Finite Group Databasebeta

← C3×#89822 #13055485

Subgroups — C3×#89822 #13055485

Order 1344.

296 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C22×C3×Dic14C211yesmaximal, characteristic
2C3×C7×C22⋊Q8C211yesmaximal, characteristic
3C2×C3×#1144C212yesmaximal, complemented
4C3×C22⋊(C7⋊C8)C211yesmaximal, characteristic
5C3×#6384C212yesmaximal, complemented
6#89822C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×Dic7—21no—
10C2×C3×Dic14—21no—
11C2×C3×Dic14—24no—
12C2×C3×Dic14C2212yes—
13C3×C7×C4⋊C4—21no—
14C3×C7×C4⋊C4C2211yescharacteristic
15C3×C7×C22⋊C4—21no—
16C3×#1144—24no—
17C2×C3×C7⋊C8C2212yes—
18C22×Dic14C2×C311yescharacteristic
19C7×C22⋊Q8C2×C311yescharacteristic
20C2×#1144C2×C312yescomplemented
21C22⋊(C7⋊C8)C2×C311yescharacteristic
22#6384C2×C312yescomplemented
23C3×#451—71nomaximal, Hall
24C23×C3×C7D411yescharacteristic
25C3×C7×Q8—22no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7D411yescharacteristic
30C2×C3×C4×C7C2311yescharacteristic
31C2×C3×Dic7—24no—
32C2×C3×Dic7—22no—
33C3×Dic14—24no—
34C3×Dic14—22no—
35C3×Dic14D414yes—
36C3×C7⋊C8—22no—
37C2×C7×Q8C22×C311yescharacteristic
38C22×C4×C7C22×C311yescharacteristic
39C22×Dic7—21no—
40C2×Dic14—21no—
41C2×Dic14—24no—
42C2×Dic14C22×C312yes—
43C7×C4⋊C4—21no—
44C7×C4⋊C4C22×C311yescharacteristic
45C7×C22⋊C4—21no—
46#1144—24no—
47C2×C7⋊C8C22×C312yes—
48C22×C3×Q8—71no—
49C2×C3×Q16—72no—
50C3×C22⋊Q8D711yescharacteristic
51C3×#103—72no—
52C3×C22⋊C8—71no—
53C22×C3×C7—21no—
54C22×C3×C7—21no—
55C22×C3×C7Q1612yescomplemented
56C22×C3×C7C2×D411yescharacteristic
57C3×C4×C7—21no—
58C3×C4×C7—41no—
59C3×C4×C7—41no—
60C3×C4×C7C2×D412yes—
61C3×Dic7—24no—
62#451—71noSylow
63C23×C7C3×D411yescharacteristic
64C7×Q8—22no—
65C2×C4×C7—22no—
66C2×C4×C7—21no—
67C2×C4×C7—21no—
68C2×C4×C7C3×D411yescharacteristic
69C2×C4×C7C23×C311yescharacteristic
70C2×Dic7—24no—
71C2×Dic7—22no—
72Dic14—24no—
73Dic14—22no—
74Dic14C3×D414yes—
75C7⋊C8—22no—
76C2×C3×Q8—72no—
77C2×C3×Q8—144no—
78C2×C3×Q8—141no—
79C2×C3×Q8C2×D711yescharacteristic
80C22×C3×C4—141no—
81C22×C3×C4C2×D711yescharacteristic
82C3×Q16—144no—
83C2×C3×C8—72no—
84C3×C4⋊C4—21no—
85C3×C4⋊C4C2×D711yescharacteristic
86C3×C22⋊C4—21no—
87C2×C3×C7—22no—
88C2×C3×C7#12011yescharacteristic
89C2×C3×C7C2×Q1611yescharacteristic
90C2×C3×C7#13711yescharacteristic
91C22×Q8—71no—
92C2×Q16—72no—
93C22⋊Q8C3×D711yescharacteristic
94#103—72no—
95C22⋊C8—71no—
96C22×C7—21no—
97C22×C7—21no—
98C22×C7C3×Q1612yescomplemented
99C22×C7C2×C3×D411yescharacteristic
100C4×C7—21no—
101C4×C7—41no—
102C4×C7—41no—
103C4×C7C2×C3×D412yes—
104Dic7—24no—
105C23×C3C22⋊D711yescharacteristic
106C3×Q8—22no—
107C3×Q8—74no—
108C3×Q8—144no—
109C3×Q8—142no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—144no—
114C2×C3×C4—142no—
115C2×C3×C4C22⋊D711yescharacteristic
116C2×C3×C4C22×D711yescharacteristic
117C3×C8—142no—
118C3×C7#45111yescharacteristic, complemented, Hall
119C2×Q8—72no—
120C2×Q8—144no—
121C2×Q8—141no—
122C2×Q8C2×C3×D711yescharacteristic
123C22×C4—141no—
124C22×C4C2×C3×D711yescharacteristic
125Q16—144no—
126C2×C8—72no—
127C4⋊C4—21no—
128C4⋊C4C2×C3×D711yescharacteristic
129C22⋊C4—21no—
130C2×C7—22no—
131C2×C7C3×#12011yescharacteristic
132C2×C7C2×C3×Q1611yescharacteristic
133C2×C7C3×#13711yescharacteristic
134C22×C3—21no—
135C22×C3—21no—
136C22×C3#114412yescomplemented
137C22×C3C2×C22⋊D711yescharacteristic
138C3×C4—21no—
139C3×C4—41no—
140C3×C4—41no—
141C3×C4—144no—
142C3×C4D4×D712yes—
143C23C3×C22⋊D711yescharacteristic
144Q8—22no—
145Q8—74no—
146Q8—144no—
147Q8—142no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—144no—
152C2×C4—142no—
153C2×C4C3×C22⋊D711yescharacteristic
154C2×C4C22×C3×D711yescharacteristic
155C8—142no—
156C7C3×#45111yescharacteristic, complemented, Sylow
157C2×C3—22no—
158C2×C3#650011yescharacteristic
159C2×C3C2×#114411yescharacteristic
160C2×C3#651411yescharacteristic
161C22—21no—
162C22—21no—
163C22C3×#114412yescomplemented
164C22C2×C3×C22⋊D711yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—144no—
169C4C3×D4×D712yes—
170C3#8982211yescharacteristic, direct product factor, Sylow
171C2—22no—
172C2C3×#650011yescharacteristic
173C2C2×C3×#114411yescharacteristic
174C2C3×#651411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC3×C22⋊Q8D7
hypocenterC7C3×#451
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#1144C2semidirect
5. C3×#6384C2semidirect
6. #89822C3direct (N × H)
20. C2×#1144C2×C3semidirect
22. #6384C2×C3semidirect
55. C22×C3×C7Q16semidirect
98. C22×C7C3×Q16semidirect
118. C3×C7#451semidirect
136. C22×C3#1144semidirect
156. C7C3×#451semidirect
163. C22C3×#1144semidirect
170. C3#89822direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic142
C3×C7×C22⋊Q82
C2×C3×#11442
C3×C22⋊(C7⋊C8)2
C3×#63842
#898223
C3×#4517