Finite Group Databasebeta

← #13054550

Subgroups — #13054550

Order 1344.

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

normal subgroups only

This group has 396 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×#66681C211yesmaximal, characteristic, complemented
2#10621421C211yesmaximal, characteristic, complemented
3C2×C7×#213C211yesmaximal, characteristic, complemented
4C2×C3×#1141C211yesmaximal, characteristic, complemented
5#10620502C211yesmaximal, characteristic, complemented
6#10620500C211yesmaximal, characteristic, complemented
7#10620492C211yesmaximal, characteristic, complemented
8#89928—31nomaximal, Hall
9C22×D3×D7—21no—
10C2×C3×C7×D4C2211yescharacteristic, complemented
11C2×#4268—21no—
12C2×C4×D21C2211yescharacteristic
13C2×C7×D12C2211yescharacteristic
14C2×C3×D28C2211yescharacteristic
15C2×#4245—21no—
16#66681—22no—
17#66681—22no—
18#66636—21no—
19C4⋊Dic21C2211yescharacteristic, complemented
20C7×#213—22no—
21C2×C7×C3⋊C8C2211yescharacteristic, complemented
22C3×#1141—22no—
23C2×C3×C7⋊C8C2211yescharacteristic, complemented
24C2×D4×D7—31no—
25C2×C7×D8—31no—
26#6501—31no—
27C2×#1141D311yescharacteristic, complemented
28#6406—31no—
29#6395—31no—
30(C2×D7)⋊C8—31no—
31#5156—71nomaximal, Hall
32C23×C3×C7—21no—
33C22×D21D411yescharacteristic
34C22×D3×C7—21no—
35C22×C3×D7—21no—
36C2×D3×D7—24no—
37C2×D3×D7—42no—
38C2×D3×D7—42no—
39C3×C7×D4—21no—
40C3×C7×D4—21no—
41C2×C3×C4×C7C2311yescharacteristic
42#4268—42no—
43C2×Dic21—21no—
44C2×Dic21D411yescharacteristic
45C4×D21—21no—
46C4×D21—21no—
47C7×D12—21no—
48C7×D12D412yescomplemented
49C3×D28—21no—
50C3×D28D412yescomplemented
51#4245—24no—
52C3×C7⋊C8—21no—
53C7×C3⋊C8—21no—
54C23×D7—61no—
55C2×C7×D4—31no—
56C2×C7×D4C2×D311yescharacteristic, complemented
57C2×C22⋊D7—61no—
58C2×C22⋊D7—61no—
59D4×D7—62no—
60D4×D7—62no—
61C2×D28C2×D311yescharacteristic
62C2×C4×D7—31no—
63C7×D8—62no—
64C2×C7×C8—31no—
65C22⋊Dic7—61no—
66#1141—22no—
67C4⋊Dic7—31no—
68C2×C7⋊C8C2×D311yescharacteristic, complemented
69C2×D3×D4—71no—
70C2×C3×D8—71no—
71#938—71no—
72C2×#213D711yescharacteristic, complemented
73#832—71no—
74#821—71no—
75(C2×D3)⋊C8—71no—
76C22×C3×C7—41no—
77C22×C3×C7—41no—
78C22×C3×C7—41no—
79C22×C3×C7C2×D411yescharacteristic
80C2×D21—21no—
81C2×D21—21no—
82C2×D21D812yescomplemented
83C2×D3×C7—22no—
84C2×D3×C7—41no—
85C2×D3×C7—41no—
86C2×C3×D7—22no—
87C2×C3×D7—41no—
88C2×C3×D7—41no—
89D3×D7—48no—
90C3×C4×C7C2×D411yescharacteristic
91C3×C4×C7C2×D411yescharacteristic
92Dic21—21no—
93Dic21—41no—
94#447—211noSylow
95C23×C7—21no—
96C23×C7—61no—
97C22×D7—21no—
98C22×D7—31no—
99C22×D7—64no—
100C22×D7—122no—
101C22×D7—122no—
102C7×D4—21no—
103C7×D4—21no—
104C7×D4—32no—
105C7×D4—61no—
106C2×C4×C7C22×D311yescharacteristic
107C22⋊D7—64no—
108C22⋊D7—122no—
109C2×Dic7—31no—
110C2×Dic7—61no—
111D28—21no—
112D28#6712yescomplemented
113C4×D7—61no—
114C4×D7—61no—
115C7×C8—61no—
116C7⋊C8—21no—
117C23×D3—141no—
118C2×C3×D4—71no—
119C2×C3×D4C2×D711yescharacteristic, complemented
120C2×#67—141no—
121C2×#67—141no—
122D3×D4—142no—
123D3×D4—142no—
124C2×D12C2×D711yescharacteristic
125C2×C4×D3—71no—
126C3×D8—142no—
127C2×C3×C8—71no—
128#217—141no—
129#213—22no—
130C4⋊Dic3—71no—
131C2×C3⋊C8C2×D711yescharacteristic, complemented
132C2×C3×C7—81no—
133C2×C3×C7#12011yescharacteristic
134C2×C3×C7C2×D811yescharacteristic
135C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
136D21—22no—
137C3×D7—42no—
138D3×C7—42no—
139C22×D4—211no—
140C2×D8—31no—
141C2×D8—71no—
142#121—211no—
143#102—211no—
144#102—211no—
145C22⋊C8—211no—
146C22×C7—41no—
147C22×C7—41no—
148C22×C7—41no—
