Finite Group Databasebeta

← #13060214

Subgroups — #13060214

Order 1344.

624 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66684C211yesmaximal, characteristic, complemented
2C2×#66683C211yesmaximal, characteristic, complemented
3#10621422C211yesmaximal, characteristic, complemented
4C7×#939C211yesmaximal, characteristic, complemented
5C3×#6503C211yesmaximal, characteristic, complemented
6#10621129C211yesmaximal, characteristic, complemented
7#10621129C211yesmaximal, characteristic, complemented
8#10621127C211yesmaximal, characteristic, complemented
9(C22⋊D7)⋊Dic3C211yesmaximal, characteristic, complemented
10Dic3×C22⋊D7C211yesmaximal, characteristic, complemented
11#10621095C211yesmaximal, characteristic, complemented
12#10620971C211yesmaximal, characteristic, complemented
13#10620970C211yesmaximal, characteristic, complemented
14#10620967C211yesmaximal, characteristic, complemented
15#10620885C211yesmaximal, characteristic, complemented
16#90290—31nomaximal, Hall
17C2×C3×C7×D4C2211yescharacteristic
18C2×#4268C2211yescharacteristic
19C2×#4268C2211yescharacteristic
20C22×Dic21C2211yescharacteristic
21C2×C7×#67C2211yescharacteristic
22C2×C7×#67C2211yescharacteristic
23C22×C7×Dic3C2211yescharacteristic
24C2×C3×C22⋊D7C2211yescharacteristic
25C2×C3×C22⋊D7C2211yescharacteristic
26C2×C3×C22⋊D7C2211yescharacteristic
27C2×C3×C22⋊D7C2211yescharacteristic
28C2×C3×D28C2211yescharacteristic
29C2×#4248C2211yescharacteristic
30C2×#4247C2211yescharacteristic
31C2×#4246C2211yescharacteristic
32C2×#4246C2211yescharacteristic
33C2×#4245C2211yescharacteristic
34C2×#4245C2211yescharacteristic
35C2×D21⋊C4C2211yescharacteristic
36C2×D3×Dic7C2211yescharacteristic
37#66684—24no—
38#66683—24no—
39C2×Dic3×D7C2211yescharacteristic
40C2×Dic3×D7C2211yescharacteristic
41#66636C2211yescharacteristic
42#66631C2211yescharacteristic
43#66629C2211yescharacteristic
44C7×#217C2211yescharacteristic
45C7×(C2×D3)⋊C4C2211yescharacteristic
46C7×Dic3⋊C4C2211yescharacteristic
47C3×C4×Dic7C2211yescharacteristic
48Dic7⋊Dic3C2211yescharacteristic
49Dic7⋊Dic3C2211yescharacteristic
50(C2×D7)⋊Dic3C2211yescharacteristic
51(C2×D7)⋊Dic3C2211yescharacteristic
52Dic3×Dic7C2211yescharacteristic
53Dic3×Dic7C2211yescharacteristic
54C2×#1159—31no—
55C2×#1157—31no—
56C7×#121—31no—
57#6503—31no—
58#6503—31no—
59#6503D311yescharacteristic, complemented
60#6502—31no—
61#6502—31no—
62C4×C22⋊D7—31no—
63#6458—31no—
64#6450—31no—
65#6447—31no—
66#6447—31no—
67(C22⋊D7)⋊C4—31no—
68(C22⋊D7)⋊C4—31no—
69#5718—71nomaximal, Hall
70C23×C3×C7C2311yescharacteristic
71C23×C3×C7C2311yescharacteristic
72C22×D21C2311yescharacteristic
73C22×D3×C7C2311yescharacteristic
74C22×C3×D7C2311yescharacteristic
75C22×C3×D7C2311yescharacteristic
76C3×C7×D4—41no—
77C2×C3×C4×C7C2311yescharacteristic
78#4268—22no—
79#4268—41no—
80C2×Dic21—22no—
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83C2×Dic21C2311yescharacteristic
84C7×#67—22no—
