Finite Group Databasebeta

← #13061726

Subgroups — #13061726

Order 1344.

484 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 147 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×C22⋊Q8C211yesmaximal, characteristic, complemented
2#10621429C212yesmaximal, complemented
3C2×#66631C211yesmaximal, characteristic
4#10621407C211yesmaximal, characteristic, complemented
5#10621380C212yesmaximal, complemented
6#10621379C212yesmaximal, complemented
7#10621379C212yesmaximal, complemented
8C22⋊C4×D21C212yesmaximal
9#10621361C212yesmaximal, complemented
10#90325—31nomaximal, Hall
11C23×D21C2211yescharacteristic
12C2×C3×C7×Q8C2211yescharacteristic, complemented
13C22×C3×C4×C7C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C2×C4×D21C2212yes—
16C2×C4×D21C2212yes—
17C2×Dic42C2212yescomplemented
18C2×Dic42C2211yescharacteristic, complemented
19C3×C7×C4⋊C4C2211yescharacteristic, complemented
20C3×C7×C4⋊C4C2212yescomplemented
21C3×C7×C22⋊C4C2212yes—
22#66636C2212yes—
23#66631C2212yes—
24#66631C2212yes—
25#66631C2212yes—
26#66631C2212yes—
27C4⋊Dic21C2212yescomplemented
28C4⋊Dic21C2211yescharacteristic, complemented
29#66629C2212yescomplemented
30#66629C2212yescomplemented
31#66629C2212yescomplemented
32C7×C22⋊Q8D311yescharacteristic, complemented
33#6509—32no—
34C2×(C2×D7)⋊C4—31no—
35#6487—31no—
36#6460—32no—
37#6459—32no—
38#6459—32no—
39D7×C22⋊C4—32no—
40#6441—32no—
41#5634—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C22×D21—41no—
44C22×D21—41no—
45C22×D21Q814yescomplemented
46C22×D21C2312yes—
47C3×C7×Q8—41no—
48C2×C3×C4×C7—41no—
49C2×C3×C4×C7C2311yescharacteristic
50C2×C3×C4×C7C2312yes—
51C2×C3×C4×C7C2312yes—
52C2×C3×C4×C7C2311yescharacteristic
53C2×Dic21—41no—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56C2×Dic21C2312yes—
57C4×D21—42no—
58C4×D21—42no—
59Dic42—41no—
60Dic42—42no—
61C23×D7—31no—
62C2×C7×Q8C2×D311yescharacteristic, complemented
63C22×C4×C7C2×D311yescharacteristic
64C22×Dic7—31no—
65C2×C4×D7—32no—
66C2×C4×D7—32no—
67C2×Dic14—32no—
68C2×Dic14—31no—
69C7×C4⋊C4C2×D311yescharacteristic, complemented
70C7×C4⋊C4C2×D312yescomplemented
71C7×C22⋊C4C2×D312yes—
72C22⋊Dic7—32no—
73(C2×D7)⋊C4—32no—
74(C2×D7)⋊C4—32no—
75(C2×D7)⋊C4—32no—
76(C2×D7)⋊C4—32no—
77C4⋊Dic7—32no—
78C4⋊Dic7—31no—
79Dic7⋊C4—32no—
80Dic7⋊C4—32no—
81Dic7⋊C4—32no—
82C3×C22⋊Q8D711yescharacteristic, complemented
83#946—72no—
84C2×(C2×D3)⋊C4—71no—
85#924—71no—
86#897—72no—
87#896—72no—
88#896—72no—
89D3×C22⋊C4—72no—
90#878—72no—
91C22×C3×C7—21no—
92C22×C3×C7—21no—
93C22×C3×C7C2×Q812yes—
