Finite Group Databasebeta

← #13061767

Subgroups — #13061767

Order 1344.

620 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 151 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×#66732C211yesmaximal, characteristic, complemented
2C2×#66730C211yesmaximal, characteristic, complemented
3C3×C7×#124C211yesmaximal, characteristic, complemented
4#10621427C211yesmaximal, characteristic, complemented
5#10621423C211yesmaximal, characteristic, complemented
6#10621367C214yesmaximal, complemented
7#10621364C214yesmaximal, complemented
8#10621358C211yesmaximal, characteristic, complemented
9C42×D21C211yesmaximal, characteristic, complemented
10#90359—31nomaximal, Hall
11C2×C3×C7×Q8C2211yescharacteristic
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2214yes—
14C2×#4268C2212yes—
15C22×Dic21C2212yes—
16#66732—24no—
17#66730—24no—
18C2×D84C2211yescharacteristic
19C2×D84C2212yes—
20C2×C4×D21C2212yes—
21C2×C4×D21C2212yes—
22C2×C4×D21C2211yescharacteristic
23C2×Dic42C2211yescharacteristic
24C3×C7×C22⋊C4C2214yes—
25C3×C42×C7C2211yescharacteristic
26#66631C2214yes—
27#66629C2214yes—
28C4×Dic21C2212yes—
29C4×Dic21C2211yescharacteristic
30C2×#1161—31no—
31C2×#1159—31no—
32C7×#124D311yescharacteristic, complemented
33#6507—31no—
34#6503—31no—
35#6447—34no—
36(C22⋊D7)⋊C4—34no—
37#6438—31no—
38C42×D7—31no—
39#5668—71nomaximal, Hall
40C23×C3×C7C2312yes—
41C22×D21C2312yes—
42C22×D21C2311yescharacteristic
43C3×C7×Q8—22no—
44C3×C7×D4—22no—
45C2×C3×C4×C7C2312yes—
46C2×C3×C4×C7C2312yes—
47C2×C3×C4×C7C2311yescharacteristic
48#4268—24no—
49#4268—28no—
50C2×Dic21—24no—
51C2×Dic21C2312yes—
52C2×Dic21C2311yescharacteristic
53C2×Dic21C2312yes—
54D84—24no—
55D84—22no—
56C4×D21—24no—
57C4×D21—24no—
58C4×D21D414yescomplemented
59Dic42—22no—
60C2×C7×Q8C2×D311yescharacteristic
61C2×C7×D4C2×D311yescharacteristic
62C2×C22⋊D7—34no—
63C2×C22⋊D7—32no—
64C22×Dic7—32no—
65#1161—64no—
66#1159—64no—
67C2×D28—31no—
68C2×D28—32no—
69C2×C4×D7—31no—
70C2×C4×D7—32no—
71C2×C4×D7—32no—
72C2×Dic14—31no—
73C7×C22⋊C4C2×D314yes—
74C42×C7C2×D311yescharacteristic
75(C2×D7)⋊C4—34no—
76Dic7⋊C4—34no—
77C4×Dic7—32no—
78C4×Dic7—31no—
79C2×#239—71no—
80C2×#237—71no—
81C3×#124D711yescharacteristic, complemented
82#944—71no—
83#940—71no—
84#884—74no—
85#67⋊C4—74no—
86#875—71no—
87C42×D3—71no—
88C22×C3×C7—22no—
89C22×C3×C7—24no—
90C22×C3×C7C2411yescharacteristic
91C2×D21—22no—
92C2×D21—22no—
93C2×D21—24no—
94C2×D21C2×D412yes—
95C3×C4×C7—22no—
96C3×C4×C7—22no—
97C3×C4×C7C2×D412yes—
98Dic21—22no—
99Dic21C2×D412yes—
100Dic21#4114yescomplemented
101#532—211noSylow
102C23×C7C22×D312yes—
103C22×D7—31no—
104C22×D7—32no—
105C7×Q8—22no—
106C7×D4—22no—
107C2×C4×C7C22×D312yes—
108C2×C4×C7C22×D312yes—
109C2×C4×C7C22×D311yescharacteristic
110C22⋊D7—68no—
111C22⋊D7—64no—
112C2×Dic7—32no—
113C2×Dic7—31no—
114C2×Dic7—32no—
115C2×Dic7—64no—
116D28—64no—
117D28—62no—
118C4×D7—34no—
119C4×D7—64no—
120C4×D7—64no—
121Dic14—62no—
122C2×C3×Q8C2×D711yescharacteristic
123C2×C3×D4C2×D711yescharacteristic
124C2×#67—74no—
125C2×#67—72no—
126C22×Dic3—72no—
127#239—144no—
128#237—144no—
129C2×D12—71no—
130C2×D12—72no—
131C2×C4×D3—71no—
132C2×C4×D3—72no—
133C2×C4×D3—72no—
134C2×Dic6—71no—
135C3×C22⋊C4C2×D714yes—
136C3×C42C2×D711yescharacteristic
