Finite Group Databasebeta

← #13054194

Subgroups — #13054194

Order 1344.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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×C7×#235C211yesmaximal, characteristic, complemented
2#10621413C211yesmaximal, characteristic, complemented
3C2×#66568C211yesmaximal, characteristic, complemented
4C2×#66562C211yesmaximal, characteristic, complemented
5C3×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620502C211yesmaximal, characteristic, complemented
8#90010—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×#4268—21no—
11C2×D84C2211yescharacteristic
12C2×C7×#67—21no—
13C7×#235—22no—
14C7×#235—22no—
15C2×C7×D12C2211yescharacteristic
16C2×C4×D3×C7—21no—
17C2×C7×Dic6C2211yescharacteristic
18#66631—21no—
19C4⋊Dic21C2211yescharacteristic
20C2×C3×C7⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22#66568—22no—
23#66562—22no—
24C2×C7×#41—31no—
25C2×#1143—31no—
26C2×#1141—31no—
27#6493—31no—
28#6409—31no—
29#6406—31no—
30C22⋊(C7⋊C8)D311yescharacteristic, complemented
31#4734—71nomaximal, Hall
32C23×C3×C7D411yescharacteristic
33C22×D21—21no—
34C22×D3×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7D411yescharacteristic
38C2×C3×C4×C7C2311yescharacteristic
39#4268—42no—
40C2×Dic21—21no—
41D84—21no—
42D84—21no—
43C7×#67—24no—
44C2×C7×Dic3—21no—
45C7×D12—21no—
46C7×D12D412yescomplemented
47C4×D3×C7—24no—
48C7×Dic6—21no—
49C7×Dic6D412yescomplemented
50C3×C7⋊C8—21no—
51C3×C7⋊C8—21no—
52C7×#41—62no—
53C7×#41—62no—
54C2×C7×Q8—31no—
55C2×C7×D4—31no—
56C2×C7×D4—61no—
57C22×C4×C7—61no—
58C22×C4×C7C2×D311yescharacteristic
59C2×C22⋊D7—61no—
60C2×D28—31no—
61#1143—62no—
62#1141—62no—
63(C2×D7)⋊C4—61no—
64C4⋊Dic7—31no—
65C2×C7⋊C8C2×D311yescharacteristic
66C2×C7⋊C8C2×D311yescharacteristic
67C2×#235D711yescharacteristic, complemented
68#930—71no—
69C2×D24—71no—
70C2×#204—71no—
71C3×C22⋊C8—71no—
72#821—71no—
73#816—71no—
74C22×C3×C7—21no—
75C22×C3×C7—21no—
76C22×C3×C7—21no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—41no—
79C2×D21—41no—
80C2×D21—41no—
81C2×D3×C7—22no—
82C2×D3×C7—41no—
83C2×D3×C7—41no—
84C3×C4×C7—21no—
85C3×C4×C7C2×D411yescharacteristic
86C3×C4×C7C2×D411yescharacteristic
87Dic21—41no—
88C7×Dic3—22no—
89#449—211noSylow
90C23×C7—61no—
91C23×C7D1211yescharacteristic
92C22×D7—61no—
93C7×Q8—32no—
94C7×Q8—61no—
95C7×D4—32no—
96C7×D4—64no—
97C7×D4—61no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—64no—
101C2×C4×C7—61no—
102C2×C4×C7D1211yescharacteristic
103C2×C4×C7C22×D311yescharacteristic
104C22⋊D7—122no—
105C2×Dic7—61no—
106D28—61no—
107D28—61no—
108C7⋊C8—21no—
109C7⋊C8—21no—
110C22×C3×C4C2×D711yescharacteristic
111C2×#67—21no—
112C2×#67—141no—
113#235—22no—
114#235—22no—
115C2×D12—71no—
116C2×D12C2×D711yescharacteristic
117C2×C4×D3—21no—
118C2×Dic6C2×D711yescharacteristic
119C2×C3×C8—71no—
120C2×C3×C8—71no—
121(C2×D3)⋊C4—141no—
122C4⋊Dic3—71no—
123D24—142no—
124#204—142no—
125C2×C3×C7—41no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129D21—81no—
130D3×C7—42no—
131C2×#41—31no—
132C2×SD16—211no—
133C2×D8—211no—
134#121—211no—
135#103—211no—
136#102—211no—
137C22⋊C8—71no—
138C22×C7—21no—
139C22×C7—21no—
