Finite Group Databasebeta

← #13054625

Subgroups — #13054625

Order 1344.

248 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×C7×#215C211yesmaximal, characteristic, complemented
2C3×#6509C211yesmaximal, characteristic, complemented
3#10621011C211yesmaximal, characteristic
4C2×D7×C3⋊C8C211yesmaximal, characteristic, complemented
5#10620674C211yesmaximal, characteristic, complemented
6#10620522C211yesmaximal, characteristic, complemented
7#10620501C211yesmaximal, characteristic, complemented
8#89943—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×D12C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12C2×#4245—21no—
13C4⋊Dic21C2211yescharacteristic
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×#215—22no—
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×(C2×D7)⋊C4—21no—
18C3×C4⋊Dic7C2211yescharacteristic
19C3×Dic7⋊C4—21no—
20(C2×D3)⋊Dic7—21no—
21D7×C3⋊C8—22no—
22C2×C7×SD16—31no—
23#6509D311yescharacteristic, complemented
24#6501—31no—
25C2×C8×D7—31no—
26#6409—31no—
27#6406—31no—
28#6391—31no—
29#5048—71nomaximal, Hall
30C22×D3×C7—21no—
31C22×C3×D7D411yescharacteristic
32C3×C7×Q8—21no—
33C3×C7×Q8—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7C2311yescharacteristic
36C2×Dic21—21no—
37C7×D12—21no—
38C7×D12—21no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7D411yescharacteristic
41C3×C4×D7—21no—
42C3×C4×D7—21no—
43#4245—42no—
44C3⋊(C7⋊C8)—21no—
45C7×C3⋊C8D412yescomplemented
46C2×C7×Q8C2×D311yescharacteristic
47C2×C7×D4—31no—
48C2×C22⋊D7—61no—
49C2×C4×D7C2×D311yescharacteristic
50C7×SD16—62no—
51C2×C7×C8—31no—
52C22⋊Dic7—61no—
53(C2×D7)⋊C4—21no—
54C4⋊Dic7—31no—
55C4⋊Dic7C2×D311yescharacteristic
56Dic7⋊C4—21no—
57C2×C7⋊C8—31no—
58C8×D7—62no—
59C3×C22⋊Q8—71no—
60C2×#215D711yescharacteristic, complemented
61#930—71no—
62C22×C3⋊C8—71no—
63#835—71no—
64#809—71no—
65#808—71no—
66C22×C3×C7C2×D411yescharacteristic
67C2×D3×C7—41no—
68C2×D3×C7—41no—
69C2×D3×C7—41no—
70C2×C3×D7—21no—
71C2×C3×D7—21no—
72C2×C3×D7SD1612yescomplemented
73C3×C4×C7—41no—
74C3×C4×C7C2×D411yescharacteristic
75C3×C4×C7#4111yescharacteristic
76Dic21—41no—
77C3×Dic7—21no—
78C3×Dic7—41no—
79#465—211noSylow
80C23×C7—61no—
81C22×D7#6711yescharacteristic
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—61no—
91C2×Dic7#6711yescharacteristic
92C4×D7—21no—
93C4×D7—21no—
94C7×C8—32no—
95C7⋊C8—61no—
96C2×C3×Q8C2×D711yescharacteristic
97C22×C3×C4—71no—
98C2×#67—141no—
99C2×D12C2×D711yescharacteristic
100C3×C4⋊C4—71no—
101C3×C4⋊C4—141no—
102C3×C22⋊C4—141no—
103#215—22no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—71no—
106C2×C3⋊C8—71no—
107C2×C3⋊C8—142no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7#12111yescharacteristic
110C2×C3×C7C2×SD1611yescharacteristic
111C2×C3×C7#13511yescharacteristic
112C3×D7—22no—
113D3×C7—81no—
114C2×SD16—31no—
