Finite Group Databasebeta

← #13054833

Subgroups — #13054833

Order 1344.

292 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#6554C211yesmaximal, characteristic, complemented
2#10621432C211yesmaximal, characteristic, complemented
3#10620936C211yesmaximal, characteristic, complemented
4#10620863C211yesmaximal, characteristic, complemented
5C7×#837C211yesmaximal, characteristic, complemented
6#10620512C211yesmaximal, characteristic, complemented
7#10620496C211yesmaximal, characteristic, complemented
8#89599—31nomaximal, Hall
9C3×C7×#41C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C3×Q8×D7—21no—
12C3×#1159—21no—
13C3×#1159—22no—
14C3×#1157—21no—
15C3×#1157C2211yescharacteristic
16C2×C3×Dic14—21no—
17C2×C3×Dic14C2211yescharacteristic
18#66634—21no—
19#66632—21no—
20#66627C2211yescharacteristic
21C4×C7×Dic3C2211yescharacteristic
22C7×#208C2211yescharacteristic
23#66575—22no—
24#66566—21no—
25#66561—21no—
26#6554D311yescharacteristic, complemented
27#6512—31no—
28#6471—31no—
29#6438—31no—
30C7×#104—31no—
31#6398—31no—
32#6379—31no—
33#5245—71nomaximal, Hall
34C22×D21—21no—
35C3×C7×Q8D411yescharacteristic
36C3×C7×D4—21no—
37C3×C7×D4D411yescharacteristic
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7C2311yescharacteristic
40D84—21no—
41D84—21no—
42C2×C7×Dic3—21no—
43C3×C22⋊D7—22no—
44C3×C22⋊D7—21no—
45C2×C3×Dic7—22no—
46C2×C3×Dic7D412yes—
47C3×D28D411yescharacteristic
48C3×C4×D7—22no—
49C3×C4×D7—21no—
50C3×Dic14—22no—
51C3×Dic14—21no—
52C3×Dic14—21no—
53C3×Dic14D411yescharacteristic
54C3⋊(C7⋊C8)—21no—
55C7×C3⋊C8—21no—
56C7×#41C2×D311yescharacteristic
57Q8×D7—21no—
58#1159—21no—
59#1159—22no—
60#1157—21no—
61#1157C2×D311yescharacteristic
62C2×D28—31no—
63C2×Dic14—21no—
64C2×Dic14C2×D311yescharacteristic
65C7×M16—31no—
66C42×C7—31no—
67#1143—61no—
68#1141—61no—
69(C2×D7)⋊C4—62no—
70#1136—31no—
71D56—61no—
72#1132—61no—
73C3×2−1+4—71no—
74#949—71no—
75#949—71no—
76#947—71no—
77#837—71no—
78#837D711yescharacteristic, complemented
79#836—71no—
80C22×C3×C7—21no—
81C22×C3×C7C2×D411yescharacteristic
82C2×D21—42no—
83C2×D21—41no—
84C2×C3×D7—21no—
85C3×C4×C7—21no—
86C3×C4×C7C2×D411yescharacteristic
87C3×C4×C7C2×D411yescharacteristic
88C3×Dic7—22no—
89C3×Dic7—21no—
90C7×Dic3—41no—
91#454—211noSylow
92C22×D7—61no—
93C7×Q8#6711yescharacteristic
94C7×D4—21no—
95C7×D4#6711yescharacteristic
96C2×C4×C7—21no—
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—22no—
100C22⋊D7—21no—
101C2×Dic7—22no—
102C2×Dic7#6712yes—
103D28—61no—
104D28—61no—
105D28#6711yescharacteristic
106C4×D7—22no—
107C4×D7—21no—
108Dic14—22no—
109Dic14—21no—
110Dic14—21no—
111Dic14#6711yescharacteristic
112C7×C8—61no—
113C7⋊C8—61no—
114C3×#41—71no—
115C3×#41—141no—
116C3×#41—142no—
117C3×#41—141no—
118C3×#41C2×D711yescharacteristic
119C2×C3×Q8—71no—
120C2×C3×Q8—141no—
121C2×C3×Q8—141no—
122C2×D12—71no—
123#215—141no—
124#215—141no—
125#213—141no—
126#213—141no—
127(C2×D3)⋊C4—142no—
128C4×Dic3C2×D711yescharacteristic
129#208—71no—
130#208C2×D711yescharacteristic
