Finite Group Databasebeta

← #13061774

Subgroups — #13061774

Order 1344.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C3×C7×#124C211yesmaximal, characteristic, complemented
2#10621429C212yesmaximal, complemented
3#10621422C212yesmaximal, complemented
4#10621369C212yesmaximal, complemented
5#10621367C212yesmaximal, complemented
6#10621366C212yesmaximal, complemented
7#10621361C212yesmaximal, complemented
8#10621358C211yesmaximal, characteristic, complemented
9#10621353C211yesmaximal, characteristic, complemented
10#90375—31nomaximal, Hall
11C2×C3×C7×Q8C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2212yes—
14C2×#4268C2212yes—
15C22×Dic21C2212yes—
16C2×D84C2211yescharacteristic
17C2×C4×D21C2212yes—
18C2×Dic42C2211yescharacteristic
19C3×C7×C22⋊C4C2212yes—
20C3×C7×C22⋊C4C2212yes—
21C3×C42×C7C2211yescharacteristic
22#66636C2212yes—
23#66631C2212yes—
24#66631C2212yes—
25#66631C2212yes—
26C4⋊Dic21C2211yescharacteristic
27C4⋊Dic21C2211yescharacteristic
28C4⋊Dic21C2212yes—
29#66629C2212yes—
30#66629C2212yes—
31#66629C2212yes—
32C7×#124D311yescharacteristic, complemented
33#6509—32no—
34#6502—32no—
35#6449—32no—
36#6447—32no—
37#6446—32no—
38#6441—32no—
39#6438—31no—
40#6433—31no—
41#5684—71nomaximal, Hall
42C23×C3×C7C2312yes—
43C22×D21C2312yes—
44C3×C7×Q8—41no—
45C3×C7×D4—41no—
46C2×C3×C4×C7C2311yescharacteristic
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2312yes—
49C2×C3×C4×C7C2311yescharacteristic
50#4268—42no—
51#4268—42no—
52C2×Dic21—42no—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56D84—41no—
57C4×D21—42no—
58Dic42—41no—
59C2×C7×Q8C2×D311yescharacteristic
60C2×C7×D4C2×D311yescharacteristic
61C2×C22⋊D7—32no—
62C2×C22⋊D7—32no—
63C22×Dic7—32no—
64C2×D28—31no—
65C2×C4×D7—32no—
66C2×Dic14—31no—
67C7×C22⋊C4C2×D312yes—
68C7×C22⋊C4C2×D312yes—
69C42×C7C2×D311yescharacteristic
70C22⋊Dic7—32no—
71(C2×D7)⋊C4—32no—
72(C2×D7)⋊C4—32no—
73(C2×D7)⋊C4—32no—
74C4⋊Dic7—31no—
75C4⋊Dic7—31no—
76C4⋊Dic7—32no—
77Dic7⋊C4—32no—
78Dic7⋊C4—32no—
79Dic7⋊C4—32no—
80C3×#124D711yescharacteristic, complemented
81#946—72no—
82#939—72no—
83#886—72no—
84#884—72no—
85#883—72no—
86#878—72no—
87#875—71no—
88#870—71no—
89C22×C3×C7—22no—
90C22×C3×C7—22no—
91C22×C3×C7—22no—
92C22×C3×C7C2411yescharacteristic
93C2×D21—22no—
94C2×D21—22no—
95C2×D21—22no—
96C3×C4×C7—21no—
97C3×C4×C7—22no—
98C3×C4×C7—21no—
99C3×C4×C7—21no—
100Dic21—22no—
101Dic21—22no—
102Dic21—22no—
103#562—211noSylow
104C23×C7C22×D312yes—
105C22×D7—32no—
106C7×Q8—41no—
107C7×D4—41no—
108C2×C4×C7C22×D311yescharacteristic
109C2×C4×C7C22×D311yescharacteristic
110C2×C4×C7C22×D312yes—
111C2×C4×C7C22×D311yescharacteristic
112C22⋊D7—122no—
113C22⋊D7—122no—
114C2×Dic7—32no—
115C2×Dic7—32no—
116C2×Dic7—32no—
117C2×Dic7—122no—
118D28—121no—
119C4×D7—122no—
120Dic14—121no—
121C2×C3×Q8C2×D711yescharacteristic
122C2×C3×D4C2×D711yescharacteristic
123C2×#67—72no—
124C2×#67—72no—
125C22×Dic3—72no—
126C2×D12—71no—
127C2×C4×D3—72no—
128C2×Dic6—71no—
129C3×C22⋊C4C2×D712yes—
130C3×C22⋊C4C2×D712yes—
131C3×C42C2×D711yescharacteristic
132#217—72no—
133(C2×D3)⋊C4—72no—
134(C2×D3)⋊C4—72no—
135(C2×D3)⋊C4—72no—
136C4⋊Dic3—71no—
137C4⋊Dic3—71no—
138C4⋊Dic3—72no—
139Dic3⋊C4—72no—
140Dic3⋊C4—72no—
141Dic3⋊C4—72no—
142C2×C3×C7—42no—
