Finite Group Databasebeta

← #13054396

Subgroups — #13054396

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×C3×#1159C211yesmaximal, characteristic, complemented
2C2×#66632C211yesmaximal, characteristic, complemented
3#10621012C211yesmaximal, characteristic, complemented
4C2×#66566C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620505C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89553—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C22×C3×Dic7—21no—
13C3×#1159—22no—
14C3×#1159—22no—
15C2×C3×C4×D7C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C2×#4246—21no—
18#66632—22no—
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×C4⋊Dic3C2211yescharacteristic
21C2×C7×C3⋊C8C2211yescharacteristic
22#66575—21no—
23#66566—22no—
24C2×#1159D311yescharacteristic, complemented
25C2×#1141—31no—
26C2×#1132—31no—
27#6458—31no—
28C7×#102—31no—
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#5238—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21—21no—
34C22×C3×D7D411yescharacteristic
35C3×C7×D4—21no—
36C3×C7×D4D412yescomplemented
37C2×C3×C4×C7C2311yescharacteristic
38D84—21no—
39D84—21no—
40C2×C7×Dic3—21no—
41C3×C22⋊D7—24no—
42C2×C3×Dic7—24no—
43C2×C3×Dic7—21no—
44C2×C3×Dic7D411yescharacteristic
45C3×C4×D7—21no—
46C3×C4×D7—21no—
47C3×Dic14—21no—
48C3×Dic14D412yescomplemented
49#4246—42no—
50C3⋊(C7⋊C8)—21no—
51C7×C3⋊C8—21no—
52C2×C7×D4C2×D311yescharacteristic
53C2×C22⋊D7—21no—
54C22×Dic7—21no—
55#1159—22no—
56#1159—22no—
57C2×D28—31no—
58C2×D28—61no—
59C2×C4×D7C2×D311yescharacteristic
60C2×Dic14C2×D311yescharacteristic
61C2×C7×C8—31no—
62C7×C4⋊C4—31no—
63#1141—62no—
64(C2×D7)⋊C4—61no—
65C2×C7⋊C8—31no—
66#1132—62no—
67C2×C3×#41—71no—
68C2×#215—71no—
69C2×#213—71no—
70#930—71no—
71#835—71no—
72#832D711yescharacteristic, complemented
73#830—71no—
74C22×C3×C7—22no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—41no—
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—22no—
87C3×Dic7—21no—
88C7×Dic3—41no—
89#449—211noSylow
90C23×C7—21no—
91C22×D7—61no—
92C22×D7#6711yescharacteristic
93C7×D4—21no—
94C7×D4#6712yescomplemented
95C2×C4×C7—61no—
96C2×C4×C7C22×D311yescharacteristic
97C22⋊D7—24no—
98C2×Dic7—24no—
99C2×Dic7—21no—
100C2×Dic7#6711yescharacteristic
101D28—61no—
102D28—61no—
103D28—122no—
104C4×D7—21no—
105C4×D7—21no—
106Dic14—21no—
107Dic14#6712yescomplemented
108C7×C8—61no—
109C7⋊C8—61no—
110C3×#41—142no—
111C3×#41—142no—
112C2×C3×Q8—71no—
113C2×C3×D4—141no—
114C2×C3×D4C2×D711yescharacteristic
115C22×C3×C4—71no—
116C22×C3×C4—141no—
117C2×#67—141no—
118C2×D12—71no—
119#215—142no—
120#213—142no—
121(C2×D3)⋊C4—141no—
122C4⋊Dic3C2×D711yescharacteristic
123C2×C3⋊C8—71no—
124C2×C3⋊C8C2×D711yescharacteristic
125C2×C3×C7—42no—
126C2×C3×C7#12011yescharacteristic
127C2×C3×C7#13511yescharacteristic
128C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
129D21—81no—
130C3×D7—41no—
131C2×#41—71no—
132C2×SD16—211no—
133C2×D8—211no—
134#121—211no—
135#103—211no—
136#102—31no—
137C22⋊C8—211no—
