Finite Group Databasebeta

← #13054418

Subgroups — #13054418

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
2C7×#938C211yesmaximal, characteristic
3#10620927C211yesmaximal, characteristic, complemented
4C2×#66558C211yesmaximal, characteristic, complemented
5C3×#6406C211yesmaximal, characteristic, complemented
6#10620524C211yesmaximal, characteristic, complemented
7#10620510C211yesmaximal, characteristic, complemented
8#89814—31nomaximal, Hall
9C2×C3×C7×D4C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C2×C7×#67—21no—
12C2×C4×D3×C7C2211yescharacteristic
13#66633—22no—
14C2×C3⋊(C7⋊C8)C2211yescharacteristic
15C7×#217—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic7C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19Dic7⋊Dic3—21no—
20(C2×D3)⋊Dic7—21no—
21#66558—22no—
22C7×#121—31no—
23C2×#1142—31no—
24#6487—31no—
25C2×#1136—31no—
26#6406D311yescharacteristic, complemented
27#6384—31no—
28#6381—31no—
29#4780—71nomaximal, Hall
30C23×C3×C7—21no—
31C22×D3×C7D411yescharacteristic
32C3×C7×D4—21no—
33C3×C7×D4—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36Dic42—21no—
37Dic42—21no—
38C7×#67—42no—
39C2×C7×Dic3—21no—
40C2×C7×Dic3D411yescharacteristic
41C4×D3×C7—21no—
42C4×D3×C7—21no—
43C2×C3×Dic7—21no—
44C3⋊(C7⋊C8)D412yescomplemented
45C3×C7⋊C8—21no—
46C2×C7×D4—61no—
47C2×C7×D4C2×D311yescharacteristic
48C22×C4×C7—31no—
49C2×Dic14—31no—
50C7×C4⋊C4—31no—
51C7×C22⋊C4—61no—
52C22⋊Dic7—61no—
53#1142—62no—
54C4⋊Dic7C2×D311yescharacteristic
55Dic7⋊C4—61no—
56#1136—62no—
57C2×C7⋊C8—31no—
58C2×C7⋊C8C2×D311yescharacteristic
59#938D711yescharacteristic
60C2×#214—71no—
61C2×ΓL(1,𝔽25)—71no—
62#897—71no—
63C3×#102—71no—
64#816—71no—
65#807—71no—
66C22×C3×C7—41no—
67C22×C3×C7—41no—
68C22×C3×C7—41no—
69C22×C3×C7C2×D411yescharacteristic
70C2×D3×C7—21no—
71C2×D3×C7—21no—
72C2×D3×C7—21no—
73C3×C4×C7C2×D411yescharacteristic
74C3×C4×C7#4111yescharacteristic
75Dic21—41no—
76C3×Dic7—41no—
77C7×Dic3—21no—
78C7×Dic3—41no—
79#468—211noSylow
80C23×C7—21no—
81C23×C7—31no—
82C7×D4—21no—
83C7×D4—21no—
84C7×D4—122no—
85C2×C4×C7—31no—
86C2×C4×C7—61no—
87C2×C4×C7—61no—
88C2×C4×C7—61no—
89C2×C4×C7C22×D311yescharacteristic
90C2×Dic7—21no—
91C2×Dic7—61no—
92Dic14—61no—
93Dic14—61no—
94C7⋊C8—21no—
95C7⋊C8—32no—
96C2×C3×D4C2×D711yescharacteristic
97C2×#67—21no—
98C2×C4×D3C2×D711yescharacteristic
99C2×Dic6—71no—
100C2×C3×C8—71no—
101C3×C4⋊C4—71no—
102#217—21no—
103#214—142no—
104(C2×D3)⋊C4—141no—
105C4⋊Dic3—141no—
106C4⋊Dic3C2×D711yescharacteristic
107C2×C3⋊C8—71no—
108ΓL(1,𝔽25)—142no—
109C2×C3×C7—81no—
110C2×C3×C7#12111yescharacteristic
111C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
112C2×C3×C7#13711yescharacteristic
113D3×C7—41no—
114C2×SD16—211no—
115C2×M16—211no—
