Finite Group Databasebeta

← C3×D7×#103 #13055233

Subgroups — C3×D7×#103 #13055233

Order 1344.

324 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 126 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×C3×Q8×D7C211yesmaximal, characteristic
2C2×C3×C8×D7C211yesmaximal, characteristic
3C3×D7×C4⋊C4C211yesmaximal, characteristic
4C3×C7×#103C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6C3×#6390C211yesmaximal, characteristic, complemented
7C3×#6384C211yesmaximal, characteristic, complemented
8D7×#103C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C3×Q8×D7—22no—
11C3×Q8×D7C414yescomplemented
12C2×C3×C4×D7—21no—
13C2×C3×C4×D7—21no—
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×Dic14—21no—
16C2×C3×Dic14C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C4⋊Dic7C2211yescharacteristic
20C3×Dic7⋊C4—21no—
21C2×C3×C7⋊C8C2211yescharacteristic
22C3×C8×D7—22no—
23C2×Q8×D7C2×C311yescharacteristic
24C2×C8×D7C2×C311yescharacteristic
25D7×C4⋊C4C2×C311yescharacteristic
26C7×#103C2×C311yescharacteristic, complemented
27#6409C2×C311yescharacteristic, complemented
28#6390C2×C311yescharacteristic, complemented
29#6384C2×C311yescharacteristic, complemented
30C2×C3×#103—71nomaximal, Hall
31C22×C3×D7D411yescharacteristic
32C3×C7×Q8—21no—
33C3×C7×Q8C2×C412yescomplemented
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×C3×Dic7—21no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7D411yescharacteristic
40C3×C4×D7—24no—
41C3×C4×D7—42no—
42C3×C4×D7C2×C412yes—
43C3×C4×D7D412yes—
44C3×Dic14—21no—
45C3×Dic14—24no—
46C3×Dic14C2×C412yescomplemented
47C3×C7×C8—21no—
48C3×C7⋊C8—21no—
49C2×C7×Q8C22×C311yescharacteristic
50Q8×D7—22no—
51Q8×D7C3×C414yescomplemented
52C2×C4×D7—21no—
53C2×C4×D7—21no—
54C2×C4×D7C22×C311yescharacteristic
55C2×Dic14—21no—
56C2×Dic14C22×C311yescharacteristic
57C2×C7×C8C22×C311yescharacteristic
58C7×C4⋊C4C22×C311yescharacteristic
59C4⋊Dic7C22×C311yescharacteristic
60Dic7⋊C4—21no—
61C2×C7⋊C8C22×C311yescharacteristic
62C8×D7—22no—
63C22×C3×Q8—71no—
64C22×C3×C8—71no—
65C2×C3×C4⋊C4—71no—
66C3×#103—71no—
67C3×#103—71no—
68C3×#103—71no—
69C3×#103D711yescharacteristic, direct product factor
70C22×C3×C7C2×D411yescharacteristic
71C2×C3×D7C22⋊C412yes—
72C2×C3×D7SD1612yes—
73C2×C3×D7Q1612yes—
74C3×C4×C7—22no—
75C3×C4×C7—41no—
76C3×C4×C7C22×C411yescharacteristic
77C3×C4×C7C2×D411yescharacteristic
78C3×Dic7—22no—
79C3×Dic7—41no—
80C3×Dic7C22⋊C412yes—
81C2×#103—71noSylow
82C22×D7C3×D411yescharacteristic
83C7×Q8—21no—
84C7×Q8C2×C3×C412yescomplemented
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7C23×C311yescharacteristic
88C2×Dic7—21no—
89C2×Dic7—21no—
90C2×Dic7C3×D411yescharacteristic
91C4×D7—24no—
92C4×D7—42no—
93C4×D7C2×C3×C412yes—
94C4×D7C3×D412yes—
95Dic14—21no—
96Dic14—24no—
97Dic14C2×C3×C412yescomplemented
98C7×C8—21no—
99C7⋊C8—21no—
100C2×C3×Q8—74no—
101C2×C3×Q8—71no—
102C2×C3×Q8—141no—
103C2×C3×Q8—142no—
104C2×C3×Q8C2×D711yescharacteristic
105C22×C3×C4—71no—
106C22×C3×C4—141no—
107C22×C3×C4—141no—
108C2×C3×C8—71no—
109C2×C3×C8—142no—
110C2×C3×C8C2×D711yescharacteristic
111C3×C4⋊C4—71no—
112C3×C4⋊C4—141no—
113C3×C4⋊C4C2×D711yescharacteristic
114C2×C3×C7C2×C22⋊C411yescharacteristic
115C2×C3×C7C2×SD1611yescharacteristic
116C2×C3×C7C2×Q1611yescharacteristic
117C3×D7#10314yesdirect product factor
118C22×Q8—71no—
119C22×C8—71no—
