Finite Group Databasebeta

← #13412843

Subgroups — #13412843

Order 1792.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731168C211yesmaximal, characteristic, complemented
2#11731090C211yesmaximal, characteristic
3C7×#1939C211yesmaximal, characteristic, complemented
4#11728237C211yesmaximal, characteristic
5#11728237C211yesmaximal, characteristic
6#11728234C211yesmaximal, characteristic
7#11728209C211yesmaximal, characteristic, complemented
8#11728209C211yesmaximal, characteristic, complemented
9#11728208C211yesmaximal, characteristic, complemented
10#11727520C211yesmaximal, characteristic, complemented
11C4×#6395C211yesmaximal, characteristic
12#11726435C211yesmaximal, characteristic
13#11726434C211yesmaximal, characteristic
14#11726378C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C2×C4×D28C2211yescharacteristic
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×C4⋊C8C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C4×C7×C8C2211yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22C2×#6395C2211yescharacteristic
23C2×#6395C2211yescharacteristic
24C2×#6395C2211yescharacteristic
25C2×#6395C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32#89762—21no—
33#89762—21no—
34#89758—21no—
35#89756—21no—
36#89711C2211yescharacteristic
37#89711C2211yescharacteristic
38#89708C2211yescharacteristic
39#89708C2211yescharacteristic
40#89708C2211yescharacteristic
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C4×C4⋊Dic7C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89438—21no—
48#89436—21no—
49#89427—21no—
50#89426—21no—
51C7×#336C2211yescharacteristic
52C7×#336C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#26094—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⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C4×D28—22no—
91C4×D28Q814yes—
92C7×C4⋊C8—22no—
93C7×C4⋊C8—22no—
94C7×C4⋊C8—22no—
95C4×C7×C8—22no—
96C7×#95D411yescharacteristic
97C7×#95D411yescharacteristic
98C7⋊#95—21no—
99C7⋊#95—21no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109#6395—22no—
110#6395—22no—
111#6395—41no—
112#6395—41no—
113#6392—22no—
114#6392—22no—
115#6392—41no—
116#6391—41no—
117#6391—41no—
118#6391—41no—
119#2570—71no—
120#2357—71no—
121#2093—71no—
122#2092—71no—
123#2065—71no—
124#2065—71no—
125#1949—71no—
126#1949—71no—
127#1948—71no—
128#1939D711yescharacteristic, complemented
129#1921—71no—
130#1918—71no—
131#1846—71no—
132#1804—71no—
133C4×#102—71no—
134C23×D7—21no—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28C2×Q812yes—
155C2×C4×D7—24no—
156C2×C4×D7—81no—
157C2×C4×D7—81no—
158C2×C7×C8—21no—
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#4112yes—
169C2×C7×C8#4112yes—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—41no—
172C42×C7—21no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7C2×Q812yes—
177C42×C7#4112yes—
178(C2×D7)⋊C4—24no—
179(C2×D7)⋊C4—42no—
180(C2×D7)⋊C4—42no—
181(C2×D7)⋊C4—81no—
182(C2×D7)⋊C4—81no—
183C4⋊Dic7—21no—
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—42no—
197C4⋊Dic7—81no—
198C4⋊Dic7C2×Q812yes—
199Dic7⋊C4—42no—
200Dic7⋊C4—81no—
201C4×Dic7—81no—
202C4×Dic7—81no—
203C4×Dic7—81no—
204C2×C4×D4—71no—
205C2×#107—71no—
206C2×#107—71no—
207C2×#107—71no—
208C2×#106—71no—
209C2×#106—71no—
210C2×#106—71no—
211C2×C4⋊C8C2×D711yescharacteristic
212C2×C4⋊C8C2×D711yescharacteristic
