Finite Group Databasebeta

← #13371148

Subgroups — #13371148

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89885C211yesmaximal, characteristic
2#11736045C211yesmaximal, characteristic
3#11736029C211yesmaximal, characteristic, complemented
4C7×#1752C211yesmaximal, characteristic, complemented
5#11729731C211yesmaximal, characteristic, complemented
6#11727597C212yesmaximal, complemented
7#11727593C212yesmaximal, complemented
8#11727208C211yesmaximal, characteristic
9#11727196C211yesmaximal, characteristic, complemented
10#11726448C212yesmaximal, complemented
11#11726400C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C2×#6489C2211yescharacteristic
15C2×#6464C2212yes—
16C2×C7×C22⋊C8C2211yescharacteristic
17C2×C7×#97C2212yescomplemented
18C4×C7×C22⋊C4C2211yescharacteristic
19#89979C2211yescharacteristic
20#89979C2211yescharacteristic
21#89885C411yescharacteristic, complemented
22#89885C411yescharacteristic, complemented
23#89885C411yescharacteristic, complemented
24#89885C411yescharacteristic, complemented
25#89885C2211yescharacteristic
26#89885C2211yescharacteristic
27#89885C2211yescharacteristic
28#89885C2211yescharacteristic
29C2×#6395C411yescharacteristic, complemented
30C2×#6395C411yescharacteristic, complemented
31#89882C2212yes—
32C2×#6393C2212yescomplemented
33C2×#6390C411yescharacteristic, complemented
34C2×#6390C411yescharacteristic, complemented
35#89874C2212yescomplemented
36#89873C2212yescomplemented
37C2×#6379C2212yes—
38C2×#6379C2212yes—
39#89499—22no—
40#89498—21no—
41D7×C22⋊C8—21no—
42#89492—22no—
43#89491—21no—
44#89490—21no—
45C7×#336C2212yescomplemented
46#89347C2212yescomplemented
47#89318C2212yescomplemented
48#89318C2212yescomplemented
49#22586—71nomaximal, Sylow
50C23×C4×C7C2311yescharacteristic
51C22×C22⋊D7—21no—
52C2×#1157—21no—
53C2×#1157—21no—
54C2×#1157—24no—
55C2×#1157—42no—
56C2×#1157C2×C411yescharacteristic
57C2×#1157C2×C411yescharacteristic
58C2×#1157C2311yescharacteristic
59C2×#1157C2311yescharacteristic
60C22×D28C2×C411yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2×C411yescharacteristic
63C22×C7×C8—42no—
64C22×C7×C8C2×C411yescharacteristic
65C22×C7×C8C2×C411yescharacteristic
66C22×C7×C8C2×C411yescharacteristic
67C22×C7×C8C2×C411yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2312yes—
71C2×C7×C22⋊C4—21no—
72C2×C42×C7C2312yes—
73C2×C22⋊Dic7—21no—
74#6489—21no—
75#6489—21no—
76#6489C2×C411yescharacteristic
77#6489C2×C411yescharacteristic
78#6489C2311yescharacteristic
79#6489C2311yescharacteristic
80C2×C4⋊Dic7C2×C411yescharacteristic
81C2×C4⋊Dic7C2×C411yescharacteristic
82C2×C4×Dic7—21no—
83C2×#1136C2312yes—
84C2×#1136C2312yes—
85#6464—24no—
86#6464—42no—
87C2×#1131—22no—
88C2×C8×D7—22no—
89C7×C22⋊C8D411yescharacteristic
90C7×C22⋊C8D411yescharacteristic
91C7×C22⋊C8D412yes—
92C7×#97—42no—
93C7×#95—21no—
94C22⋊(C7⋊C8)—21no—
95C22⋊(C7⋊C8)—21no—
96C22⋊(C7⋊C8)—21no—
97C22⋊(C7⋊C8)—21no—
98#6395C2×C411yescharacteristic
99#6395C2×C411yescharacteristic
100#6395C2×C411yescharacteristic
101#6395C2×C411yescharacteristic
102#6395C2×C411yescharacteristic
103#6395C2×C411yescharacteristic
104#6395C2×C411yescharacteristic
105#6395C2×C411yescharacteristic
106(C2×D7)⋊C8—22no—
107(C2×D7)⋊C8—22no—
108#6393—42no—
109#6390C2×C411yescharacteristic
110#6390C2×C411yescharacteristic
111#6390C2×C411yescharacteristic