120C2×C4⋊C4—71no—
121#103—71no—
122#103—71no—
123#103—71no—
124#103C3×D711yescharacteristic, direct product factor
125C22×C7C2×C3×D411yescharacteristic
126C2×D7C3×C22⋊C412yes—
127C2×D7C3×SD1612yes—
128C2×D7C3×Q1612yes—
129C4×C7—22no—
130C4×C7—41no—
131C4×C7C22×C3×C411yescharacteristic
132C4×C7C2×C3×D411yescharacteristic
133Dic7—22no—
134Dic7—41no—
135Dic7C3×C22⋊C412yes—
136C23×C3—71no—
137C3×Q8—21no—
138C3×Q8—72no—
139C3×Q8—144no—
140C3×Q8—141no—
141C3×Q8C4×D712yescomplemented
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—71no—
145C2×C3×C4—72no—
146C2×C3×C4—72no—
147C2×C3×C4—144no—
148C2×C3×C4—141no—
149C2×C3×C4—141no—
150C2×C3×C4—282no—
151C2×C3×C4C22×D711yescharacteristic
152C3×C8—21no—
153C3×C8—141no—
154C3×C7C2×#10311yescharacteristic, complemented, Hall
155C2×Q8—74no—
156C2×Q8—71no—
157C2×Q8—141no—
158C2×Q8—142no—
159C2×Q8C2×C3×D711yescharacteristic
160C22×C4—71no—
161C22×C4—141no—
162C22×C4—141no—
163C2×C8—71no—
164C2×C8—142no—
165C2×C8C2×C3×D711yescharacteristic
166C4⋊C4—71no—
167C4⋊C4—141no—
168C4⋊C4C2×C3×D711yescharacteristic
169C2×C7C2×C3×C22⋊C411yescharacteristic
170C2×C7C2×C3×SD1611yescharacteristic
171C2×C7C2×C3×Q1611yescharacteristic
172D7C3×#10314yesdirect product factor
173C22×C3—72no—
174C22×C3—72no—
175C22×C3—72no—
176C22×C3D4×D711yescharacteristic
177C3×C4—22no—
178C3×C4—41no—
179C3×C4—72no—
180C3×C4—142no—
181C3×C4—281no—
182C3×C4C2×C4×D711yescharacteristic
183C3×C4D4×D711yescharacteristic
184C23—71no—
185Q8—21no—
186Q8—72no—
187Q8—144no—
188Q8—141no—
189Q8C3×C4×D712yescomplemented
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—71no—
193C2×C4—72no—
194C2×C4—72no—
195C2×C4—144no—
196C2×C4—141no—
197C2×C4—141no—
198C2×C4—282no—
199C2×C4C22×C3×D711yescharacteristic
200C8—21no—
201C8—141no—
202C7C2×C3×#10311yescharacteristic, complemented, Sylow
203C2×C3—74no—
204C2×C3D7×C22⋊C411yescharacteristic
205C2×C3D7×SD1611yescharacteristic
206C2×C3D7×Q1611yescharacteristic
207C22—72no—
208C22—72no—
209C22—72no—
210C22C3×D4×D711yescharacteristic
211C4—22no—
212C4—41no—
213C4—72no—
214C4—142no—
215C4—281no—
216C4C2×C3×C4×D711yescharacteristic
217C4C3×D4×D711yescharacteristic
218C3D7×#10311yescharacteristic, direct product factor, Sylow
219C2—74no—
220C2C3×D7×C22⋊C411yescharacteristic
221C2C3×D7×SD1611yescharacteristic
222C2C3×D7×Q1611yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutatorC4×C7C22×C3×C4
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
cosocleC2×C4×C7C23×C3
hypercenterC3×#103D7
hypocenterC7C2×C3×#103
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
4. C3×C7×#103C2semidirect
5. C3×#6409C2semidirect
6. C3×#6390C2semidirect
7. C3×#6384C2semidirect
8. D7×#103C3direct (N × H)
11. C3×Q8×D7C4semidirect
26. C7×#103C2×C3semidirect
27. #6409C2×C3semidirect
28. #6390C2×C3semidirect
29. #6384C2×C3semidirect
33. C3×C7×Q8C2×C4semidirect
46. C3×Dic14C2×C4semidirect
51. Q8×D7C3×C4semidirect
69. C3×#103D7direct (N × H)
84. C7×Q8C2×C3×C4semidirect
97. Dic14C2×C3×C4semidirect
117. C3×D7#103direct (N × H)
124. #103C3×D7direct (N × H)
141. C3×Q8C4×D7semidirect
154. C3×C7C2×#103semidirect
172. D7C3×#103direct (N × H)
189. Q8C3×C4×D7semidirect
202. C7C2×C3×#103semidirect
218. C3D7×#103direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D72
C2×C3×C8×D72
C3×D7×C4⋊C42
C3×C7×#1032
C3×#64092
C3×#63902
C3×#63842
D7×#1033
C2×C3×#1037