138C22×C7—22no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7C2×#6711yescharacteristic
142C2×D7—21no—
143C2×D7—21no—
144C2×D7—21no—
145C2×D7—121no—
146C2×D7—121no—
147C2×D7—121no—
148C4×C7—121no—
149C4×C7C2×#6711yescharacteristic
150C4×C7C2×#6711yescharacteristic
151Dic7—22no—
152Dic7—21no—
153C23×C3—21no—
154C23×C3—71no—
155C22×D3—141no—
156C3×Q8—72no—
157C3×Q8—141no—
158C3×D4—21no—
159C3×D4—144no—
160C3×D4D2812yescomplemented
161C2×C3×C4—71no—
162C2×C3×C4—144no—
163C2×C3×C4—141no—
164C2×C3×C4—141no—
165C2×C3×C4—141no—
166C2×C3×C4C22×D711yescharacteristic
167#67—282no—
168C2×Dic3—21no—
169D12—141no—
170D12—141no—
171C3⋊C8—21no—
172C3⋊C8—141no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—142no—
175#41—142no—
176C2×Q8—71no—
177C2×D4—141no—
178C2×D4—211no—
179C2×D4—421no—
180C2×D4D3×D711yescharacteristic
181C22×C4—71no—
182C22×C4—141no—
183SD16—422no—
184D8—422no—
185C2×C8—31no—
186C2×C8—211no—
187C4⋊C4—31no—
188C22⋊C4—421no—
189C2×C7—42no—
190C2×C7#94911yescharacteristic
191C2×C7#95011yescharacteristic
192C2×C7#95311yescharacteristic
193D7—41no—
194D7—241no—
195C22×C3—22no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—141no—
199C22×C3—141no—
200C22×C3—141no—
201C22×C3D4×D711yescharacteristic
202C2×D3—281no—
203C2×D3—281no—
204C2×D3—281no—
205C3×C4—141no—
206C3×C4—142no—
207C3×C4C2×D2811yescharacteristic
208C3×C4D4×D711yescharacteristic
209Dic3—41no—
210C23—21no—
211C23—71no—
212C23—421no—
213Q8—72no—
214Q8—141no—
215D4—21no—
216D4—144no—
217D4—421no—
218D4—421no—
219D4—842no—
220D4#424612yescomplemented
221C2×C4—61no—
222C2×C4—71no—
223C2×C4—144no—
224C2×C4—141no—
225C2×C4—141no—
226C2×C4—141no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—421no—
230C7#523811yescharacteristic, complemented, Sylow
231C2×C3—42no—
232C2×C3—281no—
233C2×C3#644511yescharacteristic
234C2×C3#647411yescharacteristic
235C2×C3#647911yescharacteristic
236D3—561no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—141no—
241C22—141no—
242C22—141no—
243C22—841no—
244C22—841no—
245C22—841no—
246C22D7×#6711yescharacteristic
247C4—121no—
248C4—141no—
249C4—142no—
250C4C2×#424611yescharacteristic
251C4D7×#6711yescharacteristic
252C3#8955311yescharacteristic, complemented, Sylow
253C2—42no—
254C2—281no—
255C2—1681no—
256C2#1062104011yescharacteristic
257C2#1062105411yescharacteristic
258C2#1062112611yescharacteristic
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×C3×#1159C2semidirect
2. C2×#66632C2semidirect
3. #10621012C2semidirect
4. C2×#66566C2semidirect
5. C7×#832C2semidirect
6. #10620505C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
24. C2×#1159D3semidirect
36. C3×C7×D4D4semidirect
48. C3×Dic14D4semidirect
72. #832D7semidirect
94. C7×D4#67semidirect
107. Dic14#67semidirect
160. C3×D4D28semidirect
173. C3×C7#449semidirect
220. D4#4246semidirect
230. C7#5238semidirect
252. C3#89553semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#11592
C2×#666322
#106210122
C2×#665662
C7×#8322
#106205052
(C2×D7)⋊(C3⋊C8)2
#895533
#52387