Finite Group Databasebeta

← #13058061

Subgroups — #13058061

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
1C22×#4246C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic, complemented
3#10621365C211yesmaximal, characteristic, complemented
4C7×#877C211yesmaximal, characteristic, complemented
5C3×#6445C211yesmaximal, characteristic, complemented
6#10621134C211yesmaximal, characteristic, complemented
7#10621126C211yesmaximal, characteristic, complemented
8#10621021C212yesmaximal, complemented
9#10621020C212yesmaximal, complemented
10#10620971C212yesmaximal, complemented
11#10620915C212yesmaximal, complemented
12#90216—31nomaximal, Hall
13C23×D21C2211yescharacteristic
14C23×C3×D7C2211yescharacteristic
15C2×#4268C2211yescharacteristic
16C2×D84C2212yescomplemented
17C22×C7×Dic3C2211yescharacteristic
18C2×C3×C22⋊D7C2211yescharacteristic
19C2×C3×D28C2212yescomplemented
20C2×#4248C2211yescharacteristic, complemented
21C2×#4246—24no—
22C2×#4246C2212yes—
23C2×#4246C2212yes—
24C2×#4246C2211yescharacteristic, complemented
25C2×#4246C2212yescomplemented
26C2×#4246C2212yescomplemented
27C2×D21⋊C4C2211yescharacteristic, complemented
28#66683—24no—
29C2×Dic3×D7C2211yescharacteristic, complemented
30C3×C7×C22⋊C4C2211yescharacteristic
31#66631C2212yescomplemented
32C7×#217C2211yescharacteristic
33C7×Dic3⋊C4C2212yescomplemented
34C4×C7×Dic3C2212yescomplemented
35C3×(C2×D7)⋊C4C2212yescomplemented
36#66575C2212yescomplemented
37(C2×D7)⋊Dic3C2212yescomplemented
38C2×#1157—31no—
39C22×D28—31no—
40C7×#117—31no—
41#6458—32no—
42#6458—32no—
43#6445—31no—
44#6445—31no—
45#6445—31no—
46#6445D311yescharacteristic, complemented
47#6438—32no—
48#6437—32no—
49#5592—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21—22no—
52C22×D21—42no—
53C22×D21C2312yes—
54C22×D21C2311yescharacteristic
55C22×C3×D7—22no—
56C22×C3×D7—42no—
57C22×C3×D7C2312yes—
58C22×C3×D7C2311yescharacteristic
59C2×C3×C4×C7C2312yes—
60#4268—22no—
61C2×Dic21C2311yescharacteristic
62D84—42no—
63C2×C7×Dic3D414yescomplemented
64C2×C7×Dic3C2312yes—
65C2×C7×Dic3C2311yescharacteristic
66C2×C7×Dic3C2311yescharacteristic
67C3×C22⋊D7—22no—
68C2×C3×Dic7C2311yescharacteristic
69C3×D28—42no—
70#4248—22no—
71#4246—22no—
72#4246—24no—
73#4246—24no—
74#4246—42no—
75#4246—42no—
76D21⋊C4—22no—
77Dic3×D7—22no—
78C23×D7—31no—
79C23×D7C2×D311yescharacteristic
80C22×C4×C7—31no—
81C2×C22⋊D7—31no—
82C2×C22⋊D7C2×D311yescharacteristic
83#1157—64no—
84C2×D28—31no—
85C2×D28—32no—
86C2×D28—32no—
87C2×D28—32no—
88C2×D28—32no—
89C2×D28—32no—
90C2×D28—64no—
91C2×D28C2×D312yescomplemented
92C2×C4×D7—31no—
93C2×C4×D7—31no—
94C2×Dic14—31no—
95C7×C4⋊C4—32no—
96C7×C22⋊C4—31no—
97C7×C22⋊C4C2×D311yescharacteristic
98C42×C7—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4—32no—
101(C2×D7)⋊C4—32no—
102(C2×D7)⋊C4C2×D312yescomplemented
103C22×#67—71no—
104C2×#237—71no—
105C3×#120—71no—
106#953—71no—
107#940—72no—
108#939—72no—
109#937—71no—
110#885—72no—
111#884—72no—
112#882—71no—