85C7×#67—41no—
86C2×C7×Dic3—22no—
87C2×C7×Dic3C2311yescharacteristic
88C2×C7×Dic3C2311yescharacteristic
89C2×C7×Dic3C2311yescharacteristic
90C3×C22⋊D7—22no—
91C3×C22⋊D7—22no—
92C3×C22⋊D7—22no—
93C3×C22⋊D7—22no—
94C2×C3×Dic7C2311yescharacteristic
95C2×C3×Dic7C2311yescharacteristic
96C3×D28—41no—
97#4248D414yescomplemented
98#4247—22no—
99#4246—22no—
100#4246—41no—
101#4245—22no—
102#4245—41no—
103D21⋊C4—22no—
104D3×Dic7—22no—
105Dic3×D7—22no—
106Dic3×D7—22no—
107C2×C7×D4—31no—
108C2×C7×D4—31no—
109C2×C7×D4C2×D311yescharacteristic
110C22×C4×C7—31no—
111C2×C22⋊D7—31no—
112C2×C22⋊D7—31no—
113C2×C22⋊D7—31no—
114C2×C22⋊D7—31no—
115C2×C22⋊D7—31no—
116C2×C22⋊D7C2×D311yescharacteristic
117C2×C22⋊D7C2×D311yescharacteristic
118C2×C22⋊D7C2×D311yescharacteristic
119C2×C22⋊D7C2×D311yescharacteristic
120C22×Dic7—31no—
121C22×Dic7—31no—
122#1159—64no—
123#1157—64no—
124C2×D28—31no—
125C2×D28—31no—
126C2×D28C2×D311yescharacteristic
127C2×C4×D7—31no—
128C2×C4×D7—31no—
129C2×C4×D7—31no—
130C2×Dic14—31no—
131C7×C4⋊C4—31no—
132C7×C22⋊C4—31no—
133C7×C22⋊C4—31no—
134C22⋊Dic7—31no—
135(C2×D7)⋊C4—31no—
136(C2×D7)⋊C4—31no—
137(C2×D7)⋊C4—31no—
138Dic7⋊C4—31no—
139Dic7⋊C4—31no—
140Dic7⋊C4—31no—
141C4×Dic7—31no—
142C4×Dic7—31no—
143C4×Dic7C2×D311yescharacteristic
144C2×#237—71no—
145C2×#237—71no—
146C3×#127—71no—
147#940—71no—
148#940—71no—
149#939—71no—
150#939—71no—
151#939—71no—
152#939D711yescharacteristic, complemented
153#938—71no—
154#938—71no—
155D4×Dic3—71no—
156D4×Dic3—71no—
157C4×D12—71no—
158C4×Dic6—71no—
159C22×C3×C7—21no—
160C22×C3×C7—21no—
161C22×C3×C7—21no—
162C22×C3×C7—21no—
163C22×C3×C7—21no—
164C22×C3×C7—21no—
165C22×C3×C7C2411yescharacteristic
166C2×D21—21no—
167C2×D21—21no—
168C2×D21—21no—
169C2×D3×C7—21no—
170C2×D3×C7—21no—
171C2×D3×C7—21no—
172C2×C3×D7—21no—
173C2×C3×D7—21no—
174C2×C3×D7—21no—
175C2×C3×D7—21no—
176C2×C3×D7—21no—
177C2×C3×D7—21no—
178C3×C4×C7—21no—
179Dic21—21no—
180Dic21—21no—
181Dic21C2×D412yes—
182C3×Dic7C2×D412yes—
183C3×Dic7#4112yescomplemented
184C7×Dic3—21no—
185C7×Dic3—21no—
186C7×Dic3C2×D412yes—
187#550—211noSylow
188C23×C7—31no—
189C23×C7C22×D311yescharacteristic
190C23×C7C22×D311yescharacteristic
191C22×D7—31no—
192C22×D7C22×D311yescharacteristic
193C22×D7C22×D311yescharacteristic
194C7×D4—41no—
195C7×D4—62no—
196C7×D4—121no—
197C2×C4×C7—31no—
198C2×C4×C7—31no—
199C2×C4×C7—31no—
200C2×C4×C7—62no—
201C2×C4×C7C22×D311yescharacteristic
202C22⋊D7—22no—
203C22⋊D7—22no—
204C22⋊D7—22no—
205C22⋊D7—22no—
206C22⋊D7—62no—
207C22⋊D7—62no—
208C22⋊D7—62no—
209C22⋊D7—121no—
210C22⋊D7—121no—
211C2×Dic7—31no—
212C2×Dic7—31no—
213C2×Dic7—31no—
214C2×Dic7—62no—
215C2×Dic7—62no—
216C2×Dic7C22×D311yescharacteristic
217C2×Dic7C22×D311yescharacteristic
218D28—41no—
219D28—62no—
220D28—121no—
221C4×D7—62no—
222C4×D7—62no—
223C4×D7—62no—
224Dic14—34no—
225C2×C3×D4—71no—
226C2×C3×D4—71no—
227C2×C3×D4—71no—
228C2×C3×D4—71no—
229C2×C3×D4—71no—
230C2×C3×D4C2×D711yescharacteristic
