Finite Group Databasebeta

← #13052801

Subgroups — #13052801

Order 1344.

264 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C3×#6456C211yesmaximal, characteristic, complemented
2#10620980C211yesmaximal, characteristic, complemented
3C2×#66561C211yesmaximal, characteristic
4#10620648C211yesmaximal, characteristic, complemented
5C7×#808C211yesmaximal, characteristic, complemented
6#10620499C211yesmaximal, characteristic, complemented
7#10620487C211yesmaximal, characteristic, complemented
8#89641—31nomaximal, Hall
9C2×D84C2211yescharacteristic
10C2×C3×D28C2211yescharacteristic
11C2×C3×C4×D7—21no—
12C2×D21⋊C4—21no—
13C3×C7×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×(C2×D7)⋊C4—21no—
18C3×C4×Dic7C2211yescharacteristic
19#66575—21no—
20#66561C414yescomplemented
21C2×D56—31no—
22#6456—31no—
23#6456D311yescharacteristic, complemented
24C7×#106—31no—
25#6389—31no—
26#6383—31no—
27#6383—31no—
28#5014—71nomaximal, Hall
29C22×D21—21no—
30C22×C3×D7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7C2311yescharacteristic
33D84—21no—
34D84C2×C412yescomplemented
35C2×C7×Dic3—21no—
36C2×C3×Dic7D411yescharacteristic
37C2×C3×Dic7D411yescharacteristic
38C3×D28—21no—
39C3×D28C2×C412yescomplemented
40C3×C4×D7—42no—
41D21⋊C4—42no—
42C3⋊(C7⋊C8)—21no—
43C7×C3⋊C8C2×C412yescomplemented
44C2×D28—31no—
45C2×D28C2×D311yescharacteristic
46C2×C4×D7—21no—
47C2×C4×D7—61no—
48C2×C7×C8—31no—
49C7×C4⋊C4—31no—
50C7×C4⋊C4C2×D311yescharacteristic
51(C2×D7)⋊C4—21no—
52(C2×D7)⋊C4—61no—
53C4×Dic7C2×D311yescharacteristic
54C2×C7⋊C8—31no—
55D56—34no—
56C3×C4×D4—71no—
57C2×#213—71no—
58C4×D12—71no—
59#832—71no—
60#809—71no—
61#808D711yescharacteristic, complemented
62#803—71no—
63C22×C3×C7C2×D411yescharacteristic
64C2×D21—22no—
65C2×D21—41no—
66C2×D21—41no—
67C2×C3×D7—22no—
68C2×C3×D7—41no—
69C2×C3×D7—41no—
70C3×C4×C7—41no—
71C3×C4×C7C22×C411yescharacteristic
72C3×C4×C7#4111yescharacteristic
73C3×Dic7—21no—
74C3×Dic7—21no—
75C7×Dic3—41no—
76#442—211noSylow
77C22×D7—21no—
78C22×D7—61no—
79C2×C4×C7—21no—
80C2×C4×C7—61no—
81C2×C4×C7C22×D311yescharacteristic
82C2×Dic7#6711yescharacteristic
83C2×Dic7#6711yescharacteristic
84D28—21no—
85D28—32no—
86D28—61no—
87D28C4×D312yescomplemented
88C4×D7—42no—
89C4×D7—122no—
90C7×C8—32no—
91C7⋊C8—61no—
92C2×C3×D4—71no—
93C22×C3×C4—141no—
94C2×D12—71no—
95C2×C4×D3—141no—
96C3×C4⋊C4C2×D711yescharacteristic
97C3×C22⋊C4—141no—
98C3×C42—71no—
99#213—74no—
100(C2×D3)⋊C4—141no—
101C4⋊Dic3C2×D711yescharacteristic
102C2×C3⋊C8—71no—
103C2×C3⋊C8C2×D711yescharacteristic
104C2×C3×C7C4×D411yescharacteristic
105C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
106C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
107D21—42no—
108C3×D7—42no—
109C2×D8—211no—
110C4×D4—71no—
111C4×D4—211no—
112#106—31no—
113#102—211no—
114#102—211no—
115#97—211no—
116C22×C7C2×#6711yescharacteristic
