Finite Group Databasebeta

← #13052886

Subgroups — #13052886

Order 1344.

324 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 78 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×#66670C211yesmaximal, characteristic
2C4⋊C4×D21C211yesmaximal, characteristic, complemented
3C2×#66559C211yesmaximal, characteristic, complemented
4C7×#810C211yesmaximal, characteristic, complemented
5C3×#6383C211yesmaximal, characteristic, complemented
6#10620508C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic, complemented
8#89575—31nomaximal, Hall
9C2×C4×D21—21no—
10C2×C4×D21C2211yescharacteristic
11C2×C7×Dic6C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13C2×#4246—21no—
14C2×Dic3×D7—21no—
15#66670—22no—
16#66670C414yescomplemented
17C3×C7×C4⋊C4C2211yescharacteristic
18C4⋊Dic21C2211yescharacteristic
19#66629—21no—
20C2×C7×C3⋊C8C2211yescharacteristic
21C2×C3×C7⋊C8C2211yescharacteristic
22#66559—22no—
23C2×#1161—31no—
24C2×#1131—31no—
25D7×C4⋊C4—31no—
26C7×#103—31no—
27#6409—31no—
28#6395—31no—
29#6383D311yescharacteristic, complemented
30#4771—71nomaximal, Hall
31C22×D21D411yescharacteristic
32C22×C3×D7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C4×D21—42no—
38C4×D21C2×C412yes—
39C4×D21D412yes—
40C2×C7×Dic3—21no—
41C7×Dic6—21no—
42C7×Dic6C2×C412yescomplemented
43C3×D28—21no—
44C3×D28C2×C412yescomplemented
45#4246—24no—
46Dic3×D7—24no—
47C3×C7⋊C8—21no—
48C7×C3⋊C8—21no—
49C2×C7×Q8—31no—
50#1161—34no—
51#1161—62no—
52C2×D28—61no—
53C2×D28C2×D311yescharacteristic
54C2×C4×D7—31no—
55C2×C4×D7—61no—
56C2×C4×D7—61no—
57C2×C7×C8—31no—
58C7×C4⋊C4C2×D311yescharacteristic
59C4⋊Dic7—31no—
60Dic7⋊C4—61no—
61C2×C7⋊C8C2×D311yescharacteristic
62#1131—62no—
63C2×#237—71no—
64C2×ΓL(1,𝔽25)—71no—
65D3×C4⋊C4—71no—
66C3×#102—71no—
67#832—71no—
68#816—71no—
69#810D711yescharacteristic, complemented
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—21no—
72C2×D21—21no—
73C2×D21C22⋊C412yes—
74C2×C3×D7—22no—
75C2×C3×D7—41no—
76C2×C3×D7—41no—
77C3×C4×C7—41no—
78C3×C4×C7C22×C411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—41no—
81Dic21C22⋊C412yes—
82C7×Dic3—22no—
83#417—211noSylow
84C22×D7—21no—
85C22×D7—31no—
86C7×Q8—32no—
87C7×Q8—61no—
88C2×C4×C7—21no—
89C2×C4×C7—61no—
90C2×C4×C7C22×D311yescharacteristic
91C2×Dic7—31no—
92C2×Dic7—61no—
93D28—21no—
94D28—64no—
95D28C4×D312yescomplemented
96C4×D7—32no—
97C4×D7—32no—
98C4×D7—64no—
99C4×D7—122no—
100C7×C8—61no—
101C7⋊C8—21no—
102C2×C3×D4—71no—
103C2×#67—141no—
104C22×Dic3—141no—
105#237—74no—
106#237—142no—
107C2×C4×D3—71no—
108C2×C4×D3—141no—
109C2×Dic6C2×D711yescharacteristic
110C2×C3×C8—71no—
111C3×C4⋊C4C2×D711yescharacteristic
112C4⋊Dic3—71no—
113Dic3⋊C4—141no—
114C2×C3⋊C8C2×D711yescharacteristic
115ΓL(1,𝔽25)—142no—
116C2×C3×C7C2×C22⋊C411yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118C2×C3×C7#13711yescharacteristic
119D21—22no—
120C3×D7—42no—
121C2×#41—211no—
122C2×M16—211no—
123C2×C4⋊C4—211no—
124#103—31no—
