Finite Group Databasebeta

← #13056428

Subgroups — #13056428

Order 1344.

248 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 63 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#10621430C211yesmaximal, characteristic, complemented
2C2×#66634C211yesmaximal, characteristic, complemented
3C2×D168C211yesmaximal, characteristic
4#10621376C211yesmaximal, characteristic, complemented
5C3×C7×#103C211yesmaximal, characteristic, complemented
6#10620653C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic, complemented
8#89591—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×D84C2211yescharacteristic
12C2×C4×D21—21no—
13C2×C3×C7×C8C2211yescharacteristic
14C3×C7×C4⋊C4C2211yescharacteristic
15#66634—22no—
16#66631—21no—
17#66631—21no—
18#66631—21no—
19C4×Dic21C2211yescharacteristic
20C2×C3⋊(C7⋊C8)C2211yescharacteristic
21D168—22no—
22#6510—31no—
23C2×#1143—31no—
24C2×D56—31no—
25#6456—31no—
26C7×#103D311yescharacteristic, complemented
27Dic7⋊C8—31no—
28#6383—31no—
29#4819—71nomaximal, Hall
30C22×D21—21no—
31C22×D21—21no—
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21D411yescharacteristic
38C2×Dic21D411yescharacteristic
39D84—21no—
40D84—21no—
41D84—21no—
42D84D412yes—
43C4×D21—42no—
44C3×C7×C8—21no—
45C3⋊(C7⋊C8)—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28—31no—
48C2×D28—31no—
49C2×C4×D7—61no—
50C2×C7×C8C2×D311yescharacteristic
51C7×C4⋊C4C2×D311yescharacteristic
52#1143—62no—
53(C2×D7)⋊C4—61no—
54(C2×D7)⋊C4—61no—
55(C2×D7)⋊C4—61no—
56C4×Dic7—31no—
57C2×C7⋊C8—31no—
58D56—62no—
59#947—71no—
60C2×#215—71no—
61C2×D24—71no—
62#893—71no—
63C3×#103D711yescharacteristic, complemented
64Dic3⋊C8—71no—
65#809—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—22no—
68C2×D21—41no—
69C2×D21—41no—
70C2×D21—41no—
71C2×D21—41no—
72C2×D21—41no—
73C3×C4×C7—41no—
74C3×C4×C7—41no—
75C3×C4×C7C2×D411yescharacteristic
76C3×C4×C7#4111yescharacteristic
77Dic21—21no—
78Dic21—21no—
79#463—211noSylow
80C22×D7—61no—
81C22×D7—61no—
82C7×Q8—21no—
83C7×Q8—21no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—31no—
88C2×Dic7—31no—
89D28—32no—
90D28—61no—
91D28—61no—
92D28—61no—
93C4×D7—122no—
94C7×C8—21no—
95C7⋊C8—61no—
96C2×C3×Q8C2×D711yescharacteristic
97C2×D12—71no—
98C2×D12—71no—
99C2×C4×D3—141no—
100C2×C3×C8C2×D711yescharacteristic
101C3×C4⋊C4C2×D711yescharacteristic
102#215—142no—
103(C2×D3)⋊C4—141no—
104(C2×D3)⋊C4—141no—
105(C2×D3)⋊C4—141no—
106C4×Dic3—71no—
107C2×C3⋊C8—71no—
108D24—142no—
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7#13511yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112D21—42no—
113D21—81no—
114C2×SD16—211no—
115C2×D8—211no—
116#124—211no—
117C4×D4—211no—
118C4⋊C8—211no—
119#103D2111yescharacteristic, complemented
120#102—211no—
121C22×C7D3×D411yescharacteristic
122C2×D7—62no—
123C2×D7—121no—
124C2×D7—121no—
125C2×D7—121no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—41no—
129C4×C7—41no—
130C4×C7#23511yescharacteristic
131C4×C7D3×D411yescharacteristic
132Dic7—61no—
133Dic7—61no—
134C22×D3—141no—
135C22×D3—141no—
136C3×Q8—21no—
137C3×Q8—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4C22×D711yescharacteristic
141C2×Dic3—71no—
142C2×Dic3—71no—
143D12—72no—
144D12—141no—
145D12—141no—
146D12—141no—
147C4×D3—282no—
148C3×C8—21no—
149C3⋊C8—141no—
150C3×C7#46311yescharacteristic, complemented, Hall
151C2×Q8C2×D2111yescharacteristic
152C2×D4—211no—
153C2×D4—211no—
154C22×C4—421no—
155SD16—422no—
156D8—422no—
157C2×C8—211no—
158C2×C8C2×D2111yescharacteristic
159C4⋊C4C2×D2111yescharacteristic
160C22⋊C4—421no—
161C22⋊C4—421no—
162C22⋊C4—421no—
163C42—211no—
164C2×C7#88411yescharacteristic
165C2×C7#91411yescharacteristic
166C2×C7#91911yescharacteristic
167D7—122no—
168D7—241no—
169C22×C3D4×D711yescharacteristic
170C2×D3—142no—
171C2×D3—281no—
172C2×D3—281no—
173C2×D3—281no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4#115711yescharacteristic
179C3×C4D4×D711yescharacteristic
180Dic3—141no—
181Dic3—141no—
182C23—421no—
183C23—421no—
184Q8—21no—
185Q8—21no—
186D4—212no—
187D4—421no—
188D4—421no—
189D4—421no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—211no—
193C2×C4—211no—
194C2×C4—842no—
195C2×C4C22×D2111yescharacteristic
196C8—21no—
197C8—421no—
198C7#481911yescharacteristic, complemented, Sylow
199C2×C3#644711yescharacteristic
200C2×C3#647711yescharacteristic
201C2×C3#648211yescharacteristic
202D3—282no—
203D3—561no—
204C22—422no—
205C22—841no—
206C22—841no—
207C22—841no—
208C22—841no—
209C22—841no—
210C22D4×D2111yescharacteristic
211C4—41no—
212C4—41no—
213C4—421no—
214C4—421no—
215C4#6672811yescharacteristic
216C4D4×D2111yescharacteristic
217C3#8959111yescharacteristic, complemented, Sylow
218C2—842no—
219C2—1681no—
220C2#1062136711yescharacteristic
221C2#1062139711yescharacteristic
222C2#1062140211yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×D168C2
hypercenter#103D21
hypocenterC3×C7#463
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. #10621430C2semidirect
2. C2×#66634C2semidirect
4. #10621376C2semidirect
5. C3×C7×#103C2semidirect
6. #10620653C2semidirect
7. #10620648C2semidirect
26. C7×#103D3semidirect
63. C3×#103D7semidirect
119. #103D21semidirect
150. C3×C7#463semidirect
198. C7#4819semidirect
217. C3#89591semidirect

Maximal subgroups

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

subgroupindex
#106214302
C2×#666342
C2×D1682
#106213762
C3×C7×#1032
#106206532
#106206482
#895913
#48197