Finite Group Databasebeta

← #13056462

Subgroups — #13056462

Order 1344.

304 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621662C211yesmaximal, characteristic
2#10621434C211yesmaximal, characteristic, complemented
3#10621392C211yesmaximal, characteristic, complemented
4#10621358C211yesmaximal, characteristic, complemented
5C3×C7×#104C211yesmaximal, characteristic, complemented
6#10620662C211yesmaximal, characteristic, complemented
7#10620644C211yesmaximal, characteristic, complemented
8#89597—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10#66732—21no—
11D4×D21—22no—
12D4×D21—21no—
13#66728—21no—
14#66728C2211yescharacteristic
15C2×D84—21no—
16C2×D84C2211yescharacteristic
17C2×Dic42C2211yescharacteristic, complemented
18C3×C7×M16C2211yescharacteristic, complemented
19C3×C42×C7C2211yescharacteristic, complemented
20#66635—21no—
21#66633—21no—
22#66631—22no—
23#66627C2211yescharacteristic, complemented
24Dic84—21no—
25#66623—21no—
26#6553—31no—
27#6514—31no—
28#6472—31no—
29#6438—31no—
30C7×#104D311yescharacteristic, complemented
31#6397—31no—
32#6379—31no—
33#4825—71nomaximal, Hall
34C22×D21—22no—
35C22×D21D412yes—
36C3×C7×Q8D411yescharacteristic
37C3×C7×D4—21no—
38C3×C7×D4D411yescharacteristic
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7C2311yescharacteristic
42#4268—22no—
43#4268—21no—
44C2×Dic21—21no—
45D84—22no—
46D84—21no—
47D84—21no—
48D84D411yescharacteristic
49C4×D21—22no—
50C4×D21—21no—
51Dic42—21no—
52Dic42—21no—
53Dic42D411yescharacteristic
54C3×C7×C8—21no—
55C3⋊(C7⋊C8)—21no—
56C7×#41C2×D311yescharacteristic
57#1161—61no—
58D4×D7—62no—
59D4×D7—61no—
60#1157—31no—
61#1157—61no—
62C2×D28—31no—
63C2×D28—61no—
64C2×Dic14—31no—
65C7×M16C2×D311yescharacteristic, complemented
66C42×C7C2×D311yescharacteristic, complemented
67#1144—61no—
68#1142—61no—
69(C2×D7)⋊C4—62no—
70#1136—31no—
71Dic28—61no—
72#1132—61no—
73#1009—71no—
74#951—71no—
75#909—71no—
76#875—71no—
77C3×#104D711yescharacteristic, complemented
78#823—71no—
79#805—71no—
80C22×C3×C7—21no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—22no—
83C2×D21—21no—
84C2×D21—42no—
85C2×D21—82no—
86C3×C4×C7—21no—
87C3×C4×C7—41no—
88C3×C4×C7C2×D411yescharacteristic
89C3×C4×C7C2×D411yescharacteristic
90Dic21—21no—
91Dic21—41no—
92#455—211noSylow
93C22×D7—32no—
94C22×D7—62no—
95C7×Q8D1211yescharacteristic
96C7×D4—21no—
97C7×D4D1211yescharacteristic
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7C22×D311yescharacteristic
101C22⋊D7—62no—
102C22⋊D7—61no—
103C2×Dic7—61no—
104D28—31no—
105D28—62no—
106D28—61no—
107D28—61no—
108C4×D7—62no—
109C4×D7—61no—
110Dic14—31no—
111Dic14—61no—
112Dic14—61no—
113C7×C8—21no—
114C7⋊C8—61no—
115C3×#41C2×D711yescharacteristic
116#239—141no—
117D3×D4—142no—
118D3×D4—141no—
119#235—71no—
120#235—141no—
121C2×D12—71no—
122C2×D12—141no—
123C2×Dic6—71no—
124C3×M16C2×D711yescharacteristic, complemented
125C3×C42C2×D711yescharacteristic, complemented
126#216—141no—
127#214—141no—
128(C2×D3)⋊C4—142no—
129#208—71no—
130Dic12—141no—
131#204—141no—
132C2×C3×C7—21no—
133C2×C3×C7—41no—
134C2×C3×C7#12011yescharacteristic
135D21—42no—
136D21—41no—
137H(5,𝔽2)—211no—
138#137—211no—
139#137—211no—
140#124—211no—
141#104—211no—
