Finite Group Databasebeta

← #13052589

Subgroups — #13052589

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×D7×C4⋊C4C211yesmaximal, characteristic, complemented
2#10621011C211yesmaximal, characteristic
3#10620646C211yesmaximal, characteristic, complemented
4C7×#809C211yesmaximal, characteristic, complemented
5#10620525C211yesmaximal, characteristic, complemented
6#10620501C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#89546—31nomaximal, Hall
9C2×C7×D12C2211yescharacteristic
10C2×C3×C4×D7—21no—
11C2×C3×C4×D7C2211yescharacteristic
12C2×#4245—21no—
13C3×C7×C4⋊C4C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15C2×C3⋊(C7⋊C8)C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C3×Dic7⋊C4—21no—
19(C2×D3)⋊Dic7—21no—
20#6501—31no—
21D7×C4⋊C4D311yescharacteristic, complemented
22C7×#102—31no—
23#6406—31no—
24(C2×D7)⋊C8—31no—
25#6392—31no—
26#6381—31no—
27#4967—71nomaximal, Hall
28C22×D3×C7—21no—
29C22×C3×D7D411yescharacteristic
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C7×D12—21no—
34C7×D12—21no—
35C2×C3×Dic7—21no—
36C2×C3×Dic7D411yescharacteristic
37C3×C4×D7—21no—
38C3×C4×D7—21no—
39C3×C4×D7—42no—
40#4245—42no—
41C3⋊(C7⋊C8)—21no—
42C7×C3⋊C8—21no—
43C2×C7×D4—31no—
44C2×C22⋊D7—61no—
45C2×C4×D7—21no—
46C2×C4×D7C2×D311yescharacteristic
47C2×C7×C8—31no—
48C7×C4⋊C4C2×D311yescharacteristic
49C22⋊Dic7—61no—
50C4⋊Dic7—31no—
51C4⋊Dic7C2×D311yescharacteristic
52Dic7⋊C4—21no—
53C2×C7⋊C8—31no—
54C2×C3×C4⋊C4—71no—
55#930—71no—
56#830—71no—
57#809—71no—
58#809D711yescharacteristic, complemented
59#807—71no—
60#807—71no—
61C22×C3×C7C2×D411yescharacteristic
62C2×D3×C7—41no—
63C2×D3×C7—41no—
64C2×D3×C7—41no—
65C2×C3×D7—21no—
66C2×C3×D7—21no—
67C2×C3×D7D812yes—
68C3×C4×C7—41no—
69C3×C4×C7#4111yescharacteristic
70C3×C4×C7#4111yescharacteristic
71Dic21—41no—
72C3×Dic7—21no—
73C3×Dic7—41no—
74#480—211noSylow
75C23×C7—61no—
76C22×D7#6711yescharacteristic
77C7×D4—61no—
78C7×D4—61no—
79C2×C4×C7—21no—
80C2×C4×C7C22×D311yescharacteristic
81C22⋊D7—122no—
82C2×Dic7—21no—
83C2×Dic7—61no—
84C2×Dic7#6711yescharacteristic
85C4×D7—21no—
86C4×D7—21no—
87C4×D7—42no—
88C7×C8—61no—
89C7⋊C8—61no—
90C22×C3×C4—71no—
91C22×C3×C4—141no—
92C2×#67—141no—
93C2×D12C2×D711yescharacteristic
94C3×C4⋊C4—71no—
95C3×C4⋊C4—141no—
96C3×C4⋊C4C2×D711yescharacteristic
97(C2×D3)⋊C4—141no—
98C4⋊Dic3—71no—
99C2×C3⋊C8—71no—
100C2×C3⋊C8C2×D711yescharacteristic
101C2×C3×C7#12311yescharacteristic
102C2×C3×C7C2×D811yescharacteristic
103C2×C3×C7#13711yescharacteristic
104C3×D7—22no—
105D3×C7—81no—
106#121—211no—
107C2×C4⋊C4—71no—
108#107—211no—
109#107—211no—
110#102—31no—
111#102—211no—
112C22⋊C8—211no—
113C22×C7—121no—
114C22×C7—121no—
115C22×C7—121no—
116C22×C7C2×#6711yescharacteristic
117C2×D7—21no—
118C2×D7—21no—
119C2×D7#21312yes—
120C4×C7—41no—
121C4×C7#23511yescharacteristic
122C4×C7#23511yescharacteristic
123Dic7—21no—
124Dic7—41no—
125Dic7—121no—
126C23×C3—71no—
127C22×D3—21no—
128C2×C3×C4—21no—
129C2×C3×C4—71no—
130C2×C3×C4—141no—
131C2×C3×C4—141no—
132C2×C3×C4—141no—
133C2×C3×C4—282no—
134C2×C3×C4C22×D711yescharacteristic
135#67—282no—
136C2×Dic3—141no—
137D12—21no—
138D12—21no—
139C3⋊C8—21no—
140C3⋊C8—141no—
141C3×C7#48011yescharacteristic, complemented, Hall
142C2×D4—31no—
143C2×D4—421no—
144C22×C4—71no—
145C22×C4—141no—
146C2×C8—31no—
147C2×C8—211no—
148C4⋊C4—71no—
149C4⋊C4—141no—
150C4⋊C4—211no—
151C4⋊C4D3×D711yescharacteristic
152C22⋊C4—421no—
153C2×C7—241no—
154C2×C7#92911yescharacteristic
155C2×C7C2×#21311yescharacteristic
156C2×C7#94211yescharacteristic
157D7—22no—
158C22×C3—72no—
159C22×C3—141no—
160C22×C3—141no—
161C22×C3D4×D711yescharacteristic
162C2×D3—41no—
163C2×D3—41no—
164C2×D3—41no—
165C3×C4—41no—
166C3×C4—141no—
167C3×C4—281no—
168C3×C4#115711yescharacteristic
169C3×C4#115911yescharacteristic
170Dic3—281no—
171C23—61no—
172C23—71no—
173D4—61no—
174D4—61no—
175D4—842no—
176C2×C4—21no—
177C2×C4—71no—
178C2×C4—141no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—282no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—421no—
186C7#496711yescharacteristic, complemented, Sylow
187C2×C3—142no—
188C2×C3#644611yescharacteristic
189C2×C3D7×D811yescharacteristic
190C2×C3#647811yescharacteristic
191D3—81no—
192C22—72no—
193C22—121no—
194C22—121no—
195C22—121no—
196C22—141no—
197C22—141no—
198C22D7×#6711yescharacteristic
199C4—41no—
200C4—141no—
201C4—281no—
202C4—841no—
203C4#6667511yescharacteristic
204C4#6667611yescharacteristic
205C3#8954611yescharacteristic, complemented, Sylow
206C2—142no—
207C2—241no—
208C2#1062100911yescharacteristic
209C2D7×#21311yescharacteristic
210C2#1062107511yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radical#10621011C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#480
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×D7×C4⋊C4C2semidirect
3. #10620646C2semidirect
4. C7×#809C2semidirect
5. #10620525C2semidirect
6. #10620501C2semidirect
7. (C2×D7)⋊(C3⋊C8)C2semidirect
21. D7×C4⋊C4D3semidirect
58. #809D7semidirect
141. C3×C7#480semidirect
186. C7#4967semidirect
205. C3#89546semidirect

Maximal subgroups

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

subgroupindex
C3×D7×C4⋊C42
#106210112
#106206462
C7×#8092
#106205252
#106205012
(C2×D7)⋊(C3⋊C8)2
#895463
#49677