Finite Group Databasebeta

← #13059549

Subgroups — #13059549

Order 1344.

660 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 140 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×D21C211yesmaximal, characteristic, complemented
2C2×#66684C211yesmaximal, characteristic, complemented
3C7×#929C211yesmaximal, characteristic, complemented
4C3×#6492C211yesmaximal, characteristic, complemented
5#10621135C211yesmaximal, characteristic, complemented
6#10621134C211yesmaximal, characteristic, complemented
7#10621094C211yesmaximal, characteristic, complemented
8#10621023C212yesmaximal, complemented
9#10620968C212yesmaximal, complemented
10#10620958C212yesmaximal, complemented
11#10620939C212yesmaximal, complemented
12#90347—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C22×C3×C4×C7C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C2×C4×D21—24no—
17C2×C4×D21C2212yes—
18C2×C4×D21C2212yes—
19C2×C7×#67C2211yescharacteristic
20C2×C3×C22⋊D7C2211yescharacteristic
21C2×#4248C2211yescharacteristic, complemented
22C2×#4247C2212yes—
23C2×#4246C2212yes—
24C2×#4245C2211yescharacteristic, complemented
25C2×D3×Dic7C2211yescharacteristic, complemented
26#66684—24no—
27C2×Dic3×D7C2211yescharacteristic, complemented
28C7×#217C2211yescharacteristic
29C7×(C2×D3)⋊C4C2212yescomplemented
30C7×Dic3⋊C4C2212yescomplemented
31C3×C22⋊Dic7C2211yescharacteristic
32C3×(C2×D7)⋊C4C2212yescomplemented
33C3×Dic7⋊C4C2212yescomplemented
34#66577C2212yescomplemented
35#66575C2212yes—
36#66575C2212yes—
37Dic3×Dic7C2212yescomplemented
38C2×#1159—31no—
39C22×C4×D7—31no—
40C7×#123—31no—
41#6500—31no—
42#6492D311yescharacteristic, complemented
43#6459—32no—
44#6456—32no—
45#6447—32no—
46#6445—31no—
47(C22⋊D7)⋊C4—32no—
48#6440—31no—
49#5656—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21—22no—
52C22×D21—24no—
53C22×D21C2312yes—
54C22×D3×C7C2311yescharacteristic
55C22×C3×D7C2311yescharacteristic
56C2×C3×C4×C7—22no—
57C2×C3×C4×C7C2312yes—
58C2×Dic21D414yescomplemented
59C2×Dic21C2311yescharacteristic
60C2×Dic21C2311yescharacteristic
61C4×D21—24no—
62C4×D21—24no—
63C7×#67—22no—
64C2×C7×Dic3C2312yes—
65C2×C7×Dic3C2311yescharacteristic
66C3×C22⋊D7—22no—
67C2×C3×Dic7C2312yes—
68C2×C3×Dic7C2311yescharacteristic
69#4248—22no—
70#4247—24no—
71#4246—24no—
72#4245—22no—
73D3×Dic7—22no—
74Dic3×D7—22no—
75C23×D7—31no—
76C2×C7×D4—31no—
77C22×C4×C7C2×D311yescharacteristic
78C2×C22⋊D7—31no—
79C2×C22⋊D7—32no—
80C2×C22⋊D7C2×D311yescharacteristic
81C22×Dic7—31no—
82C22×Dic7—31no—
83#1159—64no—
84C2×D28—32no—
85C2×C4×D7—31no—
86C2×C4×D7—32no—
87C2×C4×D7—32no—
88C2×C4×D7—64no—
89C2×Dic14—31no—
90C7×C4⋊C4—32no—
91C7×C22⋊C4—31no—
92C7×C22⋊C4—32no—
93C22⋊Dic7C2×D311yescharacteristic
94(C2×D7)⋊C4—32no—
95(C2×D7)⋊C4—32no—
