Finite Group Databasebeta

← #13054510

Subgroups — #13054510

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#10621419C211yesmaximal, characteristic, complemented
2C2×C3×#1141C211yesmaximal, characteristic
3#10620996C211yesmaximal, characteristic, complemented
4C2×#66565C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620518C211yesmaximal, characteristic, complemented
7#10620510C211yesmaximal, characteristic, complemented
8#89559—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×D28C2211yescharacteristic
12C2×Dic3×D7—21no—
13#66636—21no—
14#66636—21no—
15C4×Dic21C2211yescharacteristic
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×#1141—22no—
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D7)⋊Dic3—21no—
21#66565—22no—
22#6499—31no—
23C2×#1141D311yescharacteristic
24C2×#1132—31no—
25#6456—31no—
26C7×#102—31no—
27Dic7⋊C8—31no—
28#6384—31no—
29#5153—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×C3×D7—21no—
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38Dic42—21no—
39Dic42—21no—
40C2×C7×Dic3—21no—
41C3×D28—21no—
42C3×D28D412yes—
43Dic3×D7—42no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C2×C7×D4C2×D311yescharacteristic
47C2×D28C2×D311yescharacteristic
48C2×C4×D7—61no—
49C2×Dic14—31no—
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52C22⋊Dic7—61no—
53C22⋊Dic7—61no—
54#1141—22no—
55(C2×D7)⋊C4—61no—
56C4×Dic7—31no—
57C2×C7⋊C8C2×D311yescharacteristic
58#1132—62no—
59C2×C3×D8—71no—
60#936—71no—
61D4×Dic3—71no—
62C2×#214—71no—
63#832D711yescharacteristic, complemented
64#816—71no—
65Dic3⋊C8—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×C3×D7—22no—
71C2×C3×D7—41no—
72C2×C3×D7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7#4111yescharacteristic
75Dic21—21no—
76Dic21—21no—
77Dic21—41no—
78C7×Dic3—41no—
79#463—211noSylow
80C23×C7—21no—
81C22×D7—21no—
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7—61no—
85C2×C4×C7C22×D311yescharacteristic
86C2×Dic7—31no—
87C2×Dic7—31no—
88C2×Dic7—61no—
89D28—21no—
90D28#6712yes—
91C4×D7—122no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—61no—
95C7⋊C8—21no—
96C2×C3×D4—71no—
97C2×C3×D4C2×D711yescharacteristic
98C22×Dic3—141no—
99C2×Dic6—71no—
100C3×D8—142no—
101C2×C3×C8—71no—
102#217—141no—
103#217—141no—
104#217—141no—
105#214—142no—
106C4⋊Dic3C2×D711yescharacteristic
107C4×Dic3—71no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7#13511yescharacteristic
112C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
113C3×D7—42no—
114C2×SD16—211no—
115C2×D8—71no—
116#124—211no—
117C4×D4—211no—
118C4⋊C8—211no—
119#103—211no—
120#102—31no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7D3×D411yescharacteristic
125C2×D7—22no—
126C2×D7—41no—
127C2×D7—41no—
128C4×C7—121no—
129C4×C7#23711yescharacteristic
130C4×C7C2×#6711yescharacteristic
131Dic7—61no—
132Dic7—61no—
133Dic7—121no—
134C23×C3—21no—
135C23×C3—141no—
136C3×D4—21no—
137C3×D4—21no—
138C3×D4—72no—
139C3×D4—141no—
140C2×C3×C4C22×D711yescharacteristic
141C2×Dic3—21no—
142C2×Dic3—71no—
143C2×Dic3—71no—
144C2×Dic3—141no—
145C2×Dic3—282no—
146Dic6—141no—
147Dic6—141no—
148C3×C8—141no—
149C3⋊C8—21no—
150C3×C7#46311yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—71no—
153C2×D4D3×D711yescharacteristic
154C22×C4—421no—
155SD16—422no—
156D8—142no—
157C2×C8—31no—
158C2×C8—71no—
159C4⋊C4—31no—
160C22⋊C4—421no—
161C22⋊C4—421no—
162C22⋊C4—421no—
163C42—211no—
164C2×C7—81no—
165C2×C7#91111yescharacteristic
166C2×C7#91211yescharacteristic
167C2×C7#93911yescharacteristic
168D7—42no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3—142no—
173C22×C3—281no—
174C22×C3—281no—
175C22×C3D4×D711yescharacteristic
176C3×C4#115711yescharacteristic
177C3×C4D4×D711yescharacteristic
178Dic3—41no—
179Dic3—141no—
180Dic3—141no—
181Dic3—281no—
182C23—21no—
183C23—141no—
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—72no—
189D4—141no—
190C2×C4—61no—
191C2×C4—211no—
192C2×C4—211no—
193C2×C4—421no—
194C2×C4—842no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—141no—
198C7#515311yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3—282no—
201C2×C3#644711yescharacteristic
202C2×C3#647411yescharacteristic
203C2×C3#647911yescharacteristic
204C22—41no—
205C22—41no—
206C22—41no—
207C22—142no—
208C22—281no—
209C22—281no—
210C22#6669411yescharacteristic
211C4—121no—
212C4—421no—
213C4—421no—
214C4—841no—
215C4#6668311yescharacteristic
216C4D7×#6711yescharacteristic
217C3#8955911yescharacteristic, complemented, Sylow
218C2—81no—
219C2—282no—
220C2#1062104411yescharacteristic
221C2#1062105911yescharacteristic
222C2#1062111211yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C3×#1141C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#463
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #10621419C2semidirect
3. #10620996C2semidirect
4. C2×#66565C2semidirect
5. C7×#832C2semidirect
6. #10620518C2semidirect
7. #10620510C2semidirect
63. #832D7semidirect
150. C3×C7#463semidirect
198. C7#5153semidirect
217. C3#89559semidirect

Maximal subgroups

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

subgroupindex
#106214192
C2×C3×#11412
#106209962
C2×#665652
C7×#8322
#106205182
#106205102
#895593
#51537