Finite Group Databasebeta

← #13054558

Subgroups — #13054558

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×#66672C211yesmaximal, characteristic, complemented
2#10621421C211yesmaximal, characteristic, complemented
3C2×C7×#213C211yesmaximal, characteristic, complemented
4C2×C3×#1142C211yesmaximal, characteristic, complemented
5#10620506C211yesmaximal, characteristic, complemented
6#10620502C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89930—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×#4268—21no—
11C2×C4×D21C2211yescharacteristic
12C2×C7×D12C2211yescharacteristic
13C2×C3×Dic14C2211yescharacteristic
14C2×#4247—21no—
15C2×D3×Dic7—21no—
16#66672—22no—
17#66672—22no—
18#66636—21no—
19C4⋊Dic21C2211yescharacteristic
20C7×#213—22no—
21C2×C7×C3⋊C8C2211yescharacteristic
22C3×#1142—22no—
23C2×C3×C7⋊C8C2211yescharacteristic
24C2×#1159—31no—
25C2×C7×D8—31no—
26#6501—31no—
27C2×#1142D311yescharacteristic, complemented
28#6406—31no—
29(C2×D7)⋊C8—31no—
30#6390—31no—
31#5172—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21D411yescharacteristic
34C22×D3×C7—21no—
35C3×C7×D4—21no—
36C3×C7×D4—21no—
37C2×C3×C4×C7C2311yescharacteristic
38#4268—42no—
39C2×Dic21—21no—
40C2×Dic21D411yescharacteristic
41C4×D21—21no—
42C4×D21—21no—
43C7×D12—21no—
44C7×D12D412yescomplemented
45C2×C3×Dic7—21no—
46C3×Dic14—21no—
47C3×Dic14D412yescomplemented
48#4247—24no—
49D3×Dic7—24no—
50C3×C7⋊C8—21no—
51C7×C3⋊C8—21no—
52C2×C7×D4—31no—
53C2×C7×D4C2×D311yescharacteristic
54C2×C22⋊D7—61no—
55C2×C22⋊D7—61no—
56C22×Dic7—61no—
57#1159—62no—
58#1159—62no—
59C2×C4×D7—31no—
60C2×Dic14C2×D311yescharacteristic
61C7×D8—62no—
62C2×C7×C8—31no—
63C22⋊Dic7—61no—
64#1142—22no—
65C4⋊Dic7—31no—
66C2×C7⋊C8C2×D311yescharacteristic
67C2×#239—71no—
68C2×C3×SD16—71no—
69#938—71no—
70C2×#213D711yescharacteristic, complemented
71#835—71no—
72#821—71no—
73(C2×D3)⋊C8—71no—
74C22×C3×C7—41no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7C2×D411yescharacteristic
78C2×D21—21no—
79C2×D21—21no—
80C2×D21—21no—
81C2×D3×C7—22no—
82C2×D3×C7—41no—
83C2×D3×C7—41no—
84C3×C4×C7C2×D411yescharacteristic
85C3×C4×C7C2×D411yescharacteristic
86Dic21—21no—
87Dic21—41no—
88C3×Dic7—22no—
89#449—211noSylow
90C23×C7—21no—
91C23×C7—61no—
92C22×D7—31no—
93C7×D4—21no—
94C7×D4—21no—
95C7×D4—32no—
96C7×D4—61no—
97C2×C4×C7C22×D311yescharacteristic
98C22⋊D7—64no—
99C22⋊D7—122no—
100C2×Dic7—21no—
101C2×Dic7—31no—
102C2×Dic7—64no—
103C2×Dic7—61no—
104C4×D7—61no—
105C4×D7—61no—
106Dic14—21no—
107Dic14#6712yescomplemented
108C7×C8—61no—
109C7⋊C8—21no—
110C2×C3×Q8—71no—
111C2×C3×D4C2×D711yescharacteristic
112C2×#67—141no—
113#239—142no—
114#239—142no—
115C2×D12—141no—
116C2×D12C2×D711yescharacteristic
117C2×C4×D3—71no—
118C2×C4×D3—141no—
119C3×SD16—142no—
120C2×C3×C8—71no—
121#217—141no—
122#213—22no—
123C4⋊Dic3—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—41no—
130D3×C7—42no—
131C2×#41—211no—
132C2×SD16—71no—
133C2×D8—31no—
134#121—211no—
135#103—211no—
136#102—211no—
