Finite Group Databasebeta

← #13056439

Subgroups — #13056439

Order 1344.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#66732C211yesmaximal, characteristic
2C2×#66622C211yesmaximal, characteristic, complemented
3C4⋊C4×D21C211yesmaximal, characteristic, complemented
4C3×C7×#103C211yesmaximal, characteristic, complemented
5#10620674C211yesmaximal, characteristic, complemented
6#10620660C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89575—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10#66732—22no—
11#66732C414yescomplemented
12C2×D84—21no—
13C2×D84C2211yescharacteristic
14C2×C4×D21—21no—
15C2×C4×D21—21no—
16C2×C4×D21C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C4⋊Dic21C2211yescharacteristic
20#66629—21no—
21C2×C3⋊(C7⋊C8)C2211yescharacteristic
22#66622—22no—
23C2×#1161—31no—
24C2×#1131—31no—
25D7×C4⋊C4—31no—
26C7×#103D311yescharacteristic, complemented
27#6409—31no—
28#6395—31no—
29#6383—31no—
30#4803—71nomaximal, Hall
31C22×D21—21no—
32C22×D21D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8C2×C412yescomplemented
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21D411yescharacteristic
40D84—24no—
41D84—21no—
42D84C2×C412yescomplemented
43C4×D21—24no—
44C4×D21—42no—
45C4×D21C2×C412yes—
46C4×D21D412yes—
47C3×C7×C8—21no—
48C3⋊(C7⋊C8)—21no—
49C2×C7×Q8C2×D311yescharacteristic
50#1161—34no—
51#1161—62no—
52C2×D28—31no—
53C2×D28—61no—
54C2×C4×D7—31no—
55C2×C4×D7—61no—
56C2×C4×D7—61no—
57C2×C7×C8C2×D311yescharacteristic
58C7×C4⋊C4C2×D311yescharacteristic
59C4⋊Dic7—31no—
60Dic7⋊C4—61no—
61C2×C7⋊C8—31no—
62#1131—62no—
63C2×#239—71no—
64C2×ΓL(1,𝔽25)—71no—
65D3×C4⋊C4—71no—
66C3×#103D711yescharacteristic, complemented
67#835—71no—
68#821—71no—
69#809—71no—
70C22×C3×C7C2×D411yescharacteristic
71C2×D21—21no—
72C2×D21—21no—
73C2×D21—22no—
74C2×D21—41no—
75C2×D21—41no—
76C2×D21C22⋊C412yes—
77C3×C4×C7—22no—
78C3×C4×C7—41no—
79C3×C4×C7C22×C411yescharacteristic
80C3×C4×C7C2×D411yescharacteristic
81Dic21—41no—
82Dic21C22⋊C412yes—
83#417—211noSylow
84C22×D7—31no—
85C22×D7—61no—
86C7×Q8—21no—
87C7×Q8C4×D312yescomplemented
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7C22×D311yescharacteristic
91C2×Dic7—31no—
92C2×Dic7—61no—
93D28—32no—
94D28—64no—
95D28—61no—
96C4×D7—32no—
97C4×D7—32no—
98C4×D7—64no—
99C4×D7—122no—
100C7×C8—21no—
101C7⋊C8—61no—
102C2×C3×Q8C2×D711yescharacteristic
103#239—74no—
104#239—142no—
105C2×D12—71no—
106C2×D12—141no—
107C2×C4×D3—71no—
108C2×C4×D3—141no—
109C2×C4×D3—141no—
110C2×C3×C8C2×D711yescharacteristic
111C3×C4⋊C4C2×D711yescharacteristic
112C4⋊Dic3—71no—
113Dic3⋊C4—141no—
114C2×C3⋊C8—71no—
115ΓL(1,𝔽25)—142no—
116C2×C3×C7C2×C22⋊C411yescharacteristic
117C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
118C2×C3×C7#13711yescharacteristic
119D21—22no—
120D21—42no—
121C2×#41—211no—
122C2×M16—211no—
123C2×C4⋊C4—211no—
124#103—211no—
125#103D2111yescharacteristic, complemented
126#102—211no—
127#102—211no—
128C22×C7D3×D411yescharacteristic
129C2×D7—32no—
130C2×D7—61no—
