Finite Group Databasebeta

← #13412667

Subgroups — #13412667

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739693C211yesmaximal, characteristic
2C2×#89767C211yesmaximal, characteristic
3C2×#89628C211yesmaximal, characteristic, complemented
4#11731346C211yesmaximal, characteristic
5C7×#1854C211yesmaximal, characteristic
6#11728372C211yesmaximal, characteristic, complemented
7#11728361C211yesmaximal, characteristic, complemented
8#11728325C211yesmaximal, characteristic, complemented
9#11727703C211yesmaximal, characteristic
10#11727681C211yesmaximal, characteristic
11#11727267C211yesmaximal, characteristic
12#11726614C211yesmaximal, characteristic, complemented
13#11726613C211yesmaximal, characteristic, complemented
14#11726591C211yesmaximal, characteristic, complemented
15#11726580C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6453C2211yescharacteristic
20C2×#6451C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C7×#106C2211yescharacteristic
23C2×C7×C22⋊C8C2211yescharacteristic
24C7×#380C2211yescharacteristic
25#89984C2211yescharacteristic
26C4×C22⋊Dic7C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31#89767—24no—
32#89757C2211yescharacteristic
33#89757C2211yescharacteristic
34#89749C2211yescharacteristic
35#89749C2211yescharacteristic
36#89745C2211yescharacteristic
37#89745C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89722—21no—
45#89720—21no—
46#89719—21no—
47#89714—21no—
48#89705—21no—
49#89701—21no—
50#89628—24no—
51#89433—21no—
52#89432—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#26079—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14—21no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×#117—22no—
68C7×#117—41no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C42×C7—21no—
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7D411yescharacteristic
78C2×C22⋊Dic7D411yescharacteristic
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—41no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C2×#1136—22no—
96C2×#1136—22no—
97C2×#1136Q814yescomplemented
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100#6453—22no—
101#6453—41no—
102#6451—22no—
103#6451—41no—
104C7×#107—41no—
105C7×#106—22no—
106C7×C22⋊C8Q814yescomplemented
107C7×#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95D411yescharacteristic
118C7⋊#95D411yescharacteristic
119#6392—22no—
120#6392—41no—
121#6389—22no—
122C8×Dic7—22no—
123#6382—22no—
124#6382—22no—
125#6382—22no—
126#6381—22no—
127#6381—22no—
128#6381D414yes—
129C2×#501—71no—
130C2×#428—71no—
131#2442—71no—
132#2439—71no—
133#2085—71no—
134#2085—71no—
135#2059—71no—
136#2057—71no—
137#1943—71no—
138#1942—71no—
139#1854D711yescharacteristic
140#1852—71no—
141#1846—71no—
142#1809—71no—
143#1753—71no—
144C24×C7C2×D411yescharacteristic
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×Q812yes—
155C22×C4×C7C2×Q812yes—
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—42no—
162C22×Dic7—42no—
163C22×Dic7—81no—
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C2×Dic14—42no—
169C2×Dic14—42no—
170C2×Dic14—81no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8C2×Q812yes—
176C2×C7×C8C2×Q812yes—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4C2×D412yes—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—42no—
187C42×C7—42no—
188C22⋊Dic7—41no—
189C22⋊Dic7—41no—
190C22⋊Dic7—81no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—42no—
199C4⋊Dic7—42no—
200C4⋊Dic7—81no—
201C4⋊Dic7—81no—
202C4⋊Dic7C2×D412yes—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—81no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—21no—
211#1136—28no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C2×D412yes—
215C2×C7⋊C8C2×Q812yes—
216C2×C7⋊C8C2×Q812yes—
217C2×C7⋊C8#4112yes—
218C22×M16—71no—
219C2×C4⋊Q8—71no—
220C2×#125—71no—
221C2×#117C2×D711yescharacteristic
222C22×C4⋊C4—71no—
223#501—144no—
224#428—144no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×#107—71no—
230C2×#107C2×D711yescharacteristic
231C2×#106—71no—
232C2×#106—71no—
233C2×#106—71no—
234C2×#106C2×D711yescharacteristic
235C2×C22⋊C8C2×D711yescharacteristic
236C2×#97—71no—
237C2×C4×C8—71no—
238#400—141no—
239#398—141no—
240#396—141no—
241#393—141no—
242#391—141no—
243#389—71no—
244#389—71no—
245#389—71no—
246#389—71no—
247#387—141no—
248#385—141no—
249#382—71no—
250#382—71no—
251#380—71no—
252#380—141no—
253#380C2×D711yescharacteristic
254C4×C22⋊C4—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261#336—71no—
262#336—71no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7C4⋊Q812yes—
270C23×C7C22×D411yescharacteristic
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7C4⋊Q812yes—
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×Q811yescharacteristic
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—22no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—84no—
