Finite Group Databasebeta

← #13054719

Subgroups — #13054719

Order 1344.

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

normal subgroups only

This group has 240 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#10621429C211yesmaximal, characteristic, complemented
2C2×C7×#215C211yesmaximal, characteristic, complemented
3#10621028C211yesmaximal, characteristic
4C2×#66559C211yesmaximal, characteristic, complemented
5C3×#6409C211yesmaximal, characteristic, complemented
6#10620525C211yesmaximal, characteristic, complemented
7#10620502C211yesmaximal, characteristic, complemented
8#89946—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C4×D21C2211yescharacteristic
11C2×C7×D12C2211yescharacteristic
12C2×#4247—21no—
13#66631—21no—
14C4⋊Dic21C2211yescharacteristic
15#66629—21no—
16C7×#215—22no—
17C2×C7×C3⋊C8C2211yescharacteristic
18C3×C4⋊Dic7C2211yescharacteristic
19C2×C3×C7⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic7—21no—
21#66559—22no—
22C2×C7×SD16—31no—
23#6509—31no—
24#6501—31no—
25C2×#1131—31no—
26#6409D311yescharacteristic, complemented
27#6406—31no—
28#6392—31no—
29#4812—71nomaximal, Hall
30C22×D21D411yescharacteristic
31C22×D3×C7—21no—
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21D411yescharacteristic
38C4×D21—21no—
39C4×D21—21no—
40C7×D12—21no—
41C7×D12—21no—
42C2×C3×Dic7—21no—
43#4247—42no—
44C3×C7⋊C8—21no—
45C7×C3⋊C8D412yescomplemented
46C2×C7×Q8C2×D311yescharacteristic
47C2×C7×D4—31no—
48C2×C22⋊D7—61no—
49C2×C4×D7—31no—
50C7×SD16—62no—
51C2×C7×C8—31no—
52C22⋊Dic7—61no—
53(C2×D7)⋊C4—61no—
54C4⋊Dic7—31no—
55C4⋊Dic7C2×D311yescharacteristic
56Dic7⋊C4—61no—
57C2×C7⋊C8C2×D311yescharacteristic
58#1131—62no—
59#946—71no—
60C2×#215D711yescharacteristic, complemented
61C2×ΓL(1,𝔽25)—71no—
62#895—71no—
63C3×#103—71no—
64#821—71no—
65#807—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D21—21no—
68C2×D21—21no—
69C2×D21—21no—
70C2×D3×C7—41no—
71C2×D3×C7—41no—
72C2×D3×C7—41no—
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—21no—
77Dic21—41no—
78C3×Dic7—41no—
79#468—211noSylow
80C23×C7—61no—
81C22×D7—31no—
82C7×Q8—21no—
83C7×Q8—21no—
84C7×D4—61no—
85C7×D4—61no—
86C2×C4×C7—21no—
87C2×C4×C7C22×D311yescharacteristic
88C22⋊D7—122no—
89C2×Dic7—21no—
90C2×Dic7—31no—
91C2×Dic7—61no—
92C4×D7—61no—
93C4×D7—61no—
94C7×C8—32no—
95C7⋊C8—21no—
96C2×C3×Q8C2×D711yescharacteristic
97C2×D12—141no—
98C2×D12C2×D711yescharacteristic
99C2×C4×D3—71no—
100C2×C3×C8—71no—
101C3×C4⋊C4—71no—
102#215—22no—
103(C2×D3)⋊C4—141no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—71no—
106Dic3⋊C4—141no—
107C2×C3⋊C8C2×D711yescharacteristic
108ΓL(1,𝔽25)—142no—
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
111C2×C3×C7#13711yescharacteristic
112D21—41no—
113D3×C7—81no—
114C2×SD16—31no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—211no—
118#107—211no—
119#103—71no—
120#102—211no—
121C22×C7—121no—
122C22×C7—121no—
123C22×C7—121no—
124C22×C7D3×D411yescharacteristic
125C2×D7—61no—
126C2×D7—61no—
127C2×D7—61no—
128C4×C7—41no—
129C4×C7#23511yescharacteristic
130C4×C7D3×D411yescharacteristic
131Dic7—41no—
132Dic7—61no—
133Dic7—121no—
134C22×D3—21no—
135C22×D3—71no—
136C3×Q8—21no—
137C3×Q8—21no—
138C2×C3×C4—21no—
139C2×C3×C4—141no—
140C2×C3×C4C22×D711yescharacteristic
141C2×Dic3—71no—
142C2×Dic3—141no—
143D12—21no—
144D12—21no—
145D12—282no—
146C4×D3—141no—
147C4×D3—141no—
148C3×C8—141no—
149C3⋊C8C22⋊D712yescomplemented
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8D3×D711yescharacteristic
152C2×D4—31no—
153C2×D4—421no—
154C22×C4—211no—
155SD16—62no—
156M16—422no—
157C2×C8—31no—
158C2×C8—71no—
159C4⋊C4—71no—
160C4⋊C4—211no—
161C4⋊C4—421no—
162C22⋊C4—421no—
163C22⋊C4—421no—
164C2×C7—241no—
165C2×C7#88411yescharacteristic
166C2×C7#91411yescharacteristic
167C2×C7#91811yescharacteristic
168D7—121no—
169C22×C3D4×D711yescharacteristic
170C2×D3—41no—
171C2×D3—41no—
172C2×D3—41no—
173C2×D3—141no—
174C2×D3—141no—
175C2×D3—141no—
176C3×C4—41no—
177C3×C4—281no—
178C3×C4#115911yescharacteristic
179C3×C4C2×C22⋊D711yescharacteristic
180Dic3—141no—
181Dic3—281no—
182C23—61no—
183C23—211no—
184Q8—21no—
185Q8—21no—
186D4—61no—
187D4—61no—
188D4—842no—
189C2×C4—21no—
190C2×C4—141no—
191C2×C4—211no—
192C2×C4—421no—
193C2×C4—421no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—32no—
197C8—141no—
198C7#481211yescharacteristic, complemented, Sylow
199C2×C3#647711yescharacteristic
200C2×C3#647811yescharacteristic
201C2×C3#650111yescharacteristic
202D3—81no—
203D3—281no—
204C22—121no—
205C22—121no—
206C22—121no—
207C22—421no—
208C22—421no—
209C22—421no—
210C22#6669411yescharacteristic
211C4—41no—
212C4—281no—
213C4—421no—
214C4—841no—
215C4#6668611yescharacteristic
216C4D3×C22⋊D711yescharacteristic
217C3#8994611yescharacteristic, complemented, Sylow
218C2—241no—
219C2—841no—
220C2#1062106811yescharacteristic
221C2#1062108611yescharacteristic
222C2#1062111911yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radical#10621028C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#468
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. #10621429C2semidirect
2. C2×C7×#215C2semidirect
4. C2×#66559C2semidirect
5. C3×#6409C2semidirect
6. #10620525C2semidirect
7. #10620502C2semidirect
26. #6409D3semidirect
45. C7×C3⋊C8D4semidirect
60. C2×#215D7semidirect
149. C3⋊C8C22⋊D7semidirect
150. C3×C7#468semidirect
198. C7#4812semidirect
217. C3#89946semidirect

Maximal subgroups

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

subgroupindex
#106214292
C2×C7×#2152
#106210282
C2×#665592
C3×#64092
#106205252
#106205022
#899463
#48127