Finite Group Databasebeta

← #13052623

Subgroups — #13052623

Order 1344.

212 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#10621375C211yesmaximal, characteristic, complemented
2#10621027C211yesmaximal, characteristic
3C7×#808C211yesmaximal, characteristic, complemented
4C3×#6383C211yesmaximal, characteristic, complemented
5#10620523C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89638—31nomaximal, Hall
9C2×C4×D21C2211yescharacteristic
10C2×C3×D28C2211yescharacteristic
11C2×#4246—21no—
12C3×C7×C4⋊C4C2211yescharacteristic
13#66631—21no—
14C4⋊Dic21C2211yescharacteristic
15C4×Dic21—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D7)⋊Dic3—21no—
20#6458—31no—
21#6455—31no—
22C7×#106—31no—
23#6395—31no—
24(C2×D7)⋊C8—31no—
25#6383D311yescharacteristic, complemented
26#6381—31no—
27#4784—71nomaximal, Hall
28C22×D21D411yescharacteristic
29C22×C3×D7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7C2311yescharacteristic
32C2×Dic21—21no—
33C2×Dic21D411yescharacteristic
34C4×D21—21no—
35C4×D21—21no—
36C2×C7×Dic3—21no—
37C3×D28—21no—
38C3×D28—21no—
39#4246—42no—
40C3×C7⋊C8—21no—
41C7×C3⋊C8—21no—
42C2×D28—61no—
43C2×D28C2×D311yescharacteristic
44C2×C4×D7—31no—
45C2×C7×C8—31no—
46C7×C4⋊C4—31no—
47C7×C4⋊C4C2×D311yescharacteristic
48(C2×D7)⋊C4—61no—
49(C2×D7)⋊C4—61no—
50C4⋊Dic7—31no—
51C4×Dic7—61no—
52C2×C7⋊C8C2×D311yescharacteristic
53#938—71no—
54#892—71no—
55C3×#102—71no—
56#832—71no—
57(C2×D3)⋊C8—71no—
58#818—71no—
59#808D711yescharacteristic, complemented
60C22×C3×C7C2×D411yescharacteristic
61C2×D21—21no—
62C2×D21—21no—
63C2×D21—21no—
64C2×C3×D7—41no—
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C3×C4×C7—41no—
68C3×C4×C7#4111yescharacteristic
69C3×C4×C7#4111yescharacteristic
70Dic21—21no—
71Dic21—41no—
72C7×Dic3—41no—
73#482—211noSylow
74C22×D7—21no—
75C22×D7—31no—
76C2×C4×C7—21no—
77C2×C4×C7—61no—
78C2×C4×C7C22×D311yescharacteristic
79C2×Dic7—31no—
80C2×Dic7—61no—
81D28—21no—
82D28—21no—
83D28—122no—
84C4×D7—61no—
85C4×D7—61no—
86C7×C8—61no—
87C7⋊C8—21no—
88C2×C3×D4—71no—
89C2×#67—141no—
90C2×C4×D3—71no—
91C2×C3×C8—71no—
92C3×C4⋊C4C2×D711yescharacteristic
93#217—141no—
94(C2×D3)⋊C4—141no—
95C4⋊Dic3—71no—
96C4⋊Dic3C2×D711yescharacteristic
97C4×Dic3—141no—
98C2×C3⋊C8C2×D711yescharacteristic
99C2×C3×C7#12311yescharacteristic
100C2×C3×C7#13511yescharacteristic
101C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
102D21—41no—
103C3×D7—81no—
104#121—211no—
105#117—211no—
106#107—211no—
107#106—31no—
108#102—71no—
109#102—211no—
110C22⋊C8—211no—
111C22×C7D3×D411yescharacteristic
112C2×D7—41no—
113C2×D7—41no—
114C2×D7—41no—
115C2×D7—61no—
116C2×D7—61no—
117C2×D7—61no—
118C4×C7—41no—
119C4×C7—121no—
120C4×C7#23511yescharacteristic
121C4×C7#23711yescharacteristic
122Dic7—61no—
123Dic7—121no—
124C23×C3—141no—
125C22×D3—71no—
126C3×D4—141no—
127C3×D4—141no—
128C2×C3×C4—21no—
129C2×C3×C4C22×D711yescharacteristic
130#67—282no—
131C2×Dic3—21no—
132C2×Dic3—71no—
133C2×Dic3—141no—
134C4×D3—141no—
135C4×D3—141no—
136C3×C8—141no—
137C3⋊C8—21no—
138C3×C7#48211yescharacteristic, complemented, Hall
139C2×D4—71no—
140C2×D4—421no—
141C22×C4—211no—
142C2×C8—31no—
143C2×C8—71no—
144C4⋊C4—31no—
145C4⋊C4—211no—
146C4⋊C4D3×D711yescharacteristic
147C22⋊C4—421no—
148C22⋊C4—421no—
149C42—421no—
150C2×C7#88311yescharacteristic
151C2×C7#91211yescharacteristic
152C2×C7#91411yescharacteristic
153D7—81no—
154D7—121no—
155C22×C3—281no—
156C22×C3—281no—
157C22×C3—281no—
158C22×C3D4×D711yescharacteristic
159C2×D3—141no—
160C2×D3—141no—
161C2×D3—141no—
162C3×C4—41no—
163C3×C4#115711yescharacteristic
164C3×C4#116111yescharacteristic
165Dic3—41no—
166Dic3—141no—
167Dic3—281no—
168C23—141no—
169C23—211no—
170D4—141no—
171D4—141no—
172D4—842no—
173C2×C4—21no—
174C2×C4—61no—
175C2×C4—211no—
176C2×C4—421no—
177C2×C4—421no—
178C2×C4—421no—
179C2×C4C2×D3×D711yescharacteristic
180C8—61no—
181C8—141no—
182C7#478411yescharacteristic, complemented, Sylow
183C2×C3—561no—
184C2×C3#645711yescharacteristic
185C2×C3#647711yescharacteristic
186C2×C3#647911yescharacteristic
187D3—281no—
188C22—281no—
189C22—281no—
190C22—281no—
191C22—421no—
192C22—421no—
193C22—421no—
194C22#6669411yescharacteristic
195C4—41no—
196C4—121no—
197C4—421no—
198C4—841no—
199C4#6667011yescharacteristic
200C4#6667511yescharacteristic
201C3#8963811yescharacteristic, complemented, Sylow
202C2—561no—
203C2—841no—
204C2#1062091611yescharacteristic
205C2#1062105911yescharacteristic
206C2#1062106811yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
prime radical#10621027C2
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#482
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. #10621375C2semidirect
3. C7×#808C2semidirect
4. C3×#6383C2semidirect
5. #10620523C2semidirect
6. #10620500C2semidirect
7. #10620492C2semidirect
25. #6383D3semidirect
59. #808D7semidirect
138. C3×C7#482semidirect
182. C7#4784semidirect
201. C3#89638semidirect

Maximal subgroups

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

subgroupindex
#106213752
#106210272
C7×#8082
C3×#63832
#106205232
#106205002
#106204922
#896383
#47847