117C2×D7—22no—
118C2×D7—41no—
119C2×D7—41no—
120C2×D7—62no—
121C2×D7—121no—
122C2×D7—121no—
123C4×C7—41no—
124C4×C7—121no—
125C4×C7C2×C4×D311yescharacteristic
126C4×C7#23511yescharacteristic
127Dic7—21no—
128Dic7—21no—
129C23×C3—141no—
130C22×D3—141no—
131C3×D4—72no—
132C3×D4—141no—
133C2×C3×C4—21no—
134C2×C3×C4—71no—
135C2×C3×C4—71no—
136C2×C3×C4—282no—
137C2×C3×C4C22×D711yescharacteristic
138C2×Dic3—21no—
139D12—72no—
140D12—141no—
141C4×D3—282no—
142C3⋊C8—141no—
143C3⋊C8C4×D712yescomplemented
144C3×C7#44211yescharacteristic, complemented, Hall
145C2×D4—71no—
146C2×D4—211no—
147C22×C4—141no—
148C22×C4—421no—
149D8—214no—
150C2×C8—31no—
151C2×C8—211no—
152C4⋊C4—31no—
153C4⋊C4D3×D711yescharacteristic
154C22⋊C4—141no—
155C22⋊C4—421no—
156C42—71no—
157C2×C7C4×#6711yescharacteristic
158C2×C7#93211yescharacteristic
159C2×C7#94911yescharacteristic
160D7—42no—
161D7—122no—
162C22×C3—142no—
163C22×C3—281no—
164C22×C3—281no—
165C22×C3D4×D711yescharacteristic
166C2×D3—142no—
167C2×D3—281no—
168C2×D3—281no—
169C3×C4—41no—
170C3×C4—141no—
171C3×C4—141no—
172C3×C4C2×C4×D711yescharacteristic
173C3×C4#116111yescharacteristic
174Dic3—41no—
175C23—141no—
176C23—421no—
177D4—72no—
178D4—141no—
179D4—212no—
180D4—421no—
181C2×C4—21no—
182C2×C4—61no—
183C2×C4—71no—
184C2×C4—71no—
185C2×C4—282no—
186C2×C4—842no—
187C2×C4C2×D3×D711yescharacteristic
188C8—32no—
189C8—421no—
190C7#501411yescharacteristic, complemented, Sylow
191C2×C3—282no—
192C2×C3#645611yescharacteristic
193C2×C3#647711yescharacteristic
194C2×C3#647711yescharacteristic
195D3—282no—
196C22—142no—
197C22—281no—
198C22—281no—
199C22—422no—
200C22—841no—
201C22—841no—
202C22D7×#6711yescharacteristic
203C4—41no—
204C4—121no—
205C4—141no—
206C4—141no—
207C4#6667711yescharacteristic
208C4C4×D3×D711yescharacteristic
209C3#8964111yescharacteristic, complemented, Sylow
210C2—282no—
211C2—842no—
212C2#1062100611yescharacteristic
213C2#1062105311yescharacteristic
214C2#1062106411yescharacteristic
subgroupG/H
centerC22D7×#67
commutatorC3×C4×C7C22×C4
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC2×#66561C2
cosocleC2×C3×C4×C7C23
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#442
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
1. C3×#6456C2semidirect
2. #10620980C2semidirect
4. #10620648C2semidirect
5. C7×#808C2semidirect
6. #10620499C2semidirect
7. #10620487C2semidirect
20. #66561C4semidirect
23. #6456D3semidirect
34. D84C2×C4semidirect
39. C3×D28C2×C4semidirect
43. C7×C3⋊C8C2×C4semidirect
61. #808D7semidirect
87. D28C4×D3semidirect
143. C3⋊C8C4×D7semidirect
144. C3×C7#442semidirect
190. C7#5014semidirect
209. C3#89641semidirect

Maximal subgroups

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

subgroupindex
C3×#64562
#106209802
C2×#665612
#106206482
C7×#8082
#106204992
#106204872
#896413
#50147