Finite Group Databasebeta

← #13054609

Subgroups — #13054609

Order 1344.

216 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 66 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×#66635C211yesmaximal, characteristic
2C3×#6507C211yesmaximal, characteristic
3#10620895C211yesmaximal, characteristic
4C2×#66570C211yesmaximal, characteristic
5C7×#835C211yesmaximal, characteristic
6Dic7⋊(C3⋊C8)C211yesmaximal, characteristic
7#10620505C211yesmaximal, characteristic
8#89565—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×Dic14C2211yescharacteristic
12#66635—22no—
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×C4⋊Dic3C2211yescharacteristic
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×Dic7⋊C4—21no—
17C3×Dic7⋊C4—21no—
18C3×C4×Dic7C2211yescharacteristic
19#66577—21no—
20Dic3×Dic7—21no—
21#66570—22no—
22#6507D311yescharacteristic
23C2×#1144—31no—
24C2×Dic28—31no—
25#6450—31no—
26C7×#103—31no—
27Dic7⋊C8—31no—
28#6384—31no—
29#5090—71nomaximal, Hall
30C3×C7×Q8—21no—
31C3×C7×Q8—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35Dic42—21no—
36Dic42D412yes—
37C2×C7×Dic3—21no—
38C2×C3×Dic7—21no—
39C2×C3×Dic7D411yescharacteristic
40C2×C3×Dic7D411yescharacteristic
41C3×Dic14—21no—
42C3×Dic14—21no—
43C3⋊(C7⋊C8)—21no—
44C7×C3⋊C8—21no—
45C2×C7×Q8C2×D311yescharacteristic
46C2×Dic14—31no—
47C2×Dic14C2×D311yescharacteristic
48C2×C7×C8—31no—
49C7×C4⋊C4—31no—
50#1144—62no—
51Dic7⋊C4—21no—
52Dic7⋊C4—21no—
53Dic7⋊C4—61no—
54C4×Dic7—61no—
55C4×Dic7C2×D311yescharacteristic
56C2×C7⋊C8—31no—
57Dic28—62no—
58C3×C4⋊Q8—71no—
59C2×#216—71no—
60C2×#216—71no—
61C4×Dic6—71no—
62#835—71no—
63#835D711yescharacteristic
64C4⋊(C3⋊C8)—71no—
65C22×C3×C7C2×D411yescharacteristic
66C3×C4×C7—41no—
67C3×C4×C7C2×D411yescharacteristic
68C3×C4×C7#4111yescharacteristic
69Dic21—22no—
70C3×Dic7—21no—
71C3×Dic7—41no—
72C3×Dic7Q1612yescomplemented
73C7×Dic3—41no—
74#462—211noSylow
75C7×Q8—21no—
76C7×Q8—21no—
77C2×C4×C7—21no—
78C2×C4×C7—61no—
79C2×C4×C7C22×D311yescharacteristic
80C2×Dic7—21no—
81C2×Dic7—61no—
82C2×Dic7#6711yescharacteristic
83C2×Dic7#6711yescharacteristic
84Dic14—21no—
85Dic14—21no—
86Dic14—32no—
87Dic14—61no—
88C7×C8—61no—
89C7⋊C8—61no—
90C2×C3×Q8—71no—
91C2×C3×Q8C2×D711yescharacteristic
92C2×Dic6—71no—
93C3×C4⋊C4—141no—
94C3×C4⋊C4—141no—
95C3×C42—71no—
96#216—142no—
97#216—142no—
98C4⋊Dic3C2×D711yescharacteristic
99Dic3⋊C4—141no—
100C4×Dic3—141no—
101C2×C3⋊C8—71no—
102C2×C3⋊C8C2×D711yescharacteristic
103C2×C3×C7#12111yescharacteristic
104C2×C3×C7C2×Q1611yescharacteristic
105C2×C3×C7#13711yescharacteristic
106C2×Q16—211no—
107C2×Q16—211no—
108C4⋊Q8—71no—
109C4×Q8—211no—
110C4⋊C8—211no—
111#103—31no—
112#103—211no—
113C22×C7C2×#6711yescharacteristic
114C4×C7—41no—
115C4×C7—121no—
116C4×C7D3×D411yescharacteristic
117C4×C7#23711yescharacteristic
118Dic7—21no—
119Dic7—41no—
120Dic7—62no—
121Dic7#21612yescomplemented
122C3×Q8—21no—
123C3×Q8—21no—
124C3×Q8—141no—
125C3×Q8—141no—
126C2×C3×C4—21no—
127C2×C3×C4—71no—
128C2×C3×C4—71no—
129C2×C3×C4—141no—
130C2×C3×C4C22×D711yescharacteristic
131C2×Dic3—21no—
132C2×Dic3—141no—
133Dic6—72no—
134Dic6—141no—
135C3⋊C8—21no—
136C3⋊C8—141no—
137C3×C7#46211yescharacteristic, complemented, Hall
138C2×Q8—71no—
139C2×Q8—211no—
140C2×Q8D3×D711yescharacteristic
141Q16—422no—
142Q16—422no—
143C2×C8—31no—
144C2×C8—211no—
145C4⋊C4—31no—
146C4⋊C4—141no—
147C4⋊C4—141no—
148C4⋊C4—421no—
149C42—71no—
150C42—421no—
151C2×C7#93811yescharacteristic
152C2×C7#94211yescharacteristic
153C2×C7C2×#21611yescharacteristic
154C22×C3D4×D711yescharacteristic
155C3×C4—41no—
156C3×C4—72no—
157C3×C4—141no—
158C3×C4—281no—
159C3×C4#115711yescharacteristic
160C3×C4D4×D711yescharacteristic
161Dic3—41no—
162Dic3—142no—
163Q8—21no—
164Q8—21no—
165Q8—141no—
166Q8—141no—
167Q8—212no—
168Q8—421no—
169C2×C4—21no—
170C2×C4—61no—
171C2×C4—71no—
172C2×C4—71no—
173C2×C4—141no—
174C2×C4—421no—
175C2×C4C2×D3×D711yescharacteristic
176C8—61no—
177C8—421no—
178C7#509011yescharacteristic, complemented, Sylow
179C2×C3#644711yescharacteristic
180C2×C3#647811yescharacteristic
181C2×C3D7×Q1611yescharacteristic
182C22D7×#6711yescharacteristic
183C4—41no—
184C4—72no—
185C4—121no—
186C4—141no—
187C4—281no—
188C4—422no—
189C4#6668311yescharacteristic
190C4#6669411yescharacteristic
191C3#8956511yescharacteristic, complemented, Sylow
192C2D7×#21611yescharacteristic
193C2#1062107511yescharacteristic
194C2#1062109511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socle, prime radicalC22×C3×C7C2×D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#462
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
72. C3×Dic7Q16semidirect
121. Dic7#216semidirect
137. C3×C7#462semidirect
178. C7#5090semidirect
191. C3#89565semidirect

Maximal subgroups

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

subgroupindex
C2×#666352
C3×#65072
#106208952
C2×#665702
C7×#8352
Dic7⋊(C3⋊C8)2
#106205052
#895653
#50907