Finite Group Databasebeta

← #13059579

Subgroups — #13059579

Order 1344.

428 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621407C211yesmaximal, characteristic, complemented
2C7×#926C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊D7C211yesmaximal, characteristic
4#10621110C212yesmaximal
5#10621093C212yesmaximal, complemented
6D7×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#10620955C212yesmaximal, complemented
8#10620950C211yesmaximal, characteristic, complemented
9#10620941C212yesmaximal, complemented
10#10620922C211yesmaximal, characteristic, complemented
11#10620895C211yesmaximal, characteristic, complemented
12#90220—31nomaximal, Hall
13C22×C3×C4×C7C2211yescharacteristic
14C2×Dic42C2211yescharacteristic
15C2×C3×C22⋊D7C2211yescharacteristic
16C2×C3×C4×D7C2211yescharacteristic
17C2×Dic3×D7C2212yes—
18#66636C2212yes—
19C4⋊Dic21C2211yescharacteristic
20#66629C2212yes—
21C7×#217C2212yes—
22C7×C4⋊Dic3C2211yescharacteristic
23C7×C4⋊Dic3C2211yescharacteristic
24C4×C7×Dic3C2212yes—
25C3×C22⋊Dic7C2211yescharacteristic
26C3×(C2×D7)⋊C4C2211yescharacteristic
27C3×Dic7⋊C4C2211yescharacteristic
28C3×C4×Dic7C2211yescharacteristic
29Dic3⋊Dic7C2212yes—
30Dic3⋊Dic7C2212yes—
31#66577C2212yes—
32#66577C2212yes—
33Dic7⋊Dic3C2212yes—
34(C2×D7)⋊Dic3C2212yes—
35Dic3×Dic7C2212yes—
36C7×#117—31no—
37C4×C22⋊D7D311yescharacteristic
38#6487—31no—
39#6460—31no—
40D7×C4⋊C4—31no—
41#6453—31no—
42#6450—31no—
43#6446—32no—
44#6442—32no—
45#6435—32no—
46#6433—32no—
47#5541—71nomaximal, Hall
48C23×C3×C7C2311yescharacteristic
49C22×C3×D7C2311yescharacteristic
50C2×C3×C4×C7—22no—
51C2×C3×C4×C7C2311yescharacteristic
52C2×C3×C4×C7C2311yescharacteristic
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55Dic42—22no—
56C2×C7×Dic3C2312yes—
57C2×C7×Dic3C2312yes—
58C3×C22⋊D7—22no—
59C2×C3×Dic7C2311yescharacteristic
60C2×C3×Dic7C2311yescharacteristic
61C2×C3×Dic7C2311yescharacteristic
62C3×C4×D7—22no—
63Dic3×D7—24no—
64C22×C4×C7C2×D311yescharacteristic
65C2×C22⋊D7C2×D311yescharacteristic
66C2×C4×D7—32no—
67C2×C4×D7C2×D311yescharacteristic
68C2×Dic14—31no—
69C7×C4⋊C4—31no—
70C7×C4⋊C4—31no—
71C7×C22⋊C4—32no—
72C42×C7—32no—
73C22⋊Dic7—32no—
74C22⋊Dic7C2×D311yescharacteristic
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4C2×D311yescharacteristic
77C4⋊Dic7—31no—
78C4⋊Dic7—32no—
79C4⋊Dic7—32no—
80Dic7⋊C4—32no—
81Dic7⋊C4—32no—
82Dic7⋊C4—32no—
83Dic7⋊C4—32no—
84Dic7⋊C4C2×D311yescharacteristic
85C4×Dic7—32no—
86C4×Dic7C2×D311yescharacteristic
87C3×C4×D4—71no—
88#935—72no—
89#926D711yescharacteristic, complemented
90C2×C4⋊Dic3—71no—
91#924—71no—
92#924—71no—
93#889—72no—
94#879—72no—
95#877—72no—
96#870—71no—
97C4×Dic6—71no—
98C22×C3×C7—21no—
99C22×C3×C7—21no—
100C22×C3×C7—21no—
101C22×C3×C7C2411yescharacteristic
102C2×C3×D7—21no—
103C2×C3×D7—21no—
104C2×C3×D7#4112yes—
105C3×C4×C7—21no—
106C3×C4×C7#4112yes—
107Dic21—22no—
108Dic21—22no—
109C3×Dic7—21no—
110C3×Dic7—21no—
111C3×Dic7—21no—
112C7×Dic3—22no—
113C7×Dic3—22no—
114#552—211noSylow
115C23×C7C22×D311yescharacteristic
116C22×D7C22×D311yescharacteristic
117C2×C4×C7—22no—
118C2×C4×C7—32no—
119C2×C4×C7—32no—
120C2×C4×C7C22×D311yescharacteristic
121C2×C4×C7C22×D311yescharacteristic
122C22⋊D7—22no—
123C2×Dic7—32no—
124C2×Dic7—32no—
125C2×Dic7C22×D311yescharacteristic
126C2×Dic7C22×D311yescharacteristic
127C2×Dic7C22×D311yescharacteristic
128C4×D7—22no—
129C4×D7—64no—
130Dic14—62no—
131C2×C3×D4—71no—
132C22×C3×C4—71no—
133C22×C3×C4C2×D711yescharacteristic
134C22×Dic3—72no—
135C2×Dic6—71no—
136C3×C4⋊C4—71no—
137C3×C22⋊C4—71no—
138C3×C22⋊C4—71no—
139C3×C42—71no—
140#217—72no—
141#217—72no—
142#217C2×D712yes—