231C2×#67—71no—
232C2×#67—71no—
233C2×#67—71no—
234C2×#67—71no—
235C2×#67—71no—
236C2×#67—71no—
237C2×#67C2×D711yescharacteristic
238C2×#67C2×D711yescharacteristic
239C22×Dic3—71no—
240C22×Dic3—71no—
241C22×Dic3—71no—
242C22×Dic3C2×D711yescharacteristic
243#237—144no—
244#237—144no—
245C2×D12—71no—
246C2×C4×D3—71no—
247C2×C4×D3—71no—
248C2×Dic6—71no—
249C3×C42—71no—
250#217—71no—
251#217—71no—
252#217—71no—
253#217C2×D711yescharacteristic
254(C2×D3)⋊C4—71no—
255(C2×D3)⋊C4C2×D711yescharacteristic
256C4⋊Dic3—71no—
257C4⋊Dic3—71no—
258Dic3⋊C4—71no—
259Dic3⋊C4C2×D711yescharacteristic
260C4×Dic3—71no—
261C4×Dic3—71no—
262C2×C3×C7—41no—
263C2×C3×C7—41no—
264C2×C3×C7C22×D411yescharacteristic
265C2×C3×C7C2×#4111yescharacteristic
266C2×C3×C7H(5,𝔽2)11yescharacteristic
267D21—41no—
268C3×D7—41no—
269C3×D7—41no—
270D3×C7—41no—
271C2×#41—211no—
272C2×#41—211no—
273#127—71no—
274#127—211no—
275#127—211no—
276#121—31no—
277#121—211no—
278#121—211no—
279#121—211no—
280#121—211no—
281#121—211no—
282C4×Q8—211no—
283C4×D4—211no—
284C4×D4—211no—
285C4×D4—211no—
286C22×C7—21no—
287C22×C7—21no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—21no—
291C22×C7—21no—
292C22×C7—61no—
293C22×C7—61no—
294C22×C7—61no—
295C22×C7C23×D311yescharacteristic
296C2×D7—21no—
297C2×D7—21no—
298C2×D7—21no—
299C2×D7—21no—
300C2×D7—21no—
301C2×D7—21no—
302C2×D7—61no—
303C2×D7—61no—
304C2×D7—61no—
305C4×C7—21no—
306C4×C7—32no—
307C4×C7—61no—
308C4×C7—61no—
309Dic7—32no—
310Dic7—61no—
311Dic7—61no—
312Dic7D3×D412yes—
313Dic7#23712yescomplemented
314C23×C3—71no—
315C23×C3—71no—
316C23×C3C22×D711yescharacteristic
317C23×C3C22×D711yescharacteristic
318C22×D3—71no—
319C22×D3C22×D711yescharacteristic
320C3×D4—41no—
321C3×D4—142no—
322C3×D4—142no—
323C3×D4—142no—
324C3×D4—142no—
325C3×D4—281no—
326C2×C3×C4—71no—
327C2×C3×C4—71no—
328C2×C3×C4C22×D711yescharacteristic
329#67—22no—
330#67—41no—
331#67—142no—
332#67—142no—
333#67—142no—
334#67—281no—
335#67—281no—
336#67—281no—
337C2×Dic3—22no—
338C2×Dic3—71no—
339C2×Dic3—71no—
340C2×Dic3—71no—
341C2×Dic3—142no—
342C2×Dic3—142no—
343C2×Dic3—142no—
344C2×Dic3C22×D711yescharacteristic
345C2×Dic3C22×D711yescharacteristic
346C2×Dic3C22×D711yescharacteristic
347D12—142no—
348C4×D3—142no—
349C4×D3—142no—
350Dic6—74no—
351C3×C7#55011yescharacteristic, complemented, Hall
352#41—424no—
353#41—424no—
354C2×Q8—211no—
355C2×D4—31no—
356C2×D4—31no—
357C2×D4—71no—
358C2×D4—71no—
359C2×D4—71no—
360C2×D4—71no—
361C2×D4—71no—
362C2×D4—211no—
363C2×D4—211no—
364C2×D4—211no—
365C2×D4—211no—
366C2×D4—211no—
367C2×D4—211no—
368C2×D4—211no—
369C2×D4D3×D711yescharacteristic
370C22×C4—31no—
371C22×C4—211no—
372C22×C4—211no—
373C22×C4—211no—
374C22×C4—211no—
375C22×C4—211no—
376C4⋊C4—31no—
377C4⋊C4—211no—
378C4⋊C4—211no—
379C4⋊C4—211no—
380C22⋊C4—31no—
381C22⋊C4—31no—
382C22⋊C4—211no—
383C22⋊C4—211no—
384C22⋊C4—211no—
385C22⋊C4—211no—
386C42—71no—
387C42—211no—
388C42—211no—
