Finite Group Databasebeta

← #13052631

Subgroups — #13052631

Order 1344.

240 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 62 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×#6458C211yesmaximal, characteristic
2#10620921C211yesmaximal, characteristic, complemented
3C2×#66563C211yesmaximal, characteristic, complemented
4C2×#66557C211yesmaximal, characteristic, complemented
5#10620646C211yesmaximal, characteristic, complemented
6C7×#810C211yesmaximal, characteristic, complemented
7#10620500C211yesmaximal, characteristic, complemented
8#89584—31nomaximal, Hall
9C2×C7×Dic6C2211yescharacteristic
10C2×C3×D28—21no—
11C2×C3×D28C2211yescharacteristic
12C2×C3×C4×D7C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×(C2×D7)⋊C4—21no—
18#66577—21no—
19(C2×D7)⋊Dic3—21no—
20#66563—22no—
21#66557—22no—
22#6509—31no—
23C2×#1143—31no—
24C2×#1131—31no—
25#6458D311yescharacteristic
26C7×#103—31no—
27#6395—31no—
28#6381—31no—
29#5042—71nomaximal, Hall
30C22×C3×D7—21no—
31C22×C3×D7D411yescharacteristic
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×C7×Dic3—21no—
36C7×Dic6—21no—
37C7×Dic6—21no—
38C2×C3×Dic7D411yescharacteristic
39C3×D28—21no—
40C3×D28—21no—
41C3×D28—42no—
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44C3⋊(C7⋊C8)D412yescomplemented
45C7×C3⋊C8—21no—
46C2×C7×Q8—31no—
47C2×D28—21no—
48C2×D28C2×D311yescharacteristic
49C2×C4×D7C2×D311yescharacteristic
50C2×C7×C8—31no—
51C7×C4⋊C4C2×D311yescharacteristic
52#1143—62no—
53(C2×D7)⋊C4—21no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—31no—
56Dic7⋊C4—61no—
57C2×C7⋊C8—31no—
58#1131—62no—
59C3×#121—71no—
60C2×#214—71no—
61#924—71no—
62C2×#208—71no—
63#832—71no—
64#810D711yescharacteristic, complemented
65#807—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×C3×D7—21no—
68C2×C3×D7—21no—
69C2×C3×D7—21no—
70C2×C3×D7—41no—
71C2×C3×D7—41no—
72C2×C3×D7—41no—
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—41no—
77C3×Dic7—21no—
78C7×Dic3—41no—
79#468—211noSylow
80C22×D7—21no—
81C22×D7#6711yescharacteristic
82C7×Q8—61no—
83C7×Q8—61no—
84C2×C4×C7—21no—
85C2×C4×C7—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—61no—
88C2×Dic7#6711yescharacteristic
89D28—21no—
90D28—21no—
91D28—42no—
92C4×D7—21no—
93C4×D7—21no—
94C7×C8—61no—
95C7⋊C8—32no—
96C2×C3×D4—71no—
97C2×C3×D4—141no—
98C22×C3×C4—71no—
99C2×Dic6C2×D711yescharacteristic
100C3×C4⋊C4C2×D711yescharacteristic
101C3×C22⋊C4—141no—
102#217—141no—
103#214—142no—
104C4⋊Dic3—71no—
105Dic3⋊C4—141no—
106#208—142no—
107C2×C3⋊C8—71no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
111C2×C3×C7#13711yescharacteristic
112C3×D7—41no—
113C3×D7—81no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—71no—
118#107—211no—
119#103—31no—
120#102—211no—
121C22×C7C2×#6711yescharacteristic
122C2×D7—21no—
123C2×D7—21no—
124C2×D7—21no—
125C2×D7—41no—
126C2×D7—41no—
127C2×D7—41no—
128C4×C7—41no—
129C4×C7—121no—
130C4×C7D3×D411yescharacteristic
131C4×C7#23711yescharacteristic
132Dic7—21no—
133Dic7—121no—
134C23×C3—71no—
135C23×C3—141no—
136C3×D4—141no—
137C3×D4—141no—
138C3×D4—282no—
139C2×C3×C4—21no—
140C2×C3×C4—71no—
141C2×C3×C4—141no—
142C2×C3×C4—141no—
143C2×C3×C4C22×D711yescharacteristic
144C2×Dic3—21no—
145C2×Dic3—141no—
146Dic6—21no—
147Dic6—21no—
148C3⋊C8—21no—
149C3⋊C8—72no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—31no—
152C2×D4—71no—
153C2×D4—141no—
154C22×C4—71no—
155SD16—422no—
156M16—422no—
157C2×C8—31no—
158C2×C8—211no—
159C4⋊C4—211no—
160C4⋊C4—421no—
161C4⋊C4D3×D711yescharacteristic
162C22⋊C4—141no—
163C22⋊C4—421no—
164C2×C7#93211yescharacteristic
165C2×C7#93911yescharacteristic
166C2×C7#95111yescharacteristic
167D7—41no—
168D7—81no—
169C22×C3—141no—
170C22×C3—141no—
171C22×C3—141no—
172C22×C3—281no—
173C22×C3—281no—
174C22×C3—281no—
175C22×C3D4×D711yescharacteristic
176C3×C4—41no—
177C3×C4—141no—
178C3×C4#115711yescharacteristic
179C3×C4D4×D711yescharacteristic
180Dic3—41no—
181Dic3—281no—
182C23—71no—
183C23—141no—
184Q8—61no—
185Q8—61no—
186D4—141no—
187D4—141no—
188D4—282no—
189C2×C4—21no—
190C2×C4—61no—
191C2×C4—71no—
192C2×C4—141no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—212no—
198C7#504211yescharacteristic, complemented, Sylow
199C2×C3—281no—
200C2×C3—561no—
201C2×C3#644711yescharacteristic
202C2×C3#647711yescharacteristic
203C2×C3#648111yescharacteristic
204C22—141no—
205C22—141no—
206C22—141no—
207C22—281no—
208C22—281no—
209C22—281no—
210C22D7×#6711yescharacteristic
211C4—41no—
212C4—121no—
213C4—141no—
214C4—841no—
215C4#6666911yescharacteristic
216C4#6668111yescharacteristic
217C3#8958411yescharacteristic, complemented, Sylow
218C2—281no—
219C2—561no—
220C2#1062097011yescharacteristic
221C2#1062105311yescharacteristic
222C2#1062107011yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6458C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#468
LCSG ▷ C2×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. #10620921C2semidirect
3. C2×#66563C2semidirect
4. C2×#66557C2semidirect
5. #10620646C2semidirect
6. C7×#810C2semidirect
7. #10620500C2semidirect
44. C3⋊(C7⋊C8)D4semidirect
64. #810D7semidirect
150. C3×C7#468semidirect
198. C7#5042semidirect
217. C3#89584semidirect

Maximal subgroups

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

subgroupindex
C3×#64582
#106209212
C2×#665632
C2×#665572
#106206462
C7×#8102
#106205002
#895843
#50427