137(C2×D3)⋊C4—74no—
138Dic3⋊C4—74no—
139C4×Dic3—72no—
140C4×Dic3—71no—
141C2×C3×C7—42no—
142C2×C3×C7C22×D411yescharacteristic
143C2×C3×C7C2×#4112yes—
144D21—22no—
145D21—42no—
146C2×#41—211no—
147C2×#41—211no—
148C4⋊Q8—211no—
149#127—211no—
150#124—211no—
151#124D2111yescharacteristic, complemented
152#121—214no—
153C4×D4—214no—
154C2×C42—211no—
155C22×C7—22no—
156C22×C7—24no—
157C22×C7C23×D311yescharacteristic
158C2×D7—32no—
159C2×D7—62no—
160C2×D7—62no—
161C2×D7—64no—
162C4×C7—22no—
163C4×C7—22no—
164C4×C7D3×D412yes—
165Dic7—32no—
166Dic7—34no—
167Dic7—62no—
168C23×C3C22×D712yes—
169C22×D3—71no—
170C22×D3—72no—
171C3×Q8—22no—
172C3×D4—22no—
173C2×C3×C4C22×D712yes—
174C2×C3×C4C22×D712yes—
175C2×C3×C4C22×D711yescharacteristic
176#67—148no—
177#67—144no—
178C2×Dic3—72no—
179C2×Dic3—71no—
180C2×Dic3—72no—
181C2×Dic3—144no—
182D12—144no—
183D12—142no—
184C4×D3—74no—
185C4×D3—144no—
186C4×D3—144no—
187Dic6—142no—
188C3×C7#53211yescharacteristic, complemented, Hall
189#41—424no—
190#41—424no—
191C2×Q8—211no—
192C2×Q8C2×D2111yescharacteristic
193C2×D4—214no—
194C2×D4—212no—
195C2×D4—212no—
196C2×D4—211no—
197C2×D4C2×D2111yescharacteristic
198C22×C4—211no—
199C22×C4—212no—
200C22×C4—212no—
201C22×C4—212no—
202C4⋊C4—214no—
203C22⋊C4—214no—
204C22⋊C4C2×D2114yes—
205C42—212no—
206C42—211no—
207C42C2×D2111yescharacteristic
208C2×C7—42no—
209C2×C7C2×D3×D411yescharacteristic
210C2×C7D3×#4112yes—
211D7—62no—
212D7—122no—
213C22×C3—22no—
214C22×C3—24no—
215C22×C3C23×D711yescharacteristic
216C2×D3—72no—
217C2×D3—142no—
218C2×D3—142no—
219C2×D3—144no—
220C3×C4—22no—
221C3×C4—22no—
222C3×C4D4×D712yes—
223Dic3—72no—
224Dic3—74no—
225Dic3—142no—
226C23—211no—
227C23—212no—
228C23C22×D2112yes—
229Q8—22no—
230Q8—422no—
231D4—22no—
232D4—424no—
233D4—428no—
234D4—424no—
235D4—422no—
236C2×C4—212no—
237C2×C4—211no—
238C2×C4—214no—
239C2×C4—212no—
240C2×C4—424no—
241C2×C4—424no—
242C2×C4—424no—
243C2×C4C22×D2112yes—
244C2×C4C22×D2112yes—
245C2×C4C22×D2111yescharacteristic
246C7#566811yescharacteristic, complemented, Sylow
247C2×C3—42no—
248C2×C3C2×D4×D711yescharacteristic
249C2×C3D7×#4112yes—
250D3—142no—
251D3—282no—
252C22—22no—
253C22—24no—
254C22—212no—
255C22—422no—
256C22—422no—
257C22—424no—
258C22C23×D2111yescharacteristic
259C4—22no—
260C4—22no—
261C4—212no—
262C4—214no—
263C4—422no—
264C4D4×D2112yes—
265C3#9035911yescharacteristic, complemented, Sylow
266C2—42no—
267C2—422no—
268C2—842no—
269C2C2×D4×D2111yescharacteristic
270C2#41×D2112yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
hypercenter#124D21
hypocenterC3×C7#532
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#66732C2semidirect
2. C2×#66730C2semidirect
3. C3×C7×#124C2semidirect
4. #10621427C2semidirect
5. #10621423C2semidirect
6. #10621367C2semidirect
7. #10621364C2semidirect
8. #10621358C2semidirect
9. C42×D21C2semidirect
32. C7×#124D3semidirect
58. C4×D21D4semidirect
81. C3×#124D7semidirect
100. Dic21#41semidirect
151. #124D21semidirect
188. C3×C7#532semidirect
246. C7#5668semidirect
265. C3#90359semidirect

Maximal subgroups

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

subgroupindex
C2×#667322
C2×#667302
C3×C7×#1242
#106214272
#106214232
#106213672
#106213642
#106213582
C42×D212
#903593
#56687