213C2×C4⋊C8C2×D711yescharacteristic
214C2×#102—71no—
215C2×#102—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×C4×C8C2×D711yescharacteristic
219#400—141no—
220#398—141no—
221#390—71no—
222#390—71no—
223#389—71no—
224#389—71no—
225#388—141no—
226#388—141no—
227#387—141no—
228#387—141no—
229#384—71no—
230#382—71no—
231#382—71no—
232#382—71no—
233#381—141no—
234C4×C4⋊C4—71no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4C2×D711yescharacteristic
238C4×C22⋊C4—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#336C2×D711yescharacteristic
246#336C2×D711yescharacteristic
247C23×C7C22×D411yescharacteristic
248C22×D7—22no—
249C22×D7—41no—
250C22×D7—41no—
251C22×D7—41no—
252C22×D7—41no—
253C22×D7—41no—
254C22×D7—41no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
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×C7C22⋊Q812yes—
271C2×C4×C7C22⋊Q812yes—
272C2×C4×C7#13512yes—
273C2×C4×C7C22×Q811yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7H(5,𝔽2)11yescharacteristic
278C2×C4×C72−1+411yescharacteristic
279C2×Dic7—22no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7—41no—
284C2×Dic7—41no—
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—
300D28—22no—
301D28—22no—
302D28—41no—
303D28—41no—
304C4×D7—84no—
305C7×C8—22no—
306C7×C8—22no—
307C7×C8—41no—
308C7×C8—41no—
309C22×D4—71no—
310C23×C4—141no—
311C22×C8C22×D711yescharacteristic
312C22×C8C22×D711yescharacteristic
313C22×C8C22×D711yescharacteristic
314C22×C8C22×D711yescharacteristic
315C4×D4—74no—
316C4×D4—142no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—71no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—71no—
321C2×C4⋊C4—71no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4—141no—
328C2×C4⋊C4D2811yescharacteristic
329C2×C4⋊C4D2811yescharacteristic
330C2×C22⋊C4—141no—
331C2×C22⋊C4—141no—
332C2×C22⋊C4—141no—
333C2×C22⋊C4—141no—
334C2×C22⋊C4—141no—
335C2×C42—141no—
336C2×C42—141no—
337C2×C42—141no—
338C2×C42C22×D711yescharacteristic
339C2×C42C22×D711yescharacteristic
340C2×C42C22×D711yescharacteristic
341#107—142no—
342#107—142no—
343#107—281no—
344#106—281no—
345#106—281no—
346#106—281no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350#102—142no—
351#102—142no—
352#102—281no—
353#102—281no—
354C4×C8—22no—
355#95—141no—
356#95—141no—
357#95—141no—
358#95—141no—
359#95—141no—
360#95—141no—
361#95—141no—
362#95—141no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95D2811yescharacteristic
367#95D2811yescharacteristic
368C22×C7C2×C22⋊Q811yescharacteristic
369C22×C7#54711yescharacteristic
370C22×C7#55011yescharacteristic
371C22×C7C2×#13511yescharacteristic
372C22×C7#57511yescharacteristic
373C22×C7#57611yescharacteristic
374C22×C7#57711yescharacteristic
375C2×D7—42no—
376C2×D7—42no—
377C2×D7—42no—
378C2×D7—81no—
379C2×D7—81no—
380C2×D7—81no—
381C2×D7—81no—
382C4×C7—22no—
383C4×C7—22no—
384C4×C7—22no—
385C4×C7—41no—
386C4×C7—41no—
387C4×C7—41no—
388C4×C7#53311yescharacteristic
389C4×C7#54411yescharacteristic
390C4×C7#55211yescharacteristic
391C4×C7#55311yescharacteristic
392Dic7—42no—
393Dic7—81no—
394Dic7—81no—
395Dic7—81no—
396Dic7—81no—
397Dic7—81no—
398Dic7—81no—
399C24—141no—
400C2×D4—72no—
401C2×D4—141no—
402C2×D4—141no—
403C2×D4—141no—
404C2×D4—141no—
405C2×D4—141no—
406C22×C4—141no—
407C22×C4—144no—
408C22×C4—141no—
409C22×C4—141no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—561no—
