Finite Group Databasebeta

← #89825⋊C3 #13051953

Subgroups — #89825⋊C3 #13051953

Order 1344.

320 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 66 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
1#6548⋊C3C211yesmaximal, characteristic
2ΓL(1,𝔽8)×#123C211yesmaximal, characteristic, complemented
3#6492⋊C3C212yesmaximal, complemented
4#6408⋊C3C211yesmaximal, characteristic, complemented
5#6380⋊C3C212yesmaximal, complemented
6#89825C311yesmaximal, characteristic, complemented, Hall
7C2×D4×ΓL(1,𝔽8)C2211yescharacteristic
8C22×C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
9C2×(C22⋊D7)⋊C3—21no—
10C2×(C22⋊D7)⋊C3C2212yes—
11#1159⋊C3—22no—
12D4×AGL(1,𝔽7)—22no—
13#1157⋊C3—21no—
14C4⋊C4×ΓL(1,𝔽8)—21no—
15C22⋊C4×ΓL(1,𝔽8)—21no—
16C22⋊C4×ΓL(1,𝔽8)C2211yescharacteristic, complemented
17(C22⋊Dic7)⋊C3C2212yescomplemented
18(C2×AGL(1,𝔽7))⋊C4—22no—
19(ΓL(1,𝔽8)⋊C4)⋊C4—22no—
20#6548C2×C311yescharacteristic
21C7×#123C2×C311yescharacteristic, complemented
22#6492C2×C312yescomplemented
23#6408C2×C311yescharacteristic, complemented
24#6380C2×C312yescomplemented
25C3×#458—71nomaximal, Hall
26C23×ΓL(1,𝔽8)D411yescharacteristic
27C23×ΓL(1,𝔽8)C2311yescharacteristic
28C22×AGL(1,𝔽7)—21no—
29C22×AGL(1,𝔽7)D412yes—
30D4×ΓL(1,𝔽8)—22no—
31C2×C4×ΓL(1,𝔽8)—21no—
32C2×C4×ΓL(1,𝔽8)—21no—
33C2×C4×ΓL(1,𝔽8)—22no—
34C2×C4×ΓL(1,𝔽8)D411yescharacteristic
35(C22⋊D7)⋊C3—22no—
36(C22⋊D7)⋊C3—22no—
37(C22⋊D7)⋊C3—22no—
38C2×ΓL(1,𝔽8)⋊C4—21no—
39C2×ΓL(1,𝔽8)⋊C4—22no—
40C2×ΓL(1,𝔽8)⋊C4D412yes—
41C4⋊AGL(1,𝔽7)—21no—
42C4×AGL(1,𝔽7)—22no—
43ΓL(1,𝔽8)⋊Q8—21no—
44C2×C7×D4C22×C311yescharacteristic
45C22×C4×C7C22×C311yescharacteristic, complemented
46C2×C22⋊D7—21no—
47C2×C22⋊D7C22×C312yes—
48#1159—22no—
49D4×D7—22no—
50#1157—21no—
51C7×C4⋊C4—21no—
52C7×C22⋊C4—21no—
53C7×C22⋊C4C22×C311yescharacteristic, complemented
54C22⋊Dic7C22×C312yescomplemented
55(C2×D7)⋊C4—22no—
56Dic7⋊C4—22no—
57C3×H(5,𝔽2)—71no—
58C3×#123—72no—
59C3×#123—71no—
60C3×#99—72no—
61C3×#99—71no—
62C22×ΓL(1,𝔽8)—21no—
63C22×ΓL(1,𝔽8)—41no—
64C22×ΓL(1,𝔽8)—41no—
65C22×ΓL(1,𝔽8)C2×D412yes—
66C22×ΓL(1,𝔽8)C2×D411yescharacteristic
67C2×AGL(1,𝔽7)—22no—
68C2×AGL(1,𝔽7)—42no—
69C2×AGL(1,𝔽7)—81no—
70C4×ΓL(1,𝔽8)—21no—
71C4×ΓL(1,𝔽8)—41no—
72C4×ΓL(1,𝔽8)—41no—
73ΓL(1,𝔽8)⋊C4—22no—
74ΓL(1,𝔽8)⋊C4—42no—
75#458—71noSylow
76C23×C7C3×D411yescharacteristic
77C23×C7C23×C311yescharacteristic
78C22×D7—21no—
79C22×D7C3×D412yes—
80C7×D4—22no—
81C2×C4×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7C3×D411yescharacteristic
85C22⋊D7—22no—
86C22⋊D7—22no—
87C22⋊D7—22no—
88C2×Dic7—21no—
89C2×Dic7—22no—
90C2×Dic7C3×D412yes—
91D28—21no—
92C4×D7—22no—
93Dic14—21no—
94C3×#41—141no—
95C3×#41—142no—
96C2×C3×D4—72no—
97C2×C3×D4—71no—
98C2×C3×D4—142no—
99C2×C3×D4—141no—
100C22×C3×C4—71no—
101C3×C4⋊C4—141no—
102C3×C4⋊C4—142no—
103C3×C22⋊C4—72no—
104C3×C22⋊C4—71no—
105C3×C22⋊C4—141no—
106C3×C22⋊C4—142no—
107C2×ΓL(1,𝔽8)—22no—
108C2×ΓL(1,𝔽8)—21no—
109C2×ΓL(1,𝔽8)—41no—
110C2×ΓL(1,𝔽8)#12011yescharacteristic
111AGL(1,𝔽7)—42no—
112H(5,𝔽2)—71no—
113#123—72no—