115C22×C8—211no—
116C22⋊Q8—71no—
117#121—211no—
118#106—211no—
119#103—211no—
120#102—211no—
121C22×C7—121no—
122C22×C7—121no—
123C22×C7—121no—
124C22×C7C2×#6711yescharacteristic
125C2×D7—21no—
126C2×D7—21no—
127C2×D7#21512yescomplemented
128C4×C7—41no—
129C4×C7D3×D411yescharacteristic
130C4×C7#23711yescharacteristic
131Dic7—21no—
132Dic7—41no—
133Dic7—121no—
134C23×C3—71no—
135C22×D3—21no—
136C3×Q8—21no—
137C3×Q8—21no—
138C2×C3×C4—21no—
139C2×C3×C4—71no—
140C2×C3×C4—141no—
141C2×C3×C4—141no—
142C2×C3×C4—141no—
143C2×C3×C4C22×D711yescharacteristic
144#67—282no—
145C2×Dic3—141no—
146D12—21no—
147D12—21no—
148C3⋊C8—141no—
149C3⋊C8C22⋊D712yescomplemented
150C3×C7#46511yescharacteristic, complemented, Hall
151C2×Q8D3×D711yescharacteristic
152C2×D4—31no—
153C2×D4—421no—
154C22×C4—71no—
155SD16—62no—
156C2×C8—31no—
157C2×C8—211no—
158C2×C8—422no—
159C4⋊C4—71no—
160C4⋊C4—141no—
161C4⋊C4—211no—
162C22⋊C4—141no—
163C22⋊C4—421no—
164C2×C7—241no—
165C2×C7#93911yescharacteristic
166C2×C7C2×#21511yescharacteristic
167C2×C7#95011yescharacteristic
168D7—22no—
169C22×C3—72no—
170C22×C3—141no—
171C22×C3—141no—
172C22×C3D4×D711yescharacteristic
173C2×D3—41no—
174C2×D3—41no—
175C2×D3—41no—
176C3×C4—41no—
177C3×C4—141no—
178C3×C4—281no—
179C3×C4#115911yescharacteristic
180C3×C4C2×C22⋊D711yescharacteristic
181Dic3—281no—
182C23—61no—
183C23—71no—
184Q8—21no—
185Q8—21no—
186D4—61no—
187D4—61no—
188D4—842no—
189C2×C4—21no—
190C2×C4—71no—
191C2×C4—141no—
192C2×C4—141no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—32no—
197C8—421no—
198C7#504811yescharacteristic, complemented, Sylow
199C2×C3—142no—
200C2×C3D7×SD1611yescharacteristic
201C2×C3#647911yescharacteristic
202C2×C3#650111yescharacteristic
203D3—81no—
204C22—72no—
205C22—121no—
206C22—121no—
207C22—121no—
208C22—141no—
209C22—141no—
210C22D7×#6711yescharacteristic
211C4—41no—
212C4—141no—
213C4—281no—
214C4—841no—
215C4#6668411yescharacteristic
216C4D3×C22⋊D711yescharacteristic
217C3#8994311yescharacteristic, complemented, Sylow
218C2—142no—
219C2—241no—
220C2D7×#21511yescharacteristic
221C2#1062107611yescharacteristic
222C2#1062111111yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radical#10621011C2
hypercenterC2×Q8D3×D7
hypocenterC3×C7#465
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. C2×C7×#215C2semidirect
2. C3×#6509C2semidirect
4. C2×D7×C3⋊C8C2semidirect
5. #10620674C2semidirect
6. #10620522C2semidirect
7. #10620501C2semidirect
23. #6509D3semidirect
45. C7×C3⋊C8D4semidirect
60. C2×#215D7semidirect
72. C2×C3×D7SD16semidirect
127. C2×D7#215semidirect
149. C3⋊C8C22⋊D7semidirect
150. C3×C7#465semidirect
198. C7#5048semidirect
217. C3#89943semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2152
C3×#65092
#106210112
C2×D7×C3⋊C82
#106206742
#106205222
#106205012
#899433
#50487