321Dic14—84no—
322C7×C8—22no—
323C7×C8—22no—
324C7⋊C8—22no—
325C7⋊C8C22⋊Q814yes—
326C22×Q8—141no—
327C23×C4—141no—
328C23×C4C22×D711yescharacteristic
329C2×M16—74no—
330C2×M16—142no—
331C2×M16—142no—
332C22×C8—71no—
333C22×C8—71no—
334C22×C8C22×D711yescharacteristic
335C22×C8C22×D711yescharacteristic
336C4⋊Q8—142no—
337C4⋊Q8—281no—
338#125—142no—
339#125—281no—
340#117—22no—
341#117—41no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—281no—
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—71no—
363C2×C22⋊C4—71no—
364C2×C22⋊C4—141no—
365C2×C42—21no—
366C2×C42—71no—
367C2×C42—71no—
368#107—41no—
369#107—74no—
370#107—142no—
371#107—142no—
372#107—142no—
373#107—281no—
374#106—22no—
375#106—142no—
376#106—142no—
377#106—142no—
378C22⋊C8Dic1414yescomplemented
379#97—142no—
380C4×C8—142no—
381#95—21no—
382#95—71no—
383#95—71no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7C2×C4⋊Q811yescharacteristic
401C22×C7D4211yescharacteristic
402C22×C7#54711yescharacteristic
403C22×C7#57511yescharacteristic
404C22×C7#57511yescharacteristic
405C22×C7#57611yescharacteristic
406C22×C7#57811yescharacteristic
407C4×C7—22no—
408C4×C7—42no—
409C4×C7—81no—
410C4×C7—81no—
411C4×C7—81no—
412C4×C7C2×C22⋊Q811yescharacteristic
413C4×C7#54411yescharacteristic
414C4×C7D4×Q811yescharacteristic
415C4×C7#55411yescharacteristic
416Dic7—41no—
417Dic7—41no—
418Dic7—41no—
419Dic7—42no—
420Dic7—41no—
421Dic7—81no—
422Dic7—81no—
423Dic7—81no—
424C24D4×D711yescharacteristic
425C2×Q8—282no—
426C2×Q8—282no—
427C2×Q8—561no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—71no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—282no—
445C22×C4—282no—
446C22×C4—561no—
447C22×C4C2×Dic1412yes—
448C22×C4D4×D711yescharacteristic
449C22×C4Q8×D712yes—
450C22×C4C23×D711yescharacteristic
451M16—148no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—72no—
457C2×C8—72no—
458C2×C8—72no—
459C2×C8—72no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8C2×Dic1412yes—
463C2×C8C2×Dic1412yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—72no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—282no—
480C4⋊C4—282no—
481C4⋊C4—282no—
482C4⋊C4—282no—
483C4⋊C4—282no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4—561no—
487C4⋊C4D4×D712yes—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—281no—
491C22⋊C4—281no—
492C22⋊C4—561no—
493C42—42no—
494C42—141no—
495C42—141no—
496C42—141no—
497C42—141no—
498C2×C7—42no—
499C2×C7#244011yescharacteristic
500C2×C7#316411yescharacteristic
501C2×C7#316611yescharacteristic
502C2×C7#327311yescharacteristic
503C2×C7#327711yescharacteristic
504C2×C7#334711yescharacteristic
505C2×C7#334911yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#645112yes—
513C23C2×D4×D711yescharacteristic
514Q8—564no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—142no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—564no—
558C2×C4#645112yes—
559C2×C4C22×Dic1411yescharacteristic
560C2×C4C2×D4×D711yescharacteristic
561C2×C4#654811yescharacteristic
562C2×C4C2×Q8×D711yescharacteristic
563C2×C4D7×#4111yescharacteristic
564C2×C4#655411yescharacteristic
565C8—22no—
566C8—22no—
567C8—74no—
568C8—142no—
569C7#2607911yescharacteristic, complemented, Sylow
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#645111yescharacteristic
578C22#9028111yescharacteristic
579C22#9034611yescharacteristic
580C22#9046011yescharacteristic
581C22#9046211yescharacteristic
582C22#9046411yescharacteristic
583C22#9046811yescharacteristic
584C4—22no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—281no—
590C4—282no—
591C4—281no—
592C4—281no—
593C4—281no—
594C4—561no—
595C4—561no—
596C4—561no—
597C4D4×Dic1411yescharacteristic
598C4#9023011yescharacteristic
599C4#9031411yescharacteristic
600C4D7×C22⋊Q811yescharacteristic
601C2—42no—
602C2#1173186111yescharacteristic
603C2#1173661811yescharacteristic
604C2#1173664911yescharacteristic
605C2#1173754511yescharacteristic
606C2#1173768811yescharacteristic
607C2#1173883811yescharacteristic
608C2#1173898411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1854C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1854D7
hypocenterC7#26079
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. C2×#89628C2semidirect
6. #11728372C2semidirect
7. #11728361C2semidirect
8. #11728325C2semidirect
12. #11726614C2semidirect
13. #11726613C2semidirect
14. #11726591C2semidirect
15. #11726580C2semidirect
97. C2×#1136Q8semidirect
106. C7×C22⋊C8Q8semidirect
378. C22⋊C8Dic14semidirect
569. C7#26079semidirect

Maximal subgroups

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

subgroupindex
#117396932
C2×#897672
C2×#896282
#117313462
C7×#18542
#117283722
#117283612
#117283252
#117277032
#117276812
#117272672
#117266142
#117266132
#117265912
#117265802
#260797