Finite Group Databasebeta

← #13059516

Subgroups — #13059516

Order 1344.

660 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C22×C4×D3×C7C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic, complemented
3#10621412C211yesmaximal, characteristic, complemented
4C3×#6492C211yesmaximal, characteristic, complemented
5#10621132C211yesmaximal, characteristic, complemented
6#10621130C211yesmaximal, characteristic, complemented
7#10621091C211yesmaximal, characteristic, complemented
8#10621016C212yesmaximal, complemented
9#10621008C212yesmaximal, complemented
10#10621003C212yesmaximal, complemented
11#10620914C212yesmaximal, complemented
12#90482—31nomaximal, Hall
13C23×D3×C7C2211yescharacteristic
14C22×C3×C4×C7C2211yescharacteristic
15C2×#4268C2211yescharacteristic
16C22×C7×Dic3C2211yescharacteristic
17C2×C4×D3×C7—24no—
18C2×C4×D3×C7C2212yes—
19C2×C4×D3×C7C2212yes—
20C2×C3×C22⋊D7C2211yescharacteristic
21C2×#4248C2211yescharacteristic, complemented
22C2×#4247C2212yes—
23C2×#4246C2211yescharacteristic, complemented
24C2×#4245C2212yes—
25C2×D21⋊C4C2211yescharacteristic, complemented
26#66683—24no—
27C2×Dic3×D7C2211yescharacteristic, complemented
28#66636C2211yescharacteristic
29#66631C2212yescomplemented
30#66629C2212yescomplemented
31C3×C22⋊Dic7C2211yescharacteristic
32C3×(C2×D7)⋊C4C2212yescomplemented
33C3×Dic7⋊C4C2212yescomplemented
34Dic3⋊Dic7C2212yescomplemented
35(C2×D3)⋊Dic7C2212yes—
36(C2×D3)⋊Dic7C2212yes—
37Dic3×Dic7C2212yescomplemented
38C23×C4×C7—31no—
39C2×#1157—31no—
40#6516—31no—
41#6516—31no—
42#6493—32no—
43#6492—31no—
44#6492D311yescharacteristic, complemented
45C4×C22⋊D7—32no—
46C4×C22⋊D7—32no—
47#6489—31no—
48#6487—32no—
49#5656—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×D3×C7—22no—
53C22×D3×C7—24no—
54C22×D3×C7C2312yes—
55C22×C3×D7C2311yescharacteristic
56C2×C3×C4×C7—22no—
57C2×C3×C4×C7C2312yes—
58#4268—22no—
59C2×Dic21C2312yes—
60C2×Dic21C2311yescharacteristic
61C2×C7×Dic3D414yescomplemented
62C2×C7×Dic3C2311yescharacteristic
63C2×C7×Dic3C2311yescharacteristic
64C4×D3×C7—24no—
65C4×D3×C7—24no—
66C3×C22⋊D7—22no—
67C2×C3×Dic7C2312yes—
68C2×C3×Dic7C2311yescharacteristic
69#4248—22no—
70#4247—24no—
71#4246—22no—
72#4245—24no—
73D21⋊C4—22no—
74Dic3×D7—22no—
75C24×C7—31no—
76C22×C4×C7—31no—
77C22×C4×C7—32no—
78C22×C4×C7—32no—
79C22×C4×C7—64no—
80C22×C4×C7C2×D311yescharacteristic
81C2×C22⋊D7—31no—
82C2×C22⋊D7—32no—
83C2×C22⋊D7—32no—
84C2×C22⋊D7C2×D311yescharacteristic
85#1157—64no—
86C2×D28—31no—
87C2×C4×D7—31no—
88C2×C4×D7—31no—
89C2×Dic14—31no—
90C22⋊Dic7—31no—
91C22⋊Dic7—32no—
92C22⋊Dic7—32no—
93C22⋊Dic7C2×D311yescharacteristic
94(C2×D7)⋊C4—32no—
95(C2×D7)⋊C4C2×D312yescomplemented
96C4⋊Dic7—32no—
97Dic7⋊C4—32no—
98Dic7⋊C4C2×D312yescomplemented
99C4×Dic7—32no—
100C2×#237—71no—
101C22×C4×D3D711yescharacteristic, complemented
