Finite Group Databasebeta

← #11732338

Subgroups — #11732338

Order 896.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6502C211yesmaximal, characteristic, complemented
2C2×#6500C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D7C211yesmaximal, characteristic, complemented
4C22×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
5C2×#6447C211yesmaximal, characteristic, complemented
6C2×#6445C211yesmaximal, characteristic, complemented
7C7×#386C211yesmaximal, characteristic, complemented
8#89732C211yesmaximal, characteristic, complemented
9#89731C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89718C211yesmaximal, characteristic, complemented
12#89448C211yesmaximal, characteristic, complemented
13#89444C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89437C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic, complemented
23C23×Dic7C2211yescharacteristic
24C22×D28C2211yescharacteristic
25C22×D28C2211yescharacteristic
26C22×C4×D7C2211yescharacteristic, complemented
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic, complemented
29C2×C7×C22⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic, complemented
32#6502—24no—
33#6500—24no—
34C4×C22⋊D7—24no—
35C2×(C2×D7)⋊C4—24no—
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×(C2×D7)⋊C4C2211yescharacteristic
43C2×(C2×D7)⋊C4C2211yescharacteristic
44C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×Dic7⋊C4C2211yescharacteristic, complemented
47C2×Dic7⋊C4C2211yescharacteristic, complemented
48C2×Dic7⋊C4C2211yescharacteristic, complemented
49C2×Dic7⋊C4C2211yescharacteristic, complemented
50C2×C4×Dic7C2211yescharacteristic, complemented
51#6447—24no—
52#6445—24no—
53C7×#95C2211yescharacteristic, complemented
54C7×#95C2211yescharacteristic, complemented
55C7⋊#95C2211yescharacteristic, complemented
56C7⋊#95C2211yescharacteristic, complemented
57#2424—71nomaximal, Sylow
58C24×C7C2311yescharacteristic
59C24×C7C2311yescharacteristic
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—22no—
63C23×D7—22no—
64C23×D7—22no—
65C23×D7—41no—
66C23×D7—41no—
67C23×D7C2311yescharacteristic
68C23×D7C2311yescharacteristic
69C23×D7C2311yescharacteristic
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—22no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—41no—
76C22×C4×C7C2311yescharacteristic
77C22×C4×C7C2311yescharacteristic
78C22×C4×C7C2311yescharacteristic
79C22×C4×C7C2311yescharacteristic
80C22×C4×C7C2311yescharacteristic
81C2×C22⋊D7—22no—
82C2×C22⋊D7—22no—
83C2×C22⋊D7—22no—
84C2×C22⋊D7—22no—
85C2×C22⋊D7—22no—
86C2×C22⋊D7—22no—
87C2×C22⋊D7—22no—
88C2×C22⋊D7—22no—
89C2×C22⋊D7—41no—
90C2×C22⋊D7—41no—
91C2×C22⋊D7—41no—
92C2×C22⋊D7D414yescomplemented
93C22×Dic7—22no—
94C22×Dic7—22no—
95C22×Dic7—41no—
96C22×Dic7C2311yescharacteristic
97C22×Dic7C2311yescharacteristic
98C22×Dic7C2311yescharacteristic
99C22×Dic7C2311yescharacteristic
100C22×Dic7C2311yescharacteristic
101C2×D28—22no—
102C2×D28—22no—
