Finite Group Databasebeta

← #13058067

Subgroups — #13058067

Order 1344.

668 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 136 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×#4245C211yesmaximal, characteristic, complemented
2C2×#66684C211yesmaximal, characteristic, complemented
3#10621360C211yesmaximal, characteristic, complemented
4C7×#882C211yesmaximal, characteristic, complemented
5C3×#6445C211yesmaximal, characteristic, complemented
6#10621130C211yesmaximal, characteristic, complemented
7#10621125C211yesmaximal, characteristic, complemented
8#10621026C212yesmaximal, complemented
9#10621025C212yesmaximal, complemented
10#10621015C212yesmaximal, complemented
11#10620970C212yesmaximal, complemented
12#90283—31nomaximal, Hall
13C23×D3×C7C2211yescharacteristic
14C23×C3×D7C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C2×C7×#67C2211yescharacteristic
17C2×C7×D12C2212yescomplemented
18C2×C3×C22⋊D7C2211yescharacteristic
19C2×C3×D28C2212yescomplemented
20C2×#4248C2211yescharacteristic, complemented
21C2×#4245—24no—
22C2×#4245C2212yes—
23C2×#4245C2212yes—
24C2×#4245C2211yescharacteristic, complemented
25C2×#4245C2212yescomplemented
26C2×#4245C2212yescomplemented
27C2×D3×Dic7C2211yescharacteristic, complemented
28#66684—24no—
29C2×Dic3×D7C2211yescharacteristic, complemented
30C3×C7×C22⋊C4C2211yescharacteristic
31#66636C2211yescharacteristic
32#66629C2212yescomplemented
33C4×Dic21C2212yescomplemented
34C7×(C2×D3)⋊C4C2212yescomplemented
35C3×(C2×D7)⋊C4C2212yescomplemented
36(C2×D3)⋊Dic7C2212yescomplemented
37(C2×D7)⋊Dic3C2212yescomplemented
38C22×C22⋊D7—31no—
39C2×#1159—31no—
40C7×#120—31no—
41#6516—31no—
42#6503—32no—
43#6502—32no—
44#6500—31no—
45#6448—32no—
46#6447—32no—
47#6445D311yescharacteristic, complemented
48#6440—31no—
49#5592—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D3×C7—22no—
52C22×D3×C7—42no—
53C22×D3×C7C2312yes—
54C22×D3×C7C2311yescharacteristic
55C22×C3×D7—22no—
56C22×C3×D7—42no—
57C22×C3×D7C2312yes—
58C22×C3×D7C2311yescharacteristic
59C2×C3×C4×C7C2312yes—
60C2×Dic21D414yescomplemented
61C2×Dic21C2312yes—
62C2×Dic21C2311yescharacteristic
63C2×Dic21C2311yescharacteristic
64C7×#67—22no—
65C2×C7×Dic3C2311yescharacteristic
66C7×D12—42no—
67C3×C22⋊D7—22no—
68C2×C3×Dic7C2311yescharacteristic
69C3×D28—42no—
70#4248—22no—
71#4245—22no—
72#4245—24no—
73#4245—24no—
74#4245—42no—
75#4245—42no—
76D3×Dic7—22no—
77Dic3×D7—22no—
78C24×C7—31no—
79C23×D7C2×D311yescharacteristic
80C2×C7×D4—31no—
81C2×C7×D4—32no—
82C2×C22⋊D7—32no—
83C2×C22⋊D7—32no—
84C2×C22⋊D7—31no—
85C2×C22⋊D7—32no—
86C2×C22⋊D7—32no—
87C2×C22⋊D7—64no—
88C2×C22⋊D7C2×D311yescharacteristic
89C22×Dic7—31no—
90C22×Dic7—31no—
91#1159—64no—
92C2×D28C2×D312yescomplemented
93C2×C4×D7—31no—
94C2×Dic14—31no—
95C7×C22⋊C4—32no—
96C7×C22⋊C4C2×D311yescharacteristic
97C22⋊Dic7—31no—
98C22⋊Dic7—32no—
99(C2×D7)⋊C4—32no—
100(C2×D7)⋊C4C2×D312yescomplemented
101Dic7⋊C4—32no—
102C4×Dic7—32no—
