Finite Group Databasebeta

← #13412838

Subgroups — #13412838

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#11733748C211yesmaximal, characteristic
2#11731786C211yesmaximal, characteristic, complemented
3C7×#2093C211yesmaximal, characteristic
4#11728370C211yesmaximal, characteristic, complemented
5#11728370C211yesmaximal, characteristic, complemented
6#11728363C211yesmaximal, characteristic, complemented
7#11728361C211yesmaximal, characteristic, complemented
8#11727947C211yesmaximal, characteristic
9#11727935C211yesmaximal, characteristic
10#11727908C211yesmaximal, characteristic
11Dic7×#102C211yesmaximal, characteristic
12#11727406C211yesmaximal, characteristic
13#11727322C211yesmaximal, characteristic, complemented
14#11726378C211yesmaximal, characteristic, complemented
15#11726371C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C2×C7×#107C2211yescharacteristic
18C2×C7×#107C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C7×#390C2211yescharacteristic
22C7×#384C2211yescharacteristic
23#89993C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6392C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×C8×Dic7C2211yescharacteristic
30#89757C2211yescharacteristic
31#89755—21no—
32#89754—21no—
33#89753—21no—
34#89750—21no—
35#89749C2211yescharacteristic
36#89749C2211yescharacteristic
37#89749C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43#89717—21no—
44#89716—21no—
45#89715—21no—
46C22⋊C4×Dic7—21no—
47C4×C4⋊Dic7C2211yescharacteristic
48C4×Dic7⋊C4C2211yescharacteristic
49#89700C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#26094—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×Dic7—21no—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4—21no—
71C2×C42×C7C2311yescharacteristic
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75D4×Dic7—22no—
76D4×Dic7Q814yes—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4D411yescharacteristic
84C2×Dic7⋊C4D411yescharacteristic
85C2×C4×Dic7—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7C2311yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C7×#107—22no—
93C7×#107—41no—
94C7×#102—22no—
95C7×#102—41no—
96#6406—22no—
97#6406—41no—
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—
109C7⋊#95D411yescharacteristic
110C7⋊#95D411yescharacteristic
111#6392—22no—
112Dic7⋊C8—22no—
113Dic7⋊C8—22no—
114C8×Dic7—22no—
115#6382—41no—
116#6382—41no—
117#6382—41no—
118C4⋊(C7⋊C8)—22no—
119#2570—71no—
120#2357—71no—
121#2093D711yescharacteristic
122#2092—71no—
123#2065—71no—
124#2065—71no—
125#1949—71no—
126#1949—71no—
127#1948—71no—
128#1939—71no—
129#1921—71no—
130#1918—71no—
131#1846—71no—
132#1804—71no—
133C4×#102—71no—
134C24×C7—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4C2×Q812yes—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—24no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—81no—
153C22×Dic7—81no—
154C22×Dic7C2×D411yescharacteristic
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8#4112yes—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—42no—
171C7×C4⋊C4—81no—
172C7×C22⋊C4—42no—
173C7×C22⋊C4—81no—
174C42×C7—21no—
175C42×C7—21no—
176C22⋊Dic7—24no—
177C22⋊Dic7—42no—
178C22⋊Dic7—81no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—81no—
187C4⋊Dic7C2×Q812yes—
188Dic7⋊C4—22no—
189Dic7⋊C4—41no—
190Dic7⋊C4—42no—
191C4×Dic7—21no—
192C4×Dic7—21no—
193C4×Dic7—81no—
194C4×Dic7—81no—
195C4×Dic7—81no—
196C4×Dic7C2×Q812yes—
197C4×Dic7#4112yes—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8#4112yes—
204C2×C4×D4—71no—
205C2×#107—71no—
206C2×#107C2×D711yescharacteristic
207C2×#107C2×D711yescharacteristic
208C2×#106—71no—
209C2×#106—71no—
210C2×#106—71no—
211C2×C4⋊C8—71no—
212C2×C4⋊C8—71no—
213C2×C4⋊C8—71no—
214C2×#102—71no—
215C2×#102—71no—
216C2×#102C2×D711yescharacteristic
217C2×#102C2×D711yescharacteristic
218C2×C4×C8—71no—
219#400—141no—
220#398—141no—
221#390—71no—
222#390C2×D711yescharacteristic
223#389—71no—
224#389—71no—
225#388—141no—
226#388—141no—
227#387—141no—
228#387—141no—
229#384C2×D711yescharacteristic
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⋊C4—71no—
238C4×C22⋊C4—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#336—71no—
246#336C2×D711yescharacteristic
247C23×C7—22no—
248C23×C7—41no—
249C23×C7—41no—
250C23×C7—41no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7C22×D411yescharacteristic
255C7×D4—22no—
256C7×D4—22no—
257C7×D4—41no—
258C7×D4—41no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—41no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7#13512yes—
274C2×C4×C7C22×Q811yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7H(5,𝔽2)11yescharacteristic
279C2×C4×C72−1+411yescharacteristic
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—22no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
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—84no—
303C2×Dic7C22⋊Q812yes—
304C2×Dic7C22⋊Q812yes—
305C7×C8—22no—
306C7×C8—41no—
307C7⋊C8—22no—
308C7⋊C8—41no—