142#104D2111yescharacteristic, complemented
143#100—211no—
144C22×C7—21no—
145C22×C7D3×D411yescharacteristic
146C2×D7—61no—
147C2×D7—62no—
148C2×D7—122no—
149C2×D7—242no—
150C4×C7—21no—
151C4×C7—41no—
152C4×C7C2×D1211yescharacteristic
153C4×C7D3×D411yescharacteristic
154Dic7—61no—
155Dic7—121no—
156C22×D3—72no—
157C22×D3—142no—
158C3×Q8D2811yescharacteristic
159C3×D4—21no—
160C3×D4D2811yescharacteristic
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4C22×D711yescharacteristic
164#67—142no—
165#67—141no—
166C2×Dic3—141no—
167D12—71no—
168D12—142no—
169D12—141no—
170D12—141no—
171C4×D3—142no—
172C4×D3—141no—
173Dic6—71no—
174Dic6—141no—
175Dic6—141no—
176C3×C8—21no—
177C3⋊C8—141no—
178C3×C7#45511yescharacteristic, complemented, Hall
179#41—211no—
180#41—421no—
181#41—421no—
182#41C2×D2111yescharacteristic
183C2×Q8—211no—
184C2×D4—211no—
185C2×D4—421no—
186C2×D4—422no—
187C2×D4—421no—
188Q16—421no—
189Q16—421no—
190SD16—421no—
191SD16—421no—
192M16—211no—
193M16C2×D2111yescharacteristic, complemented
194C22⋊C4—422no—
195C42C2×D2111yescharacteristic, complemented
196C2×C7—21no—
197C2×C7—41no—
198C2×C7#88211yescharacteristic
199D7—122no—
200D7—121no—
201C22×C3—21no—
202C22×C3D4×D711yescharacteristic
203C2×D3—141no—
204C2×D3—142no—
205C2×D3—282no—
206C2×D3—562no—
207C3×C4—21no—
208C3×C4—41no—
209C3×C4C2×D2811yescharacteristic
210C3×C4D4×D711yescharacteristic
211Dic3—141no—
212Dic3—281no—
213C23—212no—
214C23—422no—
215Q8—211no—
216Q8—421no—
217Q8—421no—
218Q8D8411yescharacteristic
219D4—21no—
220D4—211no—
221D4—422no—
222D4—422no—
223D4—421no—
224D4—421no—
225D4—421no—
226D4D8411yescharacteristic
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—422no—
230C2×C4—421no—
231C2×C4—421no—
232C2×C4C22×D2111yescharacteristic
233C8—21no—
234C8—421no—
235C7#482511yescharacteristic, complemented, Sylow
236C2×C3—21no—
237C2×C3—41no—
238C2×C3#644511yescharacteristic
239D3—282no—
240D3—281no—
241C22—21no—
242C22—421no—
243C22—422no—
244C22—842no—
245C22—1682no—
246C22D4×D2111yescharacteristic
247C4—21no—
248C4—41no—
249C4—421no—
250C4—841no—
251C4C2×D8411yescharacteristic
252C4D4×D2111yescharacteristic
253C3#8959711yescharacteristic, complemented, Sylow
254C2—21no—
255C2—41no—
256C2—842no—
257C2—841no—
258C2#1062136511yescharacteristic
subgroupG/H
centerC2#10621365
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#104C2
socleC2×C3×C7#120
prime radical#10621662C2
hypercenter#104D21
hypocenterC3×C7#455
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #10621434C2semidirect
3. #10621392C2semidirect
4. #10621358C2semidirect
5. C3×C7×#104C2semidirect
6. #10620662C2semidirect
7. #10620644C2semidirect
17. C2×Dic42C22semidirect
18. C3×C7×M16C22semidirect
19. C3×C42×C7C22semidirect
23. #66627C22semidirect
30. C7×#104D3semidirect
65. C7×M16C2×D3semidirect
66. C42×C7C2×D3semidirect
77. C3×#104D7semidirect
124. C3×M16C2×D7semidirect
125. C3×C42C2×D7semidirect
142. #104D21semidirect
178. C3×C7#455semidirect
193. M16C2×D21semidirect
195. C42C2×D21semidirect
235. C7#4825semidirect
253. C3#89597semidirect

Maximal subgroups

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

subgroupindex
#106216622
#106214342
#106213922
#106213582
C3×C7×#1042
#106206622
#106206442
#895973
#48257