131C2×C3×C7—21no—
132C2×C3×C7—41no—
133C2×C3×C7#12011yescharacteristic
134D21—81no—
135C3×D7—41no—
1362−1+4—71no—
137AGL(1,ℤ/8)—211no—
138AGL(1,ℤ/8)—211no—
139#124—211no—
140#104—31no—
141#104—211no—
142#101—211no—
143C22×C7—21no—
144C22×C7C2×#6711yescharacteristic
145C2×D7—21no—
146C2×D7—122no—
147C2×D7—121no—
148C4×C7—21no—
149C4×C7—121no—
150C4×C7C2×#6711yescharacteristic
151C4×C7C2×#6711yescharacteristic
152Dic7—22no—
153Dic7—21no—
154C22×D3—141no—
155C3×Q8—71no—
156C3×Q8—142no—
157C3×Q8—141no—
158C3×Q8—141no—
159C3×Q8D2811yescharacteristic
160C3×D4—21no—
161C3×D4—71no—
162C3×D4—142no—
163C3×D4—141no—
164C3×D4D2811yescharacteristic
165C2×C3×C4—21no—
166C2×C3×C4—72no—
167C2×C3×C4—142no—
168C2×C3×C4—142no—
169C2×C3×C4—141no—
170C2×C3×C4C22×D711yescharacteristic
171C2×Dic3—21no—
172D12—141no—
173D12—141no—
174C3⋊C8—21no—
175C3⋊C8—141no—
176C3×C7#45411yescharacteristic, complemented, Hall
177#41—71no—
178#41—141no—
179#41—142no—
180#41—141no—
181#41D3×D711yescharacteristic
182C2×Q8—71no—
183C2×Q8—141no—
184C2×Q8—141no—
185C2×D4—211no—
186SD16—421no—
187SD16—421no—
188D8—421no—
189D8—421no—
190M16—31no—
191M16—211no—
192C22⋊C4—422no—
193C42—31no—
194C2×C7—21no—
195C2×C7—41no—
196C2×C7#95311yescharacteristic
197D7—41no—
198D7—241no—
199C22×C3—21no—
200C22×C3—141no—
201C22×C3D4×D711yescharacteristic
202C2×D3—282no—
203C2×D3—281no—
204C3×C4—21no—
205C3×C4—142no—
206C3×C4—141no—
207C3×C4C2×D2811yescharacteristic
208C3×C4D4×D711yescharacteristic
209Dic3—41no—
210C23—421no—
211Q8—71no—
212Q8—142no—
213Q8—141no—
214Q8—141no—
215Q8#424611yescharacteristic
216D4—21no—
217D4—71no—
218D4—142no—
219D4—141no—
220D4—421no—
221D4—421no—
222D4#424611yescharacteristic
223C2×C4—21no—
224C2×C4—61no—
225C2×C4—72no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—141no—
229C2×C4C2×D3×D711yescharacteristic
230C8—61no—
231C8—421no—
232C7#524511yescharacteristic, complemented, Sylow
233C2×C3—21no—
234C2×C3—41no—
235C2×C3—281no—
236C2×C3#644511yescharacteristic
237D3—561no—
238C22—21no—
239C22—141no—
240C22—842no—
241C22—841no—
242C22D7×#6711yescharacteristic
243C4—21no—
244C4—121no—
245C4—142no—
246C4—141no—
247C4C2×#424611yescharacteristic
248C4D7×#6711yescharacteristic
249C3#8959911yescharacteristic, complemented, Sylow
250C2—21no—
251C2—41no—
252C2—281no—
253C2—1681no—
254C2#1062112611yescharacteristic
subgroupG/H
centerC2#10621126
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×#41C22
socleC2×C3×C7#120
hypercenter#41D3×D7
hypocenterC3×C7#454
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#6554C2semidirect
2. #10621432C2semidirect
3. #10620936C2semidirect
4. #10620863C2semidirect
5. C7×#837C2semidirect
6. #10620512C2semidirect
7. #10620496C2semidirect
26. #6554D3semidirect
78. #837D7semidirect
176. C3×C7#454semidirect
232. C7#5245semidirect
249. C3#89599semidirect

Maximal subgroups

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

subgroupindex
C3×#65542
#106214322
#106209362
#106208632
C7×#8372
#106205122
#106204962
#895993
#52457