96(C2×D7)⋊C4C2×D312yescomplemented
97Dic7⋊C4—32no—
98Dic7⋊C4C2×D312yescomplemented
99C4×Dic7—32no—
100C2×#237—71no—
101C22×C4×D3—71no—
102C3×#123—71no—
103#937—71no—
104#929D711yescharacteristic, complemented
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—28no—
115C2×D21—22no—
116C2×D21—22no—
117C2×D21#4114yescomplemented
118C2×D3×C7—21no—
119C2×D3×C7—22no—
120C2×C3×D7—21no—
121C2×C3×D7—22no—
122C3×C4×C7—22no—
123Dic21C2×D412yes—
124Dic21C2×D412yes—
125C3×Dic7—21no—
126C3×Dic7—22no—
127C7×Dic3—21no—
128C7×Dic3—22no—
129#525—211noSylow
130C23×C7—31no—
131C23×C7C22×D311yescharacteristic
132C22×D7—32no—
133C22×D7—64no—
134C22×D7—62no—
135C22×D7C22×D311yescharacteristic
136C7×D4—62no—
137C2×C4×C7—22no—
138C2×C4×C7—32no—
139C2×C4×C7—31no—
140C2×C4×C7C22×D312yes—
141C22⋊D7—22no—
142C22⋊D7—64no—
143C22⋊D7—62no—
144C2×Dic7—34no—
145C2×Dic7—31no—
146C2×Dic7—31no—
147C2×Dic7—62no—
148C2×Dic7C22×D312yes—
149C2×Dic7C22×D311yescharacteristic
150D28—64no—
151C4×D7—64no—
152C4×D7—64no—
153C4×D7—62no—
154Dic14—62no—
155C23×D3—71no—
156C2×C3×D4—71no—
157C22×C3×C4C2×D711yescharacteristic
158C2×#67—71no—
159C2×#67—72no—
160C2×#67C2×D711yescharacteristic
161C22×Dic3—71no—
162C22×Dic3—71no—
163#237—144no—
164C2×D12—72no—
165C2×C4×D3—71no—
166C2×C4×D3—72no—
167C2×C4×D3—72no—
168C2×C4×D3—144no—
169C2×Dic6—71no—
170C3×C4⋊C4—72no—
171C3×C22⋊C4—71no—
172C3×C22⋊C4—72no—
173#217C2×D711yescharacteristic
174(C2×D3)⋊C4—72no—
175(C2×D3)⋊C4—72no—
176(C2×D3)⋊C4C2×D712yescomplemented
177Dic3⋊C4—72no—
178Dic3⋊C4C2×D712yescomplemented
179C4×Dic3—72no—
180C2×C3×C7—22no—
181C2×C3×C7C22×D411yescharacteristic
182C2×C3×C7C2×#4112yes—
183D21—24no—
184C3×D7—41no—
185D3×C7—41no—
186C2×#41—211no—
187C23×C4—211no—
188#123—31no—
189#123—71no—
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—61no—
199C22×C7—62no—
200C22×C7D3×D412yes—
201C22×C7C23×D311yescharacteristic
202C2×D7—21no—
203C2×D7—22no—
204C2×D7—34no—
205C2×D7—68no—
206C2×D7—62no—
207C2×D7—62no—
208C4×C7—22no—
209C4×C7—61no—
210C4×C7—62no—
211Dic7—21no—
212Dic7—22no—
213Dic7—32no—
214Dic7—32no—
215C23×C3—71no—
216C23×C3C22×D711yescharacteristic
217C22×D3—72no—
218C22×D3—144no—
219C22×D3—142no—
220C22×D3C22×D711yescharacteristic
221C3×D4—142no—
222C2×C3×C4—22no—
223C2×C3×C4—72no—
224C2×C3×C4—71no—
225C2×C3×C4C22×D712yes—
226#67—22no—
227#67—144no—
228#67—142no—
229C2×Dic3—74no—
230C2×Dic3—71no—
231C2×Dic3—71no—
232C2×Dic3—142no—
233C2×Dic3C22×D712yes—
234C2×Dic3C22×D711yescharacteristic
235D12—144no—
236C4×D3—144no—
237C4×D3—144no—
238C4×D3—142no—
239Dic6—142no—
240C3×C7#52511yescharacteristic, complemented, Hall
241C24—211no—
242#41—424no—
243C2×Q8—211no—
244C2×D4—31no—
245C2×D4—71no—
246C2×D4—212no—
247C2×D4—212no—
248C2×D4—211no—
249C22×C4—211no—
