Finite Group Databasebeta

← #13060153

Subgroups — #13060153

Order 1344.

500 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10621418C211yesmaximal, characteristic, complemented
2C7×#937C211yesmaximal, characteristic, complemented
3C3×#6501C211yesmaximal, characteristic, complemented
4#10621133C211yesmaximal, characteristic, complemented
5#10621130C211yesmaximal, characteristic, complemented
6#10621127C211yesmaximal, characteristic, complemented
7#67⋊Dic7C211yesmaximal, characteristic, complemented
8D3×C22⋊Dic7C211yesmaximal, characteristic, complemented
9#10621096C211yesmaximal, characteristic, complemented
10#10621092C211yesmaximal, characteristic, complemented
11#10621087C211yesmaximal, characteristic, complemented
12#10621009C211yesmaximal, characteristic, complemented
13#10621002C211yesmaximal, characteristic, complemented
14#10620998C211yesmaximal, characteristic, complemented
15#10620903C211yesmaximal, characteristic, complemented
16#90284—31nomaximal, Hall
17C23×D3×C7C2211yescharacteristic
18C2×C3×C7×D4C2211yescharacteristic
19C22×Dic21C2211yescharacteristic, complemented
20C2×C7×#67C2211yescharacteristic
21C2×C7×#67C2211yescharacteristic
22C2×C3×C22⋊D7C2211yescharacteristic
23C2×C3×C22⋊D7C2211yescharacteristic, complemented
24C2×C3×C4×D7C2211yescharacteristic, complemented
25C2×#4248C2211yescharacteristic, complemented
26C2×#4245C2211yescharacteristic
27C2×#4245C2211yescharacteristic
28C2×D3×Dic7C2211yescharacteristic
29C2×D3×Dic7C2211yescharacteristic
30#66636C2211yescharacteristic
31#66636C2211yescharacteristic
32#66636C2211yescharacteristic, complemented
33#66629C2211yescharacteristic, complemented
34#66629C2211yescharacteristic, complemented
35C7×#217C2211yescharacteristic
36C7×(C2×D3)⋊C4C2211yescharacteristic
37C7×(C2×D3)⋊C4C2211yescharacteristic
38C3×C22⋊Dic7C2211yescharacteristic
39C3×C22⋊Dic7C2211yescharacteristic, complemented
40C3×C4⋊Dic7C2211yescharacteristic, complemented
41Dic3⋊Dic7C2211yescharacteristic, complemented
42#66577C2211yescharacteristic, complemented
43Dic7⋊Dic3C2211yescharacteristic, complemented
44(C2×D3)⋊Dic7C2211yescharacteristic
45(C2×D3)⋊Dic7C2211yescharacteristic
46(C2×D3)⋊Dic7C2211yescharacteristic
47(C2×D3)⋊Dic7C2211yescharacteristic
48(C2×D7)⋊Dic3C2211yescharacteristic, complemented
49(C2×D7)⋊Dic3C2211yescharacteristic, complemented
50Dic3×Dic7C2211yescharacteristic, complemented
51Dic3×Dic7C2211yescharacteristic, complemented
52C7×#120—31no—
53#6516—31no—
54C2×C22⋊Dic7—31no—
55#6502—31no—
56#6502—31no—
57#6501D311yescharacteristic, complemented
58#6499—31no—
59#6498—31no—
60D4×Dic7—31no—
61#6448—31no—
62#6446—31no—
63(C22⋊D7)⋊C4—31no—
64#6442—31no—
65#6442—31no—
66#6441—31no—
67#5613—71nomaximal, Hall
68C23×C3×C7C2311yescharacteristic
69C23×C3×C7C2311yescharacteristic
70C22×D3×C7—22no—
71C22×D3×C7—41no—
72C22×D3×C7—41no—
73C22×D3×C7C2311yescharacteristic
74C22×D3×C7C2311yescharacteristic
75C22×C3×D7C2311yescharacteristic