112#6390C2×C411yescharacteristic
113#6390C2×C411yescharacteristic
114#6390C2×C411yescharacteristic
115#6390C2×C411yescharacteristic
116#6390C2×C411yescharacteristic
117#6389—22no—
118Dic7⋊C8—22no—
119Dic7⋊C8—22no—
120C8×Dic7—22no—
121#6379—24no—
122#6379—42no—
123#2924—71no—
124C2×#416—71no—
125#2880—71no—
126#1938—72no—
127#1929—72no—
128#1866—72no—
129#1826—71no—
130#1810—71no—
131#1789—71no—
132#1765—72no—
133#1752D711yescharacteristic, complemented
134C24×C7#4111yescharacteristic
135C23×D7—21no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7C22×C411yescharacteristic
140C22×C4×C7C22×C411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7#4111yescharacteristic
144C22×C4×C7C2411yescharacteristic
145C2×C22⋊D7—21no—
146C2×C22⋊D7—21no—
147C2×C22⋊D7—21no—
148C2×C22⋊D7—21no—
149C2×C22⋊D7—44no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153#1157—22no—
154#1157—22no—
155#1157—22no—
156#1157—22no—
157#1157—22no—
158#1157—41no—
159#1157—41no—
160#1157—44no—
161#1157—44no—
162#1157#4114yes—
163C2×D28—22no—
164C2×D28—21no—
165C2×D28—41no—
166C2×D28C22×C411yescharacteristic
167C2×D28C22×C411yescharacteristic
168C2×C4×D7—21no—
169C2×C4×D7—21no—
170C2×C4×D7—21no—
171C2×C4×D7—21no—
172C2×C4×D7—44no—
173C2×Dic14—22no—
174C2×Dic14—21no—
175C2×Dic14—41no—
176C2×Dic14C22×C411yescharacteristic
177C2×Dic14C22×C411yescharacteristic
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8—22no—
181C2×C7×C8—22no—
182C2×C7×C8—41no—
183C2×C7×C8—41no—
184C2×C7×C8—41no—
185C2×C7×C8—41no—
186C2×C7×C8C22×C411yescharacteristic
187C2×C7×C8C22×C411yescharacteristic
188C2×C7×C8C22×C411yescharacteristic
189C2×C7×C8C22×C411yescharacteristic
190C2×C7×C8C2×D412yes—
191C2×C7×C8C2×D412yes—
192C7×C22⋊C4—42no—
193C42×C7—22no—
194C42×C7—22no—
195C22⋊Dic7—21no—
196C22⋊Dic7—21no—
197C22⋊Dic7—21no—
198C22⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7C22×C411yescharacteristic
202C4⋊Dic7C22×C411yescharacteristic
203C4⋊Dic7C22×C411yescharacteristic
204C4⋊Dic7C22×C411yescharacteristic
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209#1136—22no—
210#1136—22no—
211#1136—22no—
212#1136—22no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215#1131—44no—
216C8×D7—44no—
217C22×#41—71no—
218C2×#131—72no—
219C23×C8C2×D711yescharacteristic
220C2×#117—71no—
221#433—142no—
222#432—141no—
223#431—141no—
224C2×#108—72no—
225#424—72no—
226C2×#104—72no—
227C2×#104—72no—
228#416—71no—
229#416—71no—
230#416—71no—
231#416—71no—
232#416—71no—
233#416—71no—
234#416—71no—
235#416—71no—
236C2×#103—71no—
237C2×#103—71no—
238C2×#102—71no—
239C2×#102—71no—
240#408—72no—
241#408—142no—
242#407—71no—
243#407—71no—
244#407—141no—
245C2×C22⋊C8—141no—
246C2×C22⋊C8C2×D711yescharacteristic
247#405—72no—
248C2×#97C2×D712yescomplemented
249C4×C22⋊C4C2×D711yescharacteristic
250#341—72no—
251#339—72no—
252#339—72no—
253#336C2×D712yescomplemented
254C23×C7—22no—
255C23×C7—22no—
256C23×C7—41no—
257C23×C7C4×D411yescharacteristic
258C23×C7C4×D411yescharacteristic
259C23×C7C2×#4111yescharacteristic
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—42no—
263C22×D7—42no—
264C22×D7—81no—
265C2×C4×C7—22no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—42no—
273C2×C4×C7—42no—
274C2×C4×C7—42no—
275C2×C4×C7C4×D411yescharacteristic
276C2×C4×C7C4×D411yescharacteristic