143C4⋊Dic3—71no—
144C4⋊Dic3—72no—
145C4⋊Dic3C2×D711yescharacteristic
146C4⋊Dic3C2×D711yescharacteristic
147Dic3⋊C4—72no—
148Dic3⋊C4—72no—
149Dic3⋊C4—72no—
150Dic3⋊C4—72no—
151Dic3⋊C4—72no—
152C4×Dic3—72no—
153C4×Dic3C2×D712yes—
154C2×C3×C7—41no—
155C2×C3×C7C2×#4111yescharacteristic
156C2×C3×C7C2×#4111yescharacteristic
157C2×C3×C72−1+411yescharacteristic
158C3×D7—22no—
159#126—212no—
160#125—212no—
161#125—211no—
162#123—212no—
163C22⋊Q8—211no—
164C22⋊Q8—211no—
165C4×Q8—211no—
166C4×D4—71no—
167#117—31no—
168#117—212no—
169C2×C4⋊C4—211no—
170C22×C7—21no—
171C22×C7—21no—
172C22×C7—21no—
173C22×C7C23×D311yescharacteristic
174C2×D7—21no—
175C2×D7—21no—
176C2×D7#23512yes—
177C4×C7—21no—
178C4×C7—62no—
179C4×C7—62no—
180C4×C7#23712yes—
181Dic7—21no—
182Dic7—21no—
183Dic7—21no—
184Dic7—62no—
185Dic7—62no—
186C23×C3—71no—
187C23×C3C22×D711yescharacteristic
188C3×D4—142no—
189C2×C3×C4—22no—
190C2×C3×C4—71no—
191C2×C3×C4—71no—
192C2×C3×C4—71no—
193C2×C3×C4—142no—
194C2×C3×C4C22×D711yescharacteristic
195C2×C3×C4C22×D711yescharacteristic
196C2×Dic3—72no—
197C2×Dic3—72no—
198C2×Dic3—144no—
199C2×Dic3C22×D712yes—
200C2×Dic3C22×D712yes—
201Dic6—142no—
202C3×C7#55211yescharacteristic, complemented, Hall
203C2×Q8—211no—
204C2×D4—71no—
205C22×C4—71no—
206C22×C4—212no—
207C22×C4D3×D711yescharacteristic
208C4⋊C4—31no—
209C4⋊C4—31no—
210C4⋊C4—71no—
211C4⋊C4—212no—
212C4⋊C4—212no—
213C4⋊C4—212no—
214C4⋊C4—212no—
215C4⋊C4—212no—
216C4⋊C4—212no—
217C4⋊C4—211no—
218C22⋊C4—32no—
219C22⋊C4—71no—
220C22⋊C4—71no—
221C22⋊C4—212no—
222C22⋊C4—212no—
223C42—32no—
224C42—71no—
225C42—212no—
226C2×C7—41no—
227C2×C7C2×#23511yescharacteristic
228C2×C7C2×#23711yescharacteristic
229C2×C7#101011yescharacteristic
230D7—22no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—72no—
235C22×C3—141no—
236C22×C3—141no—
237C22×C3C23×D711yescharacteristic
238C3×C4—21no—
239C3×C4—141no—
240C3×C4—141no—
241C3×C4—141no—
242C3×C4#115712yes—
243Dic3—22no—
244Dic3—22no—
245Dic3—142no—
246Dic3—142no—
247C23—71no—
248C23C2×D3×D711yescharacteristic
249Q8—422no—
250D4—142no—
251C2×C4—22no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—71no—
255C2×C4—71no—
256C2×C4—71no—
257C2×C4—142no—
258C2×C4—212no—
259C2×C4—212no—
260C2×C4—424no—
261C2×C4C2×D3×D711yescharacteristic
262C2×C4C2×D3×D711yescharacteristic
263C7#554111yescharacteristic, complemented, Sylow
264C2×C3—41no—
265C2×C3—142no—
266C2×C3C2×#115711yescharacteristic
267C2×C3D7×#4111yescharacteristic
268C2×C3#655411yescharacteristic
269C22—21no—
270C22—21no—
271C22—21no—
272C22—72no—
273C22—141no—
274C22—141no—
275C22C22×D3×D711yescharacteristic
276C4—21no—
277C4—62no—
278C4—62no—
279C4—141no—
280C4—141no—
281C4—141no—
282C4—422no—
283C4—422no—
284C4#6668312yes—
285C3#9022011yescharacteristic, complemented, Sylow
286C2—41no—
287C2—142no—
288C2#1062154811yescharacteristic
289C2D7×#23511yescharacteristic
290C2C2×#6668311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
prime radicalC2×C3×C22⋊D7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#552
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #10621407C2semidirect
2. C7×#926C2semidirect
5. #10621093C2semidirect
6. D7×C4⋊Dic3C2semidirect
7. #10620955C2semidirect
8. #10620950C2semidirect
9. #10620941C2semidirect
10. #10620922C2semidirect
11. #10620895C2semidirect
89. #926D7semidirect
202. C3×C7#552semidirect
263. C7#5541semidirect
285. C3#90220semidirect

Maximal subgroups

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

subgroupindex
#106214072
C7×#9262
C3×C4×C22⋊D72
#106211102
#106210932
D7×C4⋊Dic32
#106209552
#106209502
#106209412
#106209222
#106208952
#902203
#55417