76C3×C7×D4—41no—
77C2×C3×C4×C7C2311yescharacteristic
78C2×Dic21—41no—
79C2×Dic21C2311yescharacteristic
80C2×Dic21C2311yescharacteristic
81C2×Dic21C2311yescharacteristic
82C2×Dic21C2311yescharacteristic
83C7×#67—22no—
84C7×#67—41no—
85C2×C7×Dic3C2311yescharacteristic
86C2×C7×Dic3C2311yescharacteristic
87C3×C22⋊D7—41no—
88C3×C22⋊D7—41no—
89C2×C3×Dic7C2311yescharacteristic
90C2×C3×Dic7C2311yescharacteristic
91C2×C3×Dic7C2311yescharacteristic
92C3×C4×D7—41no—
93#4248—41no—
94#4245—22no—
95#4245—41no—
96D3×Dic7—41no—
97D3×Dic7—41no—
98C24×C7—31no—
99C2×C7×D4—31no—
100C2×C7×D4—31no—
101C2×C7×D4C2×D311yescharacteristic
102C2×C22⋊D7—31no—
103C2×C22⋊D7—31no—
104C2×C22⋊D7C2×D311yescharacteristic
105C2×C22⋊D7C2×D311yescharacteristic, complemented
106C22×Dic7—31no—
107C22×Dic7—31no—
108C22×Dic7—31no—
109C2×C4×D7C2×D311yescharacteristic, complemented
110C2×Dic14—31no—
111C7×C22⋊C4—31no—
112C7×C22⋊C4—31no—
113C7×C22⋊C4—31no—
114C22⋊Dic7—31no—
115C22⋊Dic7—31no—
116C22⋊Dic7—31no—
117C22⋊Dic7—31no—
118C22⋊Dic7—31no—
119C22⋊Dic7—31no—
120C22⋊Dic7—31no—
121C22⋊Dic7C2×D311yescharacteristic
122C22⋊Dic7C2×D311yescharacteristic, complemented
123(C2×D7)⋊C4—31no—
124(C2×D7)⋊C4—31no—
125C4⋊Dic7—31no—
126C4⋊Dic7C2×D311yescharacteristic, complemented
127Dic7⋊C4—31no—
128Dic7⋊C4—31no—
129Dic7⋊C4—31no—
130Dic7⋊C4—31no—
131C4×Dic7—31no—
132C4×Dic7—31no—
133C3×#121—71no—
134#939—71no—
135#938—71no—
136#937—71no—
137#937D711yescharacteristic, complemented
138#936—71no—
139#935—71no—
140#929—71no—
141C4×#67—71no—
142#898—71no—
143#896—71no—
144#885—71no—
145#67⋊C4—71no—
146D3×C22⋊C4—71no—
147#879—71no—
148C22×C3×C7—21no—
149C22×C3×C7—21no—
150C22×C3×C7—21no—
151C22×C3×C7—21no—
152C22×C3×C7—21no—
153C22×C3×C7—21no—
154C22×C3×C7C2411yescharacteristic
155C2×D3×C7—21no—
156C2×D3×C7—21no—
157C2×D3×C7—21no—
158C2×D3×C7—21no—
159C2×D3×C7—21no—
160C2×D3×C7—44no—
161C2×D3×C7#4112yes—
162C2×C3×D7—21no—
163C2×C3×D7—21no—
164C2×C3×D7—21no—
165C3×C4×C7—21no—
166Dic21—21no—
167Dic21—21no—
168Dic21—21no—
169Dic21—21no—
170C3×Dic7—21no—
171C3×Dic7—21no—
172C3×Dic7—21no—
173C7×Dic3—21no—
174C7×Dic3—21no—
175#538—211noSylow
176C23×C7—31no—
177C23×C7—31no—
178C23×C7—62no—
179C23×C7—121no—
180C23×C7—121no—
181C23×C7C22×D311yescharacteristic
182C23×C7C22×D311yescharacteristic
183C22×D7C22×D311yescharacteristic
184C7×D4—41no—
185C7×D4—62no—
186C7×D4—121no—
187C2×C4×C7—31no—
188C2×C4×C7—31no—
189C2×C4×C7C22×D311yescharacteristic
190C22⋊D7—41no—
191C22⋊D7—41no—
192C22⋊D7—62no—
193C22⋊D7—121no—
194C2×Dic7—31no—
195C2×Dic7—31no—
196C2×Dic7—31no—
197C2×Dic7—31no—
198C2×Dic7—121no—
199C2×Dic7—121no—
200C2×Dic7—121no—
201C2×Dic7C22×D311yescharacteristic
202C2×Dic7C22×D311yescharacteristic
203C2×Dic7C22×D311yescharacteristic
204C4×D7—41no—
205Dic14—121no—
206C23×D3C2×D711yescharacteristic
207C2×C3×D4—71no—
208C2×C3×D4—71no—
209C2×C3×D4C2×D711yescharacteristic
210C22×C3×C4—71no—
211C2×#67—71no—
212C2×#67—71no—
213C2×#67C2×D711yescharacteristic
