Finite Group Databasebeta

← #13060107

Subgroups — #13060107

Order 1344.

668 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 144 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×C3×D4×D7C211yesmaximal, characteristic, complemented
2C2×#66677C211yesmaximal, characteristic, complemented
3#10621423C211yesmaximal, characteristic, complemented
4C7×#936C211yesmaximal, characteristic, complemented
5#10621126C214yesmaximal, complemented
6#10621095C214yesmaximal, complemented
7#10621020C211yesmaximal, characteristic, complemented
8#10620930C211yesmaximal, characteristic, complemented
9#10620919C211yesmaximal, characteristic, complemented
10#90365—31nomaximal, Hall
11C23×C3×D7C2212yes—
12C2×C3×C7×D4C2211yescharacteristic
13C2×#4268C2214yescomplemented
14C2×D84C2211yescharacteristic, complemented
15C2×C7×Dic6C2211yescharacteristic, complemented
16C2×C3×C22⋊D7C2212yes—
17C3×D4×D7—24no—
18C2×C3×D28C2211yescharacteristic
19C2×C3×C4×D7C2211yescharacteristic
20C2×#4246C2212yes—
21C2×#4246C2214yescomplemented
22C2×D21⋊C4C2212yescomplemented
23#66677—24no—
24C4×Dic21C2211yescharacteristic, complemented
25C7×#217C2214yescomplemented
26C4×C7×Dic3C2211yescharacteristic, complemented
27Dic7⋊Dic3C2212yescomplemented
28(C2×D7)⋊Dic3C2212yes—
29(C2×D7)⋊Dic3C2214yescomplemented
30C2×#1161—31no—
31C2×D4×D7D311yescharacteristic, complemented
32C7×#124—31no—
33#6510—31no—
34#6503—31no—
35#6447—34no—
36#6445—34no—
37#6437—31no—
38#6435—31no—
39#5805—71nomaximal, Hall
40C23×C3×C7C2312yes—
41C22×D21C2312yes—
42C22×C3×D7—24no—
43C22×C3×D7—44no—
44C22×C3×D7C2312yes—
45C22×C3×D7C2311yescharacteristic
46C3×C7×D4—22no—
47C2×C3×C4×C7C2311yescharacteristic
48#4268—44no—
49C2×Dic21C2312yes—
50D84—22no—
51C2×C7×Dic3C2312yes—
52C2×C7×Dic3C2312yes—
53C7×Dic6—22no—
54C3×C22⋊D7—24no—
55C2×C3×Dic7C2311yescharacteristic
56C3×D28—22no—
57C3×C4×D7D414yescomplemented
58#4246—24no—
59#4246—44no—
60D21⋊C4—24no—
61C23×D7C2×D312yes—
62C2×C7×Q8—31no—
63C2×C7×D4C2×D311yescharacteristic
64C2×C22⋊D7—34no—
65C2×C22⋊D7C2×D312yes—
66#1161—64no—
67D4×D7—24no—
68C2×D28—32no—
69C2×D28—31no—
70C2×D28—34no—
71C2×D28C2×D311yescharacteristic
72C2×C4×D7—32no—
73C2×C4×D7C2×D311yescharacteristic
74C7×C22⋊C4—34no—
75C42×C7—31no—
76(C2×D7)⋊C4—32no—
77(C2×D7)⋊C4—34no—
78Dic7⋊C4—32no—
79C4×Dic7—31no—
80C22×C3×D4—71no—
81C2×#235—71no—
82#953—74no—
83#940—71no—
84#940—71no—
85#938—74no—
86#936—71no—
87#936D711yescharacteristic, complemented
88#926—71no—
89C22×C3×C7—22no—
90C22×C3×C7—24no—
91C22×C3×C7C2411yescharacteristic
92C2×D21—22no—
93C2×D21—24no—
94C2×C3×D7—22no—
95C2×C3×D7—22no—
96C2×C3×D7—24no—
97C2×C3×D7—48no—
98C2×C3×D7C2×D412yes—
99C3×C4×C7C2×D412yes—
100Dic21—22no—
101C3×Dic7C2×D412yes—
102C7×Dic3—22no—
103C7×Dic3—22no—
104#535—211noSylow
105C23×C7C22×D312yes—
106C22×D7—24no—
107C22×D7—32no—
108C22×D7—44no—
109C22×D7C22×D312yes—
110C22×D7C22×D311yescharacteristic
111C7×Q8—62no—
112C7×D4—22no—
113C2×C4×C7—32no—
114C2×C4×C7—32no—
115C2×C4×C7C22×D311yescharacteristic
116C22⋊D7—24no—
117C22⋊D7—124no—
118C2×Dic7—32no—
119C2×Dic7C22×D311yescharacteristic
120D28—22no—
121D28—64no—
122D28—62no—
123D28—124no—
124C4×D7—64no—
125C4×D7#6714yescomplemented
126C24×C3—72no—
127C2×C3×D4—72no—
128C2×C3×D4—71no—
129C2×C3×D4—144no—
130C2×C3×D4C2×D711yescharacteristic
131C22×C3×C4—71no—
132C2×#67—74no—
133C2×#67—72no—
134C2×#67—74no—
135#235—144no—
136C2×D12—71no—
137C2×C4×D3—72no—
138C2×Dic6C2×D711yescharacteristic, complemented
139#217—72no—
140#217—74no—
141#217C2×D714yescomplemented
142C4⋊Dic3—72no—
143C4×Dic3—71no—
