Finite Group Databasebeta

← #13060104

Subgroups — #13060104

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×#66676C211yesmaximal, characteristic, complemented
3#10621419C211yesmaximal, characteristic, complemented
4C7×#940C211yesmaximal, characteristic, complemented
5#10621127C214yesmaximal, complemented
6#10621125C214yesmaximal, complemented
7#10621025C211yesmaximal, characteristic, complemented
8#10620926C211yesmaximal, characteristic, complemented
9#10620919C211yesmaximal, characteristic, complemented
10#90406—31nomaximal, Hall
11C23×C3×D7C2212yes—
12C2×C3×C7×D4C2211yescharacteristic
13C2×Dic42C2211yescharacteristic, complemented
14C2×C7×#67C2214yescomplemented
15C2×C7×D12C2211yescharacteristic, complemented
16C2×C3×C22⋊D7C2212yes—
17C3×D4×D7—24no—
18C2×C3×D28C2211yescharacteristic
19C2×C3×C4×D7C2211yescharacteristic
20C2×#4245C2212yes—
21C2×#4245C2214yescomplemented
22C2×D3×Dic7C2212yescomplemented
23#66676—24no—
24#66636C2214yescomplemented
25C4×Dic21C2211yescharacteristic, complemented
26C4×C7×Dic3C2211yescharacteristic, complemented
27Dic7⋊Dic3C2212yescomplemented
28(C2×D7)⋊Dic3C2212yes—
29(C2×D7)⋊Dic3C2214yescomplemented
30C2×#1159—31no—
31C2×D4×D7D311yescharacteristic, complemented
32C7×#127—31no—
33#6503—31no—
34#6502—34no—
35#6500—34no—
36#6499—31no—
37#6438—31no—
38#6435—31no—
39#5805—71nomaximal, Hall
40C23×C3×C7C2312yes—
41C22×D3×C7C2312yes—
42C22×C3×D7—24no—
43C22×C3×D7—44no—
44C22×C3×D7C2312yes—
45C22×C3×D7C2311yescharacteristic
46C3×C7×D4—22no—
47C2×C3×C4×C7C2311yescharacteristic
48C2×Dic21C2312yes—
49C2×Dic21C2312yes—
50Dic42—22no—
51C7×#67—44no—
52C2×C7×Dic3C2312yes—
53C7×D12—22no—
54C3×C22⋊D7—24no—
55C2×C3×Dic7C2311yescharacteristic
56C3×D28—22no—
57C3×C4×D7D414yescomplemented
58#4245—24no—
59#4245—44no—
60D3×Dic7—24no—
61C23×D7C2×D312yes—
62C2×C7×D4—34no—
63C2×C7×D4—31no—
64C2×C7×D4C2×D311yescharacteristic
65C2×C22⋊D7—32no—
66C2×C22⋊D7—34no—
67C2×C22⋊D7C2×D312yes—
68C22×Dic7—32no—
69#1159—64no—
70D4×D7—24no—
71C2×D28C2×D311yescharacteristic
72C2×C4×D7C2×D311yescharacteristic
73C2×Dic14—31no—
74C42×C7—31no—
75C22⋊Dic7—34no—
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#940D711yescharacteristic, complemented
85#938—74no—
86#936—71no—
87#936—71no—
88#926—71no—
89C22×C3×C7—22no—
90C22×C3×C7—24no—
91C22×C3×C7C2411yescharacteristic
92C2×D3×C7—22no—
93C2×D3×C7—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—
101Dic21—22no—
102C3×Dic7C2×D412yes—
103C7×Dic3—22no—
104#535—211noSylow
105C23×C7—32no—
106C23×C7C22×D312yes—
107C22×D7—24no—
108C22×D7—44no—
109C22×D7C22×D312yes—
110C22×D7C22×D311yescharacteristic
111C7×D4—22no—
112C7×D4—62no—
113C7×D4—124no—
114C2×C4×C7—32no—
115C2×C4×C7C22×D311yescharacteristic
116C22⋊D7—24no—
117C22⋊D7—64no—
118C22⋊D7—124no—
119C2×Dic7—32no—
120C2×Dic7—32no—
121C2×Dic7—64no—
122C2×Dic7C22×D311yescharacteristic
123D28—22no—
124C4×D7#6714yescomplemented
125Dic14—62no—
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—72no—
133C2×#67—74no—
134C2×#67C2×D714yescomplemented
135#235—144no—
136C2×D12C2×D711yescharacteristic, complemented
137C2×C4×D3—72no—
138C2×Dic6—71no—
139#217—74no—
140#217—72no—
141#217—74no—
142C4⋊Dic3—72no—
143C4×Dic3—71no—
