Finite Group Databasebeta

← C3×#89825 #13055486

Subgroups — C3×#89825 #13055486

Order 1344.

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

normal subgroups only

This group has 320 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
2C3×C7×#123C211yesmaximal, characteristic, complemented
3C3×#6492C212yesmaximal, complemented
4C3×#6408C211yesmaximal, characteristic, complemented
5C3×#6380C212yesmaximal, complemented
6#89825C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic, complemented
9C2×C3×C22⋊D7—21no—
10C2×C3×C22⋊D7C2212yes—
11C3×#1159—22no—
12C3×D4×D7—22no—
13C3×#1157—21no—
14C3×C7×C4⋊C4—21no—
15C3×C7×C22⋊C4—21no—
16C3×C7×C22⋊C4C2211yescharacteristic, complemented
17C3×C22⋊Dic7C2212yescomplemented
18C3×(C2×D7)⋊C4—22no—
19C3×Dic7⋊C4—22no—
20#6548C2×C311yescharacteristic
21C7×#123C2×C311yescharacteristic, complemented
22#6492C2×C312yescomplemented
23#6408C2×C311yescharacteristic, complemented
24#6380C2×C312yescomplemented
25C3×#458—71nomaximal, Hall
26C23×C3×C7D411yescharacteristic
27C23×C3×C7C2311yescharacteristic
28C22×C3×D7—21no—
29C22×C3×D7D412yes—
30C3×C7×D4—22no—
31C2×C3×C4×C7—22no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7D411yescharacteristic
35C3×C22⋊D7—22no—
36C3×C22⋊D7—22no—
37C3×C22⋊D7—22no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7—22no—
40C2×C3×Dic7D412yes—
41C3×D28—21no—
42C3×C4×D7—22no—
43C3×Dic14—21no—
44C2×C7×D4C22×C311yescharacteristic
45C22×C4×C7C22×C311yescharacteristic, complemented
46C2×C22⋊D7—21no—
47C2×C22⋊D7C22×C312yes—
48#1159—22no—
49D4×D7—22no—
50#1157—21no—
51C7×C4⋊C4—21no—
52C7×C22⋊C4—21no—
53C7×C22⋊C4C22×C311yescharacteristic, complemented
54C22⋊Dic7C22×C312yescomplemented
55(C2×D7)⋊C4—22no—
56Dic7⋊C4—22no—
57C3×H(5,𝔽2)—71no—
58C3×#123—72no—
59C3×#123D711yescharacteristic, complemented
60C3×#99—71no—
61C3×#99—72no—
62C22×C3×C7—21no—
63C22×C3×C7—41no—
64C22×C3×C7—41no—
65C22×C3×C7C2×D412yes—
66C22×C3×C7C2×D411yescharacteristic
67C2×C3×D7—22no—
68C2×C3×D7—42no—
69C2×C3×D7—81no—
70C3×C4×C7—21no—
71C3×C4×C7—41no—
72C3×C4×C7—41no—
73C3×Dic7—22no—
74C3×Dic7—42no—
75#458—71noSylow
76C23×C7C3×D411yescharacteristic
77C23×C7C23×C311yescharacteristic
78C22×D7—21no—
79C22×D7C3×D412yes—
80C7×D4—22no—
81C2×C4×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7C3×D411yescharacteristic
85C22⋊D7—22no—
86C22⋊D7—22no—
87C22⋊D7—22no—
88C2×Dic7—21no—
89C2×Dic7—22no—
90C2×Dic7C3×D412yes—
91D28—21no—
92C4×D7—22no—
93Dic14—21no—
94C3×#41—142no—
95C3×#41—141no—
96C2×C3×D4—72no—
97C2×C3×D4—142no—
98C2×C3×D4—141no—
99C2×C3×D4C2×D711yescharacteristic
100C22×C3×C4C2×D711yescharacteristic, complemented
101C3×C4⋊C4—21no—
102C3×C4⋊C4—142no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—72no—
105C3×C22⋊C4—142no—
106C3×C22⋊C4C2×D711yescharacteristic, complemented
107C2×C3×C7—22no—
108C2×C3×C7—21no—
109C2×C3×C7—41no—
110C2×C3×C7#12011yescharacteristic
111C3×D7—42no—
112H(5,𝔽2)—71no—
113#123—72no—
114#123C3×D711yescharacteristic, complemented
115#99—71no—
116#99—72no—
117C22×C7—21no—
118C22×C7—41no—
119C22×C7—41no—
