Finite Group Databasebeta

← C3×#89804 #13055465

Subgroups — C3×#89804 #13055465

Order 1344.

396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 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×#6548C211yesmaximal, characteristic, complemented
2C3×C7×#120C211yesmaximal, characteristic, complemented
3C3×#6516C212yesmaximal, complemented
4C3×#6408C211yesmaximal, characteristic, complemented
5C3×#6380C212yesmaximal, complemented
6#89804C311yesmaximal, characteristic, direct product factor, Hall
7C24×C3×C7C2211yescharacteristic, complemented
8C2×C3×C7×D4—21no—
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C2×C3×C22⋊D7—22no—
12C2×C3×C22⋊D7C2212yes—
13C3×#1159—22no—
14C3×D4×D7—22no—
15C3×#1157—21no—
16C3×C7×C22⋊C4—21no—
17C3×C7×C22⋊C4C2211yescharacteristic, complemented
18C3×C22⋊Dic7—22no—
19C3×C22⋊Dic7C2212yescomplemented
20#6548C2×C311yescharacteristic, complemented
21C7×#120C2×C311yescharacteristic, complemented
22#6516C2×C312yescomplemented
23#6408C2×C311yescharacteristic, complemented
24#6380C2×C312yescomplemented
25C3×B(4,𝔽2)—71nomaximal, Hall
26C23×C3×C7—22no—
27C23×C3×C7—21no—
28C23×C3×C7—42no—
29C23×C3×C7D411yescharacteristic, complemented
30C23×C3×C7C2311yescharacteristic
31C22×C3×D7—21no—
32C22×C3×D7D412yescomplemented
33C3×C7×D4—22no—
34C3×C7×D4—42no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7D411yescharacteristic, complemented
37C3×C22⋊D7—22no—
38C3×C22⋊D7—22no—
39C3×C22⋊D7—22no—
40C3×C22⋊D7—44no—
41C2×C3×Dic7—21no—
42C2×C3×Dic7—22no—
43C2×C3×Dic7D412yescomplemented
44C3×D28—21no—
45C3×C4×D7—22no—
46C3×Dic14—21no—
47C24×C7C22×C311yescharacteristic, complemented
48C2×C7×D4—21no—
49C2×C7×D4C22×C311yescharacteristic
50C2×C22⋊D7—21no—
51C2×C22⋊D7—22no—
52C2×C22⋊D7C22×C312yes—
53#1159—22no—
54D4×D7—22no—
55#1157—21no—
56C7×C22⋊C4—21no—
57C7×C22⋊C4C22×C311yescharacteristic, complemented
58C22⋊Dic7—22no—
59C22⋊Dic7C22×C312yescomplemented
60C3×H(5,𝔽2)—71no—
61C3×#120—72no—
62C3×#120D711yescharacteristic, complemented
63C3×#99—71no—
64C3×#99—72no—
65C22×C3×C7—24no—
66C22×C3×C7—22no—
67C22×C3×C7—21no—
68C22×C3×C7—41no—
69C22×C3×C7—41no—
70C22×C3×C7—42no—
71C22×C3×C7—41no—
72C22×C3×C7—41no—
73C22×C3×C7C2×D412yes—
74C22×C3×C7C2×D411yescharacteristic
75C2×C3×D7—22no—
76C2×C3×D7—42no—
77C2×C3×D7—81no—
78C3×C4×C7—21no—
79C3×C4×C7—41no—
80C3×Dic7—22no—
81C3×Dic7—42no—
82B(4,𝔽2)—71noSylow
83C23×C7—22no—
84C23×C7—21no—
85C23×C7—42no—
86C23×C7C3×D411yescharacteristic, complemented
87C23×C7C23×C311yescharacteristic
88C22×D7—21no—
89C22×D7C3×D412yescomplemented
90C7×D4—22no—
91C7×D4—42no—
92C2×C4×C7—21no—
93C2×C4×C7C3×D411yescharacteristic, complemented
94C22⋊D7—22no—
95C22⋊D7—22no—
96C22⋊D7—22no—
97C22⋊D7—44no—
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7C3×D412yescomplemented
101D28—21no—
102C4×D7—22no—
103Dic14—21no—
104C24×C3C2×D711yescharacteristic, complemented
105C3×#41—142no—
106C3×#41—141no—
107C2×C3×D4—21no—
108C2×C3×D4—72no—
109C2×C3×D4—142no—
110C2×C3×D4—141no—
111C2×C3×D4—142no—
112C2×C3×D4C2×D711yescharacteristic
113C3×C22⋊C4—21no—
114C3×C22⋊C4—72no—
115C3×C22⋊C4—142no—
116C3×C22⋊C4C2×D711yescharacteristic, complemented
117C2×C3×C7—22no—
118C2×C3×C7—21no—
119C2×C3×C7—41no—
120C2×C3×C7—42no—
121C2×C3×C7#12011yescharacteristic
122C3×D7—42no—
123H(5,𝔽2)—71no—
124#120—72no—
125#120C3×D711yescharacteristic, complemented
126#99—71no—
127#99—72no—
128C22×C7—24no—
129C22×C7—22no—
130C22×C7—21no—
131C22×C7—41no—
132C22×C7—41no—
133C22×C7—42no—
134C22×C7—41no—
135C22×C7—41no—
136C22×C7C2×C3×D412yes—
137C22×C7C2×C3×D411yescharacteristic