144C4×Dic3C2×D711yescharacteristic, complemented
145C2×C3×C7—42no—
146C2×C3×C7C22×D411yescharacteristic
147C2×C3×C7H(5,𝔽2)12yes—
148C3×D7—22no—
149C3×D7—42no—
150D3×C7—42no—
151C2×#41—211no—
152C22×D4—71no—
153#127—31no—
154#127—211no—
155#124—211no—
156#124—211no—
157#121—214no—
158#120—214no—
159#117—211no—
160C22×C7—24no—
161C22×C7—22no—
162C22×C7—64no—
163C22×C7—62no—
164C22×C7C23×D311yescharacteristic
165C2×D7—22no—
166C2×D7—24no—
167C2×D7—22no—
168C2×D7—48no—
169C2×D7C2×#6712yes—
170C4×C7—62no—
171C4×C7C2×#6712yes—
172Dic7—62no—
173Dic7—62no—
174Dic7C2×#6712yes—
175C23×C3—71no—
176C23×C3—72no—
177C23×C3—144no—
178C23×C3—284no—
179C23×C3C22×D712yes—
180C22×D3C22×D712yes—
181C3×D4—22no—
182C3×D4—144no—
183C3×D4—142no—
184C2×C3×C4—71no—
185C2×C3×C4—74no—
186C2×C3×C4C22×D711yescharacteristic
187#67—44no—
188#67—144no—
189#67—284no—
190C2×Dic3—72no—
191C2×Dic3—72no—
192C2×Dic3C22×D712yes—
193D12—22no—
194C4×D3—144no—
195Dic6—142no—
196C3×C7#53511yescharacteristic, complemented, Hall
197C24—72no—
198#41—424no—
199C2×Q8—211no—
200C2×D4—34no—
201C2×D4—31no—
202C2×D4—72no—
203C2×D4—71no—
204C2×D4—144no—
205C2×D4—212no—
206C2×D4—214no—
207C2×D4D3×D711yescharacteristic
208C22×C4—71no—
209C22×C4—212no—
210C4⋊C4—212no—
211C22⋊C4—212no—
212C22⋊C4—214no—
213C22⋊C4—214no—
214C42—31no—
215C42—211no—
216C2×C7—42no—
217C2×C7—122no—
218C2×C7#100412yes—
219C2×C7C22×#6711yescharacteristic
220D7—22no—
221D7—42no—
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—22no—
231C2×D3—24no—
232C3×C4—72no—
233C3×C4D4×D712yes—
234Dic3—22no—
235Dic3—142no—
236Dic3—142no—
237C23—32no—
238C23—71no—
239C23—72no—
240C23—144no—
241C23—284no—
242C23C2×D3×D712yes—
243Q8—422no—
244D4—22no—
245D4—62no—
246D4—124no—
247D4—144no—
248D4—142no—
249D4—424no—
250D4—844no—
251C2×C4—32no—
252C2×C4—71no—
253C2×C4—74no—
254C2×C4—212no—
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#654812yes—
264D3—42no—
265C22—24no—
266C22—22no—
267C22—64no—
268C22—62no—
269C22—72no—
270C22—142no—
271C22—144no—
272C22—142no—
273C22—288no—
274C22C22×D3×D711yescharacteristic
275C4—62no—
276C4—72no—
277C4—422no—
278C4—422no—
279C4D7×#6712yes—
280C3#9040611yescharacteristic, complemented, Sylow
281C2—42no—
282C2—122no—
283C2—142no—
284C2—282no—
285C2#1062157212yes—
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×#66676C2semidirect
3. #10621419C2semidirect
4. C7×#940C2semidirect
5. #10621127C2semidirect
6. #10621125C2semidirect
7. #10621025C2semidirect
8. #10620926C2semidirect
9. #10620919C2semidirect
13. C2×Dic42C22semidirect
14. C2×C7×#67C22semidirect
15. C2×C7×D12C22semidirect
21. C2×#4245C22semidirect
22. C2×D3×Dic7C22semidirect
24. #66636C22semidirect
25. C4×Dic21C22semidirect
26. C4×C7×Dic3C22semidirect
27. Dic7⋊Dic3C22semidirect
29. (C2×D7)⋊Dic3C22semidirect
31. C2×D4×D7D3semidirect
57. C3×C4×D7D4semidirect
84. #940D7semidirect
124. C4×D7#67semidirect
134. C2×#67C2×D7semidirect
136. C2×D12C2×D7semidirect
144. C4×Dic3C2×D7semidirect
196. C3×C7#535semidirect
258. C7#5805semidirect
280. C3#90406semidirect

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×#666762
#106214192
C7×#9402
#106211272
#106211252
#106210252
#106209262
#106209192
#904063
#58057