Finite Group Databasebeta

← C3×#89580 #13055246

Subgroups — C3×#89580 #13055246

Order 1344.

296 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 90 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×#1144C211yesmaximal, characteristic, complemented
3C2×C3×Dic28C211yesmaximal, characteristic, complemented
4C3×#6460C211yesmaximal, characteristic
5C3×C7×#103C211yesmaximal, characteristic, complemented
6C3×(C2×D7)⋊C8C211yesmaximal, characteristic
7C3×#6384C211yesmaximal, characteristic, complemented
8#89580C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C3×Q8×D7—22no—
11C3×Q8×D7—22no—
12C2×C3×C4×D7—21no—
13C2×C3×C4×D7C2211yescharacteristic
14C2×C3×Dic14—21no—
15C2×C3×Dic14C2211yescharacteristic
16C2×C3×Dic14C2211yescharacteristic
17C2×C3×C7×C8C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×#1144—22no—
20C3×(C2×D7)⋊C4—21no—
21C3×C4⋊Dic7—21no—
22C2×C3×C7⋊C8C2211yescharacteristic
23C3×Dic28—22no—
24C2×Q8×D7C2×C311yescharacteristic
25C2×#1144C2×C311yescharacteristic, complemented
26C2×Dic28C2×C311yescharacteristic, complemented
27#6460C2×C311yescharacteristic
28C7×#103C2×C311yescharacteristic, complemented
29(C2×D7)⋊C8C2×C311yescharacteristic
30#6384C2×C311yescharacteristic, complemented
31C3×#451—71nomaximal, Hall
32C22×C3×D7D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8D412yes—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7C2311yescharacteristic
38C2×C3×Dic7—21no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7D411yescharacteristic
41C3×C4×D7—24no—
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44C3×Dic14—24no—
45C3×Dic14—21no—
46C3×Dic14—21no—
47C3×Dic14—21no—
48C3×Dic14D412yes—
49C3×C7×C8—21no—
50C3×C7⋊C8—21no—
51C2×C7×Q8C22×C311yescharacteristic
52Q8×D7—22no—
53Q8×D7—22no—
54C2×C4×D7—21no—
55C2×C4×D7C22×C311yescharacteristic
56C2×Dic14—21no—
57C2×Dic14C22×C311yescharacteristic
58C2×Dic14C22×C311yescharacteristic
59C2×C7×C8C22×C311yescharacteristic
60C7×C4⋊C4C22×C311yescharacteristic
61#1144—22no—
62(C2×D7)⋊C4—21no—
63C4⋊Dic7—21no—
64C2×C7⋊C8C22×C311yescharacteristic
65Dic28—22no—
66C22×C3×Q8—71no—
67C2×C3×Q16—71no—
68C2×C3×Q16—71no—
69C3×C22⋊Q8—71no—
70C3×#103—71no—
71C3×#103D711yescharacteristic, complemented
72C3×C22⋊C8—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×C3×D7—21no—
75C2×C3×D7—21no—
76C2×C3×D7Q1612yescomplemented
77C3×C4×C7—22no—
78C3×C4×C7—41no—
79C3×C4×C7C2×D411yescharacteristic
80C3×C4×C7C2×D411yescharacteristic
81C3×Dic7—21no—
82C3×Dic7—22no—
83C3×Dic7—41no—
84#451—71noSylow
85C22×D7C3×D411yescharacteristic
86C7×Q8—21no—
87C7×Q8C3×D412yes—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7C23×C311yescharacteristic
91C2×Dic7—21no—
92C2×Dic7—21no—
93C2×Dic7C3×D411yescharacteristic
94C4×D7—24no—
95C4×D7—21no—
96C4×D7—21no—
97Dic14—24no—
98Dic14—21no—
99Dic14—21no—
100Dic14—21no—
101Dic14C3×D412yes—
102C7×C8—21no—
103C7⋊C8—21no—
104C2×C3×Q8—71no—
105C2×C3×Q8—71no—
106C2×C3×Q8—141no—
107C2×C3×Q8—142no—
108C2×C3×Q8—142no—
109C2×C3×Q8C2×D711yescharacteristic
110C22×C3×C4—71no—
111C22×C3×C4—141no—
112C3×Q16—142no—
113C3×Q16—142no—
114C2×C3×C8—71no—
115C2×C3×C8C2×D711yescharacteristic
116C3×C4⋊C4—141no—
