Finite Group Databasebeta

← #13061651

Subgroups — #13061651

Order 1344.

432 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C7×D4C211yesmaximal, characteristic, complemented
2#10621421C211yesmaximal, characteristic
3#10621419C211yesmaximal, characteristic, complemented
4#10621412C212yesmaximal, complemented
5#10621407C212yesmaximal, complemented
6#10621380C211yesmaximal, characteristic, complemented
7#10621371C211yesmaximal, characteristic, complemented
8#10621368C212yesmaximal, complemented
9#10621362C212yesmaximal, complemented
10#10621355C211yesmaximal, characteristic, complemented
11C4×Dic42C211yesmaximal, characteristic, complemented
12#90252—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C22×C3×C4×C7C2212yes—
15C2×#4268C2212yes—
16C2×C4×D21C2211yescharacteristic
17C2×Dic42C2211yescharacteristic
18C2×Dic42C2212yes—
19C3×C7×C4⋊C4C2211yescharacteristic
20C3×C7×C22⋊C4C2212yes—
21C3×C42×C7C2211yescharacteristic
22#66636C2212yes—
23#66636C2212yes—
24#66636C2212yes—
25#66631C2211yescharacteristic
26#66631C2211yescharacteristic
27#66631C2212yes—
28C4⋊Dic21C2211yescharacteristic
29C4⋊Dic21C2212yes—
30#66629C2212yes—
31#66629C2211yescharacteristic
32#66629C2211yescharacteristic
33#66629C2212yes—
34C4×Dic21C2212yes—
35C4×Dic21C2211yescharacteristic
36C4×C7×D4D311yescharacteristic, complemented
37#6501—31no—
38#6499—31no—
39#6492—32no—
40#6487—32no—
41#6460—31no—
42#6451—31no—
43#6448—32no—
44#6442—32no—
45#6435—31no—
46C4×Dic14—31no—
47#5561—71nomaximal, Hall
48C23×C3×C7C2312yes—
49C22×D21C2311yescharacteristic
50C3×C7×D4—22no—
51C2×C3×C4×C7—42no—
52C2×C3×C4×C7C2311yescharacteristic
53C2×C3×C4×C7C2311yescharacteristic
54C2×C3×C4×C7C2312yes—
55C2×C3×C4×C7C2311yescharacteristic
56#4268—42no—
57C2×Dic21C2311yescharacteristic
58C2×Dic21C2312yes—
59C2×Dic21C2312yes—
60C2×Dic21C2311yescharacteristic
61C2×Dic21C2311yescharacteristic
62C4×D21—22no—
63Dic42—22no—
64Dic42—42no—
65C2×C7×D4C2×D311yescharacteristic
66C22×C4×C7C2×D312yes—
67C2×C22⋊D7—32no—
68C2×C4×D7—31no—
69C2×Dic14—31no—
70C2×Dic14—32no—
71C7×C4⋊C4C2×D311yescharacteristic
72C7×C22⋊C4C2×D312yes—
73C42×C7C2×D311yescharacteristic
74C22⋊Dic7—32no—
75C22⋊Dic7—32no—
76C22⋊Dic7—32no—
77(C2×D7)⋊C4—31no—
78(C2×D7)⋊C4—31no—
79(C2×D7)⋊C4—32no—
80C4⋊Dic7—31no—
81C4⋊Dic7—32no—
82Dic7⋊C4—32no—
83Dic7⋊C4—31no—
84Dic7⋊C4—31no—
85Dic7⋊C4—32no—
86C4×Dic7—32no—
87C4×Dic7—31no—
88C3×C4×D4D711yescharacteristic, complemented
89#938—71no—
90#936—71no—
91#929—72no—
92#924—72no—
93#897—71no—
94#888—71no—
95#885—72no—
96#879—72no—
97#872—71no—
98C4×Dic6—71no—
99C22×C3×C7—22no—
100C22×C3×C7—22no—
101C22×C3×C7—22no—
102C22×C3×C7C2411yescharacteristic
103C2×D21—21no—
104C2×D21—21no—
105C2×D21—21no—
106C3×C4×C7—22no—
107C3×C4×C7—21no—
108C3×C4×C7—21no—
109C3×C4×C7#4112yes—
110Dic21—22no—
111Dic21—21no—
112Dic21—21no—
113Dic21—22no—
114Dic21—21no—
115#542—211noSylow
116C23×C7C22×D312yes—
117C22×D7—31no—
118C7×D4—22no—
119C2×C4×C7—42no—
120C2×C4×C7C22×D311yescharacteristic
121C2×C4×C7C22×D311yescharacteristic
122C2×C4×C7C22×D312yes—
123C2×C4×C7C22×D311yescharacteristic
124C22⋊D7—122no—
125C2×Dic7—31no—
126C2×Dic7—32no—
127C2×Dic7—32no—
128C2×Dic7—31no—
129C2×Dic7—31no—
130C4×D7—62no—
131Dic14—62no—
132Dic14—122no—
133C2×C3×D4C2×D711yescharacteristic
134C22×C3×C4C2×D712yes—
135C2×#67—72no—
136C2×C4×D3—71no—
137C2×Dic6—71no—
138C2×Dic6—72no—
139C3×C4⋊C4C2×D711yescharacteristic
140C3×C22⋊C4C2×D712yes—
141C3×C42C2×D711yescharacteristic
142#217—72no—
143#217—72no—
144#217—72no—
145(C2×D3)⋊C4—71no—
146(C2×D3)⋊C4—71no—
147(C2×D3)⋊C4—72no—
148C4⋊Dic3—71no—
149C4⋊Dic3—72no—
150Dic3⋊C4—72no—