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#882D711yescharacteristic, complemented
113#877—71no—
114C22×C3×C7—22no—
115C22×C3×C7C2×D412yes—
116C22×C3×C7C2411yescharacteristic
117C2×D3×C7—21no—
118C2×D3×C7—22no—
119C2×D3×C7—22no—
120C2×D3×C7—22no—
121C2×D3×C7—22no—
122C2×D3×C7—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—22no—
131Dic21C2×D412yes—
132Dic21C2×D412yes—
133C3×Dic7—21no—
134C7×Dic3—21no—
135#535—211noSylow
136C23×C7—32no—
137C23×C7—31no—
138C23×C7—62no—
139C23×C7—122no—
140C23×C7C22×D311yescharacteristic
141C22×D7—22no—
142C22×D7—42no—
143C22×D7C22×D312yes—
144C22×D7C22×D311yescharacteristic
145C7×D4—62no—
146C7×D4—122no—
147C2×C4×C7—31no—
148C2×C4×C7C22×D312yes—
149C22⋊D7—22no—
150C22⋊D7—62no—
151C22⋊D7—64no—
152C22⋊D7—64no—
153C22⋊D7—122no—
154C22⋊D7—122no—
155C2×Dic7—34no—
156C2×Dic7—31no—
157C2×Dic7—31no—
158C2×Dic7—32no—
159C2×Dic7—62no—
160C2×Dic7C22×D311yescharacteristic
161D28—42no—
162C4×D7—62no—
163Dic14—62no—
164C24×C3—71no—
165C23×D3C2×D711yescharacteristic
166C2×C3×D4—71no—
167C2×C3×D4—72no—
168C2×#67—72no—
169C2×#67—72no—
170C2×#67—71no—
171C2×#67—72no—
172C2×#67—72no—
173C2×#67—144no—
174C2×#67C2×D711yescharacteristic
175C22×Dic3—71no—
176C22×Dic3—71no—
177#237—144no—
178C2×D12C2×D712yescomplemented
179C2×C4×D3—71no—
180C2×Dic6—71no—
181C3×C22⋊C4—72no—
182C3×C22⋊C4C2×D711yescharacteristic
183#217—71no—
184#217—72no—
185(C2×D3)⋊C4—72no—
186(C2×D3)⋊C4C2×D712yescomplemented
187Dic3⋊C4—72no—
188C4×Dic3—72no—
189C2×C3×C7—22no—
190C2×C3×C7C22×D411yescharacteristic
191C2×C3×C7H(5,𝔽2)12yes—
192C3×D7—41no—
193C3×D7—42no—
194D3×C7—41no—
195D3×C7—42no—
196C2×#41—211no—
197C22×D4—211no—
198#127—212no—
199#124—212no—
200#121—212no—
201#121—212no—
202#120—31no—
203#120—71no—
204#120—211no—
205#120—211no—
206#117—211no—
207C22×C7—22no—
208C22×C7—61no—
209C22×C7—62no—
210C22×C7—62no—
211C22×C7—62no—
212C22×C7—62no—
213C22×C7—124no—
214C22×C7D3×D412yes—
215C22×C7C23×D311yescharacteristic
216C2×D7—21no—
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—22no—
220C2×D7—22no—
221C2×D7—44no—
222C4×C7—22no—
223C4×C7—61no—
224Dic7—21no—
225Dic7—32no—
226Dic7—32no—
227Dic7—62no—
228C23×C3—72no—
229C23×C3—71no—
230C23×C3—142no—
231C23×C3—282no—
232C23×C3C22×D711yescharacteristic
233C22×D3—22no—
234C22×D3—42no—
235C22×D3C22×D712yes—
236C22×D3C22×D711yescharacteristic
237C3×D4—142no—
238C3×D4—282no—
239C2×C3×C4—71no—
240C2×C3×C4C22×D712yes—
241#67—22no—
242#67—142no—
243#67—144no—
244#67—144no—
245#67—282no—
246#67—282no—
247C2×Dic3—74no—
248C2×Dic3—71no—
249C2×Dic3—71no—
250C2×Dic3—72no—
251C2×Dic3—142no—
252C2×Dic3C22×D711yescharacteristic
253D12—42no—
254C4×D3—142no—
255Dic6—142no—
256C3×C7#53511yescharacteristic, complemented, Hall
257C24—31no—
258C24—71no—
259#41—424no—
260C2×Q8—211no—
261C2×D4—31no—
262C2×D4—32no—
263C2×D4—71no—
264C2×D4—72no—
265C2×D4—212no—
266C2×D4—212no—
267C2×D4—211no—
268C2×D4—212no—
269C2×D4—212no—
270C2×D4—424no—
271C22×C4—211no—
272C22×C4—211no—