102C3×#123—71no—
103#937—71no—
104#929—71no—
105#896—72no—
106#893—72no—
107#884—72no—
108#882—71no—
109#67⋊C4—72no—
110#877—71no—
111C22×C3×C7—22no—
112C22×C3×C7C2×D412yes—
113C22×C3×C7C2411yescharacteristic
114C2×D21—21no—
115C2×D21—22no—
116C2×D3×C7—28no—
117C2×D3×C7—22no—
118C2×D3×C7—22no—
119C2×D3×C7#4114yescomplemented
120C2×C3×D7—21no—
121C2×C3×D7—22no—
122C3×C4×C7—22no—
123Dic21—21no—
124Dic21—22no—
125C3×Dic7—21no—
126C3×Dic7—22no—
127C7×Dic3C2×D412yes—
128C7×Dic3C2×D412yes—
129#525—211noSylow
130C23×C7—32no—
131C23×C7—64no—
132C23×C7—62no—
133C23×C7C22×D311yescharacteristic
134C22×D7—31no—
135C22×D7C22×D311yescharacteristic
136C2×C4×C7—22no—
137C2×C4×C7—34no—
138C2×C4×C7—31no—
139C2×C4×C7—31no—
140C2×C4×C7—64no—
141C2×C4×C7—64no—
142C2×C4×C7C22×D312yes—
143C22⋊D7—22no—
144C22⋊D7—64no—
145C22⋊D7—64no—
146C22⋊D7—62no—
147C2×Dic7—32no—
148C2×Dic7—31no—
149C2×Dic7C22×D312yes—
150C2×Dic7C22×D311yescharacteristic
151D28—62no—
152C4×D7—62no—
153C4×D7—62no—
154Dic14—62no—
155C23×D3C2×D711yescharacteristic
156C2×C3×D4—71no—
157C22×C3×C4C2×D711yescharacteristic
158C2×#67—71no—
159C2×#67—72no—
160C2×#67—71no—
161C22×Dic3—71no—
162C22×Dic3C2×D711yescharacteristic
163#237—144no—
164C2×D12—72no—
165C2×C4×D3—24no—
166C2×C4×D3—71no—
167C2×C4×D3C2×D712yes—
168C2×C4×D3C2×D712yes—
169C2×Dic6—71no—
170C3×C4⋊C4—72no—
171C3×C22⋊C4—71no—
172C3×C22⋊C4—72no—
173#217—71no—
174(C2×D3)⋊C4—72no—
175(C2×D3)⋊C4—72no—
176(C2×D3)⋊C4—72no—
177Dic3⋊C4—72no—
178Dic3⋊C4—72no—
179C4×Dic3—72no—
180C2×C3×C7—22no—
181C2×C3×C7C22×D411yescharacteristic
182C2×C3×C7C2×#4112yes—
183D21—41no—
184C3×D7—41no—
185D3×C7—24no—
186C2×#41—211no—
187C23×C4—31no—
188#123—71no—
189#123—211no—
190C22⋊Q8—212no—
191#121—212no—
192#120—211no—
193#120—211no—
194C4×D4—212no—
195C4×D4—212no—
196#117—211no—
197C22×C7—22no—
198C22×C7—34no—
199C22×C7—68no—
200C22×C7—62no—
201C22×C7—62no—
202C22×C7D3×D412yes—
203C22×C7C23×D311yescharacteristic
204C2×D7—21no—
205C2×D7—22no—
206C2×D7—61no—
207C2×D7—62no—
208C4×C7—22no—
209C4×C7—32no—
210C4×C7—32no—
211Dic7—21no—
212Dic7—22no—
213Dic7—61no—
214Dic7—62no—
215C23×C3—71no—
216C23×C3C22×D711yescharacteristic
217C22×D3—22no—
218C22×D3—24no—
219C22×D3—71no—
220C22×D3C22×D712yes—
221C3×D4—142no—
222C2×C3×C4—22no—
223C2×C3×C4—72no—
224C2×C3×C4—71no—
225C2×C3×C4C22×D712yes—
226#67—142no—
227#67—144no—
228#67—142no—
229C2×Dic3—72no—
230C2×Dic3—71no—
231C2×Dic3—142no—
232C2×Dic3C22⋊D714yescomplemented
233C2×Dic3C22×D711yescharacteristic
234C2×Dic3C22×D711yescharacteristic
235D12—144no—
236C4×D3—24no—
237C4×D3—24no—
238C4×D3—142no—
239Dic6—142no—
240C3×C7#52511yescharacteristic, complemented, Hall
241C24—31no—
242#41—424no—
243C2×Q8—211no—
244C2×D4—71no—
245C2×D4—211no—
246C2×D4—212no—
247C2×D4—212no—
248C2×D4—211no—
249C22×C4—31no—
250C22×C4—32no—
251C22×C4—32no—
