Finite Group Databasebeta

← #89974⋊C3 #13052104

Subgroups — #89974⋊C3 #13052104

Order 1344.

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

normal subgroups only

This group has 304 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#10621518C211yesmaximal, characteristic
2ΓL(1,𝔽8)×#137C211yesmaximal, characteristic, complemented
3#6510⋊C3C211yesmaximal, characteristic, complemented
4#6505⋊C3C211yesmaximal, characteristic, complemented
5#6411⋊C3C211yesmaximal, characteristic, complemented
6#6399⋊C3C211yesmaximal, characteristic, complemented
7#6397⋊C3C211yesmaximal, characteristic, complemented
8#89974C311yesmaximal, characteristic, complemented, Hall
9#41×ΓL(1,𝔽8)C2211yescharacteristic
10C2×Q8×ΓL(1,𝔽8)C2211yescharacteristic, complemented
11#66659—21no—
12D4×AGL(1,𝔽7)—22no—
13D4×AGL(1,𝔽7)—21no—
14#1157⋊C3—21no—
15#1157⋊C3C2211yescharacteristic
16C2×C4⋊AGL(1,𝔽7)—21no—
17C2×C4⋊AGL(1,𝔽7)C2211yescharacteristic
18Q16×ΓL(1,𝔽8)—21no—
19SD16×ΓL(1,𝔽8)—21no—
20M16×ΓL(1,𝔽8)C2211yescharacteristic, complemented
21#1144⋊C3—21no—
22#1143⋊C3—21no—
23(C2×AGL(1,𝔽7))⋊C4—22no—
24C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic, complemented
25#1136⋊C3C2211yescharacteristic, complemented
26#6553C2×C311yescharacteristic
27C7×#137C2×C311yescharacteristic, complemented
28#6510C2×C311yescharacteristic, complemented
29#6505C2×C311yescharacteristic, complemented
30#6411C2×C311yescharacteristic, complemented
31#6399C2×C311yescharacteristic, complemented
32#6397C2×C311yescharacteristic, complemented
33C3×#455—71nomaximal, Hall
34C22×AGL(1,𝔽7)—22no—
35C22×AGL(1,𝔽7)D412yes—
36Q8×ΓL(1,𝔽8)—21no—
37Q8×ΓL(1,𝔽8)—21no—
38Q8×ΓL(1,𝔽8)D411yescharacteristic
39D4×ΓL(1,𝔽8)—21no—
40D4×ΓL(1,𝔽8)D411yescharacteristic
41C2×C4×ΓL(1,𝔽8)—21no—
42C2×C4×ΓL(1,𝔽8)—21no—
43C2×C4×ΓL(1,𝔽8)C2311yescharacteristic
44(C22⋊D7)⋊C3—22no—
45(C22⋊D7)⋊C3—21no—
46C2×ΓL(1,𝔽8)⋊C4—21no—
47C4⋊AGL(1,𝔽7)—22no—
48C4⋊AGL(1,𝔽7)—21no—
49C4⋊AGL(1,𝔽7)—21no—
50C4⋊AGL(1,𝔽7)D411yescharacteristic
51C4×AGL(1,𝔽7)—22no—
52C4×AGL(1,𝔽7)—21no—
53ΓL(1,𝔽8)⋊Q8D411yescharacteristic
54C8×ΓL(1,𝔽8)—21no—
55ΓL(1,𝔽8)⋊C8—21no—
56C7×#41C22×C311yescharacteristic
57C2×C7×Q8C22×C311yescharacteristic, complemented
58#1161—21no—
59D4×D7—22no—
60D4×D7—21no—
61#1157—21no—
62#1157C22×C311yescharacteristic
63C2×D28—21no—
64C2×D28C22×C311yescharacteristic
65C7×Q16—21no—
66C7×SD16—21no—
67C7×M16C22×C311yescharacteristic, complemented
68#1144—21no—