415C22×C4—561no—
416C22×C4C2×D2811yescharacteristic
417C22×C4C2×D2811yescharacteristic
418C22×C4C2×D2811yescharacteristic
419C22×C4C2×D2811yescharacteristic
420C22×C4C2×D2811yescharacteristic
421C22×C4C2×D2811yescharacteristic
422C22×C4C23×D711yescharacteristic
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8#115912yes—
434C2×C8#116112yes—
435C4⋊C4—22no—
436C4⋊C4—41no—
437C4⋊C4—72no—
438C4⋊C4—141no—
439C4⋊C4—141no—
440C4⋊C4—141no—
441C4⋊C4—141no—
442C4⋊C4—141no—
443C4⋊C4—141no—
444C4⋊C4—141no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—282no—
452C4⋊C4—282no—
453C4⋊C4—561no—
454C4⋊C4—561no—
455C22⋊C4—144no—
456C22⋊C4—282no—
457C22⋊C4—282no—
458C22⋊C4—561no—
459C22⋊C4—561no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—561no—
465C42—561no—
466C42—561no—
467C42#115912yes—
468C42Q8×D712yes—
469C2×C7#244111yescharacteristic
470C2×C7#307611yescharacteristic
471C2×C7#308111yescharacteristic
472C2×C7#333711yescharacteristic
473C2×C7#335511yescharacteristic
474C2×C7#339611yescharacteristic
475C2×C7#340111yescharacteristic
476D7—82no—
477C23—142no—
478C23—281no—
479C23—281no—
480C23—281no—
481C23—281no—
482C23—281no—
483C23—281no—
484C23C22×D2811yescharacteristic
485D4—142no—
486D4—142no—
487D4—281no—
488D4—281no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—142no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—281no—
513C2×C4—281no—
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—564no—
526C2×C4#646012yes—
527C2×C4#646012yes—
528C2×C4#646712yes—
529C2×C4C2×#115911yescharacteristic
530C2×C4C2×#115911yescharacteristic
531C2×C4C2×Q8×D711yescharacteristic
532C2×C4C2×#116111yescharacteristic
533C2×C4#655311yescharacteristic
534C2×C4#655411yescharacteristic
535C8—22no—
536C8—22no—
537C8—41no—
538C8—41no—
539C7#2609411yescharacteristic, complemented, Sylow
540C22—282no—
541C22—282no—
542C22—282no—
543C22—561no—
544C22—561no—
545C22—561no—
546C22—561no—
547C22C2×#646011yescharacteristic
548C22#9024611yescharacteristic
549C22#9026811yescharacteristic
550C22C2×#646711yescharacteristic
551C22#9043911yescharacteristic
552C22#9044211yescharacteristic
553C22#9044311yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—282no—
561C4—561no—
562C4—561no—
563C4—561no—
564C4—561no—
565C4—561no—
566C4—561no—
567C4#9030911yescharacteristic
568C4#9033311yescharacteristic
569C4#9038011yescharacteristic
570C4#9041911yescharacteristic
571C2—562no—
572C2#1173344511yescharacteristic
573C2#1173647011yescharacteristic
574C2#1173653511yescharacteristic
575C2#1173685911yescharacteristic
576C2#1173697811yescharacteristic
577C2#1173736011yescharacteristic
578C2#1173739911yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1939C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1939D7
hypocenterC7#26094
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
1. #11731168C2semidirect
3. C7×#1939C2semidirect
7. #11728209C2semidirect
8. #11728209C2semidirect
9. #11728208C2semidirect
10. #11727520C2semidirect
14. #11726378C2semidirect
15. #11726375C2semidirect
128. #1939D7semidirect
539. C7#26094semidirect

Maximal subgroups

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

subgroupindex
#117311682
#117310902
C7×#19392
#117282372
#117282342
#117282092
#117282082
#117275202
C4×#63952
#117264352
#117264342
#117263782
#117263752
#260947