116C22⋊Q8—211no—
117#121—31no—
118#107—211no—
119#103—211no—
120#102—71no—
121C22×C7—41no—
122C22×C7—41no—
123C22×C7—41no—
124C22×C7—61no—
125C22×C7—61no—
126C22×C7—61no—
127C22×C7D3×D411yescharacteristic
128C4×C7—61no—
129C4×C7—121no—
130C4×C7#23511yescharacteristic
131C4×C7D3×D411yescharacteristic
132Dic7—41no—
133Dic7—121no—
134C23×C3—21no—
135C22×D3C22⋊D711yescharacteristic
136C3×D4—21no—
137C3×D4—21no—
138C2×C3×C4—141no—
139C2×C3×C4C22×D711yescharacteristic
140#67—42no—
141C2×Dic3—21no—
142C2×Dic3—141no—
143C2×Dic3C22⋊D711yescharacteristic
144C4×D3—21no—
145C4×D3—21no—
146Dic6—141no—
147Dic6—141no—
148C3×C8—141no—
149C3⋊C8—72no—
150C3×C7#46811yescharacteristic, complemented, Hall
151C2×Q8—211no—
152C2×D4—61no—
153C2×D4D3×D711yescharacteristic
154C22×C4—31no—
155SD16—422no—
156M16—422no—
157C2×C8—71no—
158C2×C8—211no—
159C4⋊C4—31no—
160C4⋊C4—71no—
161C4⋊C4—421no—
162C22⋊C4—61no—
163C22⋊C4—421no—
164C2×C7—81no—
165C2×C7—121no—
166C2×C7#88411yescharacteristic
167C2×C7#91111yescharacteristic
168C2×C7#91511yescharacteristic
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3C2×C22⋊D711yescharacteristic
173C2×D3—21no—
174C2×D3—21no—
175C2×D3—21no—
176C3×C4—281no—
177C3×C4D4×D711yescharacteristic
178C3×C4#115911yescharacteristic
179Dic3—21no—
180Dic3—41no—
181Dic3—281no—
182C23—21no—
183C23—31no—
184Q8—421no—
185Q8—421no—
186D4—21no—
187D4—21no—
188D4—122no—
189C2×C4—31no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—141no—
194C2×C4—421no—
195C2×C4C2×D3×D711yescharacteristic
196C8—141no—
197C8—212no—
198C7#478011yescharacteristic, complemented, Sylow
199C2×C3—81no—
200C2×C3#649511yescharacteristic
201C2×C3#650211yescharacteristic
202C2×C3#651411yescharacteristic
203D3—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—61no—
208C22—61no—
209C22—61no—
210C22D3×C22⋊D711yescharacteristic
211C4—61no—
212C4—121no—
213C4—281no—
214C4—841no—
215C4#6668611yescharacteristic
216C4#6669411yescharacteristic
217C3#8981411yescharacteristic, complemented, Sylow
218C2—81no—
219C2—121no—
220C2#1062104811yescharacteristic
221C2#1062105611yescharacteristic
222C2#1062112011yescharacteristic
subgroupG/H
centerC22D3×C22⋊D7
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
prime radicalC7×#938C2
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. #10620927C2semidirect
4. C2×#66558C2semidirect
5. C3×#6406C2semidirect
6. #10620524C2semidirect
7. #10620510C2semidirect
26. #6406D3semidirect
44. C3⋊(C7⋊C8)D4semidirect
150. C3×C7#468semidirect
198. C7#4780semidirect
217. C3#89814semidirect

Maximal subgroups

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

subgroupindex
C2×#666332
C7×#9382
#106209272
C2×#665582
C3×#64062
#106205242
#106205102
#898143
#47807