103C2×D28—22no—
104C2×D28—22no—
105C2×D28—41no—
106C2×D28—41no—
107C2×C4×D7—22no—
108C2×C4×D7—22no—
109C2×C4×D7—41no—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—41no—
113C22⋊Dic7—22no—
114C22⋊Dic7D414yescomplemented
115(C2×D7)⋊C4—22no—
116(C2×D7)⋊C4—22no—
117(C2×D7)⋊C4—22no—
118(C2×D7)⋊C4—22no—
119(C2×D7)⋊C4—22no—
120(C2×D7)⋊C4—22no—
121(C2×D7)⋊C4—22no—
122(C2×D7)⋊C4—22no—
123(C2×D7)⋊C4—22no—
124(C2×D7)⋊C4D414yescomplemented
125Dic7⋊C4—22no—
126Dic7⋊C4—22no—
127Dic7⋊C4—22no—
128Dic7⋊C4—41no—
129C4×Dic7—22no—
130C2×#121—71no—
131C2×#121—71no—
132C2×#120—71no—
133C2×#120—71no—
134C2×C4×D4—71no—
135C22×C22⋊C4—71no—
136#396—71no—
137#394—71no—
138#394—71no—
139#392—71no—
140#390—71no—
141#388—71no—
142#386—71no—
143#386D711yescharacteristic, complemented
144#385—71no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7C2×D412yes—
157C23×C7C2×D412yes—
158C23×C7#4112yes—
159C23×C7C2411yescharacteristic
160C22×D7—28no—
161C22×D7—21no—
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—21no—
166C22×D7—21no—
167C22×D7—21no—
168C22×D7—21no—
169C22×D7—21no—
170C22×D7—21no—
171C22×D7—21no—
172C22×D7—21no—
173C22×D7—21no—
174C22×D7—21no—
175C22×D7—21no—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—21no—
179C22×D7—44no—
180C22×D7—44no—
181C22×D7—44no—
182C22×D7—44no—
183C22×D7—44no—
184C22×D7—44no—
185C22×D7C2×D412yes—
186C22×D7C2×D412yes—
187C22×D7#4112yes—
188C7×D4—44no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—44no—
204C2×C4×C7C2×D412yes—
205C22⋊D7—28no—
206C22⋊D7—44no—
207C22⋊D7—44no—
208C22⋊D7—44no—
209C2×Dic7—21no—
210C2×Dic7—21no—
211C2×Dic7—21no—
212C2×Dic7—21no—
213C2×Dic7—21no—
214C2×Dic7—21no—
215C2×Dic7—21no—
216C2×Dic7—21no—
217C2×Dic7—21no—
218C2×Dic7—21no—
219C2×Dic7—21no—
220C2×Dic7—44no—
221C2×Dic7C2×D412yes—
222C2×Dic7C2×D412yes—
223C2×Dic7C2×D412yes—
224C2×Dic7C2×D412yes—
225D28—44no—
226D28—44no—
227C4×D7—44no—
228C25—71no—
229C22×D4—71no—
230C22×D4—71no—
231C22×D4—71no—
232C22×D4—71no—
233C22×D4—71no—
234C22×D4—71no—
235C22×D4C2×D711yescharacteristic
236C23×C4—71no—
237C23×C4—71no—
238C23×C4C2×D711yescharacteristic
239#121—144no—
240#121—144no—
241#120—144no—
242#120—144no—
243C4×D4—144no—
244C2×C4⋊C4—71no—
245C2×C4⋊C4—71no—
246C2×C4⋊C4—71no—
247C2×C4⋊C4—71no—
248C2×C22⋊C4—71no—
249C2×C22⋊C4—71no—
250C2×C22⋊C4—71no—
251C2×C22⋊C4—71no—
252C2×C22⋊C4—71no—
253C2×C22⋊C4—71no—
254C2×C22⋊C4—71no—
255C2×C22⋊C4—71no—
256C2×C22⋊C4—71no—
257C2×C22⋊C4—71no—
258C2×C22⋊C4—71no—
259C2×C22⋊C4—71no—
260C2×C22⋊C4—144no—
261C2×C22⋊C4C2×D711yescharacteristic
262C2×C22⋊C4C2×D711yescharacteristic, complemented
263C2×C22⋊C4C2×D711yescharacteristic, complemented
264C2×C42—71no—
265#95—71no—
266#95—71no—
267#95C2×D711yescharacteristic, complemented
268#95C2×D711yescharacteristic, complemented
269C22×C7—22no—
270C22×C7—22no—
271C22×C7—22no—
272C22×C7—22no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7—41no—
