Finite Group Databasebeta

← C3×#89565 #13055227

Subgroups — C3×#89565 #13055227

Order 1344.

216 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 82 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×#6507C211yesmaximal, characteristic
2C2×C3×#1144C211yesmaximal, characteristic
3C2×C3×Dic28C211yesmaximal, characteristic
4C3×#6450C211yesmaximal, characteristic
5C3×C7×#103C211yesmaximal, characteristic
6C3×Dic7⋊C8C211yesmaximal, characteristic
7C3×#6384C211yesmaximal, characteristic
8#89565C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C3×Dic14C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12C2×C3×C7×C8C2211yescharacteristic
13C3×C7×C4⋊C4C2211yescharacteristic
14C3×#1144—22no—
15C3×Dic7⋊C4—21no—
16C3×Dic7⋊C4—21no—
17C3×Dic7⋊C4—21no—
18C3×C4×Dic7—21no—
19C3×C4×Dic7C2211yescharacteristic
20C2×C3×C7⋊C8C2211yescharacteristic
21C3×Dic28—22no—
22#6507C2×C311yescharacteristic
23C2×#1144C2×C311yescharacteristic
24C2×Dic28C2×C311yescharacteristic
25#6450C2×C311yescharacteristic
26C7×#103C2×C311yescharacteristic
27Dic7⋊C8C2×C311yescharacteristic
28#6384C2×C311yescharacteristic
29C3×#462—71nomaximal, Hall
30C3×C7×Q8—21no—
31C3×C7×Q8—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×C3×Dic7—21no—
36C2×C3×Dic7—21no—
37C2×C3×Dic7D411yescharacteristic
38C2×C3×Dic7D411yescharacteristic
39C3×Dic14—21no—
40C3×Dic14—21no—
41C3×Dic14—21no—
42C3×Dic14D412yes—
43C3×C7×C8—21no—
44C3×C7⋊C8—21no—
45C2×C7×Q8C22×C311yescharacteristic
46C2×Dic14C22×C311yescharacteristic
47C2×Dic14C22×C311yescharacteristic
48C2×C7×C8C22×C311yescharacteristic
49C7×C4⋊C4C22×C311yescharacteristic
50#1144—22no—
51Dic7⋊C4—21no—
52Dic7⋊C4—21no—
53Dic7⋊C4—21no—
54C4×Dic7—21no—
55C4×Dic7C22×C311yescharacteristic
56C2×C7⋊C8C22×C311yescharacteristic
57Dic28—22no—
58C2×C3×Q16—71no—
59C2×C3×Q16—71no—
60C3×C4⋊Q8—71no—
61C3×C4×Q8—71no—
62C3×C4⋊C8—71no—
63C3×#103—71no—
64C3×#103D711yescharacteristic
65C22×C3×C7C2×D411yescharacteristic
66C3×C4×C7—41no—
67C3×C4×C7—41no—
68C3×C4×C7C2×D411yescharacteristic
69C3×C4×C7#4111yescharacteristic
70C3×Dic7—22no—
71C3×Dic7—21no—
72C3×Dic7—41no—
73C3×Dic7Q1612yescomplemented
74#462—71noSylow
75C7×Q8—21no—
76C7×Q8—21no—
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7C23×C311yescharacteristic
80C2×Dic7—21no—
81C2×Dic7—21no—
82C2×Dic7C3×D411yescharacteristic
83C2×Dic7C3×D411yescharacteristic
84Dic14—21no—
85Dic14—21no—
86Dic14—21no—
87Dic14C3×D412yes—
88C7×C8—21no—
89C7⋊C8—21no—
90C2×C3×Q8—71no—
91C2×C3×Q8—71no—
92C2×C3×Q8C2×D711yescharacteristic
93C3×Q16—142no—
94C3×Q16—142no—
95C2×C3×C8—71no—
96C2×C3×C8C2×D711yescharacteristic
97C3×C4⋊C4—141no—
98C3×C4⋊C4—141no—
99C3×C4⋊C4—141no—
