Finite Group Databasebeta

← #13054729

Subgroups — #13054729

Order 1344.

248 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 62 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×C3×#1143C211yesmaximal, characteristic, complemented
3#10620996C211yesmaximal, characteristic, complemented
4C2×#66561C211yesmaximal, characteristic
5C7×#835C211yesmaximal, characteristic, complemented
6#10620518C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic, complemented
8#89591—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C2×Dic3×D7—21no—
13#66631—21no—
14#66631—21no—
15C4×Dic21C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1143—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D7)⋊Dic3—21no—
21#66561—22no—
22#6510—31no—
23C2×#1143D311yescharacteristic, complemented
24C2×D56—31no—
25#6456—31no—
26C7×#103—31no—
27Dic7⋊C8—31no—
28#6383—31no—
29#5165—71nomaximal, Hall
30C22×D21—21no—
31C22×C3×D7—21no—
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38D84—21no—
39D84—21no—
40C2×C7×Dic3—21no—
41C3×D28—21no—
42C3×D28D412yes—
43Dic3×D7—42no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×D28—31no—
48C2×D28C2×D311yescharacteristic
49C2×C4×D7—61no—
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52#1143—22no—
53(C2×D7)⋊C4—61no—
54(C2×D7)⋊C4—61no—
55(C2×D7)⋊C4—61no—
56C4×Dic7—31no—
57C2×C7⋊C8C2×D311yescharacteristic
58D56—62no—
59C2×C3×SD16—71no—
60#947—71no—
61D4×Dic3—71no—
62C2×#213—71no—
63#835D711yescharacteristic, complemented
64#821—71no—
65Dic3⋊C8—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—41no—
68C2×D21—41no—
69C2×D21—41no—
70C2×C3×D7—22no—
71C2×C3×D7—41no—
72C2×C3×D7—41no—
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—21no—
77Dic21—21no—
78C7×Dic3—41no—
79#463—211noSylow
80C22×D7—21no—
81C22×D7—61no—
82C7×Q8—21no—
83C7×Q8—21no—
84C2×C4×C7—21no—
85C2×C4×C7—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—31no—
88C2×Dic7—31no—
89D28—21no—
90D28—61no—
91D28—61no—
92D28#6712yes—
93C4×D7—122no—
94C7×C8—61no—
95C7⋊C8—21no—
96C2×C3×Q8C2×D711yescharacteristic
97C2×C3×D4—71no—
98C22×Dic3—141no—
99C2×D12—71no—
100C3×SD16—142no—
101C2×C3×C8—71no—
102#217—141no—
103#213—142no—
104(C2×D3)⋊C4—141no—
105(C2×D3)⋊C4—141no—
106C4⋊Dic3C2×D711yescharacteristic
107C4×Dic3—71no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7#13511yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112D21—81no—
113C3×D7—42no—
114C2×SD16—71no—
115C2×D8—211no—
116#124—211no—
117C4×D4—211no—
118C4⋊C8—211no—
119#103—31no—
120#102—211no—
121C22×C7D3×D411yescharacteristic
122C2×D7—22no—
123C2×D7—41no—
124C2×D7—41no—
125C2×D7—121no—
126C2×D7—121no—
127C2×D7—121no—
128C4×C7—41no—
129C4×C7—121no—
130C4×C7#23711yescharacteristic
131C4×C7C2×#6711yescharacteristic
132Dic7—61no—
133Dic7—61no—
134C23×C3—141no—
135C22×D3—141no—
136C3×Q8—21no—
137C3×Q8—21no—
138C3×D4—72no—
139C3×D4—141no—
140C2×C3×C4—21no—
141C2×C3×C4C22×D711yescharacteristic
142C2×Dic3—21no—
143C2×Dic3—71no—
144C2×Dic3—71no—
145C2×Dic3—282no—
146D12—141no—
147D12—141no—
148C3×C8—141no—
149C3⋊C8—21no—
150C3×C7#46311yescharacteristic, complemented, Hall
151C2×Q8D3×D711yescharacteristic
152C2×D4—71no—
153C2×D4—211no—
154C22×C4—421no—
155SD16—142no—
156D8—422no—
157C2×C8—31no—
158C2×C8—71no—
159C4⋊C4—31no—
160C22⋊C4—421no—
161C22⋊C4—421no—
162C22⋊C4—421no—
163C42—211no—
164C2×C7#91411yescharacteristic
165C2×C7#91611yescharacteristic
166C2×C7#93911yescharacteristic
167D7—42no—
168D7—241no—
169C22×C3—142no—
170C22×C3—281no—
171C22×C3—281no—
172C22×C3D4×D711yescharacteristic
173C2×D3—281no—
174C2×D3—281no—
175C2×D3—281no—
176C3×C4—41no—
177C3×C4#115711yescharacteristic
178C3×C4D4×D711yescharacteristic
179Dic3—41no—
180Dic3—141no—
181Dic3—141no—
182C23—141no—
183C23—421no—
184Q8—21no—
185Q8—21no—
186D4—72no—
187D4—141no—
188D4—421no—
189D4—421no—
190C2×C4—21no—
191C2×C4—61no—
192C2×C4—211no—
193C2×C4—211no—
194C2×C4—842no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—141no—
198C7#516511yescharacteristic, complemented, Sylow
199C2×C3—282no—
200C2×C3#644711yescharacteristic
201C2×C3#647711yescharacteristic
202C2×C3#648211yescharacteristic
203D3—561no—
204C22—142no—
205C22—281no—
206C22—281no—
207C22—841no—
208C22—841no—
209C22—841no—
210C22#6669411yescharacteristic
211C4—41no—
212C4—121no—
213C4—421no—
214C4—421no—
215C4#6668311yescharacteristic
216C4D7×#6711yescharacteristic
217C3#8959111yescharacteristic, complemented, Sylow
218C2—282no—
219C2—1681no—
220C2#1062106811yescharacteristic
221C2#1062108311yescharacteristic
222C2#1062111211yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC2×#66561C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#463
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #10621430C2semidirect
2. C2×C3×#1143C2semidirect
3. #10620996C2semidirect
5. C7×#835C2semidirect
6. #10620518C2semidirect
7. #10620504C2semidirect
23. C2×#1143D3semidirect
63. #835D7semidirect
150. C3×C7#463semidirect
198. C7#5165semidirect
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×C3×#11432
#106209962
C2×#665612
C7×#8352
#106205182
#106205042
#895913
#51657