Finite Group Databasebeta

← #13058504

Subgroups — #13058504

Order 1344.

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

normal subgroups only

This group has 684 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×#66694C211yesmaximal, characteristic
2C22×D21⋊C4C211yesmaximal, characteristic, complemented
3C22⋊C4×D21C211yesmaximal, characteristic, complemented
4C7×#886C211yesmaximal, characteristic, complemented
5C3×#6449C211yesmaximal, characteristic, complemented
6#10621101C211yesmaximal, characteristic, complemented
7#10621098C211yesmaximal, characteristic, complemented
8#10621035C212yesmaximal, complemented
9#10621000C212yesmaximal, complemented
10#10620999C212yesmaximal, complemented
11#10620918C212yesmaximal, complemented
12D7×#123—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C22×D3×D7C2211yescharacteristic
15C2×C4×D21C2212yes—
16C2×C7×#67C2211yescharacteristic
17C22×C7×Dic3C2211yescharacteristic, complemented
18C2×C3×C22⋊D7C2211yescharacteristic
19C22×C3×Dic7C2211yescharacteristic, complemented
20#66694—24no—
21C2×#4247C2211yescharacteristic
22C2×#4246C2211yescharacteristic
23C2×D21⋊C4—24no—
24C2×D21⋊C4C2211yescharacteristic
25C2×D21⋊C4C2212yes—
26C2×D21⋊C4C2212yescomplemented
27C3×C7×C22⋊C4C2211yescharacteristic, complemented
28#66636C2211yescharacteristic, complemented
29#66631C2212yescomplemented
30C7×(C2×D3)⋊C4C2212yescomplemented
31C7×C4⋊Dic3C2212yescomplemented
32C3×(C2×D7)⋊C4C2212yescomplemented
33C3×C4⋊Dic7C2212yescomplemented
34#66577C2212yescomplemented
35#66577C2212yescomplemented
36(C2×D3)⋊Dic7C2212yescomplemented
37(C2×D7)⋊Dic3C2212yescomplemented
38C2×D4×D7—31no—
39C22×C4×D7—31no—
40C7×#123—31no—
41#6498—31no—
42#6492—31no—
43#6457—32no—
44D7×C4⋊C4—32no—
45#6449D311yescharacteristic, complemented
46#6446—32no—
47D7×C22⋊C4—31no—
48D7×C22⋊C4—32no—
49D3×#123—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21—24no—
52C22×D21D414yes—
53C22×D21C2311yescharacteristic
54C22×D21C2311yescharacteristic
55C22×D3×C7C2311yescharacteristic
56C22×C3×D7C2311yescharacteristic
57C2×D3×D7—22no—
58C2×D3×D7—24no—
59C2×C3×C4×C7C2312yes—
60C2×Dic21C2312yes—
61C4×D21—24no—
62C7×#67—22no—
63C2×C7×Dic3—22no—
64C2×C7×Dic3C2312yes—
65C2×C7×Dic3C2311yescharacteristic
66C3×C22⋊D7—22no—
67C2×C3×Dic7—22no—
68C2×C3×Dic7C2312yes—
69C2×C3×Dic7C2311yescharacteristic
70#4247—22no—
71#4246—22no—
72D21⋊C4—24no—
73D21⋊C4—24no—
74D21⋊C4—22no—
75C23×D7—31no—
76C23×D7—31no—
77C2×C7×D4—31no—
78C22×C4×C7—31no—
79C2×C22⋊D7—31no—
80C2×C22⋊D7C2×D311yescharacteristic
81C22×Dic7C2×D311yescharacteristic, complemented
82D4×D7—64no—
83C2×D28—31no—
84C2×C4×D7—32no—
85C2×C4×D7—32no—
86C2×C4×D7—32no—
87C2×C4×D7—31no—
88C2×C4×D7—64no—
89C7×C4⋊C4—32no—
90C7×C22⋊C4—32no—
91C7×C22⋊C4C2×D311yescharacteristic, complemented
92C22⋊Dic7—31no—
93C22⋊Dic7—32no—
94(C2×D7)⋊C4—32no—
95(C2×D7)⋊C4—32no—
96(C2×D7)⋊C4C2×D312yescomplemented
97C4⋊Dic7C2×D312yescomplemented
98Dic7⋊C4—32no—
99Dic7⋊C4—32no—
100C2×D3×D4—71no—
101C22×C4×D3—71no—
102C3×#123—71no—
103#935—71no—
