Finite Group Databasebeta

← #13052619

Subgroups — #13052619

Order 1344.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C4⋊C4×D21C211yesmaximal, characteristic, complemented
2#10621027C211yesmaximal, characteristic
3C7×#808C211yesmaximal, characteristic, complemented
4C3×#6383C211yesmaximal, characteristic, complemented
5#10620520C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89635—31nomaximal, Hall
9C2×C4×D21—21no—
10C2×C4×D21C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C2×#4246—21no—
13C3×C7×C4⋊C4C2211yescharacteristic
14C4⋊Dic21C2211yescharacteristic
15#66629—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D7)⋊Dic3—21no—
20#6458—31no—
21D7×C4⋊C4—31no—
22C7×#106—31no—
23#6395—31no—
24(C2×D7)⋊C8—31no—
25#6383D311yescharacteristic, complemented
26#6382—31no—
27#4777—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—
36C4×D21—42no—
37C2×C7×Dic3—21no—
38C3×D28—21no—
39C3×D28—21no—
40#4246—42no—
41C3×C7⋊C8—21no—
42C7×C3⋊C8—21no—
43C2×D28—61no—
44C2×D28C2×D311yescharacteristic
45C2×C4×D7—31no—
46C2×C4×D7—61no—
47C2×C7×C8—31no—
48C7×C4⋊C4—31no—
49C7×C4⋊C4C2×D311yescharacteristic
50(C2×D7)⋊C4—61no—
51C4⋊Dic7—31no—
52Dic7⋊C4—61no—
53C2×C7⋊C8C2×D311yescharacteristic
54#938—71no—
55D3×C4⋊C4—71no—
56C3×#102—71no—
57#832—71no—
58(C2×D3)⋊C8—71no—
59#817—71no—
60#808D711yescharacteristic, complemented
61C22×C3×C7C2×D411yescharacteristic
62C2×D21—21no—
63C2×D21—21no—
64C2×D21SD1612yes—
65C2×C3×D7—41no—
66C2×C3×D7—41no—
67C2×C3×D7—41no—
68C3×C4×C7—41no—
69C3×C4×C7#4111yescharacteristic
70C3×C4×C7#4111yescharacteristic
71Dic21—21no—
72Dic21—41no—
73C7×Dic3—41no—
74#481—211noSylow
75C22×D7—21no—
76C22×D7—31no—
77C2×C4×C7—21no—
78C2×C4×C7—61no—
79C2×C4×C7C22×D311yescharacteristic
80C2×Dic7—31no—
81C2×Dic7—61no—
82D28—21no—
83D28—21no—
84D28—122no—
85C4×D7—61no—
86C4×D7—61no—
87C4×D7—122no—
88C7×C8—61no—
89C7⋊C8—21no—
90C2×C3×D4—71no—
91C2×#67—141no—
92C2×C4×D3—71no—
93C2×C4×D3—141no—
94C2×C3×C8—71no—
95C3×C4⋊C4C2×D711yescharacteristic
96#217—141no—
97C4⋊Dic3—71no—
98C4⋊Dic3C2×D711yescharacteristic
99Dic3⋊C4—141no—
100C2×C3⋊C8C2×D711yescharacteristic
101C2×C3×C7#12311yescharacteristic
102C2×C3×C7C2×SD1611yescharacteristic
103C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
104D21—22no—
105C3×D7—81no—
106#121—211no—
107C2×C4⋊C4—211no—
108#106—31no—
109#106—211no—
110#102—71no—
111#102—211no—
112C22⋊C8—211no—
113C22×C7D3×D411yescharacteristic
114C2×D7—32no—
115C2×D7—41no—
116C2×D7—41no—
117C2×D7—41no—
118C2×D7—61no—
119C2×D7—61no—
120C4×C7—41no—
121C4×C7—121no—
122C4×C7#23511yescharacteristic
123C4×C7#23711yescharacteristic
124Dic7—61no—
125Dic7—121no—
126C23×C3—141no—
127C22×D3—71no—
128C3×D4—141no—
129C3×D4—141no—
130C2×C3×C4—21no—
131C2×C3×C4C22×D711yescharacteristic
132#67—282no—
133C2×Dic3—21no—
134C2×Dic3—71no—
135C2×Dic3—141no—
136C4×D3—141no—
137C4×D3—141no—
138C4×D3—282no—
139C3×C8—141no—
140C3⋊C8—21no—
141C3×C7#48111yescharacteristic, complemented, Hall
142C2×D4—71no—
143C2×D4—421no—
144C22×C4—211no—
145C22×C4—421no—
146C2×C8—31no—
147C2×C8—71no—
148C4⋊C4—31no—
149C4⋊C4—211no—
150C4⋊C4—421no—
151C4⋊C4D3×D711yescharacteristic
152C22⋊C4—421no—
153C2×C7#88311yescharacteristic
154C2×C7#91111yescharacteristic
155C2×C7D3×SD1611yescharacteristic
156D7—62no—
157D7—81no—
158C22×C3—281no—
159C22×C3—281no—
160C22×C3—281no—
161C22×C3D4×D711yescharacteristic
162C2×D3—72no—
163C2×D3—141no—
164C2×D3—141no—
165C3×C4—41no—
166C3×C4#115711yescharacteristic
167C3×C4#116111yescharacteristic
168Dic3—41no—
169Dic3—141no—
170Dic3—281no—
171C23—141no—
172C23—211no—
173D4—141no—
174D4—141no—
175D4—842no—
176C2×C4—21no—
177C2×C4—61no—
178C2×C4—211no—
179C2×C4—421no—
180C2×C4—421no—
181C2×C4—421no—
182C2×C4—842no—
183C2×C4C2×D3×D711yescharacteristic
184C8—61no—
185C8—141no—
186C7#477711yescharacteristic, complemented, Sylow
187C2×C3—561no—
188C2×C3#645711yescharacteristic
189C2×C3D7×SD1611yescharacteristic
190C2×C3#647711yescharacteristic
191D3—142no—
192C22—212no—
193C22—281no—
194C22—281no—
195C22—281no—
196C22—421no—
197C22—421no—
198C22#6669411yescharacteristic
199C4—41no—
200C4—121no—
201C4—421no—
202C4—841no—
203C4#6667011yescharacteristic
204C4#6667511yescharacteristic
205C3#8963511yescharacteristic, complemented, Sylow
206C2—422no—
207C2—561no—
208C2#1062091611yescharacteristic
209C2#1062106011yescharacteristic
210C2#1062106711yescharacteristic
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#481
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. C4⋊C4×D21C2semidirect
3. C7×#808C2semidirect
4. C3×#6383C2semidirect
5. #10620520C2semidirect
6. #10620500C2semidirect
7. #10620492C2semidirect
25. #6383D3semidirect
60. #808D7semidirect
141. C3×C7#481semidirect
186. C7#4777semidirect
205. C3#89635semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D212
#106210272
C7×#8082
C3×#63832
#106205202
#106205002
#106204922
#896353
#47777