Finite Group Databasebeta

← C3×#89752 #13055408

Subgroups — C3×#89752 #13055408

Order 1344.

300 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 122 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×C7×C4⋊C4C211yesmaximal, characteristic
2C2×C3×Dic7⋊C4C211yesmaximal, characteristic
3C2×C3×Dic7⋊C4C212yesmaximal
4C3×C7⋊#95C212yesmaximal
5C3×C7⋊#95C211yesmaximal, characteristic
6#89752C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C4×C7C2212yes—
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×Dic7C2211yescharacteristic
10C22×C3×Dic7C2211yescharacteristic
11C22×C3×Dic7C2212yes—
12C3×C7×C4⋊C4—22no—
13C3×Dic7⋊C4—22no—
14C3×Dic7⋊C4—24no—
15C2×C7×C4⋊C4C2×C311yescharacteristic
16C2×Dic7⋊C4C2×C311yescharacteristic
17C2×Dic7⋊C4C2×C312yes—
18C7⋊#95C2×C312yes—
19C7⋊#95C2×C311yescharacteristic
20C3×#398—71nomaximal, Hall
21C23×C3×C7C2311yescharacteristic
22C2×C3×C4×C7—22no—
23C2×C3×C4×C7—22no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×Dic7—22no—
28C2×C3×Dic7—21no—
29C2×C3×Dic7—22no—
30C2×C3×Dic7—21no—
31C2×C3×Dic7—21no—
32C2×C3×Dic7—22no—
33C2×C3×Dic7D412yes—
34C2×C3×Dic7Q814yes—
35C22×C4×C7C22×C312yes—
36C22×C4×C7C22×C311yescharacteristic
37C22×Dic7C22×C311yescharacteristic
38C22×Dic7C22×C311yescharacteristic
39C22×Dic7C22×C312yes—
40C7×C4⋊C4—22no—
41Dic7⋊C4—22no—
42Dic7⋊C4—24no—
43C2×C3×C4⋊C4—72no—
44C2×C3×C4⋊C4—71no—
45C2×C3×C4⋊C4D711yescharacteristic
46C3×#95—71no—
47C3×#95—72no—
48C22×C3×C7C2×D411yescharacteristic
49C22×C3×C7C2×D411yescharacteristic
50C22×C3×C7C2×Q812yes—
51C22×C3×C7#4112yes—
52C22×C3×C7#4111yescharacteristic
53C3×C4×C7—22no—
54C3×C4×C7—42no—
55C3×Dic7—22no—
56C3×Dic7—24no—
57C3×Dic7—41no—
58#398—71noSylow
59C23×C7C23×C311yescharacteristic
60C2×C4×C7—22no—
61C2×C4×C7—22no—
62C2×C4×C7—22no—
63C2×C4×C7—22no—
64C2×C4×C7C3×D412yes—
65C2×Dic7—22no—
66C2×Dic7—21no—
67C2×Dic7—22no—
68C2×Dic7—21no—
69C2×Dic7—21no—
70C2×Dic7—22no—
71C2×Dic7C3×D412yes—
72C2×Dic7C3×Q814yes—
73C22×C3×C4—72no—
74C22×C3×C4—71no—
75C22×C3×C4—71no—
76C22×C3×C4C2×D712yes—
77C22×C3×C4C2×D711yescharacteristic
78C3×C4⋊C4—22no—
79C3×C4⋊C4—144no—
80C3×C4⋊C4—142no—
81C2×C3×C7#12111yescharacteristic
82C2×C3×C7C22⋊Q812yes—
83C2×C3×C7#12311yescharacteristic
84C2×C3×C7#12512yes—
85C2×C3×C7C4⋊Q811yescharacteristic
86C2×C4⋊C4—72no—
87C2×C4⋊C4—71no—
88C2×C4⋊C4C3×D711yescharacteristic
89#95—71no—
90#95—72no—
91C22×C7C2×C3×D411yescharacteristic
92C22×C7C2×C3×D411yescharacteristic
93C22×C7C2×C3×Q812yes—
94C22×C7C3×#4112yes—
95C22×C7C3×#4111yescharacteristic
96C4×C7—22no—
97C4×C7—42no—
98Dic7—22no—
99Dic7—24no—
100Dic7—41no—
101C23×C3C22×D711yescharacteristic
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—72no—
107C2×C3×C4—74no—
108C2×C3×C4—142no—
109C2×C3×C4—142no—
110C2×C3×C4—141no—
111C2×C3×C4—141no—
112C2×C3×C4—142no—
113C2×C3×C4—141no—
114C2×C3×C4C22⋊D712yes—
115C3×C7#39811yescharacteristic, complemented, Hall
116C22×C4—72no—
117C22×C4—71no—
118C22×C4—71no—
119C22×C4C2×C3×D712yes—
120C22×C4C2×C3×D711yescharacteristic
121C4⋊C4—22no—
122C4⋊C4—144no—
123C4⋊C4—142no—
124C2×C7C3×#12111yescharacteristic
125C2×C7C3×C22⋊Q812yes—
126C2×C7C3×#12311yescharacteristic
127C2×C7C3×#12512yes—
128C2×C7C3×C4⋊Q811yescharacteristic
129C22×C3#115712yes—
130C22×C3D4×D711yescharacteristic
131C22×C3#115911yescharacteristic
132C22×C3Q8×D712yes—
133C22×C3C2×C22⋊D711yescharacteristic
134C3×C4—22no—
135C3×C4—42no—
136C3×C4—144no—
137C3×C4—142no—
138C3×C4—281no—
139C23C22×C3×D711yescharacteristic
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—72no—
145C2×C4—74no—
146C2×C4—142no—
147C2×C4—142no—
148C2×C4—141no—
149C2×C4—141no—
150C2×C4—142no—
151C2×C4—141no—
152C2×C4C3×C22⋊D712yes—
153C7C3×#39811yescharacteristic, complemented, Sylow
154C2×C3#645212yes—
155C2×C3#645912yes—
156C2×C3#649211yescharacteristic
157C2×C3#650111yescharacteristic
158C2×C3#650711yescharacteristic
159C22C3×#115712yes—
160C22C3×D4×D711yescharacteristic
161C22C3×#115911yescharacteristic
162C22C3×Q8×D712yes—
163C22C2×C3×C22⋊D711yescharacteristic
164C4—22no—
165C4—42no—
166C4—144no—
167C4—142no—
168C4—281no—
169C3#8975211yescharacteristic, direct product factor, Sylow
170C2C3×#645212yes—
171C2C3×#645912yes—
172C2C3×#649211yescharacteristic
173C2C3×#650111yescharacteristic
174C2C3×#650711yescharacteristic
subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC2×C3×C7×C4⋊C4C2
socle, prime radicalC23×C3×C7C23
hypercenterC2×C3×C4⋊C4D7
hypocenterC7C3×#398
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
6. #89752C3direct (N × H)
115. C3×C7#398semidirect
153. C7C3×#398semidirect
169. C3#89752direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
C2×C3×Dic7⋊C42
C3×C7⋊#952
#897523
C3×#3987