120C22×C7C2×C3×D412yes—
121C22×C7C2×C3×D411yescharacteristic
122C2×D7—22no—
123C2×D7—42no—
124C2×D7—81no—
125C4×C7—21no—
126C4×C7—41no—
127C4×C7—41no—
128Dic7—22no—
129Dic7—42no—
130C23×C3—72no—
131C23×C3—141no—
132C23×C3C22⋊D711yescharacteristic
133C23×C3C22×D711yescharacteristic
134C3×Q8—141no—
135C3×D4—22no—
136C3×D4—141no—
137C3×D4—142no—
138C3×D4—142no—
139C3×D4—142no—
140C2×C3×C4—22no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—72no—
144C2×C3×C4—142no—
145C2×C3×C4—141no—
146C2×C3×C4—142no—
147C2×C3×C4C22⋊D711yescharacteristic
148C3×C7#45811yescharacteristic, complemented, Hall
149#41—142no—
150#41—141no—
151C2×D4—72no—
152C2×D4—142no—
153C2×D4—141no—
154C2×D4C2×C3×D711yescharacteristic
155C22×C4C2×C3×D711yescharacteristic, complemented
156C4⋊C4—21no—
157C4⋊C4—142no—
158C22⋊C4—21no—
159C22⋊C4—72no—
160C22⋊C4—142no—
161C22⋊C4C2×C3×D711yescharacteristic, complemented
162C2×C7—22no—
163C2×C7—21no—
164C2×C7—41no—
165C2×C7C3×#12011yescharacteristic
166D7—42no—
167C22×C3—21no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—142no—
171C22×C3—282no—
172C22×C3—561no—
173C22×C3D4×D712yes—
174C22×C3C2×C22⋊D711yescharacteristic
175C3×C4—21no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—142no—
179C3×C4—282no—
180C23—72no—
181C23—141no—
182C23C3×C22⋊D711yescharacteristic
183C23C22×C3×D711yescharacteristic
184Q8—141no—
185D4—22no—
186D4—141no—
187D4—142no—
188D4—142no—
189D4—142no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—72no—
194C2×C4—142no—
195C2×C4—141no—
196C2×C4—142no—
197C2×C4C3×C22⋊D711yescharacteristic
198C7C3×#45811yescharacteristic, complemented, Sylow
199C2×C3—21no—
200C2×C3—22no—
201C2×C3—41no—
202C2×C3—282no—
203C2×C3#650011yescharacteristic
204C22—21no—
205C22—41no—
206C22—41no—
207C22—142no—
208C22—282no—
209C22—561no—
210C22C3×D4×D712yes—
211C22C2×C3×C22⋊D711yescharacteristic
212C4—21no—
213C4—41no—
214C4—41no—
215C4—142no—
216C4—282no—
217C3#8982511yescharacteristic, direct product factor, Sylow
218C2—21no—
219C2—22no—
220C2—41no—
221C2—282no—
222C2C3×#650011yescharacteristic
subgroupG/H
centerC2×C3#6500
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC3×C7×#123C2
socleC2×C3×C7#120
prime radicalC3×#6548C2
hypercenterC3×#123D7
hypocenterC7C3×#458
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
2. C3×C7×#123C2semidirect
3. C3×#6492C2semidirect
4. C3×#6408C2semidirect
5. C3×#6380C2semidirect
6. #89825C3direct (N × H)
8. C22×C3×C4×C7C22semidirect
16. C3×C7×C22⋊C4C22semidirect
17. C3×C22⋊Dic7C22semidirect
21. C7×#123C2×C3semidirect
22. #6492C2×C3semidirect
23. #6408C2×C3semidirect
24. #6380C2×C3semidirect
45. C22×C4×C7C22×C3semidirect
53. C7×C22⋊C4C22×C3semidirect
54. C22⋊Dic7C22×C3semidirect
59. C3×#123D7semidirect
100. C22×C3×C4C2×D7semidirect
106. C3×C22⋊C4C2×D7semidirect
114. #123C3×D7semidirect
148. C3×C7#458semidirect
155. C22×C4C2×C3×D7semidirect
161. C22⋊C4C2×C3×D7semidirect
198. C7C3×#458semidirect
217. C3#89825direct (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×#1232
C3×#64922
C3×#64082
C3×#63802
#898253
C3×#4587