104#929—71no—
105#894—72no—
106D3×C4⋊C4—72no—
107#886D711yescharacteristic, complemented
108#883—72no—
109D3×C22⋊C4—71no—
110D3×C22⋊C4—72no—
111C22×C3×C7—22no—
112C22×C3×C7C2×D412yes—
113C22×C3×C7C2411yescharacteristic
114C2×D21—22no—
115C2×D21—28no—
116C2×D21C2×D412yes—
117C2×D21C2×D412yes—
118C2×D21#4114yes—
119C2×D3×C7—21no—
120C2×D3×C7—22no—
121C2×C3×D7—21no—
122C2×C3×D7—22no—
123D3×D7—44no—
124C3×C4×C7—22no—
125Dic21—22no—
126C3×Dic7—22no—
127C3×Dic7—21no—
128C7×Dic3—22no—
129C7×Dic3—21no—
130C2×#123—211noSylow
131C23×C7—31no—
132C23×C7C22×D311yescharacteristic
133C22×D7—31no—
134C22×D7—34no—
135C22×D7—31no—
136C22×D7—62no—
137C22×D7—64no—
138C22×D7—64no—
139C22×D7C22×D311yescharacteristic
140C7×D4—62no—
141C2×C4×C7—31no—
142C2×C4×C7—32no—
143C2×C4×C7—62no—
144C2×C4×C7C22×D312yes—
145C22⋊D7—22no—
146C22⋊D7—62no—
147C2×Dic7—22no—
148C2×Dic7—32no—
149C2×Dic7C22×D311yescharacteristic
150C2×Dic7C22×D312yes—
151D28—62no—
152C4×D7—64no—
153C4×D7—64no—
154C4×D7—62no—
155C4×D7—64no—
156C23×D3—71no—
157C23×D3—71no—
158C2×C3×D4—71no—
159C22×C3×C4—71no—
160C2×#67—71no—
161C2×#67C2×D711yescharacteristic
162C22×Dic3C2×D711yescharacteristic, complemented
163D3×D4—144no—
164C2×D12—71no—
165C2×C4×D3—72no—
166C2×C4×D3—72no—
167C2×C4×D3—72no—
168C2×C4×D3—71no—
169C2×C4×D3—144no—
170C3×C4⋊C4—72no—
171C3×C22⋊C4—72no—
172C3×C22⋊C4C2×D711yescharacteristic, complemented
173#217—71no—
174#217—72no—
175(C2×D3)⋊C4—72no—
176(C2×D3)⋊C4—72no—
177(C2×D3)⋊C4C2×D712yescomplemented
178C4⋊Dic3C2×D712yescomplemented
179Dic3⋊C4—72no—
180Dic3⋊C4—72no—
181C2×C3×C7—22no—
182C2×C3×C7C22×D411yescharacteristic
183C2×C3×C7C2×#4112yes—
184D21—22no—
185D21#12314yescomplemented
186C3×D7—41no—
187D3×C7—41no—
188C22×D4—211no—
189C23×C4—211no—
190#123—31no—
191#123—71no—
192#123—212no—
193#123—212no—
194#123—211no—
195#123—211no—
196C2×C4⋊C4—212no—
197C2×C22⋊C4—212no—
198C2×C22⋊C4—211no—
199C22×C7—22no—
200C22×C7—61no—
201C22×C7—62no—
202C22×C7D3×D412yes—
203C22×C7C23×D311yescharacteristic
204C2×D7—21no—
205C2×D7—22no—
206C2×D7—32no—
207C2×D7—34no—
208C2×D7—32no—
209C2×D7—68no—
210C2×D7—62no—
211C2×D7—124no—
212C4×C7—22no—
213C4×C7—62no—
214C4×C7—61no—
215Dic7—22no—
216Dic7—21no—
217Dic7—62no—
218C23×C3—71no—
219C23×C3C22×D711yescharacteristic
220C22×D3—71no—
221C22×D3—74no—
222C22×D3—71no—
223C22×D3—142no—
224C22×D3—144no—
225C22×D3—144no—
226C22×D3C22×D711yescharacteristic
227C3×D4—142no—
228C2×C3×C4—71no—
229C2×C3×C4—72no—
230C2×C3×C4—142no—
231C2×C3×C4C22×D712yes—
232#67—22no—
233#67—142no—
234C2×Dic3—22no—
235C2×Dic3—72no—
236C2×Dic3C22×D711yescharacteristic
237C2×Dic3C22×D712yes—
238D12—142no—
239C4×D3—144no—
240C4×D3—144no—
241C4×D3—142no—
242C4×D3—144no—
243C3×C7C2×#12311yescharacteristic, complemented, Hall
244C24—211no—
245C24—211no—
246C2×D4—31no—
247C2×D4—71no—
248C2×D4—211no—
249C2×D4—211no—
250C2×D4—424no—
251C22×C4—31no—
252C22×C4—71no—
253C22×C4—212no—
254C22×C4—211no—
255C22×C4—212no—
256C22×C4—212no—
257C22×C4—424no—
258C4⋊C4—32no—
259C4⋊C4—72no—
260C4⋊C4—212no—
261C4⋊C4—212no—