117C3×C4⋊C4C2×D711yescharacteristic
118C3×C22⋊C4—141no—
119C2×C3×C7#12011yescharacteristic
120C2×C3×C7C2×Q1611yescharacteristic
121C2×C3×C7#13711yescharacteristic
122C3×D7—22no—
123C22×Q8—71no—
124C2×Q16—71no—
125C2×Q16—71no—
126C22⋊Q8—71no—
127#103—71no—
128#103C3×D711yescharacteristic, complemented
129C22⋊C8—71no—
130C22×C7C2×C3×D411yescharacteristic
131C2×D7—21no—
132C2×D7—21no—
133C2×D7C3×Q1612yescomplemented
134C4×C7—22no—
135C4×C7—41no—
136C4×C7C2×C3×D411yescharacteristic
137C4×C7C2×C3×D411yescharacteristic
138Dic7—21no—
139Dic7—22no—
140Dic7—41no—
141C23×C3—71no—
142C3×Q8—21no—
143C3×Q8—72no—
144C3×Q8—144no—
145C3×Q8—141no—
146C3×Q8—141no—
147C3×Q8—141no—
148C3×Q8D2812yes—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—71no—
152C2×C3×C4—144no—
153C2×C3×C4—141no—
154C2×C3×C4—141no—
155C2×C3×C4—141no—
156C2×C3×C4—141no—
157C2×C3×C4C22×D711yescharacteristic
158C3×C8—21no—
159C3×C8—141no—
160C3×C7#45111yescharacteristic, complemented, Hall
161C2×Q8—71no—
162C2×Q8—71no—
163C2×Q8—141no—
164C2×Q8—142no—
165C2×Q8—142no—
166C2×Q8C2×C3×D711yescharacteristic
167C22×C4—71no—
168C22×C4—141no—
169Q16—142no—
170Q16—142no—
171C2×C8—71no—
172C2×C8C2×C3×D711yescharacteristic
173C4⋊C4—141no—
174C4⋊C4C2×C3×D711yescharacteristic
175C22⋊C4—141no—
176C2×C7C3×#12011yescharacteristic
177C2×C7C2×C3×Q1611yescharacteristic
178C2×C7C3×#13711yescharacteristic
179D7—22no—
180C22×C3—72no—
181C22×C3—141no—
182C22×C3—141no—
183C22×C3D4×D711yescharacteristic
184C3×C4—22no—
185C3×C4—41no—
186C3×C4—141no—
187C3×C4—142no—
188C3×C4—281no—
189C3×C4C2×D2811yescharacteristic
190C3×C4D4×D711yescharacteristic
191C23—71no—
192Q8—21no—
193Q8—72no—
194Q8—144no—
195Q8—141no—
196Q8—141no—
197Q8—141no—
198Q8C3×D2812yes—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—71no—
202C2×C4—144no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—141no—
207C2×C4C22×C3×D711yescharacteristic
208C8—21no—
209C8—141no—
210C7C3×#45111yescharacteristic, complemented, Sylow
211C2×C3—142no—
212C2×C3#644511yescharacteristic
213C2×C3#647811yescharacteristic
214C2×C3D7×Q1611yescharacteristic
215C22—72no—
216C22—141no—
217C22—141no—
218C22C3×D4×D711yescharacteristic
219C4—22no—
220C4—41no—
221C4—141no—
222C4—142no—
223C4—281no—
224C4C2×C3×D2811yescharacteristic
225C4C3×D4×D711yescharacteristic
226C3#8958011yescharacteristic, direct product factor, Sylow
227C2—142no—
228C2C3×#644511yescharacteristic
229C2C3×#647811yescharacteristic
230C2C3×D7×Q1611yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
hypercenterC3×#103D7
hypocenterC7C3×#451
LCSG ▷ C2×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
2. C2×C3×#1144C2semidirect
3. C2×C3×Dic28C2semidirect
5. C3×C7×#103C2semidirect
7. C3×#6384C2semidirect
8. #89580C3direct (N × H)
25. C2×#1144C2×C3semidirect
26. C2×Dic28C2×C3semidirect
28. C7×#103C2×C3semidirect
30. #6384C2×C3semidirect
71. C3×#103D7semidirect
76. C2×C3×D7Q16semidirect
128. #103C3×D7semidirect
133. C2×D7C3×Q16semidirect
160. C3×C7#451semidirect
210. C7C3×#451semidirect
226. C3#89580direct (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×#11442
C2×C3×Dic282
C3×#64602
C3×C7×#1032
C3×(C2×D7)⋊C82
C3×#63842
#895803
C3×#4517