Finite Group Databasebeta

← #124×D21 #13061764

Subgroups — #124×D21 #13061764

Order 1344.

660 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 155 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×Q8×D21C211yesmaximal, characteristic, complemented
2C2×D4×D21C211yesmaximal, characteristic
3C3×C7×#124C211yesmaximal, characteristic, complemented
4#10621430C211yesmaximal, characteristic, complemented
5#10621419C211yesmaximal, characteristic, complemented
6#10621368C214yesmaximal, complemented
7C22⋊C4×D21C214yesmaximal, complemented
8#10621358C211yesmaximal, characteristic, complemented
9C42×D21C211yesmaximal, characteristic, complemented
10D7×#124—31nomaximal, Hall
11C23×D21C2212yes—
12C2×C3×C7×Q8C2211yescharacteristic, complemented
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15Q8×D21—24no—
16D4×D21—24no—
17C2×D84C2211yescharacteristic
18C2×C4×D21C2212yes—
19C2×C4×D21C2212yes—
20C2×C4×D21C2211yescharacteristic
21C2×Dic42C2211yescharacteristic, complemented
22C2×Dic42C2212yescomplemented
23C3×C7×C22⋊C4C2214yescomplemented
24C3×C42×C7C2211yescharacteristic, complemented
25#66636C2214yescomplemented
26#66631C2214yescomplemented
27#66631C2214yescomplemented
28C4×Dic21C2212yescomplemented
29C4×Dic21C2211yescharacteristic, complemented
30C2×Q8×D7—31no—
31C2×D4×D7—31no—
32C7×#124D311yescharacteristic, complemented
33#6510—31no—
34#6499—31no—
35#6448—34no—
36D7×C22⋊C4—34no—
37#6438—31no—
38C42×D7—31no—
39D3×#124—71nomaximal, Hall
40C23×C3×C7C2312yes—
41C22×D21—24no—
42C22×D21—28no—
43C22×D21C2312yes—
44C22×D21C2311yescharacteristic
45C3×C7×Q8—22no—
46C3×C7×D4—22no—
47C2×C3×C4×C7C2312yes—
48C2×C3×C4×C7C2312yes—
49C2×C3×C4×C7C2311yescharacteristic
50#4268—24no—
51C2×Dic21C2312yes—
52C2×Dic21C2312yes—
53C2×Dic21C2311yescharacteristic
54D84—22no—
55C4×D21—24no—
56C4×D21—24no—
57C4×D21D414yes—
58Dic42—22no—
59Dic42—24no—
60C23×D7—32no—
61C2×C7×Q8C2×D311yescharacteristic, complemented
62C2×C7×D4C2×D311yescharacteristic
63C2×C22⋊D7—32no—
64Q8×D7—64no—
65D4×D7—64no—
66C2×D28—31no—
67C2×C4×D7—31no—
68C2×C4×D7—32no—
69C2×C4×D7—32no—
70C2×Dic14—31no—
71C2×Dic14—32no—
72C7×C22⋊C4C2×D314yescomplemented
73C42×C7C2×D311yescharacteristic, complemented
74C22⋊Dic7—34no—
75(C2×D7)⋊C4—34no—
76(C2×D7)⋊C4—34no—
77C4×Dic7—32no—
78C4×Dic7—31no—
79C2×D3×Q8—71no—
80C2×D3×D4—71no—
81C3×#124D711yescharacteristic, complemented
82#947—71no—
83#936—71no—
84#885—74no—
85D3×C22⋊C4—74no—
86#875—71no—
87C42×D3—71no—
88C22×C3×C7—22no—
89C22×C3×C7—24no—
90C22×C3×C7C2411yescharacteristic
91C2×D21—22no—
92C2×D21—24no—
93C2×D21—48no—
94C2×D21C2×D412yes—
95C2×D21#4114yes—
96C3×C4×C7—22no—
97C3×C4×C7—22no—
98C3×C4×C7C2×D412yes—
99Dic21—22no—
100Dic21—22no—
101Dic21C2×D412yes—
102C2×#124—211noSylow
103C23×C7C22×D312yes—
104C22×D7—31no—
105C22×D7—32no—
106C22×D7—64no—
107C22×D7—68no—
108C7×Q8—22no—
109C7×D4—22no—
110C2×C4×C7C22×D312yes—
111C2×C4×C7C22×D312yes—
112C2×C4×C7C22×D311yescharacteristic
113C22⋊D7—64no—
114C2×Dic7—31no—
115C2×Dic7—32no—
116C2×Dic7—32no—
117D28—62no—
118C4×D7—34no—
119C4×D7—64no—
120C4×D7—64no—
121Dic14—64no—
122Dic14—62no—
123C23×D3—72no—
124C2×C3×Q8C2×D711yescharacteristic, complemented
125C2×C3×D4C2×D711yescharacteristic
126C2×#67—72no—
127D3×Q8—144no—
128D3×D4—144no—
129C2×D12—71no—
130C2×C4×D3—71no—
131C2×C4×D3—72no—
132C2×C4×D3—72no—
133C2×Dic6—71no—
134C2×Dic6—72no—
135C3×C22⋊C4C2×D714yescomplemented
136C3×C42C2×D711yescharacteristic, complemented
137#217—74no—
138(C2×D3)⋊C4—74no—
139(C2×D3)⋊C4—74no—
140C4×Dic3—72no—
141C4×Dic3—71no—
142C2×C3×C7—42no—
143C2×C3×C7C22×D411yescharacteristic
144C2×C3×C7C2×#4112yes—
145D21—42no—
146D21#12414yesdirect product factor