277C2×C4×C7C23×C411yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C2×#4112yes—
280C2×C4×C7H(5,𝔽2)12yes—
281C22⋊D7—24no—
282C22⋊D7—42no—
283C22⋊D7—42no—
284C22⋊D7—42no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—42no—
292C2×Dic7—42no—
293C2×Dic7—81no—
294D28—21no—
295D28—21no—
296D28—21no—
297D28—42no—
298D28#11712yes—
299C4×D7—24no—
300C4×D7—42no—
301C4×D7—42no—
302C4×D7—42no—
303Dic14—21no—
304Dic14—21no—
305Dic14—21no—
306Dic14—42no—
307Dic14#11712yes—
308C7×C8—21no—
309C7×C8—21no—
310C7×C8—21no—
311C7×C8—21no—
312C7×C8—22no—
313C7×C8—22no—
314C7⋊C8—42no—
315C7⋊C8—42no—
316C2×#41—71no—
317C2×#41—71no—
318C2×#41—71no—
319C2×#41—71no—
320C2×#41—144no—
321C2×#41—141no—
322C2×#41—141no—
323C2×#41—282no—
324C22×Q8—71no—
325C22×D4—71no—
326C22×D4—141no—
327C23×C4—141no—
328C23×C4C22×D711yescharacteristic
329#131—144no—
330#131—282no—
331C2×M16—72no—
332C2×M16—72no—
333C2×M16—142no—
334C22×C8—42no—
335C22×C8—142no—
336C22×C8C4×D711yescharacteristic
337C22×C8C4×D711yescharacteristic
338C22×C8C4×D711yescharacteristic
339C22×C8C4×D711yescharacteristic
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D712yes—
343#117—71no—
344#117—71no—
345#117—71no—
346#117—71no—
347#117—141no—
348#117—141no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—141no—
353C2×C42—141no—
354C2×C42C22×D712yes—
355#108—282no—
356C4⋊C8—142no—
357C4⋊C8—142no—
358#104—144no—
359#104—282no—
360#103—71no—
361#103—71no—
362#103—71no—
363#103—71no—
364#103—71no—
365#103—71no—
366#103—71no—
367#103—71no—
368#102—71no—
369#102—71no—
370#102—71no—
371#102—71no—
372#102—71no—
373#102—71no—
374#102—71no—
375#102—71no—
376C22⋊C8—141no—
377C22⋊C8—141no—
378C22⋊C8—141no—
379C22⋊C8—141no—
380C22⋊C8—142no—
381C22⋊C8—142no—
382C22⋊C8D2811yescharacteristic
383C22⋊C8D2811yescharacteristic
384C22⋊C8D2812yes—
385#97—42no—
386#97—142no—
387C4×C8—142no—
388#95—21no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7—21no—
392C22×C7—21no—
393C22×C7—21no—
394C22×C7—21no—
395C22×C7—21no—
396C22×C7—44no—
397C22×C7#44412yes—
398C22×C7C2×C4×D411yescharacteristic
399C22×C7#55112yes—
400C2×D7—22no—
401C2×D7—41no—
402C2×D7—41no—
403C2×D7—41no—
404C2×D7—41no—
405C2×D7—41no—
406C2×D7—84no—
407C4×C7—21no—
408C4×C7—21no—
409C4×C7—82no—
410C4×C7C2×#11711yescharacteristic
411C4×C7#51811yescharacteristic
412C4×C7#54612yes—
413Dic7—22no—
414Dic7—41no—
415Dic7—41no—
416Dic7—41no—
417Dic7—41no—
418Dic7—41no—
419C24—141no—
420C24#115711yescharacteristic
421#41—74no—
422#41—142no—
423#41—142no—
424#41—142no—
425#41—142no—
426#41—142no—
427#41—284no—
428#41—281no—
429#41—281no—
430#41—284no—
431C2×Q8—71no—
432C2×Q8—71no—
433C2×Q8—142no—
434C2×Q8—141no—
435C2×Q8—281no—
436C2×D4—71no—
437C2×D4—71no—
438C2×D4—142no—
439C2×D4—141no—
440C2×D4—141no—
441C2×D4—141no—
442C2×D4—141no—
443C2×D4—141no—
444C2×D4—281no—
445C2×D4—284no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—284no—
457C22×C4C2×C4×D711yescharacteristic
458C22×C4C2×C4×D711yescharacteristic
459C22×C4C2×D2811yescharacteristic
460C22×C4C2×D2811yescharacteristic
461C22×C4#115711yescharacteristic
462C22×C4C23×D711yescharacteristic
463M16—142no—
464M16—142no—
465M16—142no—
466M16—142no—
467M16—284no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8—284no—
479C2×C8C2×C4×D711yescharacteristic
480C2×C8C2×C4×D711yescharacteristic