94C22×C3×C7C2411yescharacteristic
95C2×D21—22no—
96C2×D21—22no—
97C2×D21—44no—
98C2×D21C2×Q814yes—
99C3×C4×C7—22no—
100C3×C4×C7—21no—
101C3×C4×C7—22no—
102C3×C4×C7—21no—
103Dic21—22no—
104Dic21—22no—
105Dic21—22no—
106#543—211noSylow
107C23×C7C22×D311yescharacteristic
108C22×D7—32no—
109C22×D7—34no—
110C22×D7—121no—
111C22×D7—121no—
112C7×Q8—41no—
113C2×C4×C7—41no—
114C2×C4×C7C22×D311yescharacteristic
115C2×C4×C7C22×D312yes—
116C2×C4×C7C22×D312yes—
117C2×C4×C7C22×D311yescharacteristic
118C2×Dic7—32no—
119C2×Dic7—32no—
120C2×Dic7—32no—
121C2×Dic7—121no—
122C4×D7—122no—
123C4×D7—122no—
124Dic14—121no—
125Dic14—122no—
126C23×D3—71no—
127C2×C3×Q8C2×D711yescharacteristic, complemented
128C22×C3×C4C2×D711yescharacteristic
129C22×Dic3—71no—
130C2×C4×D3—72no—
131C2×C4×D3—72no—
132C2×Dic6—72no—
133C2×Dic6—71no—
134C3×C4⋊C4C2×D711yescharacteristic, complemented
135C3×C4⋊C4C2×D712yescomplemented
136C3×C22⋊C4C2×D712yes—
137#217—72no—
138(C2×D3)⋊C4—72no—
139(C2×D3)⋊C4—72no—
140(C2×D3)⋊C4—72no—
141(C2×D3)⋊C4—72no—
142C4⋊Dic3—72no—
143C4⋊Dic3—71no—
144Dic3⋊C4—72no—
145Dic3⋊C4—72no—
146Dic3⋊C4—72no—
147C2×C3×C7—22no—
148C2×C3×C7C22×Q811yescharacteristic
149C2×C3×C7H(5,𝔽2)11yescharacteristic
150C2×C3×C7H(5,𝔽2)11yescharacteristic
151D21—24no—
152C22⋊Q8—212no—
153C22⋊Q8—211no—
154C22⋊Q8—212no—
155C22⋊Q8—212no—
156C22⋊Q8—212no—
157C22⋊Q8—212no—
158C22⋊Q8D2111yescharacteristic, complemented
159C2×C22⋊C4—211no—
160C2×C22⋊C4—212no—
161C22×C7—21no—
162C22×C7—21no—
163C22×C7D3×Q812yes—
164C22×C7C23×D311yescharacteristic
165C2×D7—34no—
166C2×D7—62no—
167C2×D7—62no—
168C2×D7—124no—
169C4×C7—21no—
170C4×C7—22no—
171C4×C7—21no—
172C4×C7—22no—
173Dic7—62no—
174Dic7—62no—
175Dic7—62no—
176C23×C3C22×D711yescharacteristic
177C22×D3—72no—
178C22×D3—74no—
179C22×D3—281no—
180C22×D3—281no—
181C3×Q8—41no—
182C2×C3×C4—41no—
183C2×C3×C4C22×D711yescharacteristic
184C2×C3×C4C22×D712yes—
185C2×C3×C4C22×D712yes—
186C2×C3×C4C22×D711yescharacteristic
187C2×Dic3—72no—
188C2×Dic3—72no—
189C2×Dic3—72no—
190C2×Dic3—281no—
191C4×D3—282no—
192C4×D3—282no—
193Dic6—281no—
194Dic6—282no—
195C3×C7#54311yescharacteristic, complemented, Hall
196C24—211no—
197C2×Q8—211no—
198C2×Q8—212no—
199C2×Q8C2×D2111yescharacteristic, complemented
200C22×C4—211no—
201C22×C4—212no—
202C22×C4—212no—
203C22×C4C2×D2111yescharacteristic
204C4⋊C4—212no—
205C4⋊C4—212no—
206C4⋊C4—212no—
207C4⋊C4—211no—
208C4⋊C4—212no—
209C4⋊C4C2×D2111yescharacteristic, complemented
210C4⋊C4C2×D2112yescomplemented
211C22⋊C4—212no—
212C22⋊C4—212no—
213C22⋊C4—212no—
214C22⋊C4—212no—
