Finite Group Databasebeta

← #13054326

Subgroups — #13054326

Order 1344.

228 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 64 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×#6501C211yesmaximal, characteristic
2D7×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#10620671C211yesmaximal, characteristic, complemented
4C7×#832C211yesmaximal, characteristic, complemented
5#10620522C211yesmaximal, characteristic, complemented
6#10620521C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89549—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×C3×C22⋊D7—21no—
11C2×C3×C4×D7C2211yescharacteristic
12C2×Dic3×D7—21no—
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C22⋊Dic7—21no—
18C3×C4⋊Dic7C2211yescharacteristic
19Dic7⋊Dic3—21no—
20#6501D311yescharacteristic
21D7×C4⋊C4—31no—
22C7×#102—31no—
23#6406—31no—
24(C2×D7)⋊C8—31no—
25#6391—31no—
26#6382—31no—
27#5034—71nomaximal, Hall
28C23×C3×C7—21no—
29C22×C3×D7D411yescharacteristic
30C3×C7×D4—21no—
31C3×C7×D4—21no—
32C2×C3×C4×C7C2311yescharacteristic
33C2×Dic21—21no—
34C2×C7×Dic3—21no—
35C3×C22⋊D7—42no—
36C2×C3×Dic7—21no—
37C2×C3×Dic7D411yescharacteristic
38C3×C4×D7—21no—
39C3×C4×D7—21no—
40Dic3×D7—42no—
41C3⋊(C7⋊C8)—21no—
42C7×C3⋊C8—21no—
43C2×C7×D4C2×D311yescharacteristic
44C2×C22⋊D7—21no—
45C2×C4×D7—61no—
46C2×C4×D7C2×D311yescharacteristic
47C2×C7×C8—31no—
48C7×C4⋊C4—31no—
49C22⋊Dic7—21no—
50C4⋊Dic7—31no—
51C4⋊Dic7C2×D311yescharacteristic
52Dic7⋊C4—61no—
53C2×C7⋊C8—31no—
54C3×#121—71no—
55C2×C4⋊Dic3—71no—
56#832—71no—
57#832D711yescharacteristic, complemented
58#830—71no—
59#808—71no—
60#808—71no—
61C22×C3×C7—41no—
62C22×C3×C7—41no—
63C22×C3×C7—41no—
64C22×C3×C7C2×D411yescharacteristic
65C2×C3×D7—21no—
66C2×C3×D7—21no—
67C2×C3×D7SD1612yes—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—41no—
71C3×Dic7—21no—
72C3×Dic7—41no—
73C7×Dic3—41no—
74#481—211noSylow
75C23×C7—21no—
76C22×D7#6711yescharacteristic
77C7×D4—21no—
78C7×D4—21no—
79C2×C4×C7—61no—
80C2×C4×C7C22×D311yescharacteristic
81C22⋊D7—42no—
82C2×Dic7—21no—
83C2×Dic7—61no—
84C2×Dic7#6711yescharacteristic
85C4×D7—21no—
86C4×D7—21no—
87C4×D7—122no—
88C7×C8—61no—
89C7⋊C8—61no—
90C2×C3×D4—141no—
91C2×C3×D4C2×D711yescharacteristic
92C22×C3×C4—71no—
93C22×Dic3—141no—
94C3×C4⋊C4—71no—
95C3×C22⋊C4—141no—
96C4⋊Dic3—71no—
97C4⋊Dic3—141no—
98C4⋊Dic3C2×D711yescharacteristic
99C2×C3⋊C8—71no—
100C2×C3⋊C8C2×D711yescharacteristic
101C2×C3×C7—81no—
102C2×C3×C7#12311yescharacteristic
103C2×C3×C7C2×SD1611yescharacteristic
104C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
105C3×D7—22no—
106#121—71no—
107C2×C4⋊C4—211no—
108#106—211no—
109#106—211no—
110#102—31no—
111#102—211no—
112C22⋊C8—211no—
113C22×C7—41no—
114C22×C7—41no—
115C22×C7—41no—
116C22×C7C2×#6711yescharacteristic
117C2×D7—21no—
118C2×D7—21no—
119C2×D7#21412yes—
120C4×C7—121no—
121C4×C7#23711yescharacteristic
122C4×C7#23711yescharacteristic
123Dic7—21no—
124Dic7—41no—
125Dic7—121no—
126C23×C3—21no—
127C23×C3—71no—
128C3×D4—21no—
129C3×D4—21no—
130C3×D4—282no—
131C2×C3×C4—71no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4—141no—
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—141no—
138C2×Dic3—282no—
139C3⋊C8—21no—
140C3⋊C8—141no—
141C3×C7#48111yescharacteristic, complemented, Hall
142C2×D4—141no—
143C2×D4D3×D711yescharacteristic
144C22×C4—71no—
145C22×C4—421no—
146C2×C8—31no—
147C2×C8—211no—
148C4⋊C4—31no—
149C4⋊C4—71no—
150C4⋊C4—211no—
151C4⋊C4—421no—
152C22⋊C4—141no—
153C2×C7—81no—
154C2×C7C2×#21411yescharacteristic
155C2×C7#93511yescharacteristic
156C2×C7#94911yescharacteristic
157D7—22no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3—72no—
162C22×C3—141no—
163C22×C3—141no—
164C22×C3D4×D711yescharacteristic
165C3×C4—141no—
166C3×C4—281no—
167C3×C4#115711yescharacteristic
168C3×C4#115911yescharacteristic
169Dic3—41no—
170Dic3—281no—
171C23—21no—
172C23—71no—
173D4—21no—
174D4—21no—
175D4—282no—
176C2×C4—61no—
177C2×C4—71no—
178C2×C4—141no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—421no—
182C2×C4—842no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—421no—
186C7#503411yescharacteristic, complemented, Sylow
187C2×C3—81no—
188C2×C3—142no—
189C2×C3#644611yescharacteristic
190C2×C3#647411yescharacteristic
191C2×C3D7×SD1611yescharacteristic
192C22—41no—
193C22—41no—
194C22—41no—
195C22—72no—
196C22—141no—
197C22—141no—
198C22D7×#6711yescharacteristic
199C4—121no—
200C4—141no—
201C4—281no—
202C4—841no—
203C4#6668311yescharacteristic
204C4#6668411yescharacteristic
205C3#8954911yescharacteristic, complemented, Sylow
206C2—81no—
207C2—142no—
208C2#1062104011yescharacteristic
209C2D7×#21411yescharacteristic
210C2#1062111011yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6501C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#481
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D7×C4⋊Dic3C2semidirect
3. #10620671C2semidirect
4. C7×#832C2semidirect
5. #10620522C2semidirect
6. #10620521C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
57. #832D7semidirect
141. C3×C7#481semidirect
186. C7#5034semidirect
205. C3#89549semidirect

Maximal subgroups

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

subgroupindex
C3×#65012
D7×C4⋊Dic32
#106206712
C7×#8322
#106205222
#106205212
(C2×D7)⋊(C3⋊C8)2
#895493
#50347