Finite Group Databasebeta

← #13054414

Subgroups — #13054414

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
1C2×#66633C211yesmaximal, characteristic, complemented
2C3×#6501C211yesmaximal, characteristic
3#10620922C211yesmaximal, characteristic, complemented
4C2×#66557C211yesmaximal, characteristic, complemented
5C7×#832C211yesmaximal, characteristic, complemented
6#10620524C211yesmaximal, characteristic, complemented
7#10620509C211yesmaximal, characteristic, complemented
8#89552—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C3×C22⋊D7—21no—
12C2×C3×C4×D7C2211yescharacteristic
13#66633—22no—
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×C4⋊Dic3C2211yescharacteristic
16C2×C7×C3⋊C8C2211yescharacteristic
17C3×C22⋊Dic7—21no—
18C3×C4⋊Dic7C2211yescharacteristic
19Dic3⋊Dic7—21no—
20(C2×D7)⋊Dic3—21no—
21#66557—22no—
22#6501D311yescharacteristic
23C2×#1142—31no—
24C2×#1131—31no—
25#6460—31no—
26C7×#102—31no—
27#6390—31no—
28#6381—31no—
29#5042—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×C3×D7D411yescharacteristic
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36Dic42—21no—
37Dic42—21no—
38C2×C7×Dic3—21no—
39C3×C22⋊D7—42no—
40C2×C3×Dic7—21no—
41C2×C3×Dic7D411yescharacteristic
42C3×C4×D7—21no—
43C3×C4×D7—21no—
44C3⋊(C7⋊C8)D412yescomplemented
45C7×C3⋊C8—21no—
46C2×C7×D4C2×D311yescharacteristic
47C2×C22⋊D7—21no—
48C2×C4×D7C2×D311yescharacteristic
49C2×Dic14—31no—
50C2×C7×C8—31no—
51C7×C4⋊C4—31no—
52C22⋊Dic7—21no—
53#1142—62no—
54(C2×D7)⋊C4—61no—
55C4⋊Dic7—61no—
56C4⋊Dic7C2×D311yescharacteristic
57C2×C7⋊C8—31no—
58#1131—62no—
59C3×#121—71no—
60C2×#214—71no—
61#924—71no—
62C2×#208—71no—
63#832D711yescharacteristic, complemented
64#810—71no—
65#807—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×C3×D7—21no—
71C2×C3×D7—21no—
72C2×C3×D7—21no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7#4111yescharacteristic
75Dic21—41no—
76C3×Dic7—21no—
77C3×Dic7—41no—
78C7×Dic3—41no—
79#468—211noSylow
80C23×C7—21no—
81C22×D7#6711yescharacteristic
82C7×D4—21no—
83C7×D4—21no—
84C2×C4×C7—61no—
85C2×C4×C7C22×D311yescharacteristic
86C22⋊D7—42no—
87C2×Dic7—21no—
88C2×Dic7—61no—
89C2×Dic7#6711yescharacteristic
90C4×D7—21no—
91C4×D7—21no—
92Dic14—61no—
93Dic14—61no—
94C7×C8—61no—
95C7⋊C8—32no—
96C2×C3×D4—141no—
97C2×C3×D4C2×D711yescharacteristic
98C22×C3×C4—71no—
99C2×Dic6—71no—
100C3×C4⋊C4—71no—
101C3×C22⋊C4—141no—
102#217—141no—
103#214—142no—
104C4⋊Dic3C2×D711yescharacteristic
105Dic3⋊C4—141no—
106#208—142no—
107C2×C3⋊C8—71no—
108C2×C3⋊C8C2×D711yescharacteristic
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112C2×C3×C7#13711yescharacteristic
113C3×D7—41no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—71no—
118#107—211no—
119#103—211no—
120#102—31no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7C2×#6711yescharacteristic
125C2×D7—21no—
126C2×D7—21no—
127C2×D7—21no—
128C4×C7—121no—
129C4×C7D3×D411yescharacteristic
130C4×C7#23711yescharacteristic
131Dic7—21no—
132Dic7—41no—
133Dic7—121no—
134C23×C3—21no—
135C23×C3—71no—
136C3×D4—21no—
137C3×D4—21no—
138C3×D4—282no—
139C2×C3×C4—71no—
140C2×C3×C4—141no—
141C2×C3×C4—141no—
142C2×C3×C4—141no—
143C2×C3×C4C22×D711yescharacteristic
144C2×Dic3—21no—
145C2×Dic3—141no—
146Dic6—141no—
147Dic6—141no—
148C3⋊C8—21no—
149C3⋊C8—72no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—141no—
153C2×D4D3×D711yescharacteristic
154C22×C4—71no—
155SD16—422no—
156M16—422no—
157C2×C8—31no—
158C2×C8—211no—
159C4⋊C4—31no—
160C4⋊C4—71no—
161C4⋊C4—421no—
162C22⋊C4—141no—
163C22⋊C4—421no—
164C2×C7—81no—
165C2×C7#93211yescharacteristic
166C2×C7#93911yescharacteristic
167C2×C7#95111yescharacteristic
168D7—41no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3—141no—
173C22×C3—141no—
174C22×C3—141no—
175C22×C3D4×D711yescharacteristic
176C3×C4—141no—
177C3×C4—281no—
178C3×C4#115711yescharacteristic
179C3×C4D4×D711yescharacteristic
180Dic3—41no—
181Dic3—281no—
182C23—21no—
183C23—71no—
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—282no—
189C2×C4—61no—
190C2×C4—71no—
191C2×C4—141no—
192C2×C4—141no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—61no—
197C8—212no—
198C7#504211yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3—281no—
201C2×C3#644711yescharacteristic
202C2×C3#647411yescharacteristic
203C2×C3#647811yescharacteristic
204C22—41no—
205C22—41no—
206C22—41no—
207C22—141no—
208C22—141no—
209C22—141no—
210C22D7×#6711yescharacteristic
211C4—121no—
212C4—141no—
213C4—281no—
214C4—841no—
215C4#6668311yescharacteristic
216C4#6669411yescharacteristic
217C3#8955211yescharacteristic, complemented, Sylow
218C2—81no—
219C2—281no—
220C2#1062104611yescharacteristic
221C2#1062105111yescharacteristic
222C2#1062111211yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC3×#6501C2
hypercenterC2×D4D3×D7
hypocenterC3×C7#468
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. C2×#66633C2semidirect
3. #10620922C2semidirect
4. C2×#66557C2semidirect
5. C7×#832C2semidirect
6. #10620524C2semidirect
7. #10620509C2semidirect
44. C3⋊(C7⋊C8)D4semidirect
63. #832D7semidirect
150. C3×C7#468semidirect
198. C7#5042semidirect
217. C3#89552semidirect

Maximal subgroups

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

subgroupindex
C2×#666332
C3×#65012
#106209222
C2×#665572
C7×#8322
#106205242
#106205092
#895523
#50427