Finite Group Databasebeta

← #13056464

Subgroups — #13056464

Order 1344.

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

normal subgroups only

This group has 292 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#10621663C211yesmaximal, characteristic, complemented
2#10621432C211yesmaximal, characteristic, complemented
3#10621391C211yesmaximal, characteristic, complemented
4#10621358C211yesmaximal, characteristic, complemented
5C3×C7×#104C211yesmaximal, characteristic, complemented
6#10620663C211yesmaximal, characteristic, complemented
7#10620644C211yesmaximal, characteristic, complemented
8#89599—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10Q8×D21—21no—
11#66730—21no—
12#66730—22no—
13#66728—21no—
14#66728C2211yescharacteristic
15C2×D84C2211yescharacteristic
16C2×Dic42—21no—
17C2×Dic42C2211yescharacteristic
18C3×C7×M16C2211yescharacteristic
19C3×C42×C7C2211yescharacteristic
20#66634—21no—
21#66632—21no—
22#66631—22no—
23#66627C2211yescharacteristic
24D168—21no—
25#66623—21no—
26#6554—31no—
27#6512—31no—
28#6471—31no—
29#6438—31no—
30C7×#104D311yescharacteristic, complemented
31#6398—31no—
32#6379—31no—
33#4827—71nomaximal, Hall
34C22×D21—21no—
35C3×C7×Q8D411yescharacteristic
36C3×C7×D4—21no—
37C3×C7×D4D411yescharacteristic
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7C2311yescharacteristic
41#4268—22no—
42#4268—21no—
43C2×Dic21—22no—
44C2×Dic21D412yes—
45D84—21no—
46D84—21no—
47D84D411yescharacteristic
48C4×D21—22no—
49C4×D21—21no—
50Dic42—22no—
51Dic42—21no—
52Dic42—21no—
53Dic42D411yescharacteristic
54C3×C7×C8—21no—
55C3⋊(C7⋊C8)—21no—
56C7×#41C2×D311yescharacteristic
57Q8×D7—61no—
58#1159—62no—
59#1159—61no—
60#1157—31no—
61#1157—61no—
62C2×D28—31no—
63C2×Dic14—31no—
64C2×Dic14—61no—
65C7×M16C2×D311yescharacteristic
66C42×C7C2×D311yescharacteristic
67#1143—61no—
68#1141—61no—
69(C2×D7)⋊C4—62no—
70#1136—31no—
71D56—61no—
72#1132—61no—
73#1010—71no—
74#949—71no—
75#908—71no—
76#875—71no—
77C3×#104D711yescharacteristic, complemented
78#824—71no—
79#805—71no—
80C22×C3×C7—21no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—21no—
83C2×D21—42no—
84C2×D21—41no—
85C3×C4×C7—21no—
86C3×C4×C7—41no—
87C3×C4×C7C2×D411yescharacteristic
88C3×C4×C7C2×D411yescharacteristic
89Dic21—22no—
90Dic21—21no—
91#454—211noSylow
92C22×D7—61no—
93C7×Q8D1211yescharacteristic
94C7×D4—21no—
95C7×D4D1211yescharacteristic
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—62no—
100C22⋊D7—61no—
101C2×Dic7—32no—
102C2×Dic7—62no—
103D28—31no—
104D28—61no—
105D28—61no—
106C4×D7—62no—
107C4×D7—61no—
108Dic14—31no—
109Dic14—62no—
110Dic14—61no—
111Dic14—61no—
112C7×C8—21no—
113C7⋊C8—61no—
114C3×#41C2×D711yescharacteristic
115D3×Q8—141no—
116#237—142no—
117#237—141no—
118#235—71no—
119#235—141no—
120C2×D12—71no—
121C2×Dic6—71no—
122C2×Dic6—141no—
123C3×M16C2×D711yescharacteristic
124C3×C42C2×D711yescharacteristic
125#215—141no—
126#213—141no—
127(C2×D3)⋊C4—142no—
128#208—71no—
129D24—141no—
130#204—141no—
131C2×C3×C7—21no—
132C2×C3×C7—41no—
133C2×C3×C7#12011yescharacteristic
