—
53C22×ΓL(1,𝔽8)C22⋊C412yes—
54C22×ΓL(1,𝔽8)C22×C411yescharacteristic
55C2×AGL(1,𝔽7)—28no—
56C2×AGL(1,𝔽7)—22no—
57C2×AGL(1,𝔽7)—22no—
58C2×AGL(1,𝔽7)M1614yescomplemented
59C4×ΓL(1,𝔽8)C22⋊C412yes—
60C4×ΓL(1,𝔽8)C2×D411yescharacteristic
61C4×ΓL(1,𝔽8)C2×D411yescharacteristic
62ΓL(1,𝔽8)⋊C4—22no—
63#407—71noSylow
64C23×C7C2×C3×C411yescharacteristic
65C22×D7—22no—
66C22×D7—24no—
67C22×D7C2×C3×C412yes—
68C2×C4×C7C2×C3×C411yescharacteristic
69C2×C4×C7C3×D412yes—
70C2×C4×C7C3×D412yes—
71C2×C4×C7C23×C311yescharacteristic
72C2×Dic7—22no—
73C2×Dic7C2×C3×C412yes—
74C4×D7—24no—
75C4×D7—22no—
76C4×D7—22no—
77C7×C8—22no—
78C7⋊C8—22no—
79C24×C3—71no—
80C22×C3×C4—72no—
81C22×C3×C4—72no—
82C22×C3×C4—71no—
83C22×C3×C4—71no—
84C22×C3×C4—142no—
85C22×C3×C4—142no—
86C3×M16—142no—
87C3×M16—142no—
88C2×C3×C8—72no—
89C2×C3×C8—72no—
90C2×ΓL(1,𝔽8)—22no—
91C2×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
92C2×ΓL(1,𝔽8)C2×M1612yes—
93AGL(1,𝔽7)—24no—
94C23×C4—71no—
95C2×M16—71no—
96C2×M16AGL(1,𝔽7)11yescharacteristic, complemented
97C22⋊C8—74no—
98C22×C7—22no—
99C22×C7C3×C22⋊C412yes—
100C22×C7C22×C3×C411yescharacteristic
101C2×D7—28no—
102C2×D7—22no—
103C2×D7—22no—
104C2×D7C3×M1614yescomplemented
105C4×C7C3×C22⋊C412yes—
106C4×C7C2×C3×D411yescharacteristic
107C4×C7C2×C3×D411yescharacteristic
108Dic7—22no—
109C23×C3—72no—
110C23×C3—71no—
111C23×C3—142no—
112C23×C3—144no—
113C2×C3×C4—72no—
114C2×C3×C4—71no—
115C2×C3×C4—72no—
116C2×C3×C4—72no—
117C2×C3×C4—71no—
118C2×C3×C4—142no—
119C2×C3×C4—142no—
120C2×C3×C4—144no—
121C2×C3×C4—142no—
122C3×C8—142no—
123C3×C8—142no—
124ΓL(1,𝔽8)#40711yescharacteristic, complemented, Hall
125C24—71no—
126C22×C4—71no—
127C22×C4—72no—
128C22×C4—72no—
129C22×C4—142no—
130C22×C4—142no—
131C22×C4C2×AGL(1,𝔽7)11yescharacteristic
132M16—22no—
133M16—142no—
134C2×C8—72no—
135C2×C8C2×AGL(1,𝔽7)12yescomplemented
136C2×C7—22no—
137C2×C7C2×C3×C22⋊C411yescharacteristic
138C2×C7C2×C3×M1612yes—
139D7—24no—
140C22×C3—74no—
141C22×C3—71no—
142C22×C3—72no—
143C22×C3—142no—
144C22×C3—142no—
145C22×C3—142no—
146C22×C3—148no—
147C3×C4—71no—
148C3×C4—71no—
149C3×C4—72no—
150C3×C4—142no—
151C23—72no—
152C23—144no—
153C23—142no—
154C23C4×AGL(1,𝔽7)11yescharacteristic
155C2×C4—72no—
156C2×C4—142no—
157C2×C4—142no—
158C2×C4—142no—
159C2×C4—144no—
160C2×C4C4×AGL(1,𝔽7)11yescharacteristic
161C2×C4C4⋊AGL(1,𝔽7)12yes—
162C2×C4(C22⋊D7)⋊C312yes—
163C2×C4C22×AGL(1,𝔽7)11yescharacteristic
164C8—22no—
165C8—142no—
166C7C3×#40711yescharacteristic, complemented, Sylow
167C2×C3—72no—
168C2×C3—71no—
169C2×C3—142no—
170C2×C3—144no—
171C22—22no—
172C22—74no—
173C22—148no—
174C22—142no—
175C22—142no—
176C22(C2×AGL(1,𝔽7))⋊C412yes—
177C22C2×C4×AGL(1,𝔽7)11yescharacteristic
178C4—142no—
179C4(C2×AGL(1,𝔽7))⋊C412yes—
180C4C2×C4⋊AGL(1,𝔽7)11yescharacteristic
181C4C2×(C22⋊D7)⋊C311yescharacteristic
182C3—71noSylow
183C2—22no—
184C2—144no—
185C2M16×AGL(1,𝔽7)12yes—
186C2C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×AGL(1,𝔽7)
commutator, socleC22×C7C22×C3×C4
FittingC2×C7×M16C2×C3
prime radicalC23×AGL(1,𝔽7)C4
cosocleC2×C4×C7C23×C3
hypercenterC2×M16AGL(1,𝔽7)
hypocenterC7C3×#407
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×M16×ΓL(1,𝔽8)C2semidirect
3. C2×#1136⋊C3C2semidirect
4. (C2×AGL(1,𝔽7))⋊C8C2semidirect
5. #89898C3semidirect
14. C2×C8×ΓL(1,𝔽8)C22semidirect
16. C2×ΓL(1,𝔽8)⋊C8C22semidirect
18. C2×C7×M16C2×C3semidirect
19. C2×#1136C2×C3semidirect
20. (C2×D7)⋊C8C2×C3semidirect
45. C2×C7×C8C22×C3semidirect
47. C2×C7⋊C8C22×C3semidirect
58. C2×AGL(1,𝔽7)M16semidirect
96. C2×M16AGL(1,𝔽7)semidirect
104. C2×D7C3×M16semidirect
124. ΓL(1,𝔽8)#407semidirect
135. C2×C8C2×AGL(1,𝔽7)semidirect
166. C7C3×#407semidirect

Maximal subgroups

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

subgroupindex
C22×C4×AGL(1,𝔽7)2
C2×M16×ΓL(1,𝔽8)2
C2×#1136⋊C32
(C2×AGL(1,𝔽7))⋊C82
#898983
C3×#4077