114#123AGL(1,𝔽7)11yescharacteristic, complemented
115#99—71no—
116#99—72no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7C2×C3×D412yes—
121C22×C7C2×C3×D411yescharacteristic
122C2×D7—22no—
123C2×D7—42no—
124C2×D7—81no—
125C4×C7—21no—
126C4×C7—41no—
127C4×C7—41no—
128Dic7—22no—
129Dic7—42no—
130C23×C3—72no—
131C23×C3—71no—
132C23×C3—71no—
133C23×C3—141no—
134C3×Q8—141no—
135C3×D4—142no—
136C3×D4—142no—
137C3×D4—142no—
138C3×D4—141no—
139C3×D4—142no—
140C2×C3×C4—72no—
141C2×C3×C4—71no—
142C2×C3×C4—141no—
143C2×C3×C4—142no—
144C2×C3×C4—141no—
145C2×C3×C4—142no—
146C2×C3×C4—141no—
147C2×C3×C4—142no—
148ΓL(1,𝔽8)#45811yescharacteristic, complemented, Hall
149#41—142no—
150#41—141no—
151C2×D4—72no—
152C2×D4—142no—
153C2×D4—141no—
154C2×D4C2×AGL(1,𝔽7)11yescharacteristic
155C22×C4C2×AGL(1,𝔽7)11yescharacteristic, complemented
156C4⋊C4—21no—
157C4⋊C4—142no—
158C22⋊C4—21no—
159C22⋊C4—72no—
160C22⋊C4—142no—
161C22⋊C4C2×AGL(1,𝔽7)11yescharacteristic, complemented
162C2×C7—22no—
163C2×C7—21no—
164C2×C7—41no—
165C2×C7C3×#12011yescharacteristic
166D7—42no—
167C22×C3—72no—
168C22×C3—71no—
169C22×C3—141no—
170C22×C3—142no—
171C22×C3—281no—
172C22×C3—281no—
173C22×C3—282no—
174C22×C3—561no—
175C3×C4—141no—
176C3×C4—142no—
177C3×C4—281no—
178C3×C4—281no—
179C3×C4—282no—
180C23—72no—
181C23—141no—
182C23(C22⋊D7)⋊C311yescharacteristic
183C23C22×AGL(1,𝔽7)11yescharacteristic
184Q8—141no—
185D4—22no—
186D4—141no—
187D4—142no—
188D4—142no—
189D4—142no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—72no—
194C2×C4—142no—
195C2×C4—141no—
196C2×C4—142no—
197C2×C4(C22⋊D7)⋊C311yescharacteristic
198C7C3×#45811yescharacteristic, complemented, Sylow
199C2×C3—71no—
200C2×C3—141no—
201C2×C3—142no—
202C2×C3—281no—
203C2×C3—282no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—142no—
208C22—282no—
209C22—561no—
210C22D4×AGL(1,𝔽7)12yes—
211C22C2×(C22⋊D7)⋊C311yescharacteristic
212C4—21no—
213C4—41no—
214C4—41no—
215C4—142no—
216C4—282no—
217C3—71noSylow
218C2—21no—
219C2—22no—
220C2—41no—
221C2—282no—
222C2#6500⋊C311yescharacteristic
subgroupG/H
centerC2#6500⋊C3
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×AGL(1,𝔽7)
FittingC7×#123C2×C3
socleC2×C7C3×#120
prime radical#6548⋊C3C2
hypercenter#123AGL(1,𝔽7)
hypocenterC7C3×#458
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×#123C2semidirect
3. #6492⋊C3C2semidirect
4. #6408⋊C3C2semidirect
5. #6380⋊C3C2semidirect
6. #89825C3semidirect
8. C22×C4×ΓL(1,𝔽8)C22semidirect
16. C22⋊C4×ΓL(1,𝔽8)C22semidirect
17. (C22⋊Dic7)⋊C3C22semidirect
21. C7×#123C2×C3semidirect
22. #6492C2×C3semidirect
23. #6408C2×C3semidirect
24. #6380C2×C3semidirect
45. C22×C4×C7C22×C3semidirect
53. C7×C22⋊C4C22×C3semidirect
54. C22⋊Dic7C22×C3semidirect
114. #123AGL(1,𝔽7)semidirect
148. ΓL(1,𝔽8)#458semidirect
155. C22×C4C2×AGL(1,𝔽7)semidirect
161. C22⋊C4C2×AGL(1,𝔽7)semidirect
198. C7C3×#458semidirect

Maximal subgroups

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

subgroupindex
#6548⋊C32
ΓL(1,𝔽8)×#1232
#6492⋊C32
#6408⋊C32
#6380⋊C32
#898253
C3×#4587