131C2×D7—61no—
132C2×D7—62no—
133C2×D7—121no—
134C2×D7—121no—
135C4×C7—22no—
136C4×C7—41no—
137C4×C7C2×C4×D311yescharacteristic
138C4×C7D3×D411yescharacteristic
139Dic7—32no—
140Dic7—121no—
141C22×D3—71no—
142C22×D3—141no—
143C3×Q8—21no—
144C3×Q8C4×D712yescomplemented
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4C22×D711yescharacteristic
148C2×Dic3—71no—
149C2×Dic3—141no—
150D12—72no—
151D12—144no—
152D12—141no—
153C4×D3—72no—
154C4×D3—72no—
155C4×D3—144no—
156C4×D3—282no—
157C3×C8—21no—
158C3⋊C8—141no—
159C3×C7#41711yescharacteristic, complemented, Hall
160#41—214no—
161#41—422no—
162C2×Q8C2×D2111yescharacteristic
163C2×D4—211no—
164C2×D4—421no—
165C22×C4—211no—
166C22×C4—421no—
167C22×C4—421no—
168M16—422no—
169C2×C8—211no—
170C2×C8C2×D2111yescharacteristic
171C4⋊C4—211no—
172C4⋊C4—421no—
173C4⋊C4C2×D2111yescharacteristic
174C2×C7D3×C22⋊C411yescharacteristic
175C2×C7#91411yescharacteristic
176C2×C7#91811yescharacteristic
177D7—62no—
178D7—122no—
179C22×C3D4×D711yescharacteristic
180C2×D3—72no—
181C2×D3—141no—
182C2×D3—141no—
183C2×D3—142no—
184C2×D3—281no—
185C2×D3—281no—
186C3×C4—22no—
187C3×C4—41no—
188C3×C4C2×C4×D711yescharacteristic
189C3×C4D4×D711yescharacteristic
190Dic3—72no—
191Dic3—281no—
192C23—211no—
193C23—421no—
194Q8—21no—
195Q8C4×D2112yescomplemented
196D4—212no—
197D4—424no—
198D4—421no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—211no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—424no—
205C2×C4—421no—
206C2×C4—842no—
207C2×C4C22×D2111yescharacteristic
208C8—21no—
209C8—421no—
210C7#480311yescharacteristic, complemented, Sylow
211C2×C3D7×C22⋊C411yescharacteristic
212C2×C3#647711yescharacteristic
213C2×C3#648111yescharacteristic
214D3—142no—
215D3—282no—
216C22—212no—
217C22—421no—
218C22—421no—
219C22—422no—
220C22—841no—
221C22—841no—
222C22D4×D2111yescharacteristic
223C4—22no—
224C4—41no—
225C4—212no—
226C4—841no—
227C4C2×C4×D2111yescharacteristic
228C4D4×D2111yescharacteristic
229C3#8957511yescharacteristic, complemented, Sylow
230C2—422no—
231C2—842no—
232C2C22⋊C4×D2111yescharacteristic
233C2#1062139711yescharacteristic
234C2#1062140111yescharacteristic
subgroupG/H
centerC22D4×D21
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×#66732C2
cosocleC2×C3×C4×C7C23
hypercenter#103D21
hypocenterC3×C7#417
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#66622C2semidirect
3. C4⋊C4×D21C2semidirect
4. C3×C7×#103C2semidirect
5. #10620674C2semidirect
6. #10620660C2semidirect
7. #10620648C2semidirect
11. #66732C4semidirect
26. C7×#103D3semidirect
34. C3×C7×Q8C2×C4semidirect
42. D84C2×C4semidirect
66. C3×#103D7semidirect
87. C7×Q8C4×D3semidirect
125. #103D21semidirect
144. C3×Q8C4×D7semidirect
159. C3×C7#417semidirect
195. Q8C4×D21semidirect
210. C7#4803semidirect
229. C3#89575semidirect

Maximal subgroups

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

subgroupindex
C2×#667322
C2×#666222
C4⋊C4×D212
C3×C7×#1032
#106206742
#106206602
#106206482
#895753
#48037