276C22×C7—41no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7—41no—
281C22×C7—41no—
282C22×C7#12114yescomplemented
283C22×C7C22×D411yescharacteristic
284C22×C7C22×D411yescharacteristic
285C22×C7C22×D411yescharacteristic
286C22×C7C2×#4111yescharacteristic
287C22×C7C2×#4111yescharacteristic
288C22×C7H(5,𝔽2)11yescharacteristic
289C22×C7H(5,𝔽2)11yescharacteristic
290C2×D7—22no—
291C2×D7—22no—
292C2×D7—22no—
293C2×D7—22no—
294C2×D7—22no—
295C2×D7—22no—
296C2×D7—22no—
297C2×D7—22no—
298C2×D7—22no—
299C2×D7—416no—
300C2×D7—41no—
301C2×D7—41no—
302C2×D7—41no—
303C2×D7—41no—
304C2×D7—41no—
305C2×D7—41no—
306C2×D7—41no—
307C2×D7—41no—
308C2×D7—41no—
309C2×D7—41no—
310C2×D7—41no—
311C2×D7—41no—
312C4×C7—22no—
313C4×C7—41no—
314C4×C7—41no—
315C4×C7—41no—
316C4×C7—41no—
317Dic7—22no—
318Dic7—22no—
319Dic7—22no—
320Dic7—22no—
321Dic7—41no—
322C24—71no—
323C24—71no—
324C24—71no—
325C24—142no—
326C24—142no—
327C24—142no—
328C24—142no—
329C24—142no—
330C24—281no—
331C24—281no—
332C24C22×D711yescharacteristic
333C24C22×D711yescharacteristic
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—74no—
338C2×D4—142no—
339C2×D4—142no—
340C2×D4—142no—
341C2×D4—142no—
342C2×D4—142no—
343C2×D4—142no—
344C2×D4—142no—
345C2×D4—142no—
346C2×D4—142no—
347C2×D4—142no—
348C2×D4—142no—
349C2×D4—142no—
350C2×D4—281no—
351C2×D4—281no—
352C2×D4—281no—
353C2×D4—281no—
354C2×D4—281no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—71no—
359C22×C4—71no—
360C22×C4—71no—
361C22×C4—71no—
362C22×C4—71no—
363C22×C4—142no—
364C22×C4—142no—
365C22×C4—142no—
366C22×C4—142no—
367C22×C4—281no—
368C22×C4—281no—
369C22×C4C22×D711yescharacteristic
370C22×C4C22×D711yescharacteristic
371C22×C4C22×D711yescharacteristic
372C22×C4C22×D711yescharacteristic
373C22×C4C22×D711yescharacteristic
374C4⋊C4—142no—
375C4⋊C4—142no—
376C4⋊C4—142no—
377C4⋊C4—281no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—41no—
381C22⋊C4—74no—
382C22⋊C4—74no—
383C22⋊C4—142no—
384C22⋊C4—142no—
385C22⋊C4—142no—
386C22⋊C4—142no—
387C22⋊C4—142no—
388C22⋊C4—142no—
389C22⋊C4—142no—
390C22⋊C4—142no—
391C22⋊C4—142no—
392C22⋊C4—142no—
393C42—142no—
394C2×C7—24no—
395C2×C7—81no—
396C2×C7C2×#12111yescharacteristic
397C2×C7#52511yescharacteristic
398C2×C7#53811yescharacteristic
399C2×C7D4211yescharacteristic
400C2×C7D4211yescharacteristic
401C2×C7#54611yescharacteristic
402C2×C7#54611yescharacteristic
403D7—42no—
404D7—42no—
405D7—42no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—72no—
418C23—72no—
419C23—72no—
420C23—148no—
421C23—141no—
422C23—141no—
423C23—141no—
424C23—141no—
425C23—141no—
426C23—141no—
427C23—141no—
428C23—141no—
429C23—141no—
430C23—141no—
431C23—141no—
432C23—141no—
433C23—141no—
434C23—141no—
435C23—141no—
436C23—141no—
437C23—141no—
438C23—141no—
439C23—284no—
440C23—284no—
441C23—284no—
442C23—284no—
443C23—284no—
444C23—284no—
445C23#115712yes—
446C23D4×D712yes—
447C23D4×D712yes—
448C23C23×D711yescharacteristic
449D4—44no—
450D4—148no—