140C22×C7—21no—
141C22×C7—62no—
142C22×C7—121no—
143C22×C7—121no—
144C22×C7C2×D1211yescharacteristic
145C2×D7—121no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—21no—
149C4×C7—62no—
150C4×C7D3×D411yescharacteristic
151C4×C7D3×D411yescharacteristic
152Dic7—121no—
153C23×C3C22⋊D711yescharacteristic
154C22×D3—21no—
155C22×D3—141no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4C22⋊D711yescharacteristic
159C2×C3×C4C22×D711yescharacteristic
160#67—24no—
161#67—282no—
162C2×Dic3—21no—
163C2×Dic3—141no—
164D12—21no—
165D12—141no—
166D12—141no—
167D12C22⋊D712yescomplemented
168C4×D3—24no—
169Dic6—21no—
170Dic6C22⋊D712yescomplemented
171C3×C8—141no—
172C3×C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—62no—
175#41—62no—
176C2×Q8—31no—
177C2×D4—31no—
178C2×D4—61no—
179C2×D4—211no—
180C2×D4—421no—
181C22×C4—61no—
182C22×C4D3×D711yescharacteristic
183SD16—422no—
184D8—422no—
185C2×C8—71no—
186C2×C8—71no—
187C4⋊C4—211no—
188C22⋊C4—421no—
189C2×C7—41no—
190C2×C7—122no—
191C2×C7#88211yescharacteristic
192C2×C7#90411yescharacteristic
193C2×C7#90811yescharacteristic
194D7—241no—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3C2×C22⋊D711yescharacteristic
199C2×D3—22no—
200C2×D3—41no—
201C2×D3—41no—
202C2×D3—281no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—21no—
206C3×C4C2×C22⋊D711yescharacteristic
207C3×C4C2×C22⋊D711yescharacteristic
208Dic3—22no—
209Dic3—281no—
210C23—61no—
211C23—421no—
212C23#424711yescharacteristic
213Q8—32no—
214Q8—61no—
215D4—32no—
216D4—64no—
217D4—61no—
218D4—421no—
219D4—421no—
220D4—842no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—64no—
224C2×C4—61no—
225C2×C4—421no—
226C2×C4#424711yescharacteristic
227C2×C4C2×D3×D711yescharacteristic
228C8—141no—
229C8—141no—
230C7#473411yescharacteristic, complemented, Sylow
231C2×C3—41no—
232C2×C3#651211yescharacteristic
233C2×C3#651311yescharacteristic
234C2×C3#651611yescharacteristic
235D3—42no—
236D3—561no—
237C22—21no—
238C22—21no—
239C22—21no—
240C22—62no—
241C22—121no—
242C22—121no—
243C22—841no—
244C22—841no—
245C22—841no—
246C22C2×#424711yescharacteristic
247C4—21no—
248C4—62no—
249C4—841no—
250C4D3×C22⋊D711yescharacteristic
251C4D3×C22⋊D711yescharacteristic
252C3#9001011yescharacteristic, complemented, Sylow
253C2—41no—
254C2—122no—
255C2—1681no—
256C2#1062086511yescharacteristic
257C2#1062086711yescharacteristic
258C2#1062113211yescharacteristic
subgroupG/H
centerC22C2×#4247
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC22×C3×C4×C7C22
socleC22×C3×C7C2×D4
hypercenterC22×C4D3×D7
hypocenterC3×C7#449
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C7×#235C2semidirect
2. #10621413C2semidirect
3. C2×#66568C2semidirect
4. C2×#66562C2semidirect
5. C3×C22⋊(C7⋊C8)C2semidirect
6. #10620508C2semidirect
7. #10620502C2semidirect
30. C22⋊(C7⋊C8)D3semidirect
46. C7×D12D4semidirect
49. C7×Dic6D4semidirect
67. C2×#235D7semidirect
167. D12C22⋊D7semidirect
170. Dic6C22⋊D7semidirect
173. C3×C7#449semidirect
230. C7#4734semidirect
252. C3#90010semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2352
#106214132
C2×#665682
C2×#665622
C3×C22⋊(C7⋊C8)2
#106205082
#106205022
#900103
#47347