Finite Group Databasebeta

← #89957⋊C3 #13052082

Subgroups — #89957⋊C3 #13052082

Order 1344.

224 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 68 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×Q16×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
2#6510⋊C3C211yesmaximal, characteristic
3Q8×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
4C2×#1143⋊C3C211yesmaximal, characteristic
5#6409⋊C3C211yesmaximal, characteristic, complemented
6#6395⋊C3C211yesmaximal, characteristic
7(ΓL(1,𝔽8)⋊C4)⋊C8C211yesmaximal, characteristic, complemented
8#89957C311yesmaximal, characteristic, complemented, Hall
9C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic
10C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic
11C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
12Q16×ΓL(1,𝔽8)—22no—
13C2×C8×ΓL(1,𝔽8)C2211yescharacteristic
14#1143⋊C3—22no—
15(C2×AGL(1,𝔽7))⋊C4—21no—
16(C2×AGL(1,𝔽7))⋊C4—21no—
17C4⋊(ΓL(1,𝔽8)⋊C4)—21no—
18C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic
19C4×ΓL(1,𝔽8)⋊C4—21no—
20C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic
21C2×ΓL(1,𝔽8)⋊C8C2211yescharacteristic
22C2×C7×Q16C2×C311yescharacteristic, complemented
23#6510C2×C311yescharacteristic
24Q8×Dic7C2×C311yescharacteristic, complemented
25C2×#1143C2×C311yescharacteristic
26#6409C2×C311yescharacteristic, complemented
27#6395C2×C311yescharacteristic
28Dic7⋊C8C2×C311yescharacteristic, complemented
29C3×#464—71nomaximal, Hall
30C22×AGL(1,𝔽7)—21no—
31Q8×ΓL(1,𝔽8)—21no—
32Q8×ΓL(1,𝔽8)—21no—
33Q8×ΓL(1,𝔽8)—21no—
34Q8×ΓL(1,𝔽8)D412yes—
35C2×C4×ΓL(1,𝔽8)—21no—
36C2×C4×ΓL(1,𝔽8)—21no—
37C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
38C2×ΓL(1,𝔽8)⋊C4—21no—
39C2×ΓL(1,𝔽8)⋊C4D411yescharacteristic
40C2×ΓL(1,𝔽8)⋊C4D411yescharacteristic
41C4⋊AGL(1,𝔽7)—21no—
42C4⋊AGL(1,𝔽7)—21no—
43C8×ΓL(1,𝔽8)—21no—
44ΓL(1,𝔽8)⋊C8—21no—
45C2×C7×Q8C22×C311yescharacteristic
46C2×C7×Q8C22×C311yescharacteristic
47C2×D28C22×C311yescharacteristic
48C7×Q16—22no—
49C2×C7×C8C22×C311yescharacteristic
50#1143—22no—
51(C2×D7)⋊C4—21no—
52(C2×D7)⋊C4—21no—
53C4⋊Dic7—21no—
54C4⋊Dic7C22×C311yescharacteristic
55C4×Dic7—21no—
56C4×Dic7C22×C311yescharacteristic
57C2×C7⋊C8C22×C311yescharacteristic
58C2×C3×Q16—71no—
59C2×C3×SD16—71no—
60C3×#124—71no—
61C3×C4×Q8—71no—
62C3×C4⋊C8—71no—
63C3×#103—71no—
64C3×#102—71no—
65C22×ΓL(1,𝔽8)C2×D411yescharacteristic
66C2×AGL(1,𝔽7)—41no—
67C2×AGL(1,𝔽7)—41no—
68C2×AGL(1,𝔽7)—41no—
69C4×ΓL(1,𝔽8)—22no—
70C4×ΓL(1,𝔽8)—41no—
71C4×ΓL(1,𝔽8)C2×D411yescharacteristic
72C4×ΓL(1,𝔽8)#4111yescharacteristic
73ΓL(1,𝔽8)⋊C4—21no—
74ΓL(1,𝔽8)⋊C4—21no—
75ΓL(1,𝔽8)⋊C4—41no—
76#464—71noSylow
77C22×D7—21no—
78C7×Q8—21no—
79C7×Q8—21no—
80C7×Q8—21no—
81C7×Q8C3×D412yes—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7C23×C311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7C3×D411yescharacteristic
87C2×Dic7C3×D411yescharacteristic
88D28—21no—
89D28—21no—
90C7×C8—21no—
91C7⋊C8—21no—
92C2×C3×Q8—71no—
93C2×C3×Q8—71no—
94C2×C3×D4—71no—