69#1143—21no—
70(C2×D7)⋊C4—22no—
71C4×Dic7C22×C311yescharacteristic, complemented
72#1136C22×C311yescharacteristic, complemented
73C3×H(5,𝔽2)—71no—
74C3×#137—71no—
75C3×#137—71no—
76C3×#124—71no—
77C3×#104—71no—
78C3×#104—71no—
79C3×#100—71no—
80C22×ΓL(1,𝔽8)—21no—
81C22×ΓL(1,𝔽8)C2×D411yescharacteristic
82C2×AGL(1,𝔽7)—22no—
83C2×AGL(1,𝔽7)—21no—
84C2×AGL(1,𝔽7)—42no—
85C2×AGL(1,𝔽7)—82no—
86C4×ΓL(1,𝔽8)—21no—
87C4×ΓL(1,𝔽8)—41no—
88C4×ΓL(1,𝔽8)C2×D411yescharacteristic
89C4×ΓL(1,𝔽8)C2×D411yescharacteristic
90ΓL(1,𝔽8)⋊C4—21no—
91ΓL(1,𝔽8)⋊C4—41no—
92#455—71noSylow
93C22×D7—22no—
94C22×D7C3×D412yes—
95C7×Q8—21no—
96C7×Q8—21no—
97C7×Q8C3×D411yescharacteristic
98C7×D4—21no—
99C7×D4C3×D411yescharacteristic
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7C23×C311yescharacteristic
103C22⋊D7—22no—
104C22⋊D7—21no—
105C2×Dic7—21no—
106D28—22no—
107D28—21no—
108D28—21no—
109D28C3×D411yescharacteristic
110C4×D7—22no—
111C4×D7—21no—
112Dic14C3×D411yescharacteristic
113C7×C8—21no—
114C7⋊C8—21no—
115C3×#41—71no—
116C3×#41—71no—
117C3×#41—141no—
118C3×#41—141no—
119C2×C3×Q8—71no—
120C2×C3×D4—71no—
121C2×C3×D4—141no—
122C2×C3×D4—141no—
123C2×C3×D4—142no—
124C3×Q16—141no—
125C3×Q16—141no—
126C3×SD16—141no—
127C3×SD16—141no—
128C3×M16—71no—
129C3×M16—71no—
130C3×C22⋊C4—142no—
131C3×C42—71no—
132C2×ΓL(1,𝔽8)—21no—
133C2×ΓL(1,𝔽8)—41no—
134C2×ΓL(1,𝔽8)#12011yescharacteristic
135AGL(1,𝔽7)—42no—
136AGL(1,𝔽7)—41no—
137H(5,𝔽2)—71no—
138#137—71no—
139#137AGL(1,𝔽7)11yescharacteristic, complemented
140#124—71no—
141#104—71no—
142#104—71no—
143#100—71no—
144C22×C7—21no—
145C22×C7C2×C3×D411yescharacteristic
146C2×D7—22no—
147C2×D7—21no—
148C2×D7—42no—
149C2×D7—82no—
150C4×C7—21no—
151C4×C7—41no—
152C4×C7C2×C3×D411yescharacteristic
153C4×C7C2×C3×D411yescharacteristic
154Dic7—21no—
155Dic7—41no—
156C23×C3—72no—
157C23×C3—142no—
158C3×Q8—71no—
159C3×Q8—71no—
160C3×Q8—141no—
161C3×Q8—141no—
162C3×D4—71no—
163C3×D4—71no—
164C3×D4—141no—
165C3×D4—141no—
166C3×D4—141no—
167C3×D4—141no—
168C3×D4—142no—
169C3×D4—142no—
170C2×C3×C4—71no—
171C2×C3×C4—141no—
172C2×C3×C4—141no—
173C2×C3×C4—141no—
174C2×C3×C4—142no—
175C2×C3×C4—141no—
176C3×C8—141no—
177C3×C8—141no—
178ΓL(1,𝔽8)#45511yescharacteristic, complemented, Hall
179#41—71no—
180#41—141no—
181#41—141no—
182#41C2×AGL(1,𝔽7)11yescharacteristic
183C2×Q8C2×AGL(1,𝔽7)11yescharacteristic, complemented
184C2×D4—71no—
185C2×D4—141no—
186C2×D4—142no—
187C2×D4—141no—
188Q16—21no—
