Finite Group Databasebeta

← #13054520

Subgroups — #13054520

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×C7×#213C211yesmaximal, characteristic
3#10620997C211yesmaximal, characteristic, complemented
4C2×#66567C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6#10620518C211yesmaximal, characteristic, complemented
7#10620509C211yesmaximal, characteristic, complemented
8#89925—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×D3×Dic7—21no—
13#66636—21no—
14#66636—21no—
15C4×Dic21C2211yescharacteristic
16C7×#213—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21#66567—22no—
22C2×C7×D8—31no—
23#6499—31no—
24D4×Dic7—31no—
25C2×#1142—31no—
26#6406D311yescharacteristic, complemented
27#6390—31no—
28Dic7⋊C8—31no—
29#4787—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D3×C7—21no—
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21D411yescharacteristic
37C2×Dic21D411yescharacteristic
38Dic42—21no—
39Dic42—21no—
40C7×D12—21no—
41C7×D12D412yes—
42C2×C3×Dic7—21no—
43D3×Dic7—42no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8—21no—
46C2×C7×D4—31no—
47C2×C7×D4C2×D311yescharacteristic
48C22×Dic7—61no—
49C2×Dic14—31no—
50C7×D8—62no—
51C2×C7×C8—31no—
52C22⋊Dic7—61no—
53C22⋊Dic7—61no—
54C22⋊Dic7—61no—
55#1142—62no—
56C4⋊Dic7C2×D311yescharacteristic
57C4×Dic7—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59#936—71no—
60C2×#213D711yescharacteristic
61C2×#204—71no—
62#893—71no—
63C3×#102—71no—
64Dic3⋊C8—71no—
65#810—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D3×C7—22no—
71C2×D3×C7—41no—
72C2×D3×C7—41no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7#4111yescharacteristic
75Dic21—21no—
76Dic21—21no—
77Dic21—41no—
78C3×Dic7—41no—
79#463—211noSylow
80C23×C7—21no—
81C23×C7—61no—
82C7×D4—21no—
83C7×D4—21no—
84C7×D4—32no—
85C7×D4—61no—
86C2×C4×C7C22×D311yescharacteristic
87C2×Dic7—21no—
88C2×Dic7—31no—
89C2×Dic7—31no—
90C2×Dic7—61no—
91C2×Dic7—122no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—61no—
95C7⋊C8—21no—
96C2×C3×D4C2×D711yescharacteristic
97C2×D12C2×D711yescharacteristic
98C2×C4×D3—141no—
99C2×Dic6—71no—
100C2×C3×C8—71no—
101C3×C4⋊C4—71no—
102#217—141no—
103#217—141no—
104#213—22no—
105(C2×D3)⋊C4—141no—
106C4×Dic3—71no—
107C2×C3⋊C8C2×D711yescharacteristic
108#204—142no—
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7#13511yescharacteristic
112C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
113D3×C7—42no—
114C2×SD16—211no—
115C2×D8—31no—
116#124—211no—
117C4×D4—211no—
118C4⋊C8—211no—
119#103—211no—
120#102—71no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7—62no—
125C22×C7—121no—
126C22×C7—121no—
127C22×C7D3×D411yescharacteristic
128C4×C7#23511yescharacteristic
129C4×C7D3×D411yescharacteristic
130Dic7—41no—
131Dic7—61no—
132Dic7—61no—
133Dic7—121no—
134C23×C3—21no—
135C22×D3—21no—
136C3×D4—21no—
137C3×D4—21no—
138C2×C3×C4—141no—
139C2×C3×C4C22×D711yescharacteristic
140C2×Dic3—71no—
141C2×Dic3—71no—
142C2×Dic3—141no—
143D12—21no—
144D12C22⋊D712yes—
145C4×D3—282no—
146Dic6—141no—
147Dic6—141no—
148C3×C8—141no—
149C3⋊C8—21no—
150C3×C7#46311yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—31no—
153C2×D4D3×D711yescharacteristic
154C22×C4—421no—
155SD16—422no—
156D8—62no—
157C2×C8—31no—
158C2×C8—71no—
159C4⋊C4—71no—
160C22⋊C4—421no—
161C22⋊C4—421no—
162C22⋊C4—421no—
163C42—211no—
164C2×C7—81no—
165C2×C7—122no—
166C2×C7#88411yescharacteristic
167C2×C7#91111yescharacteristic
168C2×C7#91611yescharacteristic
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3D4×D711yescharacteristic
173C2×D3—22no—
174C2×D3—41no—
175C2×D3—41no—
176C3×C4—281no—
177C3×C4#115911yescharacteristic
178C3×C4C2×C22⋊D711yescharacteristic
179Dic3—141no—
180Dic3—141no—
181Dic3—281no—
182C23—21no—
183C23—61no—
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—32no—
189D4—61no—
190C2×C4—141no—
191C2×C4—211no—
192C2×C4—211no—
193C2×C4—421no—
194C2×C4—842no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—141no—
198C7#478711yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3#647411yescharacteristic
201C2×C3#647511yescharacteristic
202C2×C3#650211yescharacteristic
203D3—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—62no—
208C22—121no—
209C22—121no—
210C22#6669411yescharacteristic
211C4—281no—
212C4—421no—
213C4—421no—
214C4—841no—
215C4#6668611yescharacteristic
216C4D3×C22⋊D711yescharacteristic
217C3#8992511yescharacteristic, complemented, Sylow
218C2—81no—
219C2—122no—
220C2#1062104411yescharacteristic
221C2#1062106111yescharacteristic
222C2#1062112011yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC2×C7×#213C2
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. #10620997C2semidirect
4. C2×#66567C2semidirect
5. C3×#6406C2semidirect
6. #10620518C2semidirect
7. #10620509C2semidirect
26. #6406D3semidirect
150. C3×C7#463semidirect
198. C7#4787semidirect
217. C3#89925semidirect

Maximal subgroups

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

subgroupindex
#106214192
C2×C7×#2132
#106209972
C2×#665672
C3×#64062
#106205182
#106205092
#899253
#47877