95C3×Q16—142no—
96C3×SD16—142no—
97C2×C3×C8—71no—
98C2×C3×C8—71no—
99C3×C4⋊C4—71no—
100C3×C4⋊C4—141no—
101C3×C22⋊C4—141no—
102C3×C22⋊C4—141no—
103C3×C42—71no—
104C3×C42—141no—
105C2×ΓL(1,𝔽8)#12111yescharacteristic
106C2×ΓL(1,𝔽8)#13511yescharacteristic
107C2×ΓL(1,𝔽8)#13711yescharacteristic
108AGL(1,𝔽7)—81no—
109C2×Q16AGL(1,𝔽7)11yescharacteristic, complemented
110C2×SD16—71no—
111#124—71no—
112C4×Q8—71no—
113C4⋊C8—71no—
114#103—71no—
115#102—71no—
116C22×C7C2×C3×D411yescharacteristic
117C2×D7—41no—
118C2×D7—41no—
119C2×D7—41no—
120C4×C7—22no—
121C4×C7—41no—
122C4×C7C2×C3×D411yescharacteristic
123C4×C7C3×#4111yescharacteristic
124Dic7—21no—
125Dic7—21no—
126Dic7—41no—
127C23×C3—141no—
128C3×Q8—72no—
129C3×Q8—141no—
130C3×Q8—141no—
131C3×Q8—141no—
132C3×D4—141no—
133C3×D4—141no—
134C2×C3×C4—71no—
135C2×C3×C4—71no—
136C2×C3×C4—71no—
137C2×C3×C4—141no—
138C2×C3×C4—141no—
139C2×C3×C4—141no—
140C3×C8—141no—
141C3×C8—141no—
142ΓL(1,𝔽8)#46411yescharacteristic, complemented, Hall
143C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
144C2×Q8C2×AGL(1,𝔽7)11yescharacteristic
145C2×D4—71no—
146Q16—22no—
147SD16—142no—
148C2×C8—71no—
149C2×C8C2×AGL(1,𝔽7)11yescharacteristic
150C4⋊C4—71no—
151C4⋊C4—141no—
152C22⋊C4—141no—
153C22⋊C4—141no—
154C42—71no—
155C42—141no—
156C2×C7C3×#12111yescharacteristic
157C2×C7C3×#13511yescharacteristic
158C2×C7C3×#13711yescharacteristic
159D7—81no—
160C22×C3—71no—
161C22×C3—281no—
162C22×C3—281no—
163C22×C3—281no—
164C3×C4—71no—
165C3×C4—71no—
166C3×C4—142no—
167C3×C4—141no—
168C3×C4—141no—
169C3×C4—281no—
170C3×C4—281no—
171C23—141no—
172Q8—21no—
173Q8—21no—
174Q8—21no—
175Q8(C22⋊D7)⋊C312yes—
176D4—141no—
177D4—141no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—71no—
181C2×C4—71no—
182C2×C4—141no—
183C2×C4C22×AGL(1,𝔽7)11yescharacteristic
184C8—21no—
185C8—141no—
186C7C3×#46411yescharacteristic, complemented, Sylow
187C2×C3—71no—
188C2×C3—71no—
189C2×C3—71no—
190C2×C3—561no—
191C22—281no—
192C22—281no—
193C22—281no—
194C22D4×AGL(1,𝔽7)11yescharacteristic
195C4—22no—
196C4—41no—
197C4—141no—
198C4—141no—
199C4—281no—
200C4#1159⋊C311yescharacteristic
201C4C2×(C22⋊D7)⋊C311yescharacteristic
202C3—71noSylow
203C2—561no—
204C2#6481⋊C311yescharacteristic
205C2#6482⋊C311yescharacteristic
206C2#6502⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC2×C7×Q16C2×C3
socleC22×C7C2×C3×D4
prime radicalC2×C4⋊AGL(1,𝔽7)C22
hypercenterC2×Q16AGL(1,𝔽7)
hypocenterC7C3×#464
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×ΓL(1,𝔽8)C2semidirect
3. Q8×ΓL(1,𝔽8)⋊C4C2semidirect
5. #6409⋊C3C2semidirect
7. (ΓL(1,𝔽8)⋊C4)⋊C8C2semidirect
8. #89957C3semidirect
22. C2×C7×Q16C2×C3semidirect
24. Q8×Dic7C2×C3semidirect
26. #6409C2×C3semidirect
28. Dic7⋊C8C2×C3semidirect
109. C2×Q16AGL(1,𝔽7)semidirect
142. ΓL(1,𝔽8)#464semidirect
186. C7C3×#464semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×ΓL(1,𝔽8)2
#6510⋊C32
Q8×ΓL(1,𝔽8)⋊C42
C2×#1143⋊C32
#6409⋊C32
#6395⋊C32
(ΓL(1,𝔽8)⋊C4)⋊C82
#899573
C3×#4647