Finite Group Databasebeta

← C3×#89751 #13055407

Subgroups — C3×#89751 #13055407

Order 1344.

364 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 134 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
2C2×C3×C7×C4⋊C4C211yesmaximal, characteristic
3C2×C3×Dic7⋊C4C212yesmaximal
4C3×C7⋊#95C212yesmaximal
5C3×C7⋊#95C211yesmaximal, characteristic
6#89751C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C4×C7C2211yescharacteristic
8C22×C3×C4×C7C2211yescharacteristic
9C22×C3×C4×C7C2211yescharacteristic
10C22×C3×Dic7C2212yes—
11C22×C3×Dic7C2212yes—
12C2×C3×Dic14—22no—
13C2×C3×Dic14—24no—
14C3×C7×C4⋊C4—22no—
15C3×Dic7⋊C4—24no—
16C22×Dic14C2×C311yescharacteristic
17C2×C7×C4⋊C4C2×C311yescharacteristic
18C2×Dic7⋊C4C2×C312yes—
19C7⋊#95C2×C312yes—
20C7⋊#95C2×C311yescharacteristic
21C3×#395—71nomaximal, Hall
22C23×C3×C7C2311yescharacteristic
23C2×C3×C4×C7—21no—
24C2×C3×C4×C7—22no—
25C2×C3×C4×C7—21no—
26C2×C3×C4×C7—22no—
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7D412yes—
29C2×C3×C4×C7Q812yescomplemented
30C2×C3×Dic7—22no—
31C2×C3×Dic7—22no—
32C2×C3×Dic7—22no—
33C2×C3×Dic7—22no—
34C2×C3×Dic7D414yes—
35C2×C3×Dic7Q814yescomplemented
36C3×Dic14—28no—
37C22×C4×C7C22×C311yescharacteristic
38C22×C4×C7C22×C311yescharacteristic
39C22×C4×C7C22×C311yescharacteristic
40C22×Dic7C22×C312yes—
41C22×Dic7C22×C312yes—
42C2×Dic14—22no—
43C2×Dic14—24no—
44C7×C4⋊C4—22no—
45Dic7⋊C4—24no—
46C22×C3×Q8—71no—
47C2×C3×C4⋊C4—72no—
48C2×C3×C4⋊C4D711yescharacteristic
49C3×#95—71no—
50C3×#95—72no—
51C22×C3×C7C2×D412yes—
52C22×C3×C7C2×D411yescharacteristic
53C22×C3×C7C2×Q811yescharacteristic
54C22×C3×C7C2×Q812yes—
55C22×C3×C7#4111yescharacteristic
56C3×C4×C7—22no—
57C3×C4×C7—22no—
58C3×C4×C7—41no—
59C3×Dic7—24no—
60C3×Dic7—24no—
61#395—71noSylow
62C23×C7C23×C311yescharacteristic
63C2×C4×C7—21no—
64C2×C4×C7—22no—
65C2×C4×C7—21no—
66C2×C4×C7—22no—
67C2×C4×C7—21no—
68C2×C4×C7C3×D412yes—
69C2×C4×C7C3×Q812yescomplemented
70C2×Dic7—22no—
71C2×Dic7—22no—
72C2×Dic7—22no—
73C2×Dic7—22no—
74C2×Dic7C3×D414yes—
75C2×Dic7C3×Q814yescomplemented
76Dic14—28no—
77C2×C3×Q8—144no—
78C2×C3×Q8—142no—
79C22×C3×C4—72no—
80C22×C3×C4—72no—
81C22×C3×C4C2×D711yescharacteristic
82C22×C3×C4C2×D711yescharacteristic
83C22×C3×C4C2×D711yescharacteristic
84C3×C4⋊C4—22no—
85C3×C4⋊C4—144no—
86C2×C3×C7#12011yescharacteristic
87C2×C3×C7C22⋊Q812yes—
88C2×C3×C7C22⋊Q811yescharacteristic
89C2×C3×C7C4⋊Q812yes—
90C2×C3×C7C4⋊Q811yescharacteristic
91C22×Q8—71no—
92C2×C4⋊C4—72no—
93C2×C4⋊C4C3×D711yescharacteristic