134D21—41no—
135D21—81no—
1362−1+4—211no—
137AGL(1,ℤ/8)—211no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—211no—
141#104D2111yescharacteristic, complemented
142#101—211no—
143C22×C7—21no—
144C22×C7D3×D411yescharacteristic
145C2×D7—61no—
146C2×D7—122no—
147C2×D7—121no—
148C4×C7—21no—
149C4×C7—41no—
150C4×C7C2×D1211yescharacteristic
151C4×C7D3×D411yescharacteristic
152Dic7—61no—
153Dic7—62no—
154C22×D3—141no—
155C3×Q8D2811yescharacteristic
156C3×D4—21no—
157C3×D4D2811yescharacteristic
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4C22×D711yescharacteristic
161#67—142no—
162#67—141no—
163C2×Dic3—72no—
164C2×Dic3—142no—
165D12—71no—
166D12—141no—
167D12—141no—
168C4×D3—142no—
169C4×D3—141no—
170Dic6—71no—
171Dic6—142no—
172Dic6—141no—
173Dic6—141no—
174C3×C8—21no—
175C3⋊C8—141no—
176C3×C7#45411yescharacteristic, complemented, Hall
177#41—211no—
178#41—421no—
179#41—422no—
180#41—421no—
181#41C2×D2111yescharacteristic
182C2×Q8—211no—
183C2×Q8—421no—
184C2×Q8—421no—
185C2×D4—211no—
186SD16—421no—
187SD16—421no—
188D8—421no—
189D8—421no—
190M16—211no—
191M16C2×D2111yescharacteristic
192C22⋊C4—422no—
193C42C2×D2111yescharacteristic
194C2×C7—21no—
195C2×C7—41no—
196C2×C7#88211yescharacteristic
197D7—121no—
198D7—241no—
199C22×C3—21no—
200C22×C3D4×D711yescharacteristic
201C2×D3—141no—
202C2×D3—282no—
203C2×D3—281no—
204C3×C4—21no—
205C3×C4—41no—
206C3×C4C2×D2811yescharacteristic
207C3×C4D4×D711yescharacteristic
208Dic3—141no—
209Dic3—142no—
210C23—421no—
211Q8—211no—
212Q8—422no—
213Q8—421no—
214Q8—421no—
215Q8D8411yescharacteristic
216D4—21no—
217D4—211no—
218D4—422no—
219D4—421no—
220D4—421no—
221D4—421no—
222D4D8411yescharacteristic
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—212no—
226C2×C4—422no—
227C2×C4—422no—
228C2×C4—421no—
229C2×C4C22×D2111yescharacteristic
230C8—21no—
231C8—421no—
232C7#482711yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—41no—
235C2×C3#644511yescharacteristic
236D3—281no—
237D3—561no—
238C22—21no—
239C22—421no—
240C22—842no—
241C22—841no—
242C22D4×D2111yescharacteristic
243C4—21no—
244C4—41no—
245C4—422no—
246C4—421no—
247C4C2×D8411yescharacteristic
248C4D4×D2111yescharacteristic
249C3#8959911yescharacteristic, complemented, Sylow
250C2—21no—
251C2—41no—
252C2—841no—
253C2—1681no—
254C2#1062136511yescharacteristic
subgroupG/H
centerC2#10621365
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#104C2
socleC2×C3×C7#120
hypercenter#104D21
hypocenterC3×C7#454
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
1. #10621663C2semidirect
2. #10621432C2semidirect
3. #10621391C2semidirect
4. #10621358C2semidirect
5. C3×C7×#104C2semidirect
6. #10620663C2semidirect
7. #10620644C2semidirect
30. C7×#104D3semidirect
77. C3×#104D7semidirect
141. #104D21semidirect
176. C3×C7#454semidirect
232. C7#4827semidirect
249. C3#89599semidirect

Maximal subgroups

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

subgroupindex
#106216632
#106214322
#106213912
#106213582
C3×C7×#1042
#106206632
#106206442
#895993
#48277