151Dic3⋊C4—71no—
152Dic3⋊C4—71no—
153Dic3⋊C4—72no—
154C4×Dic3—72no—
155C4×Dic3—71no—
156C2×C3×C7—42no—
157C2×C3×C7C2×#4111yescharacteristic
158C2×C3×C7H(5,𝔽2)11yescharacteristic
159C2×C3×C72−1+411yescharacteristic
160D21—41no—
161C4⋊Q8—211no—
162#126—212no—
163#124—211no—
164#124—212no—
165#123—212no—
166C22⋊Q8—212no—
167C22⋊Q8—211no—
168#121—211no—
169C4×Q8—211no—
170C4×D4D2111yescharacteristic, complemented
171#117—211no—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—22no—
175C22×C7C23×D311yescharacteristic
176C2×D7—61no—
177C2×D7—61no—
178C2×D7—61no—
179C4×C7—22no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7#23512yes—
183Dic7—62no—
184Dic7—61no—
185Dic7—61no—
186Dic7—62no—
187Dic7—61no—
188C23×C3C22×D712yes—
189C22×D3—71no—
190C3×D4—22no—
191C2×C3×C4—42no—
192C2×C3×C4C22×D711yescharacteristic
193C2×C3×C4C22×D711yescharacteristic
194C2×C3×C4C22×D712yes—
195C2×C3×C4C22×D711yescharacteristic
196#67—282no—
197C2×Dic3—71no—
198C2×Dic3—72no—
199C2×Dic3—72no—
200C2×Dic3—71no—
201C2×Dic3—71no—
202C4×D3—142no—
203Dic6—142no—
204Dic6—282no—
205C3×C7#54211yescharacteristic, complemented, Hall
206C2×Q8—211no—
207C2×Q8—212no—
208C2×D4—212no—
209C2×D4C2×D2111yescharacteristic
210C22×C4—211no—
211C22×C4C2×D2112yes—
212C4⋊C4—211no—
213C4⋊C4—211no—
214C4⋊C4—212no—
215C4⋊C4—212no—
216C4⋊C4—211no—
217C4⋊C4—212no—
218C4⋊C4C2×D2111yescharacteristic
219C22⋊C4—212no—
220C22⋊C4—212no—
221C22⋊C4—212no—
222C22⋊C4—211no—
223C22⋊C4—211no—
224C22⋊C4—212no—
225C22⋊C4C2×D2112yes—
226C42—211no—
227C42—212no—
228C42C2×D2111yescharacteristic
229C2×C7—42no—
230C2×C7C2×#23511yescharacteristic
231C2×C7#100411yescharacteristic
232C2×C7#101011yescharacteristic
233D7—121no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3C23×D711yescharacteristic
238C2×D3—141no—
239C2×D3—141no—
240C2×D3—141no—
241C3×C4—22no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4#115712yes—
245Dic3—142no—
246Dic3—141no—
247Dic3—141no—
248Dic3—142no—
249Dic3—141no—
250C23—211no—
251C23C22×D2112yes—
252Q8—422no—
253Q8—842no—
254D4—22no—
255D4—842no—
256C2×C4—42no—
257C2×C4—211no—
258C2×C4—212no—
259C2×C4—211no—
260C2×C4—212no—
261C2×C4—211no—
262C2×C4—422no—
263C2×C4C22×D2111yescharacteristic
264C2×C4C22×D2111yescharacteristic
265C2×C4C22×D2112yes—
266C2×C4C22×D2111yescharacteristic
267C7#556111yescharacteristic, complemented, Sylow
268C2×C3—42no—
269C2×C3C2×#115711yescharacteristic
270C2×C3#654811yescharacteristic
271C2×C3#655411yescharacteristic
272D3—281no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—421no—
277C22—421no—
278C22—421no—
279C22C23×D2111yescharacteristic
280C4—21no—
281C4—22no—
282C4—21no—
283C4—421no—
284C4—422no—
285C4—422no—
286C4—421no—
287C4—421no—
288C4#6672812yes—
289C3#9025211yescharacteristic, complemented, Sylow
290C2—42no—
291C2—841no—
292C2C2×#6672811yescharacteristic
293C2#1062165711yescharacteristic
294C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C4×C7×D4C2
prime radical#10621421C2
hypercenterC4×D4D21
hypocenterC3×C7#542
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
3. #10621419C2semidirect
4. #10621412C2semidirect
5. #10621407C2semidirect
6. #10621380C2semidirect
7. #10621371C2semidirect
8. #10621368C2semidirect
9. #10621362C2semidirect
10. #10621355C2semidirect
11. C4×Dic42C2semidirect
36. C4×C7×D4D3semidirect
88. C3×C4×D4D7semidirect
170. C4×D4D21semidirect
205. C3×C7#542semidirect
267. C7#5561semidirect
289. C3#90252semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C7×D42
#106214212
#106214192
#106214122
#106214072
#106213802
#106213712
#106213682
#106213622
#106213552
C4×Dic422
#902523
#55617