Finite Group Databasebeta

← #13052809

Subgroups — #13052809

Order 1344.

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

normal subgroups only

This group has 260 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#10621378C211yesmaximal, characteristic, complemented
2#10621027C211yesmaximal, characteristic, complemented
3C2×#66561C211yesmaximal, characteristic, complemented
4C2×#66559C211yesmaximal, characteristic, complemented
5C7×#808C211yesmaximal, characteristic, complemented
6C3×#6383C211yesmaximal, characteristic, complemented
7#10620504C211yesmaximal, characteristic, complemented
8#89637—31nomaximal, Hall
9C2×D84—21no—
10C2×D84C2211yescharacteristic
11C2×C4×D21C2211yescharacteristic
12C2×C3×D28C2211yescharacteristic
13C2×#4246—21no—
14C3×C7×C4⋊C4C2211yescharacteristic
15#66631—21no—
16C7×C4⋊Dic3C2211yescharacteristic
17C2×C7×C3⋊C8C2211yescharacteristic
18C2×C3×C7⋊C8C2211yescharacteristic
19(C2×D7)⋊Dic3—21no—
20#66561—22no—
21#66559—22no—
22C2×D56—31no—
23C2×#1131—31no—
24#6458—31no—
25#6458—31no—
26C7×#106—31no—
27#6383—31no—
28#6383D311yescharacteristic, complemented
29#4782—71nomaximal, Hall
30C22×D21—21no—
31C22×D21D411yescharacteristic
32C22×C3×D7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21D411yescharacteristic
36D84—21no—
37D84—21no—
38D84—42no—
39C4×D21—21no—
40C4×D21—21no—
41C2×C7×Dic3—21no—
42C3×D28—21no—
43C3×D28—21no—
44#4246—42no—
45C3×C7⋊C8—21no—
46C7×C3⋊C8D412yescomplemented
47C2×D28—31no—
48C2×D28—61no—
49C2×D28—61no—
50C2×D28C2×D311yescharacteristic
51C2×C4×D7—31no—
52C2×C7×C8—31no—
53C7×C4⋊C4—31no—
54C7×C4⋊C4C2×D311yescharacteristic
55(C2×D7)⋊C4—61no—
56(C2×D7)⋊C4—61no—
57C2×C7⋊C8C2×D311yescharacteristic
58D56—62no—
59#1131—62no—
60#938—71no—
61C2×#213—71no—
62C2×ΓL(1,𝔽25)—71no—
63#895—71no—
64C3×#102—71no—
65#821—71no—
66#808D711yescharacteristic, complemented
67C22×C3×C7C2×D411yescharacteristic
68C2×D21—21no—
69C2×D21—21no—
70C2×D21—21no—
71C2×D21—41no—
72C2×D21—41no—
73C2×D21—41no—
74C2×C3×D7—41no—
75C2×C3×D7—41no—
76C2×C3×D7—41no—
77C3×C4×C7—41no—
78C3×C4×C7C2×D411yescharacteristic
79C3×C4×C7#4111yescharacteristic
80Dic21—21no—
81C7×Dic3—41no—
82#469—211noSylow
83C22×D7—21no—
84C22×D7—31no—
85C22×D7—61no—
86C2×C4×C7—21no—
87C2×C4×C7—61no—
88C2×C4×C7C22×D311yescharacteristic
89C2×Dic7—31no—
90D28—21no—
91D28—21no—
92D28—61no—
93D28—61no—
94D28—122no—
95D28—122no—
96C4×D7—61no—
97C4×D7—61no—
98C7×C8—32no—
99C7⋊C8—21no—
100C2×C3×D4—71no—
101C2×#67—141no—
102C2×D12—71no—
103C2×D12—141no—
104C2×C4×D3—71no—
105C2×C3×C8—71no—
106C3×C4⋊C4C2×D711yescharacteristic
107#217—141no—
108#213—142no—
109(C2×D3)⋊C4—141no—
110C4⋊Dic3C2×D711yescharacteristic
111C2×C3⋊C8C2×D711yescharacteristic
112ΓL(1,𝔽25)—142no—
113C2×C3×C7#12111yescharacteristic
114C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
115C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
116D21—41no—
117D21—81no—
118C3×D7—81no—
119C2×D8—211no—
120C2×M16—211no—
121#121—211no—