389C2×C7—41no—
390C2×C7—41no—
391C2×C7—121no—
392C2×C7C2×D3×D411yescharacteristic
393C2×C7C2×#23711yescharacteristic
394C2×C7#100411yescharacteristic
395D7—41no—
396D7—41no—
397D7—121no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—141no—
405C22×C3—141no—
406C22×C3—141no—
407C22×C3—141no—
408C22×C3—141no—
409C22×C3—141no—
410C22×C3C23×D711yescharacteristic
411C2×D3—21no—
412C2×D3—21no—
413C2×D3—21no—
414C2×D3—141no—
415C2×D3—141no—
416C2×D3—141no—
417C3×C4—21no—
418C3×C4—72no—
419C3×C4—72no—
420Dic3—21no—
421Dic3—21no—
422Dic3—72no—
423Dic3—141no—
424Dic3—141no—
425Dic3D4×D712yes—
426C23—31no—
427C23—71no—
428C23—71no—
429C23—211no—
430C23C2×D3×D711yescharacteristic
431C23C2×D3×D711yescharacteristic
432Q8—214no—
433D4—41no—
434D4—62no—
435D4—121no—
436D4—142no—
437D4—142no—
438D4—142no—
439D4—142no—
440D4—281no—
441D4—422no—
442D4—422no—
443D4—422no—
444D4—422no—
445D4—841no—
446D4—841no—
447D4—841no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—62no—
452C2×C4—71no—
453C2×C4—71no—
454C2×C4—211no—
455C2×C4—211no—
456C2×C4—211no—
457C2×C4—422no—
458C2×C4—422no—
459C2×C4—422no—
460C2×C4—422no—
461C2×C4—422no—
462C2×C4C2×D3×D711yescharacteristic
463C7#571811yescharacteristic, complemented, Sylow
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—281no—
467C2×C3—281no—
468C2×C3C2×D4×D711yescharacteristic
469C2×C3#654811yescharacteristic
470C2×C3D7×#4111yescharacteristic
471D3—41no—
472D3—281no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—61no—
480C22—61no—
481C22—61no—
482C22—141no—
483C22—141no—
484C22—141no—
485C22—141no—
486C22—141no—
487C22—141no—
488C22—421no—
489C22—421no—
490C22—421no—
491C22C22×D3×D711yescharacteristic
492C4—21no—
493C4—32no—
494C4—61no—
495C4—61no—
496C4—72no—
497C4—72no—
498C4—212no—
499C4—421no—
500C4—421no—
501C3#9029011yescharacteristic, complemented, Sylow
502C2—41no—
503C2—41no—
504C2—121no—
505C2—281no—
506C2—281no—
507C2—841no—
508C2D3×D4×D711yescharacteristic
509C2D7×#23711yescharacteristic
510C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#550
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66684C2semidirect
2. C2×#66683C2semidirect
3. #10621422C2semidirect
4. C7×#939C2semidirect
5. C3×#6503C2semidirect
6. #10621129C2semidirect
7. #10621129C2semidirect
8. #10621127C2semidirect
9. (C22⋊D7)⋊Dic3C2semidirect
10. Dic3×C22⋊D7C2semidirect
11. #10621095C2semidirect
12. #10620971C2semidirect
13. #10620970C2semidirect
14. #10620967C2semidirect
15. #10620885C2semidirect
59. #6503D3semidirect
97. #4248D4semidirect
152. #939D7semidirect
183. C3×Dic7#41semidirect
313. Dic7#237semidirect
351. C3×C7#550semidirect
463. C7#5718semidirect
501. C3#90290semidirect

Maximal subgroups

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

subgroupindex
C2×#666842
C2×#666832
#106214222
C7×#9392
C3×#65032
#106211292
#106211272
(C22⋊D7)⋊Dic32
Dic3×C22⋊D72
#106210952
#106209712
#106209702
#106209672
#106208852
#902903
#57187