189Q16—141no—
190SD16—21no—
191SD16—141no—
192M16—71no—
193M16C2×AGL(1,𝔽7)11yescharacteristic, complemented
194C22⋊C4—142no—
195C42—71no—
196C2×C7—21no—
197C2×C7—41no—
198C2×C7C3×#12011yescharacteristic
199D7—42no—
200D7—41no—
201C22×C3—71no—
202C22×C3—141no—
203C22×C3—141no—
204C22×C3—142no—
205C22×C3—282no—
206C22×C3—562no—
207C3×C4—71no—
208C3×C4—71no—
209C3×C4—141no—
210C3×C4—141no—
211C3×C4—281no—
212C3×C4—281no—
213C23—72no—
214C23—142no—
215Q8—21no—
216Q8—21no—
217Q8—71no—
218Q8(C22⋊D7)⋊C311yescharacteristic
219D4—21no—
220D4—71no—
221D4—142no—
222D4—142no—
223D4—141no—
224D4—141no—
225D4—141no—
226D4(C22⋊D7)⋊C311yescharacteristic
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—142no—
230C2×C4—141no—
231C2×C4—141no—
232C2×C4C22×AGL(1,𝔽7)11yescharacteristic
233C8—21no—
234C8—141no—
235C7C3×#45511yescharacteristic, complemented, Sylow
236C2×C3—71no—
237C2×C3—141no—
238C2×C3—281no—
239C2×C3—282no—
240C2×C3—281no—
241C22—21no—
242C22—141no—
243C22—142no—
244C22—282no—
245C22—562no—
246C22D4×AGL(1,𝔽7)11yescharacteristic
247C4—21no—
248C4—41no—
249C4—141no—
250C4—281no—
251C4D4×AGL(1,𝔽7)11yescharacteristic
252C4C2×(C22⋊D7)⋊C311yescharacteristic
253C3—71noSylow
254C2—21no—
255C2—41no—
256C2—282no—
257C2—281no—
258C2#6500⋊C311yescharacteristic
subgroupG/H
centerC2#6500⋊C3
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×AGL(1,𝔽7)
FittingC7×#137C2×C3
socleC2×C7C3×#120
prime radical#10621518C2
hypercenter#137AGL(1,𝔽7)
hypocenterC7C3×#455
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. ΓL(1,𝔽8)×#137C2semidirect
3. #6510⋊C3C2semidirect
4. #6505⋊C3C2semidirect
5. #6411⋊C3C2semidirect
6. #6399⋊C3C2semidirect
7. #6397⋊C3C2semidirect
8. #89974C3semidirect
10. C2×Q8×ΓL(1,𝔽8)C22semidirect
20. M16×ΓL(1,𝔽8)C22semidirect
24. C4×ΓL(1,𝔽8)⋊C4C22semidirect
25. #1136⋊C3C22semidirect
27. C7×#137C2×C3semidirect
28. #6510C2×C3semidirect
29. #6505C2×C3semidirect
30. #6411C2×C3semidirect
31. #6399C2×C3semidirect
32. #6397C2×C3semidirect
57. C2×C7×Q8C22×C3semidirect
67. C7×M16C22×C3semidirect
71. C4×Dic7C22×C3semidirect
72. #1136C22×C3semidirect
139. #137AGL(1,𝔽7)semidirect
178. ΓL(1,𝔽8)#455semidirect
183. C2×Q8C2×AGL(1,𝔽7)semidirect
193. M16C2×AGL(1,𝔽7)semidirect
235. C7C3×#455semidirect

Maximal subgroups

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

subgroupindex
#106215182
ΓL(1,𝔽8)×#1372
#6510⋊C32
#6505⋊C32
#6411⋊C32
#6399⋊C32
#6397⋊C32
#899743
C3×#4557