144C4×Dic3C2×D711yescharacteristic, complemented
145C2×C3×C7—42no—
146C2×C3×C7C22×D411yescharacteristic
147C2×C3×C7H(5,𝔽2)12yes—
148D21—42no—
149C3×D7—22no—
150C3×D7—42no—
151C2×#41—211no—
152C22×D4—71no—
153#127—211no—
154#127—211no—
155#124—31no—
156#124—211no—
157#121—214no—
158#120—214no—
159#117—211no—
160C22×C7—24no—
161C22×C7—22no—
162C22×C7C23×D311yescharacteristic
163C2×D7—22no—
164C2×D7—24no—
165C2×D7—22no—
166C2×D7—48no—
167C2×D7—64no—
168C2×D7—62no—
169C2×D7C2×#6712yes—
170C4×C7—62no—
171C4×C7—62no—
172C4×C7C2×#6712yes—
173Dic7—62no—
174Dic7C2×#6712yes—
175C23×C3—71no—
176C23×C3—72no—
177C23×C3—144no—
178C23×C3—284no—
179C23×C3C22×D712yes—
180C22×D3—72no—
181C3×D4—22no—
182C3×D4—144no—
183C3×D4—142no—
184C2×C3×C4—71no—
185C2×C3×C4—74no—
186C2×C3×C4C22×D711yescharacteristic
187#67—144no—
188#67—284no—
189#67—284no—
190C2×Dic3—72no—
191C2×Dic3C22×D712yes—
192C2×Dic3C22×D712yes—
193D12—142no—
194C4×D3—144no—
195Dic6—22no—
196C3×C7#53511yescharacteristic, complemented, Hall
197C24—72no—
198#41—424no—
199C2×Q8—31no—
200C2×D4—72no—
201C2×D4—71no—
202C2×D4—144no—
203C2×D4—212no—
204C2×D4—214no—
205C2×D4—214no—
206C2×D4—211no—
207C2×D4D3×D711yescharacteristic
208C22×C4—71no—
209C22×C4—212no—
210C4⋊C4—212no—
211C22⋊C4—34no—
212C22⋊C4—212no—
213C22⋊C4—214no—
214C42—31no—
215C42—211no—
216C2×C7—42no—
217C2×C7#100412yes—
218C2×C7C22×#6711yescharacteristic
219D7—22no—
220D7—42no—
221D7—122no—
222C22×C3—22no—
223C22×C3—24no—
224C22×C3—72no—
225C22×C3—142no—
226C22×C3—142no—
227C22×C3—144no—
228C22×C3—288no—
229C22×C3C23×D711yescharacteristic
230C2×D3—142no—
231C2×D3—144no—
232C3×C4—72no—
233C3×C4D4×D712yes—
234Dic3—22no—
235Dic3—22no—
236Dic3—142no—
237C23—71no—
238C23—72no—
239C23—144no—
240C23—212no—
241C23—284no—
242C23C2×D3×D712yes—
243Q8—62no—
244D4—22no—
245D4—144no—
246D4—142no—
247D4—424no—
248D4—422no—
249D4—844no—
250D4—844no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—71no—
254C2×C4—74no—
255C2×C4—212no—
256C2×C4—424no—
257C2×C4C2×D3×D711yescharacteristic
258C7#580511yescharacteristic, complemented, Sylow
259C2×C3—42no—
260C2×C3—142no—
261C2×C3—282no—
262C2×C3C2×D4×D711yescharacteristic
263C2×C3#655312yes—
264D3—282no—
265C22—24no—
266C22—22no—
267C22—72no—
268C22—142no—
269C22—144no—
270C22—142no—
271C22—288no—
272C22—424no—
273C22—422no—
274C22C22×D3×D711yescharacteristic
275C4—62no—
276C4—62no—
277C4—72no—
278C4—422no—
279C4D7×#6712yes—
280C3#9036511yescharacteristic, complemented, Sylow
281C2—42no—
282C2—142no—
283C2—282no—
284C2—842no—
285C2#1062157312yes—
286C2C2×D7×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#535
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×D7C2semidirect
2. C2×#66677C2semidirect
3. #10621423C2semidirect
4. C7×#936C2semidirect
5. #10621126C2semidirect
6. #10621095C2semidirect
7. #10621020C2semidirect
8. #10620930C2semidirect
9. #10620919C2semidirect
13. C2×#4268C22semidirect
14. C2×D84C22semidirect
15. C2×C7×Dic6C22semidirect
21. C2×#4246C22semidirect
22. C2×D21⋊C4C22semidirect
24. C4×Dic21C22semidirect
25. C7×#217C22semidirect
26. C4×C7×Dic3C22semidirect
27. Dic7⋊Dic3C22semidirect
29. (C2×D7)⋊Dic3C22semidirect
31. C2×D4×D7D3semidirect
57. C3×C4×D7D4semidirect
87. #936D7semidirect
125. C4×D7#67semidirect
138. C2×Dic6C2×D7semidirect
141. #217C2×D7semidirect
144. C4×Dic3C2×D7semidirect
196. C3×C7#535semidirect
258. C7#5805semidirect
280. C3#90365semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D72
C2×#666772
#106214232
C7×#9362
#106211262
#106210952
#106210202
#106209302
#106209192
#903653
#58057