Finite Group Databasebeta

← #13061655

Subgroups — #13061655

Order 1344.

476 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C7×D4C211yesmaximal, characteristic, complemented
2#10621422C212yesmaximal, complemented
3#10621413C211yesmaximal, characteristic, complemented
4#10621413C211yesmaximal, characteristic, complemented
5#10621409C211yesmaximal, characteristic, complemented
6C2×C4⋊Dic21C211yesmaximal, characteristic, complemented
7#10621381C212yesmaximal, complemented
8#10621366C212yesmaximal, complemented
9#10621364C212yesmaximal, complemented
10C4×D84C211yesmaximal, characteristic, complemented
11#10621353C211yesmaximal, characteristic, complemented
12#90256—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C22×C3×C4×C7C2211yescharacteristic
15C22×C3×C4×C7C2211yescharacteristic
16C2×#4268C2212yes—
17C2×#4268C2212yes—
18C22×Dic21C2212yes—
19C2×D84C2211yescharacteristic
20C2×C4×D21C2212yes—
21C3×C7×C4⋊C4C2211yescharacteristic
22C3×C7×C22⋊C4C2211yescharacteristic
23C3×C7×C22⋊C4C2211yescharacteristic
24C3×C42×C7C2211yescharacteristic
25#66636C2212yes—
26#66631C2212yes—
27#66631C2212yes—
28#66631C2212yes—
29C4⋊Dic21C2211yescharacteristic
30C4⋊Dic21C2211yescharacteristic
31C4⋊Dic21C2211yescharacteristic
32C4⋊Dic21C2212yes—
33#66629C2212yes—
34#66629C2212yes—
35C4×Dic21C2212yes—
36C4×C7×D4D311yescharacteristic, complemented
37#6502—32no—
38#6493—31no—
39#6493—31no—
40#6489—31no—
41C2×C4⋊Dic7—31no—
42#6461—32no—
43#6446—32no—
44(C22⋊D7)⋊C4—32no—
45C4×D28—31no—
46#6433—31no—
47#5565—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C23×C3×C7C2311yescharacteristic
50C22×D21C2312yes—
51C3×C7×D4—22no—
52C2×C3×C4×C7—22no—
53C2×C3×C4×C7—22no—
54C2×C3×C4×C7C2311yescharacteristic
55C2×C3×C4×C7C2311yescharacteristic
56C2×C3×C4×C7C2311yescharacteristic
57C2×C3×C4×C7C2311yescharacteristic
58C2×C3×C4×C7C2311yescharacteristic
59#4268—24no—
60#4268—42no—
61C2×Dic21—42no—
62C2×Dic21C2312yes—
63C2×Dic21C2312yes—
64C2×Dic21C2312yes—
65D84—22no—
66C4×D21—42no—
67C2×C7×D4C2×D311yescharacteristic
68C22×C4×C7C2×D311yescharacteristic
69C22×C4×C7C2×D311yescharacteristic
70C2×C22⋊D7—32no—
71C2×C22⋊D7—32no—
72C22×Dic7—32no—
73C2×D28—31no—
74C2×C4×D7—32no—
75C7×C4⋊C4C2×D311yescharacteristic
76C7×C22⋊C4C2×D311yescharacteristic
77C7×C22⋊C4C2×D311yescharacteristic
78C42×C7C2×D311yescharacteristic
79C22⋊Dic7—32no—
80(C2×D7)⋊C4—32no—
81(C2×D7)⋊C4—32no—
82(C2×D7)⋊C4—32no—
83C4⋊Dic7—31no—
84C4⋊Dic7—31no—
85C4⋊Dic7—31no—
86C4⋊Dic7—32no—
87Dic7⋊C4—32no—
88Dic7⋊C4—32no—
89C4×Dic7—32no—
90C3×C4×D4D711yescharacteristic, complemented
91#939—72no—
92#930—71no—
93#930—71no—
94#926—71no—
95C2×C4⋊Dic3—71no—
96#898—72no—
97#883—72no—
98#67⋊C4—72no—
99C4×D12—71no—
100#870—71no—
101C22×C3×C7—21no—
102C22×C3×C7—21no—
103C22×C3×C7—21no—
104C22×C3×C7—21no—
105C22×C3×C7—21no—
106C22×C3×C7#4112yes—
107C22×C3×C7C2411yescharacteristic
108C2×D21—22no—
109C2×D21—22no—
110C2×D21—22no—
111C3×C4×C7—21no—
112C3×C4×C7—21no—
113C3×C4×C7—21no—
114C3×C4×C7—21no—
115C3×C4×C7#4112yes—
116Dic21—22no—
117Dic21—22no—
118Dic21—22no—
119#553—211noSylow
120C23×C7C22×D311yescharacteristic
121C23×C7C22×D311yescharacteristic
122C22×D7—32no—
123C7×D4—22no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7C22×D311yescharacteristic
127C2×C4×C7C22×D311yescharacteristic
128C2×C4×C7C22×D311yescharacteristic
129C2×C4×C7C22×D311yescharacteristic
130C2×C4×C7C22×D311yescharacteristic
131C22⋊D7—64no—
132C22⋊D7—122no—
133C2×Dic7—32no—
134C2×Dic7—32no—
135C2×Dic7—32no—
136C2×Dic7—122no—
137D28—62no—
138C4×D7—122no—
139C2×C3×D4C2×D711yescharacteristic
140C22×C3×C4C2×D711yescharacteristic
141C22×C3×C4C2×D711yescharacteristic
142C2×#67—72no—
143C2×#67—72no—
144C22×Dic3—72no—
145C2×D12—71no—
146C2×C4×D3—72no—
147C3×C4⋊C4C2×D711yescharacteristic
148C3×C22⋊C4C2×D711yescharacteristic
149C3×C22⋊C4C2×D711yescharacteristic
150C3×C42C2×D711yescharacteristic
151#217—72no—
152(C2×D3)⋊C4—72no—
153(C2×D3)⋊C4—72no—
154(C2×D3)⋊C4—72no—
155C4⋊Dic3—71no—