451D4—284no—
452D4—284no—
453D4—284no—
454D4—284no—
455D4—284no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—44no—
471C2×C4—72no—
472C2×C4—72no—
473C2×C4—72no—
474C2×C4—72no—
475C2×C4—141no—
476C2×C4—141no—
477C2×C4—141no—
478C2×C4—141no—
479C2×C4—141no—
480C2×C4—141no—
481C2×C4—141no—
482C2×C4—141no—
483C2×C4—141no—
484C2×C4—141no—
485C2×C4—141no—
486C2×C4—284no—
487C2×C4—284no—
488C2×C4D4×D712yes—
489C7#242411yescharacteristic, complemented, Sylow
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—142no—
504C22—142no—
505C22—142no—
506C22—142no—
507C22—142no—
508C22—142no—
509C22—142no—
510C22—142no—
511C22—142no—
512C22—2816no—
513C22—281no—
514C22—281no—
515C22—281no—
516C22—281no—
517C22—281no—
518C22—281no—
519C22—281no—
520C22—281no—
521C22—281no—
522C22—281no—
523C22—281no—
524C22—281no—
525C22#644714yescomplemented
526C22C2×#115711yescharacteristic
527C22C2×D4×D711yescharacteristic
528C22C2×D4×D711yescharacteristic
529C22C2×D4×D711yescharacteristic
530C22#654811yescharacteristic
531C22D7×#4111yescharacteristic
532C22#655311yescharacteristic
533C4—22no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—142no—
539C4—142no—
540C4—142no—
541C4—142no—
542C4—281no—
543C2—24no—
544C2—81no—
545C2—282no—
546C2—282no—
547C2—282no—
548C2C2×#644711yescharacteristic
549C2#9023811yescharacteristic
550C2#9025111yescharacteristic
551C2#9028111yescharacteristic
552C2#9030011yescharacteristic
553C2#9030111yescharacteristic
554C2#9034511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
hypercenter#386D7
hypocenterC7#2424
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#6502C2semidirect
2. C2×#6500C2semidirect
3. C2×C4×C22⋊D7C2semidirect
4. C22×(C2×D7)⋊C4C2semidirect
5. C2×#6447C2semidirect
6. C2×#6445C2semidirect
7. C7×#386C2semidirect
8. #89732C2semidirect
9. #89731C2semidirect
10. #89729C2semidirect
11. #89718C2semidirect
12. #89448C2semidirect
13. #89444C2semidirect
14. #89441C2semidirect
15. #89437C2semidirect
22. C22×C22⋊D7C22semidirect
26. C22×C4×D7C22semidirect
28. C2×C7×C22⋊C4C22semidirect
29. C2×C7×C22⋊C4C22semidirect
31. C2×C22⋊Dic7C22semidirect
44. C2×(C2×D7)⋊C4C22semidirect
45. C2×(C2×D7)⋊C4C22semidirect
46. C2×Dic7⋊C4C22semidirect
47. C2×Dic7⋊C4C22semidirect
48. C2×Dic7⋊C4C22semidirect
49. C2×Dic7⋊C4C22semidirect
50. C2×C4×Dic7C22semidirect
53. C7×#95C22semidirect
54. C7×#95C22semidirect
55. C7⋊#95C22semidirect
56. C7⋊#95C22semidirect
92. C2×C22⋊D7D4semidirect
114. C22⋊Dic7D4semidirect
124. (C2×D7)⋊C4D4semidirect
143. #386D7semidirect
262. C2×C22⋊C4C2×D7semidirect
263. C2×C22⋊C4C2×D7semidirect
267. #95C2×D7semidirect
268. #95C2×D7semidirect
282. C22×C7#121semidirect
489. C7#2424semidirect
525. C22#6447semidirect

Maximal subgroups

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

subgroupindex
C2×#65022
C2×#65002
C2×C4×C22⋊D72
C22×(C2×D7)⋊C42
C2×#64472
C2×#64452
C7×#3862
#897322
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