273C22×C4—211no—
274C4⋊C4—212no—
275C22⋊C4—32no—
276C22⋊C4—72no—
277C22⋊C4—212no—
278C22⋊C4—212no—
279C22⋊C4—211no—
280C22⋊C4D3×D711yescharacteristic
281C42—212no—
282C2×C7—22no—
283C2×C7—122no—
284C2×C7—121no—
285C2×C7C2×D3×D411yescharacteristic
286C2×C7#100412yes—
287D7—42no—
288D7—41no—
289C22×C3—22no—
290C22×C3—141no—
291C22×C3—142no—
292C22×C3—142no—
293C22×C3—142no—
294C22×C3—142no—
295C22×C3—284no—
296C22×C3D4×D712yes—
297C22×C3C23×D711yescharacteristic
298C2×D3—21no—
299C2×D3—22no—
300C2×D3—22no—
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—44no—
304C3×C4—22no—
305C3×C4—141no—
306Dic3—21no—
307Dic3—72no—
308Dic3—72no—
309Dic3—142no—
310C23—32no—
311C23—31no—
312C23—62no—
313C23—72no—
314C23—71no—
315C23—122no—
316C23—142no—
317C23—282no—
318C23C2×D3×D711yescharacteristic
319Q8—422no—
320D4—62no—
321D4—122no—
322D4—142no—
323D4—282no—
324D4—424no—
325D4—424no—
326D4—422no—
327D4—842no—
328D4—842no—
329C2×C4—31no—
330C2×C4—71no—
331C2×C4—214no—
332C2×C4—211no—
333C2×C4—211no—
334C2×C4—212no—
335C2×C4—422no—
336C2×C4—422no—
337C2×C4C2×D3×D712yes—
338C7#559211yescharacteristic, complemented, Sylow
339C2×C3—22no—
340C2×C3—282no—
341C2×C3—281no—
342C2×C3C2×D4×D711yescharacteristic
343C2×C3#654812yes—
344D3—42no—
345D3—41no—
346C22—22no—
347C22—61no—
348C22—62no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22—124no—
353C22—141no—
354C22—142no—
355C22—142no—
356C22—142no—
357C22—142no—
358C22—284no—
359C22#6668112yes—
360C22C22×D3×D711yescharacteristic
361C4—22no—
362C4—61no—
363C4—141no—
364C4—212no—
365C4—212no—
366C4—422no—
367C3#9028311yescharacteristic, complemented, Sylow
368C2—22no—
369C2—122no—
370C2—121no—
371C2—282no—
372C2—281no—
373C2C2×#6668111yescharacteristic
374C2#1062157212yes—
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×#4245C2semidirect
2. C2×#66684C2semidirect
3. #10621360C2semidirect
4. C7×#882C2semidirect
5. C3×#6445C2semidirect
6. #10621130C2semidirect
7. #10621125C2semidirect
8. #10621026C2semidirect
9. #10621025C2semidirect
10. #10621015C2semidirect
11. #10620970C2semidirect
17. C2×C7×D12C22semidirect
19. C2×C3×D28C22semidirect
20. C2×#4248C22semidirect
24. C2×#4245C22semidirect
25. C2×#4245C22semidirect
26. C2×#4245C22semidirect
27. C2×D3×Dic7C22semidirect
29. C2×Dic3×D7C22semidirect
32. #66629C22semidirect
33. C4×Dic21C22semidirect
34. C7×(C2×D3)⋊C4C22semidirect
35. C3×(C2×D7)⋊C4C22semidirect
36. (C2×D3)⋊Dic7C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
47. #6445D3semidirect
60. C2×Dic21D4semidirect
92. C2×D28C2×D3semidirect
100. (C2×D7)⋊C4C2×D3semidirect
112. #882D7semidirect
178. C2×D12C2×D7semidirect
186. (C2×D3)⋊C4C2×D7semidirect
256. C3×C7#535semidirect
338. C7#5592semidirect
367. C3#90283semidirect

Maximal subgroups

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

subgroupindex
C22×#42452
C2×#666842
#106213602
C7×#8822
C3×#64452
#106211302
#106211252
#106210262
#106210252
#106210152
#106209702
#902833
#55927