214C2×#67C2×D711yescharacteristic
215C22×Dic3—71no—
216C2×C4×D3—71no—
217C2×C4×D3—71no—
218C2×Dic6—71no—
219C3×C4⋊C4—71no—
220C3×C22⋊C4—71no—
221C3×C22⋊C4—71no—
222#217—71no—
223#217—71no—
224#217—71no—
225#217—71no—
226#217—71no—
227#217C2×D711yescharacteristic
228(C2×D3)⋊C4—71no—
229(C2×D3)⋊C4—71no—
230(C2×D3)⋊C4—71no—
231(C2×D3)⋊C4—71no—
232(C2×D3)⋊C4C2×D711yescharacteristic
233(C2×D3)⋊C4C2×D711yescharacteristic
234C4⋊Dic3—71no—
235Dic3⋊C4—71no—
236Dic3⋊C4—71no—
237Dic3⋊C4—71no—
238Dic3⋊C4—71no—
239C4×Dic3—71no—
240C4×Dic3—71no—
241C2×C3×C7—41no—
242C2×C3×C7—41no—
243C2×C3×C7C2×#4111yescharacteristic
244C2×C3×C7H(5,𝔽2)11yescharacteristic
245C2×C3×C7H(5,𝔽2)11yescharacteristic
246C3×D7—41no—
247D3×C7—22no—
248D3×C7—41no—
249#126—211no—
250#126—211no—
251#124—211no—
252#124—211no—
253#123—211no—
254#123—211no—
255C22⋊Q8—211no—
256#121—71no—
257#121—211no—
258#121—211no—
259#120—31no—
260#120—211no—
261C4×D4—211no—
262C4×D4—211no—
263C2×C22⋊C4—211no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—32no—
271C22×C7—61no—
272C22×C7—61no—
273C22×C7—61no—
274C22×C7—61no—
275C22×C7—61no—
276C22×C7—124no—
277C22×C7C23×D311yescharacteristic
278C2×D7—21no—
279C2×D7—21no—
280C2×D7—21no—
281C4×C7—21no—
282C4×C7—61no—
283C4×C7—61no—
284Dic7—21no—
285Dic7—21no—
286Dic7—21no—
287Dic7—61no—
288Dic7—61no—
289Dic7—61no—
290Dic7—61no—
291C23×C3—71no—
292C23×C3C22×D711yescharacteristic
293C23×C3C22×D711yescharacteristic
294C22×D3—22no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3C22×D711yescharacteristic
298C22×D3C22×D711yescharacteristic
299C3×D4—41no—
300C3×D4—281no—
301C3×D4—281no—
302C2×C3×C4—71no—
303C2×C3×C4—71no—
304C2×C3×C4—71no—
305C2×C3×C4—281no—
306C2×C3×C4C22×D711yescharacteristic
307#67—22no—
308#67—41no—
309#67—142no—
310#67—281no—
311C2×Dic3—71no—
312C2×Dic3—71no—
313C2×Dic3—71no—
314C2×Dic3—71no—
315C2×Dic3—281no—
316C2×Dic3C22×D711yescharacteristic
317C2×Dic3C22×D711yescharacteristic
318C4×D3—281no—
319C4×D3—281no—
320Dic6—281no—
321C3×C7#53811yescharacteristic, complemented, Hall
322C24—31no—
323C2×Q8—211no—
324C2×D4—31no—
325C2×D4—31no—
326C2×D4—71no—
327C2×D4—71no—
328C2×D4—211no—
329C2×D4—211no—
330C2×D4D3×D711yescharacteristic
331C22×C4—71no—
332C22×C4—211no—
333C22×C4—211no—
334C22×C4—211no—
335C4⋊C4—71no—
336C4⋊C4—211no—
337C4⋊C4—211no—
338C4⋊C4—211no—
339C4⋊C4—211no—
340C4⋊C4—211no—
341C22⋊C4—31no—
342C22⋊C4—31no—
343C22⋊C4—31no—
344C22⋊C4—71no—
345C22⋊C4—71no—
346C22⋊C4—211no—
347C22⋊C4—211no—
348C22⋊C4—211no—
349C22⋊C4—211no—
350C22⋊C4—211no—
351C22⋊C4—211no—
352C22⋊C4—211no—
353C22⋊C4—211no—
354C22⋊C4—211no—
355C42—211no—
356C42—211no—
357C2×C7—41no—
358C2×C7—41no—
359C2×C7—62no—
360C2×C7—121no—
361C2×C7#100411yescharacteristic
362C2×C7#100411yescharacteristic
363C2×C7D3×#4111yescharacteristic
364D7—41no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—141no—
372C22×C3—141no—
373C22×C3—141no—
374C22×C3C23×D711yescharacteristic
375C2×D3—21no—