252C22×C4—64no—
253C22×C4—211no—
254C22×C4—211no—
255C22×C4D3×D711yescharacteristic
256C4⋊C4—72no—
257C4⋊C4—212no—
258C4⋊C4—212no—
259C22⋊C4—71no—
260C22⋊C4—72no—
261C22⋊C4—212no—
262C22⋊C4—212no—
263C22⋊C4—211no—
264C22⋊C4—212no—
265C42—212no—
266C2×C7—22no—
267C2×C7—64no—
268C2×C7C2×D3×D411yescharacteristic
269C2×C7D3×#4112yes—
270D7—41no—
271D7—121no—
272C22×C3—22no—
273C22×C3—142no—
274C22×C3—141no—
275C22×C3C2×C22⋊D712yes—
276C22×C3C23×D711yescharacteristic
277C2×D3—28no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—142no—
281C2×D3—141no—
282C2×D3#115714yescomplemented
283C3×C4—22no—
284C3×C4—142no—
285C3×C4—141no—
286Dic3—142no—
287Dic3—141no—
288Dic3C2×C22⋊D712yes—
289Dic3C2×C22⋊D712yes—
290C23—32no—
291C23—64no—
292C23—62no—
293C23—71no—
294C23—211no—
295C23C2×D3×D711yescharacteristic
296Q8—422no—
297D4—142no—
298D4—422no—
299D4—424no—
300D4—424no—
301D4—422no—
302C2×C4—22no—
303C2×C4—34no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—64no—
307C2×C4—64no—
308C2×C4—72no—
309C2×C4—71no—
310C2×C4—212no—
311C2×C4—211no—
312C2×C4—422no—
313C2×C4—422no—
314C2×C4C2×D3×D712yes—
315C7#565611yescharacteristic, complemented, Sylow
316C2×C3—22no—
317C2×C3—281no—
318C2×C3C2×#115712yes—
319C2×C3C22×C22⋊D711yescharacteristic
320D3—24no—
321D3—281no—
322C22—22no—
323C22—34no—
324C22—68no—
325C22—62no—
326C22—62no—
327C22—142no—
328C22—141no—
329C22—422no—
330C22—421no—
331C22D3×C22⋊D712yes—
332C22C22×D3×D711yescharacteristic
333C4—22no—
334C4—32no—
335C4—32no—
336C4—142no—
337C4—141no—
338C4—422no—
339C4—421no—
340C3#9048211yescharacteristic, complemented, Sylow
341C2—22no—
342C2—64no—
343C2—281no—
344C2—841no—
345C2D3×#115712yes—
346C2C2×D3×C22⋊D711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#525
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×D3×C7C2semidirect
2. C2×#66683C2semidirect
3. #10621412C2semidirect
4. C3×#6492C2semidirect
5. #10621132C2semidirect
6. #10621130C2semidirect
7. #10621091C2semidirect
8. #10621016C2semidirect
9. #10621008C2semidirect
10. #10621003C2semidirect
11. #10620914C2semidirect
21. C2×#4248C22semidirect
23. C2×#4246C22semidirect
25. C2×D21⋊C4C22semidirect
27. C2×Dic3×D7C22semidirect
29. #66631C22semidirect
30. #66629C22semidirect
32. C3×(C2×D7)⋊C4C22semidirect
33. C3×Dic7⋊C4C22semidirect
34. Dic3⋊Dic7C22semidirect
37. Dic3×Dic7C22semidirect
44. #6492D3semidirect
61. C2×C7×Dic3D4semidirect
95. (C2×D7)⋊C4C2×D3semidirect
98. Dic7⋊C4C2×D3semidirect
101. C22×C4×D3D7semidirect
119. C2×D3×C7#41semidirect
232. C2×Dic3C22⋊D7semidirect
240. C3×C7#525semidirect
282. C2×D3#1157semidirect
315. C7#5656semidirect
340. C3#90482semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3×C72
C2×#666832
#106214122
C3×#64922
#106211322
#106211302
#106210912
#106210162
#106210082
#106210032
#106209142
#904823
#56567