149C22×C7—62no—
150C22×C7—121no—
151C22×C7—121no—
152C22×C7D3×D411yescharacteristic
153C2×D7—22no—
154C2×D7—32no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—61no—
158C2×D7—61no—
159C2×D7—128no—
160C4×C7D3×D411yescharacteristic
161C4×C7C2×#6711yescharacteristic
162Dic7—61no—
163Dic7—121no—
164C23×C3—21no—
165C23×C3—141no—
166C22×D3—21no—
167C22×D3—71no—
168C22×D3—144no—
169C22×D3—282no—
170C22×D3—282no—
171C3×D4—21no—
172C3×D4—21no—
173C3×D4—72no—
174C3×D4—141no—
175C2×C3×C4C22×D711yescharacteristic
176#67—144no—
177#67—282no—
178C2×Dic3—71no—
179C2×Dic3—141no—
180D12—21no—
181D12C22⋊D712yescomplemented
182C4×D3—141no—
183C4×D3—141no—
184C3×C8—141no—
185C3⋊C8—21no—
186C3×C7#44711yescharacteristic, complemented, Hall
187C24—421no—
188C2×D4—31no—
189C2×D4—71no—
190C2×D4—421no—
191C2×D4—422no—
192C2×D4—422no—
193C2×D4—421no—
194C2×D4D3×D711yescharacteristic, complemented
195C22×C4—211no—
196D8—62no—
197D8—142no—
198C2×C8—31no—
199C2×C8—71no—
200C4⋊C4—211no—
201C22⋊C4—421no—
202C2×C7—81no—
203C2×C7—122no—
204C2×C7D3×D811yescharacteristic
205C2×C7#91111yescharacteristic
206C2×C7#93711yescharacteristic
207D7—42no—
208D7—62no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—142no—
213C22×C3—281no—
214C22×C3—281no—
215C22×C3D4×D711yescharacteristic
216C2×D3—22no—
217C2×D3—41no—
218C2×D3—41no—
219C2×D3—72no—
220C2×D3—141no—
221C2×D3—141no—
222C2×D3—288no—
223C3×C4D4×D711yescharacteristic
224C3×C4C2×C22⋊D711yescharacteristic
225Dic3—141no—
226Dic3—281no—
227C23—21no—
228C23—61no—
229C23—141no—
230C23—211no—
231C23—424no—
232C23—842no—
233C23—842no—
234D4—21no—
235D4—21no—
236D4—32no—
237D4—61no—
238D4—72no—
239D4—141no—
240D4—424no—
241D4—842no—
242C2×C4—211no—
243C2×C4—421no—
244C2×C4—421no—
245C2×C4—421no—
246C2×C4C2×D3×D711yescharacteristic
247C8—61no—
248C8—141no—
249C7#515611yescharacteristic, complemented, Sylow
250C2×C3—81no—
251C2×C3—282no—
252C2×C3D7×D811yescharacteristic
253C2×C3#647411yescharacteristic
254C2×C3#650011yescharacteristic
255D3—42no—
256D3—142no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—62no—
261C22—121no—
262C22—121no—
263C22—142no—
264C22—212no—
265C22—281no—
266C22—281no—
267C22—421no—
268C22—421no—
269C22—848no—
270C22#6669411yescharacteristic
271C4—421no—
272C4—841no—
273C4D7×#6711yescharacteristic
274C4D3×C22⋊D711yescharacteristic
275C3#8992811yescharacteristic, complemented, Sylow
276C2—81no—
277C2—122no—
278C2—282no—
279C2—422no—
280C2#1062104311yescharacteristic
281C2#1062104411yescharacteristic
282C2#1062113911yescharacteristic
subgroupG/H
centerC22#66694
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×D4C22
socleC22×C3×C7C2×D4
hypercenterC2×D4D3×D7
hypocenterC3×C7#447
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×#66681C2semidirect
2. #10621421C2semidirect
3. C2×C7×#213C2semidirect
4. C2×C3×#1141C2semidirect
5. #10620502C2semidirect
6. #10620500C2semidirect
7. #10620492C2semidirect
10. C2×C3×C7×D4C22semidirect
19. C4⋊Dic21C22semidirect
21. C2×C7×C3⋊C8C22semidirect
23. C2×C3×C7⋊C8C22semidirect
27. C2×#1141D3semidirect
48. C7×D12D4semidirect
50. C3×D28D4semidirect
56. C2×C7×D4C2×D3semidirect
68. C2×C7⋊C8C2×D3semidirect
72. C2×#213D7semidirect
82. C2×D21D8semidirect
112. D28#67semidirect
119. C2×C3×D4C2×D7semidirect
131. C2×C3⋊C8C2×D7semidirect
181. D12C22⋊D7semidirect
186. C3×C7#447semidirect
194. C2×D4D3×D7semidirect
249. C7#5156semidirect
275. C3#89928semidirect

Maximal subgroups

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

subgroupindex
C2×#666812
#106214212
C2×C7×#2132
C2×C3×#11412
#106205022
#106205002
#106204922
#899283
#51567