113#877D711yescharacteristic, complemented
114C22×C3×C7—22no—
115C22×C3×C7C2×D412yes—
116C22×C3×C7C2411yescharacteristic
117C2×D21—21no—
118C2×D21—22no—
119C2×D21—22no—
120C2×D21—22no—
121C2×D21—22no—
122C2×D21—44no—
123C2×C3×D7—21no—
124C2×C3×D7—22no—
125C2×C3×D7—22no—
126C2×C3×D7—22no—
127C2×C3×D7—22no—
128C2×C3×D7—44no—
129C3×C4×C7—22no—
130Dic21—21no—
131C3×Dic7—21no—
132C7×Dic3—22no—
133C7×Dic3C2×D412yes—
134C7×Dic3C2×D412yes—
135#535—211noSylow
136C23×C7C22×D311yescharacteristic
137C22×D7—22no—
138C22×D7—32no—
139C22×D7—31no—
140C22×D7—42no—
141C22×D7—62no—
142C22×D7—122no—
143C22×D7C22×D312yes—
144C22×D7C22×D311yescharacteristic
145C2×C4×C7—34no—
146C2×C4×C7—31no—
147C2×C4×C7—31no—
148C2×C4×C7—32no—
149C2×C4×C7C22×D312yes—
150C22⋊D7—22no—
151C22⋊D7—62no—
152C2×Dic7—31no—
153C2×Dic7C22×D311yescharacteristic
154D28—42no—
155D28—64no—
156D28—64no—
157D28—62no—
158D28—122no—
159D28—122no—
160D28—122no—
161C4×D7—62no—
162C4×D7—62no—
163Dic14—62no—
164C24×C3—71no—
165C23×D3—71no—
166C2×C3×D4—71no—
167C2×C3×D4—72no—
168C2×#67—71no—
169C2×#67—72no—
170C2×#67—72no—
171C2×#67—71no—
172C2×#67—72no—
173C2×#67—72no—
174C2×#67—144no—
175C22×Dic3—71no—
176C22×Dic3C2×D711yescharacteristic
177#237—144no—
178C2×D12—72no—
179C2×C4×D3—71no—
180C2×Dic6—71no—
181C3×C22⋊C4—72no—
182C3×C22⋊C4C2×D711yescharacteristic
183#217—72no—
184#217C2×D711yescharacteristic
185(C2×D3)⋊C4—72no—
186(C2×D3)⋊C4—72no—
187Dic3⋊C4C2×D712yescomplemented
188C4×Dic3C2×D712yescomplemented
189C2×C3×C7—22no—
190C2×C3×C7C22×D411yescharacteristic
191C2×C3×C7H(5,𝔽2)12yes—
192D21—41no—
193D21—42no—
194C3×D7—41no—
195C3×D7—42no—
196C2×#41—211no—
197C22×D4—211no—
198#127—212no—
199#124—212no—
200#121—212no—
201#121—212no—
202#120—71no—
203#120—211no—
204#120—211no—
205#120—211no—
206#117—31no—
207C22×C7—22no—
208C22×C7D3×D412yes—
209C22×C7C23×D311yescharacteristic
210C2×D7—21no—
211C2×D7—22no—
212C2×D7—22no—
213C2×D7—22no—
214C2×D7—22no—
215C2×D7—44no—
216C2×D7—61no—
217C2×D7—62no—
218C2×D7—62no—
219C2×D7—62no—
220C2×D7—62no—
221C2×D7—124no—
222C4×C7—22no—
223C4×C7—32no—
224C4×C7—32no—
225C4×C7—62no—
226Dic7—21no—
227Dic7—61no—
228C23×C3—72no—
229C23×C3—71no—
230C23×C3—142no—
231C23×C3—282no—
232C23×C3C22×D711yescharacteristic
233C22×D3—72no—
234C22×D3—71no—
235C22×D3—142no—
236C22×D3—282no—
237C3×D4—142no—
238C3×D4—282no—
239C2×C3×C4—71no—
240C2×C3×C4C22×D712yes—
241#67—142no—
242#67—142no—
243#67—144no—
244#67—144no—
245#67—282no—
246#67—282no—
247C2×Dic3—71no—
248C2×Dic3—142no—
249C2×Dic3D2814yescomplemented
250C2×Dic3C22×D712yes—
251C2×Dic3C22×D711yescharacteristic
252C2×Dic3C22×D711yescharacteristic
253D12—282no—
254C4×D3—142no—
255Dic6—142no—
256C3×C7#53511yescharacteristic, complemented, Hall
257C24—71no—
258C24—211no—
259#41—424no—
260C2×Q8—211no—
261C2×D4—71no—
262C2×D4—72no—
263C2×D4—212no—
264C2×D4—212no—
265C2×D4—211no—
266C2×D4—212no—
267C2×D4—212no—
268C2×D4—211no—
269C2×D4—212no—
270C2×D4—424no—
271C22×C4—31no—
272C22×C4—211no—
273C22×C4—211no—
274C4⋊C4—32no—
275C22⋊C4—31no—
