Finite Group Databasebeta

← #13054557

Subgroups — #13054557

Order 1344.

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

normal subgroups only

This group has 324 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
1C2×#66677C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3C2×C7×#214C211yesmaximal, characteristic, complemented
4C3×#6501C211yesmaximal, characteristic, complemented
5#10620507C211yesmaximal, characteristic, complemented
6#10620503C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89944—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C2×C3×C22⋊D7—21no—
13C2×C3×C4×D7C2211yescharacteristic
14C2×#4246—21no—
15C2×D21⋊C4—21no—
16#66677—22no—
17#66677—22no—
18#66632—22no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×#214—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×C22⋊Dic7—21no—
23C3×C4⋊Dic7C2211yescharacteristic
24C2×#1161—31no—
25C2×C7×SD16—31no—
26#6501D311yescharacteristic, complemented
27C2×#1141—31no—
28#6409—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5037—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21—21no—
34C22×C3×D7D411yescharacteristic
35C3×C7×D4—21no—
36C3×C7×D4—21no—
37C2×C3×C4×C7C2311yescharacteristic
38D84—21no—
39D84D412yescomplemented
40C2×C7×Dic3—21no—
41C7×Dic6—21no—
42C7×Dic6D412yescomplemented
43C3×C22⋊D7—42no—
44C2×C3×Dic7—21no—
45C2×C3×Dic7D411yescharacteristic
46C3×C4×D7—21no—
47C3×C4×D7—21no—
48#4246—24no—
49D21⋊C4—24no—
50C3⋊(C7⋊C8)—21no—
51C7×C3⋊C8—21no—
52C2×C7×Q8—31no—
53C2×C7×D4C2×D311yescharacteristic
54C2×C22⋊D7—21no—
55#1161—62no—
56#1161—62no—
57C2×D28—31no—
58C2×D28—61no—
59C2×C4×D7—61no—
60C2×C4×D7C2×D311yescharacteristic
61C7×SD16—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—21no—
64#1141—62no—
65C4⋊Dic7C2×D311yescharacteristic
66C2×C7⋊C8—31no—
67C2×#235—71no—
68C3×#121—71no—
69C2×#214D711yescharacteristic, complemented
70C2×#213—71no—
71#830—71no—
72#810—71no—
73#809—71no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—22no—
79C2×D21—41no—
80C2×D21—41no—
81C2×C3×D7—21no—
82C2×C3×D7—21no—
83C2×C3×D7—21no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86C3×Dic7—21no—
87C3×Dic7—41no—
88C7×Dic3—22no—
89#449—211noSylow
90C23×C7—21no—
91C22×D7—61no—
92C22×D7#6711yescharacteristic
93C7×Q8—32no—
94C7×Q8—61no—
95C7×D4—21no—
96C7×D4—21no—
97C2×C4×C7—61no—
98C2×C4×C7C22×D311yescharacteristic
99C22⋊D7—42no—
100C2×Dic7—21no—
101C2×Dic7#6711yescharacteristic
102D28—32no—
103D28—64no—
104D28—61no—
105C4×D7—21no—
106C4×D7—21no—
107C4×D7—64no—
108C7×C8—61no—
109C7⋊C8—61no—
110C2×C3×D4—141no—
111C2×C3×D4C2×D711yescharacteristic
112C22×C3×C4—71no—
113C2×#67—141no—
114#235—142no—
115#235—142no—
116C2×D12—71no—
117C2×C4×D3—141no—
118C2×Dic6C2×D711yescharacteristic
119C3×C4⋊C4—71no—
120C3×C22⋊C4—141no—
121#214—22no—
122#213—142no—
123C2×C3⋊C8—71no—
124C2×C3⋊C8C2×D711yescharacteristic
125C2×C3×C7—81no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129D21—42no—
130C3×D7—41no—
131C2×#41—211no—
132C2×SD16—31no—
133C2×D8—211no—
134#121—71no—
135#103—211no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—41no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7C2×#6711yescharacteristic
142C2×D7—21no—
143C2×D7—21no—
144C2×D7—21no—
145C2×D7—62no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—62no—
149C4×C7D3×D411yescharacteristic
150C4×C7D3×D411yescharacteristic
151Dic7—21no—
152Dic7—41no—
153C23×C3—21no—
154C23×C3—71no—
155C22×D3—141no—
156C3×D4—21no—
157C3×D4—21no—
158C3×D4—282no—
159C2×C3×C4—71no—
160C2×C3×C4—141no—
161C2×C3×C4—141no—
162C2×C3×C4—141no—
163C2×C3×C4C22×D711yescharacteristic
164#67—144no—
165C2×Dic3—21no—
166D12—72no—
167D12—141no—
168C4×D3—144no—
169Dic6—21no—
170Dic6C22⋊D712yescomplemented
171C3⋊C8—21no—
172C3⋊C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—31no—
177C2×D4—141no—
178C2×D4—211no—
179C2×D4—421no—
180C2×D4D3×D711yescharacteristic
181C22×C4—71no—
182C22×C4—421no—
183SD16—62no—
184D8—422no—
185C2×C8—31no—
186C2×C8—211no—
187C4⋊C4—71no—
188C22⋊C4—141no—
189C2×C7—81no—
190C2×C7#93211yescharacteristic
191C2×C7#93711yescharacteristic
192C2×C7#95011yescharacteristic
193D7—41no—
194D7—122no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—141no—
199C22×C3—141no—
200C22×C3—141no—
201C22×C3D4×D711yescharacteristic
202C2×D3—142no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—141no—
206C3×C4—281no—
207C3×C4D4×D711yescharacteristic
208C3×C4C2×C22⋊D711yescharacteristic
209Dic3—22no—
210C23—21no—
211C23—71no—
212C23—421no—
213Q8—32no—
214Q8—61no—
215D4—21no—
216D4—21no—
217D4—212no—
218D4—282no—
219D4—424no—
220D4—421no—
221C2×C4—61no—
222C2×C4—71no—
223C2×C4—141no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—424no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—421no—
230C7#503711yescharacteristic, complemented, Sylow
231C2×C3—81no—
232C2×C3—281no—
233C2×C3#647711yescharacteristic
234C2×C3#647911yescharacteristic
235C2×C3#650011yescharacteristic
236D3—282no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—141no—
241C22—141no—
242C22—141no—
243C22—422no—
244C22—841no—
245C22—841no—
246C22D7×#6711yescharacteristic
247C4—62no—
248C4—141no—
249C4—281no—
250C4D3×C22⋊D711yescharacteristic
251C4#6669411yescharacteristic
252C3#8994411yescharacteristic, complemented, Sylow
253C2—81no—
254C2—281no—
255C2—842no—
256C2#1062105311yescharacteristic
257C2#1062105411yescharacteristic
258C2#1062113911yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66677C2semidirect
2. C2×#66632C2semidirect
3. C2×C7×#214C2semidirect
4. C3×#6501C2semidirect
5. #10620507C2semidirect
6. #10620503C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
26. #6501D3semidirect
39. D84D4semidirect
42. C7×Dic6D4semidirect
69. C2×#214D7semidirect
170. Dic6C22⋊D7semidirect
173. C3×C7#449semidirect
230. C7#5037semidirect
252. C3#89944semidirect

Maximal subgroups

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

subgroupindex
C2×#666772
C2×#666322
C2×C7×#2142
C3×#65012
#106205072
#106205032
(C2×D7)⋊(C3⋊C8)2
#899443
#50377