Finite Group Databasebeta

← #13412963

Subgroups — #13412963

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731162C211yesmaximal, characteristic, complemented
2#11731090C211yesmaximal, characteristic
3C7×#1940C211yesmaximal, characteristic, complemented
4#11728238C211yesmaximal, characteristic
5#11728237C211yesmaximal, characteristic
6#11728233C211yesmaximal, characteristic
7#11728209C211yesmaximal, characteristic, complemented
8#11728208C211yesmaximal, characteristic, complemented
9#11728207C211yesmaximal, characteristic, complemented
10#11727570C211yesmaximal, characteristic, complemented
11#11727566C211yesmaximal, characteristic
12#11726436C211yesmaximal, characteristic
13#11726435C211yesmaximal, characteristic
14#11726378C211yesmaximal, characteristic, complemented
15#11726376C211yesmaximal, characteristic, complemented
16C2×#6453—21no—
17C2×C4×D28C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×C4⋊C8C2211yescharacteristic
21C2×C7×#97C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23C2×#6395C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6395C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×#6391C2211yescharacteristic
33#89762—21no—
34#89762—21no—
35#89757—21no—
36#89711C2211yescharacteristic
37#89711C2211yescharacteristic
38#89708C2211yescharacteristic
39#89707C2211yescharacteristic
40#89707C2211yescharacteristic
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C4×C4⋊Dic7C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89438—21no—
48#89436—21no—
49#89433—21no—
50#89433—21no—
51C7×#336C2211yescharacteristic
52C7×#336C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#26102—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4D411yescharacteristic
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C42×C7C2311yescharacteristic
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2311yescharacteristic
71C2×(C2×D7)⋊C4—21no—
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94#6453—44no—
95C4×D28—22no—
96C4×D28Q814yes—
97C7×C4⋊C8—22no—
98C7×C4⋊C8—22no—
99C7×C4⋊C8—22no—
100C7×#97—22no—
101C7×#95D411yescharacteristic
102C7×#95D411yescharacteristic
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110#6395—22no—
111#6395—22no—
112#6395—41no—
113#6395—41no—
114#6392—22no—
115#6392—41no—
116#6392—41no—
117#6391—22no—
118#6391—41no—
119#6391—41no—
120#2552—71no—
121#2357—71no—
122#2094—71no—
123#2093—71no—
124#2067—71no—
125#2065—71no—
126#1949—71no—
127#1948—71no—
128#1947—71no—
129#1940D711yescharacteristic, complemented
130#1921—71no—
131#1919—71no—
132#1847—71no—
133#1802—71no—
134#1762—71no—
135C23×D7—21no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2411yescharacteristic
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28C2×Q812yes—
156C2×C4×D7—24no—
157C2×C4×D7—81no—
158C2×C4×D7—81no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8#4112yes—
170C2×C7×C8#4112yes—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7C2×Q812yes—
178C42×C7#4112yes—
179(C2×D7)⋊C4—24no—
180(C2×D7)⋊C4—42no—
181(C2×D7)⋊C4—42no—
182(C2×D7)⋊C4—81no—
183(C2×D7)⋊C4—81no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—42no—
198C4⋊Dic7—42no—
199C4⋊Dic7—42no—
200C4⋊Dic7—81no—
201C4⋊Dic7—81no—
202C4⋊Dic7—81no—
203C4⋊Dic7C2×Q812yes—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206C4×Dic7—42no—
207C4×Dic7—81no—
208C4×Dic7—81no—
209C2×#125—141no—
210C2×C4×D4—71no—
211C2×#107—71no—
212C2×#107—71no—
213C2×#107—71no—
214C2×#106—71no—
215C2×#106—71no—
216C2×#106—71no—
217C2×C4⋊C8C2×D711yescharacteristic
218C2×C4⋊C8C2×D711yescharacteristic
219C2×C4⋊C8C2×D711yescharacteristic
220C2×#102—71no—
221C2×#102—71no—
222C2×#102—71no—
223C2×#102—71no—
224C2×#97C2×D711yescharacteristic
225#398—141no—
226#398—141no—
227#390—71no—
228#390—71no—
229#389—71no—
230#389—71no—
231#389—141no—
232#388—141no—
233#388—141no—
234#384—71no—
235#384—71no—
236#384—71no—
237#382—71no—
238#381—141no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C22⋊C4—141no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#336C2×D711yescharacteristic
251#336C2×D711yescharacteristic
252C23×C7C22×D411yescharacteristic
253C22×D7—22no—
254C22×D7—41no—
255C22×D7—41no—
256C22×D7—41no—
257C22×D7—41no—
258C22×D7—41no—
259C22×D7—41no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7C22⋊Q812yes—
275C2×C4×C7C22⋊Q812yes—
276C2×C4×C7AGL(1,ℤ/8)12yes—
277C2×C4×C7#13712yes—
278C2×C4×C7C22×Q811yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7H(5,𝔽2)11yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C2×Dic7—22no—
285C2×Dic7—41no—
286C2×Dic7—41no—
287C2×Dic7—41no—
288C2×Dic7—41no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305D28—22no—
306D28—22no—
307D28—41no—
308D28—41no—
309C4×D7—84no—
310C7×C8—22no—
311C7×C8—22no—
312C7×C8—41no—
313C7×C8—41no—
314C22×D4—71no—
315C23×C4—141no—
316C22×C8C22×D711yescharacteristic
317C22×C8C22×D711yescharacteristic