309C22×D4C22×D711yescharacteristic
310C23×C4—141no—
311C22×C8—71no—
312C22×C8—71no—
313C22×C8C22×D711yescharacteristic
314C22×C8C22×D711yescharacteristic
315C4×D4—74no—
316C4×D4—142no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—71no—
320C2×C4⋊C4—71no—
321C2×C4⋊C4—71no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—141no—
326C2×C4⋊C4—141no—
327C2×C4⋊C4C22×D711yescharacteristic
328C2×C4⋊C4C22×D711yescharacteristic
329C2×C4⋊C4C22×D711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—141no—
333C2×C22⋊C4—141no—
334C2×C22⋊C4—141no—
335C2×C42—71no—
336C2×C42—71no—
337C2×C42—141no—
338C2×C42—141no—
339C2×C42—141no—
340C2×C42C22×D711yescharacteristic
341#107—22no—
342#107—41no—
343#107—142no—
344#106—281no—
345#106—281no—
346#106—281no—
347C4⋊C8—142no—
348C4⋊C8—142no—
349C4⋊C8—142no—
350#102—22no—
351#102—41no—
352#102—142no—
353#102—281no—
354C4×C8—142no—
355#95—71no—
356#95—71no—
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#95—141no—
367#95—141no—
368C22×C7—42no—
369C22×C7—42no—
370C22×C7—42no—
371C22×C7—81no—
372C22×C7—81no—
373C22×C7—81no—
374C22×C7—81no—
375C22×C7C2×C22⋊Q811yescharacteristic
376C22×C7#54711yescharacteristic
377C22×C7#55011yescharacteristic
378C22×C7C2×#13511yescharacteristic
379C22×C7#57511yescharacteristic
380C22×C7#57611yescharacteristic
381C22×C7#57711yescharacteristic
382C4×C7—22no—
383C4×C7—41no—
384C4×C7—81no—
385C4×C7—81no—
386C4×C7—81no—
387C4×C7#53311yescharacteristic
388C4×C7#54411yescharacteristic
389C4×C7#55211yescharacteristic
390C4×C7#55311yescharacteristic
391Dic7—22no—
392Dic7—22no—
393Dic7—42no—
394Dic7—41no—
395Dic7—41no—
396Dic7—81no—
397Dic7—81no—
398Dic7—81no—
399C24—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4Q8×D712yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—71no—
410C22×C4—71no—
411C22×C4—71no—
412C22×C4—71no—
413C22×C4—144no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—561no—
419C22×C4—561no—
420C22×C4D4×D711yescharacteristic
421C22×C4D4×D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—72no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8#115912yes—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—42no—
442C4⋊C4—72no—
443C4⋊C4—81no—
444C4⋊C4—142no—
445C4⋊C4—141no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—281no—
453C4⋊C4—282no—
454C4⋊C4—561no—
455C22⋊C4—42no—
456C22⋊C4—81no—
457C22⋊C4—144no—
458C22⋊C4—282no—
459C22⋊C4—561no—
460C42—21no—
461C42—21no—
462C42—72no—
463C42—72no—
464C42—141no—
465C42—141no—
466C42—561no—
467C42—561no—
468C42—561no—
469C2×C7—82no—
470C2×C7#244111yescharacteristic
471C2×C7#307611yescharacteristic
472C2×C7#308111yescharacteristic
473C2×C7#333711yescharacteristic
474C2×C7#335511yescharacteristic
475C2×C7#339611yescharacteristic
476C2×C7#340111yescharacteristic
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C2×D4×D711yescharacteristic
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—72no—
504C2×C4—72no—
505C2×C4—141no—
506C2×C4—141no—
507C2×C4—141no—
508C2×C4—141no—
509C2×C4—141no—
510C2×C4—141no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—142no—
514C2×C4—141no—
515C2×C4—141no—
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—564no—
528C2×C4#647512yes—
529C2×C4C2×#115911yescharacteristic
530C2×C4#654811yescharacteristic
531C2×C4C2×Q8×D711yescharacteristic
532C2×C4D7×#4111yescharacteristic
533C2×C4D7×#4111yescharacteristic
534C2×C4#655411yescharacteristic
535C8—22no—
536C8—41no—
537C8—142no—
538C8—281no—
539C7#2609411yescharacteristic, complemented, Sylow
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22D7×C22⋊Q811yescharacteristic
548C22#9034611yescharacteristic
549C22#9040911yescharacteristic
550C22C2×#647511yescharacteristic
551C22#9044811yescharacteristic
552C22#9046811yescharacteristic
553C22#9046911yescharacteristic
554C4—22no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—142no—
560C4—142no—
561C4—282no—
562C4—281no—
563C4—281no—
564C4—561no—
565C4—561no—
566C4—561no—
567C4#9031311yescharacteristic
568C4#9031411yescharacteristic
569C4#9034111yescharacteristic
570C4#9035211yescharacteristic
571C2—82no—
572C2#1173546611yescharacteristic
573C2#1173840911yescharacteristic
574C2#1173841211yescharacteristic
575C2#1173873711yescharacteristic
576C2#1173874211yescharacteristic
577C2#1173922411yescharacteristic
578C2#1173931111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2093C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2093D7
hypocenterC7#26094
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #11731786C2semidirect
4. #11728370C2semidirect
5. #11728370C2semidirect
6. #11728363C2semidirect
7. #11728361C2semidirect
13. #11727322C2semidirect
14. #11726378C2semidirect
15. #11726371C2semidirect
539. C7#26094semidirect

Maximal subgroups

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

subgroupindex
#117337482
#117317862
C7×#20932
#117283702
#117283632
#117283612
#117279472
#117279352
#117279082
Dic7×#1022
#117274062
#117273222
#117263782
#117263712
#260947