143C2×C3×C7H(5,𝔽2)11yescharacteristic
144C2×C3×C7H(5,𝔽2)11yescharacteristic
145C2×C3×C72−1+411yescharacteristic
146D21—42no—
147#125—211no—
148#124—211no—
149#124D2111yescharacteristic, complemented
150#123—212no—
151#123—212no—
152C22⋊Q8—212no—
153C22⋊Q8—212no—
154#121—212no—
155#121—212no—
156C22×C7—22no—
157C22×C7—22no—
158C22×C7—22no—
159C22×C7C23×D311yescharacteristic
160C2×D7—62no—
161C2×D7—62no—
162C2×D7—62no—
163C4×C7—22no—
164C4×C7—21no—
165C4×C7—21no—
166C4×C7—21no—
167Dic7—62no—
168Dic7—62no—
169Dic7—62no—
170C23×C3C22×D712yes—
171C22×D3—72no—
172C3×Q8—41no—
173C3×D4—41no—
174C2×C3×C4C22×D711yescharacteristic
175C2×C3×C4C22×D711yescharacteristic
176C2×C3×C4C22×D712yes—
177C2×C3×C4C22×D711yescharacteristic
178#67—282no—
179#67—282no—
180C2×Dic3—72no—
181C2×Dic3—72no—
182C2×Dic3—72no—
183C2×Dic3—282no—
184D12—281no—
185C4×D3—282no—
186Dic6—281no—
187C3×C7#56211yescharacteristic, complemented, Hall
188C2×Q8—211no—
189C2×Q8C2×D2111yescharacteristic
190C2×D4—212no—
191C2×D4—212no—
192C2×D4—211no—
193C2×D4C2×D2111yescharacteristic
194C22×C4—212no—
195C22×C4—212no—
196C4⋊C4—212no—
197C4⋊C4—212no—
198C4⋊C4—212no—
199C4⋊C4—212no—
200C4⋊C4—211no—
201C4⋊C4—211no—
202C22⋊C4—212no—
203C22⋊C4—212no—
204C22⋊C4—212no—
205C22⋊C4—212no—
206C22⋊C4C2×D2112yes—
207C22⋊C4C2×D2112yes—
208C42C2×D2111yescharacteristic
209C2×C7—42no—
210C2×C7#100411yescharacteristic
211C2×C7#100911yescharacteristic
212C2×C7#101011yescharacteristic
213D7—122no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3C23×D711yescharacteristic
218C2×D3—142no—
219C2×D3—142no—
220C2×D3—142no—
221C3×C4—22no—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—21no—
225Dic3—142no—
226Dic3—142no—
227Dic3—142no—
228C23—212no—
229C23C22×D2112yes—
230Q8—41no—
231Q8—841no—
232D4—41no—
233D4—841no—
234D4—842no—
235D4—842no—
236C2×C4—212no—
237C2×C4—212no—
238C2×C4—212no—
239C2×C4—842no—
240C2×C4—842no—
241C2×C4C22×D2111yescharacteristic
242C2×C4C22×D2111yescharacteristic
243C2×C4C22×D2111yescharacteristic
244C2×C4C22×D2112yes—
245C7#568411yescharacteristic, complemented, Sylow
246C2×C3—42no—
247C2×C3#654811yescharacteristic
248C2×C3#655311yescharacteristic
249C2×C3#655411yescharacteristic
250D3—282no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—422no—
255C22—422no—
256C22—422no—
257C22C23×D2111yescharacteristic
258C4—22no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—422no—
263C4—422no—
264C4—422no—
265C3#9037511yescharacteristic, complemented, Sylow
266C2—42no—
267C2—842no—
268C2#1062165711yescharacteristic
269C2#1062166211yescharacteristic
270C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
hypercenter#124D21
hypocenterC3×C7#562
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C3×C7×#124C2semidirect
2. #10621429C2semidirect
3. #10621422C2semidirect
4. #10621369C2semidirect
5. #10621367C2semidirect
6. #10621366C2semidirect
7. #10621361C2semidirect
8. #10621358C2semidirect
9. #10621353C2semidirect
32. C7×#124D3semidirect
80. C3×#124D7semidirect
149. #124D21semidirect
187. C3×C7#562semidirect
245. C7#5684semidirect
265. C3#90375semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1242
#106214292
#106214222
#106213692
#106213672
#106213662
#106213612
#106213582
#106213532
#903753
#56847