100C3×C4⋊C4C2×D711yescharacteristic
101C3×C42—71no—
102C3×C42—141no—
103C2×C3×C7#12111yescharacteristic
104C2×C3×C7C2×Q1611yescharacteristic
105C2×C3×C7#13711yescharacteristic
106C2×Q16—71no—
107C2×Q16—71no—
108C4⋊Q8—71no—
109C4×Q8—71no—
110C4⋊C8—71no—
111#103—71no—
112#103C3×D711yescharacteristic
113C22×C7C2×C3×D411yescharacteristic
114C4×C7—41no—
115C4×C7—41no—
116C4×C7C2×C3×D411yescharacteristic
117C4×C7C3×#4111yescharacteristic
118Dic7—22no—
119Dic7—21no—
120Dic7—41no—
121Dic7C3×Q1612yescomplemented
122C3×Q8—21no—
123C3×Q8—21no—
124C3×Q8—72no—
125C3×Q8—141no—
126C3×Q8—141no—
127C3×Q8—141no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—71no—
131C2×C3×C4—71no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4C22×D711yescharacteristic
135C3×C8—21no—
136C3×C8—141no—
137C3×C7#46211yescharacteristic, complemented, Hall
138C2×Q8—71no—
139C2×Q8—71no—
140C2×Q8C2×C3×D711yescharacteristic
141Q16—142no—
142Q16—142no—
143C2×C8—71no—
144C2×C8C2×C3×D711yescharacteristic
145C4⋊C4—141no—
146C4⋊C4—141no—
147C4⋊C4—141no—
148C4⋊C4C2×C3×D711yescharacteristic
149C42—71no—
150C42—141no—
151C2×C7C3×#12111yescharacteristic
152C2×C7C2×C3×Q1611yescharacteristic
153C2×C7C3×#13711yescharacteristic
154C22×C3D4×D711yescharacteristic
155C3×C4—41no—
156C3×C4—41no—
157C3×C4—72no—
158C3×C4—142no—
159C3×C4—141no—
160C3×C4—281no—
161C3×C4#115711yescharacteristic
162C3×C4D4×D711yescharacteristic
163Q8—21no—
164Q8—21no—
165Q8—72no—
166Q8—141no—
167Q8—141no—
168Q8—141no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—71no—
172C2×C4—71no—
173C2×C4—141no—
174C2×C4—141no—
175C2×C4C22×C3×D711yescharacteristic
176C8—21no—
177C8—141no—
178C7C3×#46211yescharacteristic, complemented, Sylow
179C2×C3#644711yescharacteristic
180C2×C3#647811yescharacteristic
181C2×C3D7×Q1611yescharacteristic
182C22C3×D4×D711yescharacteristic
183C4—41no—
184C4—41no—
185C4—72no—
186C4—142no—
187C4—141no—
188C4—281no—
189C4C3×#115711yescharacteristic
190C4C3×D4×D711yescharacteristic
191C3#8956511yescharacteristic, direct product factor, Sylow
192C2C3×#644711yescharacteristic
193C2C3×#647811yescharacteristic
194C2C3×D7×Q1611yescharacteristic
subgroupG/H
centerC22×C3D4×D7
commutator, cosocleC2×C4×C7C23×C3
FrattiniC2×C4C22×C3×D7
FittingC3×C7×#103C2
socle, prime radicalC22×C3×C7C2×D4
hypercenterC3×#103D7
hypocenterC7C3×#462
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
8. #89565C3direct (N × H)
73. C3×Dic7Q16semidirect
121. Dic7C3×Q16semidirect
137. C3×C7#462semidirect
178. C7C3×#462semidirect
191. C3#89565direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#65072
C2×C3×#11442
C2×C3×Dic282
C3×#64502
C3×C7×#1032
C3×Dic7⋊C82
C3×#63842
#895653
C3×#4627