262C22⋊C4—32no—
263C22⋊C4—72no—
264C22⋊C4—212no—
265C22⋊C4—212no—
266C22⋊C4—211no—
267C22⋊C4—212no—
268C22⋊C4D3×D711yescharacteristic, complemented
269C2×C7—22no—
270C2×C7—121no—
271C2×C7C2×D3×D411yescharacteristic
272C2×C7D3×#4112yes—
273D7—34no—
274D7—41no—
275D7—62no—
276C22×C3—22no—
277C22×C3—141no—
278C22×C3—142no—
279C22×C3D4×D712yes—
280C22×C3C23×D711yescharacteristic
281C2×D3—21no—
282C2×D3—22no—
283C2×D3—72no—
284C2×D3—74no—
285C2×D3—72no—
286C2×D3—148no—
287C2×D3—142no—
288C2×D3—284no—
289C3×C4—22no—
290C3×C4—142no—
291C3×C4—141no—
292Dic3—22no—
293Dic3—21no—
294Dic3—142no—
295C23—31no—
296C23—71no—
297C23—211no—
298C23—214no—
299C23—211no—
300C23—424no—
301C23—422no—
302C23—424no—
303C23C2×D3×D711yescharacteristic
304D4—62no—
305D4—142no—
306D4—422no—
307D4—422no—
308C2×C4—32no—
309C2×C4—31no—
310C2×C4—62no—
311C2×C4—72no—
312C2×C4—71no—
313C2×C4—142no—
314C2×C4—212no—
315C2×C4—424no—
316C2×C4—424no—
317C2×C4—424no—
318C2×C4—422no—
319C2×C4C2×D3×D712yes—
320C7D3×#12311yescharacteristic, complemented, Sylow
321C2×C3—22no—
322C2×C3—281no—
323C2×C3C2×D4×D711yescharacteristic
324C2×C3D7×#4112yes—
325D3—41no—
326D3—74no—
327D3—142no—
328C22—22no—
329C22—61no—
330C22—62no—
331C22—141no—
332C22—142no—
333C22—212no—
334C22—214no—
335C22—212no—
336C22—428no—
337C22—422no—
338C22—844no—
339C22#6668112yes—
340C22C22×D3×D711yescharacteristic
341C4—22no—
342C4—62no—
343C4—61no—
344C4—142no—
345C4—141no—
346C4—422no—
347C3D7×#12311yescharacteristic, complemented, Sylow
348C2—22no—
349C2—121no—
350C2—214no—
351C2—281no—
352C2—422no—
353C2C2×#6668111yescharacteristic
354C2#1062157112yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
prime radicalC2×#66694C2
hypercenterC22⋊C4D3×D7
hypocenterC3×C7C2×#123
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×D21⋊C4C2semidirect
3. C22⋊C4×D21C2semidirect
4. C7×#886C2semidirect
5. C3×#6449C2semidirect
6. #10621101C2semidirect
7. #10621098C2semidirect
8. #10621035C2semidirect
9. #10621000C2semidirect
10. #10620999C2semidirect
11. #10620918C2semidirect
17. C22×C7×Dic3C22semidirect
19. C22×C3×Dic7C22semidirect
26. C2×D21⋊C4C22semidirect
27. C3×C7×C22⋊C4C22semidirect
28. #66636C22semidirect
29. #66631C22semidirect
30. C7×(C2×D3)⋊C4C22semidirect
31. C7×C4⋊Dic3C22semidirect
32. C3×(C2×D7)⋊C4C22semidirect
33. C3×C4⋊Dic7C22semidirect
34. #66577C22semidirect
35. #66577C22semidirect
36. (C2×D3)⋊Dic7C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
45. #6449D3semidirect
81. C22×Dic7C2×D3semidirect
91. C7×C22⋊C4C2×D3semidirect
96. (C2×D7)⋊C4C2×D3semidirect
97. C4⋊Dic7C2×D3semidirect
107. #886D7semidirect
162. C22×Dic3C2×D7semidirect
172. C3×C22⋊C4C2×D7semidirect
177. (C2×D3)⋊C4C2×D7semidirect
178. C4⋊Dic3C2×D7semidirect
185. D21#123semidirect
243. C3×C7C2×#123semidirect
268. C22⋊C4D3×D7semidirect
320. C7D3×#123semidirect
347. C3D7×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#666942
C22×D21⋊C42
C22⋊C4×D212
C7×#8862
C3×#64492
#106211012
#106210982
#106210352
#106210002
#106209992
#106209182
D7×#1233
D3×#1237