215C22⋊C4—212no—
216C22⋊C4C2×D2112yes—
217C2×C7—22no—
218C2×C7#100411yescharacteristic
219C2×C7C2×D3×Q811yescharacteristic
220C2×C7#100911yescharacteristic
221D7—64no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3Q8×D712yes—
225C22×C3C23×D711yescharacteristic
226C2×D3—74no—
227C2×D3—142no—
228C2×D3—142no—
229C2×D3—284no—
230C3×C4—21no—
231C3×C4—22no—
232C3×C4—21no—
233C3×C4—22no—
234Dic3—142no—
235Dic3—142no—
236Dic3—142no—
237C23—212no—
238C23—214no—
239C23—841no—
240C23—841no—
241C23C22×D2111yescharacteristic
242Q8—41no—
243Q8—841no—
244Q8—842no—
245C2×C4—41no—
246C2×C4—212no—
247C2×C4—212no—
248C2×C4—212no—
249C2×C4—842no—
250C2×C4—842no—
251C2×C4—841no—
252C2×C4C22×D2111yescharacteristic
253C2×C4C22×D2111yescharacteristic
254C2×C4C22×D2112yes—
255C2×C4C22×D2112yes—
256C7#563411yescharacteristic, complemented, Sylow
257C2×C3—22no—
258C2×C3#654811yescharacteristic
259C2×C3C2×Q8×D711yescharacteristic
260C2×C3#655311yescharacteristic
261D3—144no—
262C22—21no—
263C22—21no—
264C22—214no—
265C22—422no—
266C22—422no—
267C22—844no—
268C22Q8×D2112yes—
269C22C23×D2111yescharacteristic
270C4—21no—
271C4—21no—
272C4—22no—
273C4—22no—
274C4—422no—
275C4—422no—
276C4—422no—
277C3#9032511yescharacteristic, complemented, Sylow
278C2—22no—
279C2—424no—
280C2#1062165711yescharacteristic
281C2C2×Q8×D2111yescharacteristic
282C2#1062166211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊Q8C2
prime radicalC23×D21C22
hypercenterC22⋊Q8D21
hypocenterC3×C7#543
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C7×C22⋊Q8C2semidirect
2. #10621429C2semidirect
4. #10621407C2semidirect
5. #10621380C2semidirect
6. #10621379C2semidirect
7. #10621379C2semidirect
9. #10621361C2semidirect
12. C2×C3×C7×Q8C22semidirect
17. C2×Dic42C22semidirect
18. C2×Dic42C22semidirect
19. C3×C7×C4⋊C4C22semidirect
20. C3×C7×C4⋊C4C22semidirect
27. C4⋊Dic21C22semidirect
28. C4⋊Dic21C22semidirect
29. #66629C22semidirect
30. #66629C22semidirect
31. #66629C22semidirect
32. C7×C22⋊Q8D3semidirect
45. C22×D21Q8semidirect
62. C2×C7×Q8C2×D3semidirect
69. C7×C4⋊C4C2×D3semidirect
70. C7×C4⋊C4C2×D3semidirect
82. C3×C22⋊Q8D7semidirect
127. C2×C3×Q8C2×D7semidirect
134. C3×C4⋊C4C2×D7semidirect
135. C3×C4⋊C4C2×D7semidirect
158. C22⋊Q8D21semidirect
195. C3×C7#543semidirect
199. C2×Q8C2×D21semidirect
209. C4⋊C4C2×D21semidirect
210. C4⋊C4C2×D21semidirect
256. C7#5634semidirect
277. C3#90325semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214292
C2×#666312
#106214072
#106213802
#106213792
C22⋊C4×D212
#106213612
#903253
#56347