137C22⋊C8—211no—
138C22×C7—41no—
139C22×C7—41no—
140C22×C7—41no—
141C22×C7—62no—
142C22×C7—121no—
143C22×C7—121no—
144C22×C7D3×D411yescharacteristic
145C2×D7—61no—
146C2×D7—61no—
147C2×D7—61no—
148C4×C7D3×D411yescharacteristic
149C4×C7C2×#6711yescharacteristic
150Dic7—22no—
151Dic7—61no—
152Dic7—121no—
153C23×C3—21no—
154C22×D3—21no—
155C22×D3—71no—
156C3×Q8—72no—
157C3×Q8—141no—
158C3×D4—21no—
159C3×D4—21no—
160C2×C3×C4—141no—
161C2×C3×C4C22×D711yescharacteristic
162#67—282no—
163C2×Dic3—71no—
164C2×Dic3—141no—
165D12—21no—
166D12—144no—
167D12C22⋊D712yescomplemented
168C4×D3—144no—
169C4×D3—141no—
170C4×D3—141no—
171C3×C8—141no—
172C3⋊C8—21no—
173C3×C7#44911yescharacteristic, complemented, Hall
174#41—422no—
175#41—422no—
176C2×Q8—71no—
177C2×D4—31no—
178C2×D4—421no—
179C2×D4—421no—
180C2×D4D3×D711yescharacteristic
181C22×C4—211no—
182C22×C4—421no—
183SD16—142no—
184D8—62no—
185C2×C8—31no—
186C2×C8—71no—
187C4⋊C4—211no—
188C22⋊C4—421no—
189C2×C7—81no—
190C2×C7—122no—
191C2×C7#91411yescharacteristic
192C2×C7#91611yescharacteristic
193C2×C7#93711yescharacteristic
194D7—121no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3D4×D711yescharacteristic
199C2×D3—22no—
200C2×D3—41no—
201C2×D3—41no—
202C2×D3—141no—
203C2×D3—141no—
204C2×D3—141no—
205C3×C4—142no—
206C3×C4D4×D711yescharacteristic
207C3×C4C2×C22⋊D711yescharacteristic
208Dic3—141no—
209Dic3—281no—
210C23—21no—
211C23—61no—
212C23—211no—
213Q8—72no—
214Q8—141no—
215D4—21no—
216D4—21no—
217D4—32no—
218D4—61no—
219D4—424no—
220D4—842no—
221C2×C4—141no—
222C2×C4—211no—
223C2×C4—424no—
224C2×C4—421no—
225C2×C4—421no—
226C2×C4—421no—
227C2×C4C2×D3×D711yescharacteristic
228C8—61no—
229C8—141no—
230C7#517211yescharacteristic, complemented, Sylow
231C2×C3—81no—
232C2×C3#647411yescharacteristic
233C2×C3#647511yescharacteristic
234C2×C3#650011yescharacteristic
235D3—42no—
236D3—281no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—62no—
241C22—121no—
242C22—121no—
243C22—421no—
244C22—421no—
245C22—421no—
246C22#6669411yescharacteristic
247C4—142no—
248C4—421no—
249C4—841no—
250C4D7×#6711yescharacteristic
251C4D3×C22⋊D711yescharacteristic
252C3#8993011yescharacteristic, complemented, Sylow
253C2—81no—
254C2—122no—
255C2—841no—
256C2#1062106111yescharacteristic
257C2#1062106211yescharacteristic
258C2#1062113911yescharacteristic
subgroupG/H
centerC22#66694
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×#66672C2semidirect
2. #10621421C2semidirect
3. C2×C7×#213C2semidirect
4. C2×C3×#1142C2semidirect
5. #10620506C2semidirect
6. #10620502C2semidirect
7. #10620492C2semidirect
27. C2×#1142D3semidirect
44. C7×D12D4semidirect
47. C3×Dic14D4semidirect
70. C2×#213D7semidirect
107. Dic14#67semidirect
167. D12C22⋊D7semidirect
173. C3×C7#449semidirect
230. C7#5172semidirect
252. C3#89930semidirect

Maximal subgroups

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

subgroupindex
C2×#666722
#106214212
C2×C7×#2132
C2×C3×#11422
#106205062
#106205022
#106204922
#899303
#51727