122#121—211no—
123#106—31no—
124#102—71no—
125#102—211no—
126C22×C7D3×D411yescharacteristic
127C2×D7—41no—
128C2×D7—41no—
129C2×D7—41no—
130C2×D7—61no—
131C2×D7—61no—
132C2×D7—61no—
133C2×D7—121no—
134C2×D7—121no—
135C2×D7—121no—
136C4×C7—41no—
137C4×C7—121no—
138C4×C7#23511yescharacteristic
139C4×C7D3×D411yescharacteristic
140Dic7—61no—
141C23×C3—141no—
142C22×D3—71no—
143C22×D3—141no—
144C3×D4—141no—
145C3×D4—141no—
146C2×C3×C4—21no—
147C2×C3×C4C22×D711yescharacteristic
148#67—282no—
149C2×Dic3—21no—
150C2×Dic3—71no—
151D12—141no—
152D12—141no—
153D12—282no—
154C4×D3—141no—
155C4×D3—141no—
156C3×C8—141no—
157C3⋊C8D2812yescomplemented
158C3×C7#46911yescharacteristic, complemented, Hall
159C2×D4—71no—
160C2×D4—211no—
161C2×D4—421no—
162C2×D4—421no—
163C22×C4—211no—
164D8—422no—
165M16—422no—
166C2×C8—31no—
167C2×C8—71no—
168C4⋊C4—31no—
169C4⋊C4D3×D711yescharacteristic
170C22⋊C4—421no—
171C22⋊C4—421no—
172C2×C7#88411yescharacteristic
173C2×C7#91111yescharacteristic
174C2×C7#91411yescharacteristic
175D7—81no—
176D7—121no—
177D7—241no—
178C22×C3—281no—
179C22×C3—281no—
180C22×C3—281no—
181C22×C3D4×D711yescharacteristic
182C2×D3—141no—
183C2×D3—141no—
184C2×D3—141no—
185C2×D3—281no—
186C2×D3—281no—
187C2×D3—281no—
188C3×C4—41no—
189C3×C4C2×D2811yescharacteristic
190C3×C4#116111yescharacteristic
191Dic3—41no—
192Dic3—141no—
193C23—141no—
194C23—211no—
195C23—421no—
196D4—141no—
197D4—141no—
198D4—421no—
199D4—421no—
200D4—842no—
201D4—842no—
202C2×C4—21no—
203C2×C4—61no—
204C2×C4—211no—
205C2×C4—421no—
206C2×C4—421no—
207C2×C4C2×D3×D711yescharacteristic
208C8—32no—
209C8—141no—
210C7#478211yescharacteristic, complemented, Sylow
211C2×C3—561no—
212C2×C3#645811yescharacteristic
213C2×C3#647711yescharacteristic
214C2×C3#647711yescharacteristic
215D3—281no—
216D3—561no—
217C22—281no—
218C22—281no—
219C22—281no—
220C22—421no—
221C22—421no—
222C22—421no—
223C22—841no—
224C22—841no—
225C22—841no—
226C22#6669411yescharacteristic
227C4—41no—
228C4—121no—
229C4—421no—
230C4#6667711yescharacteristic
231C4D3×D2811yescharacteristic
232C3#8963711yescharacteristic, complemented, Sylow
233C2—561no—
234C2—841no—
235C2—1681no—
236C2#1062102111yescharacteristic
237C2#1062106011yescharacteristic
238C2#1062106811yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC3×C7×C4⋊C4C22
socleC22×C3×C7C2×D4
hypercenterC4⋊C4D3×D7
hypocenterC3×C7#469
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10621378C2semidirect
2. #10621027C2semidirect
3. C2×#66561C2semidirect
4. C2×#66559C2semidirect
5. C7×#808C2semidirect
6. C3×#6383C2semidirect
7. #10620504C2semidirect
28. #6383D3semidirect
46. C7×C3⋊C8D4semidirect
66. #808D7semidirect
157. C3⋊C8D28semidirect
158. C3×C7#469semidirect
210. C7#4782semidirect
232. C3#89637semidirect

Maximal subgroups

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

subgroupindex
#106213782
#106210272
C2×#665612
C2×#665592
C7×#8082
C3×#63832
#106205042
#896373
#47827