125#103—211no—
126#102—71no—
127#102—211no—
128C22×C7D3×D411yescharacteristic
129C2×D7—22no—
130C2×D7—32no—
131C2×D7—41no—
132C2×D7—41no—
133C2×D7—61no—
134C2×D7—61no—
135C4×C7—41no—
136C4×C7—62no—
137C4×C7C2×C4×D311yescharacteristic
138C4×C7D3×D411yescharacteristic
139Dic7—32no—
140Dic7—121no—
141C23×C3—141no—
142C22×D3—71no—
143C3×D4—72no—
144C3×D4—141no—
145C2×C3×C4—21no—
146C2×C3×C4C22×D711yescharacteristic
147#67—144no—
148C2×Dic3—21no—
149C2×Dic3—71no—
150C2×Dic3—144no—
151C2×Dic3—141no—
152C4×D3—72no—
153C4×D3—72no—
154C4×D3—282no—
155Dic6—21no—
156Dic6C4×D712yescomplemented
157C3×C8—141no—
158C3⋊C8—21no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—214no—
161#41—422no—
162C2×Q8—31no—
163C2×D4—71no—
164C2×D4—421no—
165C22×C4—211no—
166C22×C4—421no—
167C22×C4—421no—
168M16—422no—
169C2×C8—31no—
170C2×C8—71no—
171C4⋊C4—211no—
172C4⋊C4—421no—
173C4⋊C4D3×D711yescharacteristic
174C2×C7D3×C22⋊C411yescharacteristic
175C2×C7#91111yescharacteristic
176C2×C7#91511yescharacteristic
177D7—42no—
178D7—62no—
179C22×C3—142no—
180C22×C3—281no—
181C22×C3—281no—
182C22×C3D4×D711yescharacteristic
183C2×D3—72no—
184C2×D3—141no—
185C2×D3—141no—
186C3×C4—41no—
187C3×C4C2×C4×D711yescharacteristic
188C3×C4D4×D711yescharacteristic
189Dic3—22no—
190Dic3—72no—
191Dic3—281no—
192C23—141no—
193C23—211no—
194Q8—32no—
195Q8—61no—
196D4—72no—
197D4—141no—
198D4—424no—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—211no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—424no—
205C2×C4—421no—
206C2×C4—842no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—141no—
210C7#477111yescharacteristic, complemented, Sylow
211C2×C3—282no—
212C2×C3D7×C22⋊C411yescharacteristic
213C2×C3#647711yescharacteristic
214C2×C3#648111yescharacteristic
215D3—142no—
216C22—142no—
217C22—212no—
218C22—281no—
219C22—281no—
220C22—421no—
221C22—421no—
222C22#6669411yescharacteristic
223C4—41no—
224C4—62no—
225C4—212no—
226C4—841no—
227C4C4×D3×D711yescharacteristic
228C4#6668111yescharacteristic
229C3#8957511yescharacteristic, complemented, Sylow
230C2—282no—
231C2—422no—
232C2#1062103511yescharacteristic
233C2#1062106011yescharacteristic
234C2#1062108411yescharacteristic
subgroupG/H
centerC22#66694
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66670C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#417
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C4⋊C4×D21C2semidirect
3. C2×#66559C2semidirect
4. C7×#810C2semidirect
5. C3×#6383C2semidirect
6. #10620508C2semidirect
7. #10620500C2semidirect
16. #66670C4semidirect
29. #6383D3semidirect
42. C7×Dic6C2×C4semidirect
44. C3×D28C2×C4semidirect
69. #810D7semidirect
95. D28C4×D3semidirect
156. Dic6C4×D7semidirect
159. C3×C7#417semidirect
210. C7#4771semidirect
229. C3#89575semidirect

Maximal subgroups

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

subgroupindex
C2×#666702
C4⋊C4×D212
C2×#665592
C7×#8102
C3×#63832
#106205082
#106205002
#895753
#47717