276C22⋊C4—72no—
277C22⋊C4—212no—
278C22⋊C4—212no—
279C22⋊C4—212no—
280C22⋊C4D3×D711yescharacteristic
281C42—32no—
282C2×C7—22no—
283C2×C7C2×D3×D411yescharacteristic
284C2×C7#100412yes—
285D7—42no—
286D7—41no—
287D7—122no—
288D7—121no—
289C22×C3—22no—
290C22×C3—141no—
291C22×C3—142no—
292C22×C3—142no—
293C22×C3—142no—
294C22×C3—142no—
295C22×C3—284no—
296C22×C3C2×D2812yes—
297C22×C3C23×D711yescharacteristic
298C2×D3—141no—
299C2×D3—142no—
300C2×D3—142no—
301C2×D3—142no—
302C2×D3—142no—
303C2×D3—284no—
304C3×C4—22no—
305C3×C4—141no—
306Dic3—22no—
307Dic3—141no—
308Dic3C2×D2812yes—
309Dic3C2×D2812yes—
310C23—72no—
311C23—71no—
312C23—142no—
313C23—212no—
314C23—211no—
315C23—282no—
316C23—422no—
317C23—842no—
318C23C2×D3×D711yescharacteristic
319Q8—422no—
320D4—142no—
321D4—282no—
322D4—424no—
323D4—424no—
324D4—422no—
325D4—422no—
326D4—842no—
327D4—842no—
328D4—842no—
329C2×C4—34no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—32no—
333C2×C4—71no—
334C2×C4—211no—
335C2×C4—422no—
336C2×C4—422no—
337C2×C4C2×D3×D712yes—
338C7#559211yescharacteristic, complemented, Sylow
339C2×C3—22no—
340C2×C3—282no—
341C2×C3—281no—
342C2×C3C22×D2811yescharacteristic
343C2×C3#655312yes—
344D3—282no—
345D3—281no—
346C22—22no—
347C22—141no—
348C22—142no—
349C22—142no—
350C22—142no—
351C22—142no—
352C22—284no—
353C22—421no—
354C22—422no—
355C22—422no—
356C22—422no—
357C22—422no—
358C22—844no—
359C22D3×D2812yes—
360C22C22×D3×D711yescharacteristic
361C4—22no—
362C4—32no—
363C4—32no—
364C4—62no—
365C4—141no—
366C4—421no—
367C3#9021611yescharacteristic, complemented, Sylow
368C2—22no—
369C2—282no—
370C2—281no—
371C2—842no—
372C2—841no—
373C2C2×D3×D2811yescharacteristic
374C2#1062157312yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#535
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×#4246C2semidirect
2. C2×#66683C2semidirect
3. #10621365C2semidirect
4. C7×#877C2semidirect
5. C3×#6445C2semidirect
6. #10621134C2semidirect
7. #10621126C2semidirect
8. #10621021C2semidirect
9. #10621020C2semidirect
10. #10620971C2semidirect
11. #10620915C2semidirect
16. C2×D84C22semidirect
19. C2×C3×D28C22semidirect
20. C2×#4248C22semidirect
24. C2×#4246C22semidirect
25. C2×#4246C22semidirect
26. C2×#4246C22semidirect
27. C2×D21⋊C4C22semidirect
29. C2×Dic3×D7C22semidirect
31. #66631C22semidirect
33. C7×Dic3⋊C4C22semidirect
34. C4×C7×Dic3C22semidirect
35. C3×(C2×D7)⋊C4C22semidirect
36. #66575C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
46. #6445D3semidirect
63. C2×C7×Dic3D4semidirect
91. C2×D28C2×D3semidirect
102. (C2×D7)⋊C4C2×D3semidirect
113. #877D7semidirect
187. Dic3⋊C4C2×D7semidirect
188. C4×Dic3C2×D7semidirect
249. C2×Dic3D28semidirect
256. C3×C7#535semidirect
338. C7#5592semidirect
367. C3#90216semidirect

Maximal subgroups

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

subgroupindex
C22×#42462
C2×#666832
#106213652
C7×#8772
C3×#64452
#106211342
#106211262
#106210212
#106210202
#106209712
#106209152
#902163
#55927