481C2×C8C2×C4×D711yescharacteristic
482C2×C8C2×C4×D711yescharacteristic
483C2×C8C2×D2812yes—
484C2×C8C2×D2812yes—
485C4⋊C4—71no—
486C4⋊C4—71no—
487C4⋊C4—71no—
488C4⋊C4—71no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C22⋊C4—42no—
492C22⋊C4—141no—
493C22⋊C4—141no—
494C22⋊C4—141no—
495C22⋊C4—141no—
496C42—22no—
497C42—22no—
498C42—141no—
499C42—141no—
500C42—141no—
501C42—141no—
502C2×C7—22no—
503C2×C7—22no—
504C2×C7—41no—
505C2×C7#234211yescharacteristic
506C2×C7C2×#44411yescharacteristic
507C2×C7#295711yescharacteristic
508D7—42no—
509D7—81no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23—141no—
514C23—141no—
515C23—282no—
516C23—282no—
517C23—561no—
518C23C4×D2811yescharacteristic
519C23C4×D2811yescharacteristic
520C23C2×#115711yescharacteristic
521Q8—72no—
522Q8—141no—
523Q8—141no—
524Q8—141no—
525Q8—282no—
526D4—72no—
527D4—144no—
528D4—141no—
529D4—141no—
530D4—141no—
531D4—282no—
532D4—282no—
533D4—282no—
534D4—282no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—141no—
546C2×C4—144no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—282no—
553C2×C4—282no—
554C2×C4—282no—
555C2×C4—282no—
556C2×C4—282no—
557C2×C4—561no—
558C2×C4C4×D2811yescharacteristic
559C2×C4C4×D2811yescharacteristic
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C22×D2811yescharacteristic
562C2×C4#654812yes—
563C2×C4D7×#4112yes—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—22no—
569C8—22no—
570C8—282no—
571C8—282no—
572C7#2258611yescharacteristic, complemented, Sylow
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—44no—
581C22—142no—
582C22—281no—
583C22—281no—
584C22—281no—
585C22—281no—
586C22—281no—
587C22—564no—
588C22#8948912yes—
589C22C2×C4×D2811yescharacteristic
590C22#9024712yes—
591C4—21no—
592C4—21no—
593C4—82no—
594C4—142no—
595C4—281no—
596C4—281no—
597C4—281no—
598C4—281no—
599C4—281no—
600C4#9017711yescharacteristic
601C4D7×#11711yescharacteristic
602C4#9024512yes—
603C2—22no—
604C2—22no—
605C2—41no—
606C2—282no—
607C2—561no—
608C2#1173102011yescharacteristic
609C2C2×#8948911yescharacteristic
610C2#1173602511yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1752C2
socleC22×C7C2×C4×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1752D7
hypocenterC7#22586
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ #1752

Complements

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

NH ≅ G/Ntype
3. #11736029C2semidirect
4. C7×#1752C2semidirect
5. #11729731C2semidirect
6. #11727597C2semidirect
7. #11727593C2semidirect
9. #11727196C2semidirect
10. #11726448C2semidirect
11. #11726400C2semidirect
17. C2×C7×#97C22semidirect
21. #89885C4semidirect
22. #89885C4semidirect
23. #89885C4semidirect
24. #89885C4semidirect
29. C2×#6395C4semidirect
30. C2×#6395C4semidirect
32. C2×#6393C22semidirect
33. C2×#6390C4semidirect
34. C2×#6390C4semidirect
35. #89874C22semidirect
36. #89873C22semidirect
45. C7×#336C22semidirect
46. #89347C22semidirect
47. #89318C22semidirect
48. #89318C22semidirect
133. #1752D7semidirect
248. C2×#97C2×D7semidirect
253. #336C2×D7semidirect
572. C7#22586semidirect

Maximal subgroups

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

subgroupindex
C2×#898852
#117360452
#117360292
C7×#17522
#117297312
#117275972
#117275932
#117272082
#117271962
#117264482
#117264002
#225867