376C2×D3—21no—
377C2×D3—21no—
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—44no—
381C2×D3#115912yes—
382C3×C4—21no—
383C3×C4—141no—
384C3×C4—141no—
385C3×C4—141no—
386Dic3—21no—
387Dic3—21no—
388Dic3—141no—
389Dic3—141no—
390Dic3—141no—
391Dic3—141no—
392C23—31no—
393C23—31no—
394C23—62no—
395C23—71no—
396C23—121no—
397C23—121no—
398C23C2×D3×D711yescharacteristic
399C23C2×D3×D711yescharacteristic
400Q8—841no—
401D4—41no—
402D4—62no—
403D4—121no—
404D4—281no—
405D4—281no—
406D4—422no—
407D4—841no—
408C2×C4—31no—
409C2×C4—31no—
410C2×C4—71no—
411C2×C4—71no—
412C2×C4—71no—
413C2×C4—211no—
414C2×C4—211no—
415C2×C4—211no—
416C2×C4—211no—
417C2×C4—281no—
418C2×C4—841no—
419C2×C4—841no—
420C2×C4—841no—
421C2×C4C2×D3×D711yescharacteristic
422C7#561311yescharacteristic, complemented, Sylow
423C2×C3—41no—
424C2×C3—41no—
425C2×C3—281no—
426C2×C3C2×#115911yescharacteristic
427C2×C3#654811yescharacteristic
428C2×C3#654811yescharacteristic
429D3—22no—
430D3—41no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—32no—
438C22—61no—
439C22—61no—
440C22—61no—
441C22—61no—
442C22—61no—
443C22—124no—
444C22—141no—
445C22—141no—
446C22—141no—
447C22C22×D3×D711yescharacteristic
448C4—21no—
449C4—61no—
450C4—61no—
451C4—141no—
452C4—141no—
453C4—141no—
454C4—421no—
455C4—421no—
456C4—421no—
457C4—421no—
458C3#9028411yescharacteristic, complemented, Sylow
459C2—41no—
460C2—41no—
461C2—62no—
462C2—121no—
463C2—281no—
464C2D3×#115911yescharacteristic
465C2#1062157211yescharacteristic
466C2#1062159611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#538
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. #10621418C2semidirect
2. C7×#937C2semidirect
3. C3×#6501C2semidirect
4. #10621133C2semidirect
5. #10621130C2semidirect
6. #10621127C2semidirect
7. #67⋊Dic7C2semidirect
8. D3×C22⋊Dic7C2semidirect
9. #10621096C2semidirect
10. #10621092C2semidirect
11. #10621087C2semidirect
12. #10621009C2semidirect
13. #10621002C2semidirect
14. #10620998C2semidirect
15. #10620903C2semidirect
19. C22×Dic21C22semidirect
23. C2×C3×C22⋊D7C22semidirect
24. C2×C3×C4×D7C22semidirect
25. C2×#4248C22semidirect
32. #66636C22semidirect
33. #66629C22semidirect
34. #66629C22semidirect
39. C3×C22⋊Dic7C22semidirect
40. C3×C4⋊Dic7C22semidirect
41. Dic3⋊Dic7C22semidirect
42. #66577C22semidirect
43. Dic7⋊Dic3C22semidirect
48. (C2×D7)⋊Dic3C22semidirect
49. (C2×D7)⋊Dic3C22semidirect
50. Dic3×Dic7C22semidirect
51. Dic3×Dic7C22semidirect
57. #6501D3semidirect
105. C2×C22⋊D7C2×D3semidirect
109. C2×C4×D7C2×D3semidirect
122. C22⋊Dic7C2×D3semidirect
126. C4⋊Dic7C2×D3semidirect
137. #937D7semidirect
321. C3×C7#538semidirect
422. C7#5613semidirect
458. C3#90284semidirect

Maximal subgroups

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

subgroupindex
#106214182
C7×#9372
C3×#65012
#106211332
#106211302
#106211272
#67⋊Dic72
D3×C22⋊Dic72
#106210962
#106210922
#106210872
#106210092
#106210022
#106209982
#106209032
#902843
#56137