156C4⋊Dic3—71no—
157C4⋊Dic3—71no—
158C4⋊Dic3—72no—
159Dic3⋊C4—72no—
160Dic3⋊C4—72no—
161C4×Dic3—72no—
162C2×C3×C7—22no—
163C2×C3×C7—41no—
164C2×C3×C7C2×#4111yescharacteristic
165C2×C3×C7C2×#4111yescharacteristic
166C2×C3×C7H(5,𝔽2)11yescharacteristic
167D21—42no—
168#126—212no—
169#125—211no—
170#123—212no—
171#121—212no—
172#121—211no—
173#121—211no—
174C4×D4—212no—
175C4×D4—211no—
176C4×D4D2111yescharacteristic, complemented
177#117—211no—
178C2×C4⋊C4—211no—
179C22×C7—21no—
180C22×C7—21no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7#23512yes—
185C22×C7C23×D311yescharacteristic
186C2×D7—62no—
187C2×D7—62no—
188C2×D7—62no—
189C4×C7—21no—
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—21no—
193C4×C7#23712yes—
194Dic7—62no—
195Dic7—62no—
196Dic7—62no—
197C23×C3C22×D711yescharacteristic
198C23×C3C22×D711yescharacteristic
199C22×D3—72no—
200C3×D4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4C22×D711yescharacteristic
204C2×C3×C4C22×D711yescharacteristic
205C2×C3×C4C22×D711yescharacteristic
206C2×C3×C4C22×D711yescharacteristic
207C2×C3×C4C22×D711yescharacteristic
208#67—144no—
209#67—282no—
210C2×Dic3—72no—
211C2×Dic3—72no—
212C2×Dic3—72no—
213C2×Dic3—282no—
214D12—142no—
215C4×D3—282no—
216C3×C7#55311yescharacteristic, complemented, Hall
217C2×D4—212no—
218C2×D4—212no—
219C2×D4—211no—
220C2×D4C2×D2111yescharacteristic
221C22×C4—212no—
222C22×C4—212no—
223C22×C4C2×D2111yescharacteristic
224C22×C4C2×D2111yescharacteristic
225C4⋊C4—212no—
226C4⋊C4—212no—
227C4⋊C4—211no—
228C4⋊C4—211no—
229C4⋊C4—212no—
230C4⋊C4—211no—
231C4⋊C4C2×D2111yescharacteristic
232C22⋊C4—212no—
233C22⋊C4—212no—
234C22⋊C4—212no—
235C22⋊C4—212no—
236C22⋊C4C2×D2111yescharacteristic
237C22⋊C4C2×D2111yescharacteristic
238C42—212no—
239C42C2×D2111yescharacteristic
240C2×C7—22no—
241C2×C7—41no—
242C2×C7C2×#23511yescharacteristic
243C2×C7C2×#23711yescharacteristic
244C2×C7#100911yescharacteristic
245D7—122no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3#115712yes—
252C22×C3C23×D711yescharacteristic
253C2×D3—142no—
254C2×D3—142no—
255C2×D3—142no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4#115912yes—
261Dic3—142no—
262Dic3—142no—
263Dic3—142no—
264C23—212no—
265C23C22×D2111yescharacteristic
266C23C22×D2111yescharacteristic
267D4—22no—
268D4—424no—
269D4—422no—
270D4—842no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—212no—
274C2×C4—212no—
275C2×C4—212no—
276C2×C4—842no—
277C2×C4—842no—
278C2×C4C22×D2111yescharacteristic
279C2×C4C22×D2111yescharacteristic
280C2×C4C22×D2111yescharacteristic
281C2×C4C22×D2111yescharacteristic
282C2×C4C22×D2111yescharacteristic
283C7#556511yescharacteristic, complemented, Sylow
284C2×C3—22no—
285C2×C3—41no—
286C2×C3C2×#115711yescharacteristic
287C2×C3C2×#115911yescharacteristic
288C2×C3#655311yescharacteristic
289D3—282no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—422no—
296C22—422no—
297C22—422no—
298C22#6672812yes—
299C22C23×D2111yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—422no—
305C4—422no—
306C4—422no—
307C4#6673012yes—
308C3#9025611yescharacteristic, complemented, Sylow
309C2—22no—
310C2—41no—
311C2—842no—
312C2C2×#6672811yescharacteristic
313C2C2×#6673011yescharacteristic
314C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C4×C7×D4C2
hypercenterC4×D4D21
hypocenterC3×C7#553
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C7×D4C2semidirect
2. #10621422C2semidirect
3. #10621413C2semidirect
4. #10621413C2semidirect
5. #10621409C2semidirect
6. C2×C4⋊Dic21C2semidirect
7. #10621381C2semidirect
8. #10621366C2semidirect
9. #10621364C2semidirect
10. C4×D84C2semidirect
11. #10621353C2semidirect
36. C4×C7×D4D3semidirect
90. C3×C4×D4D7semidirect
176. C4×D4D21semidirect
216. C3×C7#553semidirect
283. C7#5565semidirect
308. C3#90256semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C7×D42
#106214222
#106214132
#106214092
C2×C4⋊Dic212
#106213812
#106213662
#106213642
C4×D842
#106213532
#902563
#55657