138C2×D7—22no—
139C2×D7—42no—
140C2×D7—81no—
141C4×C7—21no—
142C4×C7—41no—
143Dic7—22no—
144Dic7—42no—
145C23×C3—22no—
146C23×C3—21no—
147C23×C3—42no—
148C23×C3—72no—
149C23×C3—141no—
150C23×C3C22⋊D711yescharacteristic, complemented
151C23×C3C22×D711yescharacteristic
152C3×Q8—141no—
153C3×D4—22no—
154C3×D4—42no—
155C3×D4—141no—
156C3×D4—142no—
157C3×D4—142no—
158C3×D4—142no—
159C3×D4—284no—
160C2×C3×C4—21no—
161C2×C3×C4—72no—
162C2×C3×C4—142no—
163C2×C3×C4—141no—
164C2×C3×C4—142no—
165C2×C3×C4C22⋊D711yescharacteristic, complemented
166C3×C7B(4,𝔽2)11yescharacteristic, complemented, Hall
167C24C2×C3×D711yescharacteristic, complemented
168#41—142no—
169#41—141no—
170C2×D4—21no—
171C2×D4—72no—
172C2×D4—142no—
173C2×D4—141no—
174C2×D4—142no—
175C2×D4C2×C3×D711yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—72no—
178C22⋊C4—142no—
179C22⋊C4C2×C3×D711yescharacteristic, complemented
180C2×C7—22no—
181C2×C7—21no—
182C2×C7—41no—
183C2×C7—42no—
184C2×C7C3×#12011yescharacteristic
185D7—42no—
186C22×C3—22no—
187C22×C3—24no—
188C22×C3—21no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—142no—
195C22×C3—282no—
196C22×C3—561no—
197C22×C3D4×D712yes—
198C22×C3C2×C22⋊D711yescharacteristic
199C3×C4—21no—
200C3×C4—41no—
201C3×C4—142no—
202C3×C4—282no—
203C23—22no—
204C23—21no—
205C23—42no—
206C23—72no—
207C23—141no—
208C23C3×C22⋊D711yescharacteristic, complemented
209C23C22×C3×D711yescharacteristic
210Q8—141no—
211D4—22no—
212D4—42no—
213D4—141no—
214D4—142no—
215D4—142no—
216D4—142no—
217D4—284no—
218C2×C4—21no—
219C2×C4—72no—
220C2×C4—142no—
221C2×C4—141no—
222C2×C4—142no—
223C2×C4C3×C22⋊D711yescharacteristic, complemented
224C7C3×B(4,𝔽2)11yescharacteristic, complemented, Sylow
225C2×C3—21no—
226C2×C3—22no—
227C2×C3—42no—
228C2×C3—41no—
229C2×C3—282no—
230C2×C3#650011yescharacteristic
231C22—22no—
232C22—24no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—142no—
240C22—282no—
241C22—561no—
242C22C3×D4×D712yes—
243C22C2×C3×C22⋊D711yescharacteristic
244C4—21no—
245C4—41no—
246C4—142no—
247C4—282no—
248C3#8980411yescharacteristic, direct product factor, Sylow
249C2—21no—
250C2—22no—
251C2—42no—
252C2—41no—
253C2—282no—
254C2C3×#650011yescharacteristic
subgroupG/H
centerC2×C3#6500
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC3×C7×#120C2
socleC2×C3×C7#120
hypercenterC3×#120D7
hypocenterC7C3×B(4,𝔽2)
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
1. C3×#6548C2semidirect
2. C3×C7×#120C2semidirect
3. C3×#6516C2semidirect
4. C3×#6408C2semidirect
5. C3×#6380C2semidirect
6. #89804C3direct (N × H)
7. C24×C3×C7C22semidirect
17. C3×C7×C22⋊C4C22semidirect
19. C3×C22⋊Dic7C22semidirect
20. #6548C2×C3semidirect
21. C7×#120C2×C3semidirect
22. #6516C2×C3semidirect
23. #6408C2×C3semidirect
24. #6380C2×C3semidirect
29. C23×C3×C7D4semidirect
32. C22×C3×D7D4semidirect
36. C2×C3×C4×C7D4semidirect
43. C2×C3×Dic7D4semidirect
47. C24×C7C22×C3semidirect
57. C7×C22⋊C4C22×C3semidirect
59. C22⋊Dic7C22×C3semidirect
62. C3×#120D7semidirect
86. C23×C7C3×D4semidirect
89. C22×D7C3×D4semidirect
93. C2×C4×C7C3×D4semidirect
100. C2×Dic7C3×D4semidirect
104. C24×C3C2×D7semidirect
116. C3×C22⋊C4C2×D7semidirect
125. #120C3×D7semidirect
150. C23×C3C22⋊D7semidirect
165. C2×C3×C4C22⋊D7semidirect
166. C3×C7B(4,𝔽2)semidirect
167. C24C2×C3×D7semidirect
179. C22⋊C4C2×C3×D7semidirect
208. C23C3×C22⋊D7semidirect
223. C2×C4C3×C22⋊D7semidirect
224. C7C3×B(4,𝔽2)semidirect
248. C3#89804direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#65482
C3×C7×#1202
C3×#65162
C3×#64082
C3×#63802
#898043
C3×B(4,𝔽2)7