318C22×C8C22×D711yescharacteristic
319C22×C8C22×D711yescharacteristic
320#125—284no—
321C4×D4—74no—
322C4×D4—142no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4D2811yescharacteristic
339C2×C4⋊C4D2811yescharacteristic
340C2×C22⋊C4—141no—
341C2×C22⋊C4—141no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C42—141no—
346C2×C42—141no—
347C2×C42—141no—
348C2×C42C22×D711yescharacteristic
349C2×C42C22×D711yescharacteristic
350C2×C42C22×D711yescharacteristic
351#107—142no—
352#107—281no—
353#107—281no—
354#106—142no—
355#106—281no—
356#106—281no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360#102—142no—
361#102—142no—
362#102—281no—
363#102—281no—
364#97—22no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95D2811yescharacteristic
373#95D2811yescharacteristic
374C22×C7C2×C22⋊Q811yescharacteristic
375C22×C7#54711yescharacteristic
376C22×C7#55011yescharacteristic
377C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
378C22×C7C2×#13711yescharacteristic
379C22×C7#57611yescharacteristic
380C22×C7#57711yescharacteristic
381C2×D7—42no—
382C2×D7—42no—
383C2×D7—42no—
384C2×D7—81no—
385C2×D7—81no—
386C2×D7—81no—
387C2×D7—81no—
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7#53311yescharacteristic
395C4×C7#54411yescharacteristic
396C4×C7#55211yescharacteristic
397C4×C7#55311yescharacteristic
398Dic7—42no—
399Dic7—81no—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24—141no—
406C2×D4—72no—
407C2×D4—141no—
408C2×D4—141no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—141no—
412C22×C4—141no—
413C22×C4—144no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—561no—
421C22×C4—561no—
422C22×C4C2×D2811yescharacteristic
423C22×C4C2×D2811yescharacteristic
424C22×C4C2×D2811yescharacteristic
425C22×C4C2×D2811yescharacteristic
426C22×C4C2×D2811yescharacteristic
427C22×C4C2×D2811yescharacteristic
428C22×C4C23×D711yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8#115912yes—
440C2×C8#116112yes—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—72no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—282no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4—282no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4—561no—
465C22⋊C4—144no—
466C22⋊C4—282no—
467C22⋊C4—282no—
468C22⋊C4—561no—
469C22⋊C4—561no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—282no—
475C42—561no—
476C42—561no—
477C42#115912yes—
478C42Q8×D712yes—
479C2×C7#244111yescharacteristic
480C2×C7#307911yescharacteristic
481C2×C7#308111yescharacteristic
482C2×C7#334511yescharacteristic
483C2×C7#334611yescharacteristic
484C2×C7#339911yescharacteristic
485C2×C7#340011yescharacteristic
486D7—82no—
487C23—142no—
488C23—281no—
489C23—281no—
490C23—281no—
491C23—281no—
492C23—281no—
493C23—281no—
494C23C22×D2811yescharacteristic
495D4—142no—
496D4—142no—
497D4—281no—
498D4—281no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—142no—
514C2×C4—281no—
515C2×C4—281no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—564no—
535C2×C4#646012yes—
536C2×C4#646012yes—
537C2×C4#647112yes—
538C2×C4#647212yes—
539C2×C4C2×#115911yescharacteristic
540C2×C4C2×#115911yescharacteristic
541C2×C4C2×Q8×D711yescharacteristic
542C2×C4C2×#116111yescharacteristic
543C2×C4#655311yescharacteristic
544C2×C4#655411yescharacteristic
545C8—22no—
546C8—22no—
547C8—41no—
548C8—41no—
549C7#2610211yescharacteristic, complemented, Sylow
550C22—282no—
551C22—282no—
552C22—282no—
553C22—561no—
554C22—561no—
555C22—561no—
556C22—561no—
557C22C2×#646011yescharacteristic
558C22#9024611yescharacteristic
559C22#9026811yescharacteristic
560C22C2×#647111yescharacteristic
561C22C2×#647211yescharacteristic
562C22#9044211yescharacteristic
563C22#9044311yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—282no—
571C4—561no—
572C4—561no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9030911yescharacteristic
578C4#9033311yescharacteristic
579C4#9038011yescharacteristic
580C4#9041911yescharacteristic
581C2—562no—
582C2#1173344511yescharacteristic
583C2#1173650511yescharacteristic
584C2#1173653511yescharacteristic
585C2#1173691611yescharacteristic
586C2#1173691711yescharacteristic
587C2#1173739711yescharacteristic
588C2#1173739811yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1940C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1940D7
hypocenterC7#26102
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
1. #11731162C2semidirect
3. C7×#1940C2semidirect
7. #11728209C2semidirect
8. #11728208C2semidirect
9. #11728207C2semidirect
10. #11727570C2semidirect
14. #11726378C2semidirect
15. #11726376C2semidirect
129. #1940D7semidirect
549. C7#26102semidirect

Maximal subgroups

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

subgroupindex
#117311622
#117310902
C7×#19402
#117282382
#117282372
#117282332
#117282092
#117282082
#117282072
#117275702
#117275662
#117264362
#117264352
#117263782
#117263762
#261027