94#95—71no—
95#95—72no—
96C22×C7C2×C3×D412yes—
97C22×C7C2×C3×D411yescharacteristic
98C22×C7C2×C3×Q811yescharacteristic
99C22×C7C2×C3×Q812yes—
100C22×C7C3×#4111yescharacteristic
101C4×C7—22no—
102C4×C7—22no—
103C4×C7—41no—
104Dic7—24no—
105Dic7—24no—
106C23×C3C22×D711yescharacteristic
107C3×Q8—148no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—21no—
113C2×C3×C4—74no—
114C2×C3×C4—74no—
115C2×C3×C4—142no—
116C2×C3×C4—142no—
117C2×C3×C4—142no—
118C2×C3×C4—142no—
119C2×C3×C4Dic1412yescomplemented
120C2×C3×C4C22⋊D712yes—
121C3×C7#39511yescharacteristic, complemented, Hall
122C2×Q8—144no—
123C2×Q8—142no—
124C22×C4—72no—
125C22×C4—72no—
126C22×C4C2×C3×D711yescharacteristic
127C22×C4C2×C3×D711yescharacteristic
128C22×C4C2×C3×D711yescharacteristic
129C4⋊C4—22no—
130C4⋊C4—144no—
131C2×C7C3×#12011yescharacteristic
132C2×C7C3×C22⋊Q812yes—
133C2×C7C3×C22⋊Q811yescharacteristic
134C2×C7C3×C4⋊Q812yes—
135C2×C7C3×C4⋊Q811yescharacteristic
136C22×C3C2×Dic1411yescharacteristic
137C22×C3#115711yescharacteristic
138C22×C3D4×D712yes—
139C22×C3Q8×D712yes—
140C22×C3C2×C22⋊D711yescharacteristic
141C3×C4—22no—
142C3×C4—22no—
143C3×C4—41no—
144C3×C4—144no—
145C3×C4—144no—
146C23C22×C3×D711yescharacteristic
147Q8—148no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—74no—
154C2×C4—74no—
155C2×C4—142no—
156C2×C4—142no—
157C2×C4—142no—
158C2×C4—142no—
159C2×C4C3×Dic1412yescomplemented
160C2×C4C3×C22⋊D712yes—
161C7C3×#39511yescharacteristic, complemented, Sylow
162C2×C3#645112yes—
163C2×C3#645912yes—
164C2×C3#648711yescharacteristic
165C2×C3#650011yescharacteristic
166C2×C3#650711yescharacteristic
167C22C2×C3×Dic1411yescharacteristic
168C22C3×#115711yescharacteristic
169C22C3×D4×D712yes—
170C22C3×Q8×D712yes—
171C22C2×C3×C22⋊D711yescharacteristic
172C4—22no—
173C4—22no—
174C4—41no—
175C4—144no—
176C4—144no—
177C3#8975111yescharacteristic, direct product factor, Sylow
178C2C3×#645112yes—
179C2C3×#645912yes—
180C2C3×#648711yescharacteristic
181C2C3×#650011yescharacteristic
182C2C3×#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×#395
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. #89751C3direct (N × H)
29. C2×C3×C4×C7Q8semidirect
35. C2×C3×Dic7Q8semidirect
69. C2×C4×C7C3×Q8semidirect
75. C2×Dic7C3×Q8semidirect
119. C2×C3×C4Dic14semidirect
121. C3×C7#395semidirect
159. C2×C4C3×Dic14semidirect
161. C7C3×#395semidirect
177. C3#89751direct (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
C2×C3×C7×C4⋊C42
C2×C3×Dic7⋊C42
C3×C7⋊#952
#897513
C3×#3957