147C22×Q8—211no—
148C22×D4—211no—
149#124—211no—
150#124—211no—
151#124—214no—
152#124—211no—
153#124D2111yescharacteristic, direct product factor
154C2×C22⋊C4—214no—
155C2×C42—211no—
156C22×C7—22no—
157C22×C7—24no—
158C22×C7C23×D311yescharacteristic
159C2×D7—32no—
160C2×D7—34no—
161C2×D7—62no—
162C2×D7—64no—
163C2×D7—128no—
164C4×C7—22no—
165C4×C7—22no—
166C4×C7D3×D412yes—
167Dic7—32no—
168Dic7—62no—
169Dic7—62no—
170C23×C3C22×D712yes—
171C22×D3—71no—
172C22×D3—72no—
173C22×D3—144no—
174C22×D3—148no—
175C3×Q8—22no—
176C3×D4—22no—
177C2×C3×C4C22×D712yes—
178C2×C3×C4C22×D712yes—
179C2×C3×C4C22×D711yescharacteristic
180#67—144no—
181C2×Dic3—71no—
182C2×Dic3—72no—
183C2×Dic3—72no—
184D12—142no—
185C4×D3—74no—
186C4×D3—144no—
187C4×D3—144no—
188Dic6—144no—
189Dic6—142no—
190C3×C7C2×#12411yescharacteristic, complemented, Hall
191C24—212no—
192C2×Q8—212no—
193C2×Q8—211no—
194C2×Q8—424no—
195C2×Q8C2×D2111yescharacteristic, complemented
196C2×D4—212no—
197C2×D4—211no—
198C2×D4—424no—
199C2×D4C2×D2111yescharacteristic
200C22×C4—211no—
201C22×C4—212no—
202C22×C4—212no—
203C22⋊C4—214no—
204C22⋊C4—214no—
205C22⋊C4—214no—
206C22⋊C4C2×D2114yescomplemented
207C42—212no—
208C42—211no—
209C42C2×D2111yescharacteristic, complemented
210C2×C7—42no—
211C2×C7C2×D3×D411yescharacteristic
212C2×C7D3×#4112yes—
213D7—34no—
214D7—122no—
215C22×C3—22no—
216C22×C3—24no—
217C22×C3C23×D711yescharacteristic
218C2×D3—72no—
219C2×D3—74no—
220C2×D3—142no—
221C2×D3—144no—
222C2×D3—288no—
223C3×C4—22no—
224C3×C4—22no—
225C3×C4D4×D712yes—
226Dic3—72no—
227Dic3—142no—
228Dic3—142no—
229C23—211no—
230C23—212no—
231C23—424no—
232C23—428no—
233C23C22×D2112yes—
234Q8—22no—
235Q8—424no—
236Q8—422no—
237D4—22no—
238D4—424no—
239D4—422no—
240C2×C4—211no—
241C2×C4—214no—
242C2×C4—212no—
243C2×C4—212no—
244C2×C4—424no—
245C2×C4—424no—
246C2×C4C22×D2112yes—
247C2×C4C22×D2112yes—
248C2×C4C22×D2111yescharacteristic
249C7D3×#12411yescharacteristic, complemented, Sylow
250C2×C3—42no—
251C2×C3C2×D4×D711yescharacteristic
252C2×C3D7×#4112yes—
253D3—74no—
254D3—282no—
255C22—22no—
256C22—24no—
257C22—212no—
258C22—214no—
259C22—422no—
260C22—424no—
261C22—848no—
262C22C23×D2111yescharacteristic
263C4—22no—
264C4—22no—
265C4—212no—
266C4—422no—
267C4—422no—
268C4D4×D2112yes—
269C3D7×#12411yescharacteristic, complemented, Sylow
270C2—42no—
271C2—214no—
272C2—842no—
273C2C2×D4×D2111yescharacteristic
274C2#41×D2112yes—
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×#124C2
prime radicalC2×D4×D21C2
hypercenter#124D21
hypocenterC3×C7C2×#124
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×Q8×D21C2semidirect
3. C3×C7×#124C2semidirect
4. #10621430C2semidirect
5. #10621419C2semidirect
6. #10621368C2semidirect
7. C22⋊C4×D21C2semidirect
8. #10621358C2semidirect
9. C42×D21C2semidirect
12. C2×C3×C7×Q8C22semidirect
21. C2×Dic42C22semidirect
22. C2×Dic42C22semidirect
23. C3×C7×C22⋊C4C22semidirect
24. C3×C42×C7C22semidirect
25. #66636C22semidirect
26. #66631C22semidirect
27. #66631C22semidirect
28. C4×Dic21C22semidirect
29. C4×Dic21C22semidirect
32. C7×#124D3semidirect
61. C2×C7×Q8C2×D3semidirect
72. C7×C22⋊C4C2×D3semidirect
73. C42×C7C2×D3semidirect
81. C3×#124D7semidirect
124. C2×C3×Q8C2×D7semidirect
135. C3×C22⋊C4C2×D7semidirect
136. C3×C42C2×D7semidirect
146. D21#124direct (N × H)
153. #124D21direct (N × H)
190. C3×C7C2×#124semidirect
195. C2×Q8C2×D21semidirect
206. C22⋊C4C2×D21semidirect
209. C42C2×D21semidirect
249. C7D3×#124semidirect
269. C3D7×#124semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D212
C2×D4×D212
C3×C7×#1242
#106214302
#106214192
#106213682
C22⋊C4×D212
#106213582
C42×D212
D7×#1243
D3×#1247