250C22×C4—211no—
251C22×C4—211no—
252C22×C4—212no—
253C22×C4—212no—
254C22×C4—424no—
255C22×C4D3×D711yescharacteristic
256C4⋊C4—32no—
257C4⋊C4—72no—
258C4⋊C4—212no—
259C22⋊C4—31no—
260C22⋊C4—32no—
261C22⋊C4—71no—
262C22⋊C4—72no—
263C22⋊C4—212no—
264C22⋊C4—212no—
265C42—212no—
266C2×C7—22no—
267C2×C7—121no—
268C2×C7C2×D3×D411yescharacteristic
269C2×C7D3×#4112yes—
270D7—41no—
271D7—64no—
272C22×C3—22no—
273C22×C3—141no—
274C22×C3—142no—
275C22×C3D4×D712yes—
276C22×C3C23×D711yescharacteristic
277C2×D3—21no—
278C2×D3—22no—
279C2×D3—74no—
280C2×D3—148no—
281C2×D3—142no—
282C2×D3—142no—
283C3×C4—22no—
284C3×C4—141no—
285C3×C4—142no—
286Dic3—21no—
287Dic3—22no—
288Dic3—72no—
289Dic3—72no—
290C23—31no—
291C23—71no—
292C23—212no—
293C23—424no—
294C23—422no—
295C23C2×D3×D711yescharacteristic
296Q8—422no—
297D4—62no—
298D4—142no—
299D4—424no—
300D4—424no—
301D4—422no—
302C2×C4—22no—
303C2×C4—32no—
304C2×C4—31no—
305C2×C4—72no—
306C2×C4—71no—
307C2×C4—214no—
308C2×C4—211no—
309C2×C4—211no—
310C2×C4—424no—
311C2×C4—424no—
312C2×C4—422no—
313C2×C4—422no—
314C2×C4C2×D3×D712yes—
315C7#565611yescharacteristic, complemented, Sylow
316C2×C3—22no—
317C2×C3—281no—
318C2×C3C2×D4×D711yescharacteristic
319C2×C3D7×#4112yes—
320D3—41no—
321D3—144no—
322C22—22no—
323C22—62no—
324C22—61no—
325C22—142no—
326C22—141no—
327C22—214no—
328C22—428no—
329C22—422no—
330C22—422no—
331C22#6669412yes—
332C22C22×D3×D711yescharacteristic
333C4—22no—
334C4—62no—
335C4—61no—
336C4—142no—
337C4—141no—
338C4—212no—
339C4—212no—
340C3#9034711yescharacteristic, complemented, Sylow
341C2—22no—
342C2—121no—
343C2—281no—
344C2—424no—
345C2#1062156312yes—
346C2C2×#6669411yescharacteristic
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×D21C2semidirect
2. C2×#66684C2semidirect
3. C7×#929C2semidirect
4. C3×#6492C2semidirect
5. #10621135C2semidirect
6. #10621134C2semidirect
7. #10621094C2semidirect
8. #10621023C2semidirect
9. #10620968C2semidirect
10. #10620958C2semidirect
11. #10620939C2semidirect
21. C2×#4248C22semidirect
24. C2×#4245C22semidirect
25. C2×D3×Dic7C22semidirect
27. C2×Dic3×D7C22semidirect
29. C7×(C2×D3)⋊C4C22semidirect
30. C7×Dic3⋊C4C22semidirect
32. C3×(C2×D7)⋊C4C22semidirect
33. C3×Dic7⋊C4C22semidirect
34. #66577C22semidirect
37. Dic3×Dic7C22semidirect
42. #6492D3semidirect
58. C2×Dic21D4semidirect
96. (C2×D7)⋊C4C2×D3semidirect
98. Dic7⋊C4C2×D3semidirect
104. #929D7semidirect
117. C2×D21#41semidirect
176. (C2×D3)⋊C4C2×D7semidirect
178. Dic3⋊C4C2×D7semidirect
240. C3×C7#525semidirect
315. C7#5656semidirect
340. C3#90347semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D212
C2×#666842
C7×#9292
C3×#64922
#106211352
#106211342
#106210942
#106210232
#106209682
#106209582
#106209392
#903473
#56567