Finite Group Databasebeta

← C3×#89525 #13055186

Subgroups — C3×#89525 #13055186

Order 1344.

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

normal subgroups only

This group has 292 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×#6551C211yesmaximal, characteristic, complemented
2C3×#6510C211yesmaximal, characteristic, complemented
3C3×#6471C212yesmaximal, complemented
4C3×C7×#101C211yesmaximal, characteristic, complemented
5C3×#6399C212yesmaximal, complemented
6#89525C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C7×Q8C2211yescharacteristic
8C3×#1161—22no—
9C3×Q8×D7—22no—
10C3×#1157—22no—
11C3×#1157C2212yes—
12C2×C3×D28C2211yescharacteristic
13C3×C7×M16C2212yes—
14C3×(C2×D7)⋊C4—22no—
15C3×C4×Dic7C2211yescharacteristic
16C3×D56—22no—
17C3×#1132—22no—
18#6551C2×C311yescharacteristic, complemented
19#6510C2×C311yescharacteristic, complemented
20#6471C2×C312yescomplemented
21C7×#101C2×C311yescharacteristic, complemented
22#6399C2×C312yescomplemented
23C3×#454—71nomaximal, Hall
24C22×C3×D7—21no—
25C3×C7×Q8—22no—
26C2×C3×C4×C7D412yes—
27C2×C3×C4×C7C2311yescharacteristic
28C3×C22⋊D7—22no—
29C2×C3×Dic7—21no—
30C3×D28—22no—
31C3×D28—22no—
32C3×D28D412yes—
33C3×C4×D7—24no—
34C3×C4×D7—22no—
35C3×Dic14—22no—
36C3×Dic14D412yes—
37C3×C7×C8—22no—
38C2×C7×Q8C22×C311yescharacteristic
39#1161—22no—
40Q8×D7—22no—
41#1157—22no—
42#1157C22×C312yes—
43C2×D28C22×C311yescharacteristic
44C7×M16C22×C312yes—
45(C2×D7)⋊C4—22no—
46C4×Dic7C22×C311yescharacteristic
47D56—22no—
48#1132—22no—
49C3×2−1+4—71no—
50C3×AGL(1,ℤ/8)—72no—
51C3×#124—71no—
52C3×#104—72no—
53C3×#101D711yescharacteristic, complemented
54C22×C3×C7C2×D411yescharacteristic
55C2×C3×D7—22no—
56C2×C3×D7—42no—
57C2×C3×D7—41no—
58C3×C4×C7—22no—
59C3×C4×C7C2×D412yes—
60C3×Dic7—22no—
61C3×Dic7—41no—
62#454—71noSylow
63C22×D7—21no—
64C7×Q8—22no—
65C2×C4×C7C3×D412yes—
66C2×C4×C7C23×C311yescharacteristic
67C22⋊D7—22no—
68C2×Dic7—21no—
69D28—22no—
70D28—22no—
71D28C3×D412yes—
72C4×D7—24no—
73C4×D7—22no—
74Dic14—22no—
75Dic14C3×D412yes—
76C7×C8—22no—
77C3×#41—72no—
78C3×#41—142no—
79C3×#41—142no—
80C2×C3×Q8—142no—
81C2×C3×Q8C2×D711yescharacteristic
82C2×C3×D4—71no—
83C3×SD16—142no—
84C3×D8—142no—
85C3×M16C2×D712yes—
86C3×C22⋊C4—142no—
87C3×C42—71no—
88C2×C3×C7—21no—
89C2×C3×C7#12011yescharacteristic
90C3×D7—42no—
91C3×D7—81no—
922−1+4—71no—
93AGL(1,ℤ/8)—72no—
94#124—71no—
95#104—72no—
96#101C3×D711yescharacteristic, complemented
97C22×C7C2×C3×D411yescharacteristic
98C2×D7—22no—
99C2×D7—42no—
100C2×D7—41no—
101C4×C7—22no—
102C4×C7C2×C3×D412yes—
103Dic7—22no—
104Dic7—41no—
105C23×C3—141no—
106C3×Q8—22no—
107C3×Q8—72no—
108C3×Q8—142no—
109C3×D4—72no—
110C3×D4—142no—
111C3×D4—142no—
112C3×D4—142no—
113C2×C3×C4—144no—
114C2×C3×C4—142no—
115C2×C3×C4—141no—
116C2×C3×C4D2812yes—
117C2×C3×C4C22×D711yescharacteristic
118C3×C8—22no—
119C3×C7#45411yescharacteristic, complemented, Hall
120#41—72no—
121#41—142no—
122#41—142no—
123C2×Q8—142no—
124C2×Q8C2×C3×D711yescharacteristic
125C2×D4—71no—
126SD16—142no—
127D8—142no—
128M16C2×C3×D712yes—
129C22⋊C4—142no—
130C42—71no—
131C2×C7—21no—
132C2×C7C3×#12011yescharacteristic
133D7—42no—
134D7—81no—
135C22×C3—142no—
136C22×C3—282no—
137C22×C3—281no—
138C22×C3C2×D2811yescharacteristic
139C3×C4—22no—
140C3×C4—142no—
141C3×C4—281no—
142C3×C4D4×D712yes—
143C23—141no—
144Q8—22no—
145Q8—72no—
146Q8—142no—
147D4—72no—
148D4—142no—
149D4—142no—
150D4—142no—
151C2×C4—144no—
152C2×C4—142no—
153C2×C4—141no—
154C2×C4C3×D2812yes—
155C2×C4C22×C3×D711yescharacteristic
156C8—22no—
157C7C3×#45411yescharacteristic, complemented, Sylow
158C2×C3—21no—
159C2×C3—282no—
160C2×C3—561no—
161C2×C3#644511yescharacteristic
162C22—142no—
163C22—282no—
164C22—281no—
165C22C2×C3×D2811yescharacteristic
166C4—22no—
167C4—142no—
168C4—281no—
169C4C3×D4×D712yes—
170C3#8952511yescharacteristic, direct product factor, Sylow
171C2—21no—
172C2—282no—
173C2—561no—
174C2C3×#644511yescharacteristic
subgroupG/H
centerC2×C3#6445
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#101C2
socleC2×C3×C7#120
hypercenterC3×#101D7
hypocenterC7C3×#454
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#101

Complements

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

NH ≅ G/Ntype
1. C3×#6551C2semidirect
2. C3×#6510C2semidirect
3. C3×#6471C2semidirect
4. C3×C7×#101C2semidirect
5. C3×#6399C2semidirect
6. #89525C3direct (N × H)
18. #6551C2×C3semidirect
19. #6510C2×C3semidirect
20. #6471C2×C3semidirect
21. C7×#101C2×C3semidirect
22. #6399C2×C3semidirect
53. C3×#101D7semidirect
96. #101C3×D7semidirect
119. C3×C7#454semidirect
157. C7C3×#454semidirect
170. C3#89525direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#65512
C3×#65102
C3×#64712
C3×C7×#1012
C3×#63992
#895253
C3×#4547