Finite Group Databasebeta

← #13382752

Subgroups — #13382752

Order 1792.

888 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×#89904C211yesmaximal, characteristic
2C2×#89896C211yesmaximal, characteristic, complemented
3#11739658C211yesmaximal, characteristic
4#11739658C211yesmaximal, characteristic
5C2×C7×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6C2×#89328C211yesmaximal, characteristic, complemented
7C2×#89328C211yesmaximal, characteristic, complemented
8#11726463C214yesmaximal, complemented
9#11726460C214yesmaximal, complemented
10C22×#1157C2211yescharacteristic
11C22×C7×M16C2211yescharacteristic
12C2×C7×#117C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×#1136C2211yescharacteristic
15C22×#1136C2211yescharacteristic
16C22×#1136C2211yescharacteristic
17C2×#6398C2212yes—
18#89904—22no—
19#89904—22no—
20C2×#6397C2212yes—
21#89896—22no—
22#89896C2214yes—
23C2×#6396—22no—
24#89776C2212yes—
25#89776C2212yes—
26#89774—22no—
27#89774—22no—
28#89773C2212yes—
29#89773C2212yes—
30#89738—22no—
31#89738—22no—
32C7×H(3,ℤ/4)C2214yes—
33#89328C2214yes—
34#89328C2214yes—
35#23652—71nomaximal, Sylow
36C23×C4×C7C2311yescharacteristic
37C22×C22⋊D7—21no—
38C2×#1157—21no—
39C2×#1157—21no—
40C2×#1157—21no—
41C2×#1157—21no—
42C2×#1157—21no—
43C2×#1157—21no—
44C2×#1157—21no—
45C2×#1157—21no—
46C2×#1157—21no—
47C2×#1157—21no—
48C2×#1157D412yes—
49C2×#1157D412yes—
50C22×D28C2311yescharacteristic
51C22×C4×D7—21no—
52C22×Dic14C2311yescharacteristic
53C2×C7×M16—21no—
54C2×C7×M16—21no—
55C2×C7×M16—21no—
56C2×C7×M16—21no—
57C2×C7×M16—21no—
58C2×C7×M16C2312yes—
59C22×C7×C8—21no—
60C7×#117—21no—
61C7×#117—21no—
62C7×#117—21no—
63C7×#117—21no—
64C7×#117—21no—
65C7×#117—21no—
66C7×#117C2312yes—
67C7×#117C2312yes—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C42×C7—21no—
73C2×C42×C7—21no—
74C2×#1136—21no—
75C2×#1136—21no—
76C2×#1136—21no—
77C2×#1136—21no—
78C2×#1136—21no—
79C2×#1136—21no—
80C2×#1136—21no—
81C2×#1136—21no—
82C2×#1136—21no—
83C2×#1136—21no—
84C2×#1136—21no—
85C2×#1136—21no—
86C2×#1136—21no—
87C2×#1136—21no—
88C2×#1136D412yes—
89C2×#1136C2312yes—
90C2×#1136C2312yes—
91C2×#1136C2312yes—
92C22×C7⋊C8—21no—
93C22×C7⋊C8—21no—
94C22×C7⋊C8D412yes—
95#6398—24no—
96#6397—24no—
97#6396—28no—
98#6384—24no—
99#6384—24no—
100#6383—24no—
101#6383—24no—
102C2×#430—71no—
103#2896—71no—
104#2896—71no—
105C2×#413—71no—
106#2082—74no—
107#2081—74no—
108C2×#344—71no—
109C2×#344—71no—
110C2×H(3,ℤ/4)D711yescharacteristic, complemented
111C24×C7#4111yescharacteristic
112C23×D7—21no—
113C22×C4×C7—41no—
114C22×C4×C7—41no—
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7C2×D411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D411yescharacteristic
119C22×C4×C7#4111yescharacteristic
120C22×C4×C7#4111yescharacteristic
121C22×C4×C7#4111yescharacteristic
122C22×C4×C7#4111yescharacteristic
123C22×C4×C7#4111yescharacteristic
124C22×C4×C7#4111yescharacteristic
125C22×C4×C7#4111yescharacteristic
126C22×C4×C7#4111yescharacteristic
127C22×C4×C7#4111yescharacteristic
128C22×C4×C7C2411yescharacteristic
129C2×C22⋊D7—22no—
130C2×C22⋊D7—22no—
131C2×C22⋊D7—41no—
132C2×C22⋊D7—41no—
133C2×C22⋊D7—41no—
134C2×C22⋊D7—41no—
135C22×Dic7—21no—
136#1157—22no—
137#1157—22no—
138#1157—22no—
139#1157—22no—
140#1157—22no—
141#1157—22no—
142#1157—22no—
143#1157—22no—
144#1157—41no—
145#1157—41no—
146#1157—41no—
147#1157—41no—
148#1157—41no—
149#1157—41no—
150#1157—41no—
151#1157—41no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×D28C2×D412yes—
158C2×C4×D7—22no—
159C2×C4×D7—22no—
160C2×C4×D7—41no—
161C2×C4×D7—41no—
162C2×C4×D7—41no—
163C2×C4×D7—41no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14C2×D412yes—
170C7×M16—22no—
171C7×M16—22no—
172C7×M16—41no—
173C7×M16—41no—
174C2×C7×C8—22no—
175C2×C7×C8—41no—
176C2×C7×C8—41no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—41no—
180C7×C4⋊C4—41no—
181C7×C22⋊C4—22no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—41no—
184C7×C22⋊C4—41no—
185C42×C7—22no—
186C42×C7—22no—
187C42×C7—41no—
188C42×C7—41no—
189#1136—22no—
190#1136—22no—
191#1136—22no—
192#1136—22no—
193#1136—22no—
194#1136—22no—
195#1136—22no—
196#1136—22no—
197#1136—41no—
198#1136—41no—
199#1136—41no—
200#1136—41no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—41no—
206C2×C7⋊C8—41no—
207C2×C7⋊C8—41no—
208C2×C7⋊C8—41no—
209C2×C7⋊C8C2×D414yes—
210C22×#41—71no—
211C22×M16—71no—
212C22×M16—71no—
213C22×M16—71no—
214C22×M16C2×D711yescharacteristic
215C2×#117C2×D711yescharacteristic
216C2×#117C2×D711yescharacteristic
217#430—74no—
218#430—142no—
219C2×#108—142no—
220#419—72no—
221#419—72no—
222#418—72no—
223#418—72no—
224#417—142no—
225#417—142no—
226#416—142no—
227#416—142no—
228#413—142no—
229#413—142no—
230C2×#101—72no—
231C2×#100—72no—
232#344—74no—
233#344—74no—
234H(3,ℤ/4)C2×D714yes—
235C23×C7—81no—
236C23×C7#12311yescharacteristic
237C23×C7#12311yescharacteristic
238C23×C7#12411yescharacteristic
239C23×C7#12411yescharacteristic
240C23×C7#12611yescharacteristic
241C23×C7#12611yescharacteristic
242C23×C7C2×#4111yescharacteristic
243C22×D7—22no—
244C22×D7—41no—
245C22×D7—41no—
246C22×D7—82no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—82no—
250C2×C4×C7—82no—
251C2×C4×C7#12111yescharacteristic
252C2×C4×C7#12111yescharacteristic
253C2×C4×C7#12111yescharacteristic
254C2×C4×C7#12111yescharacteristic
255C2×C4×C7#12311yescharacteristic
256C2×C4×C7#12311yescharacteristic
257C2×C4×C7#12311yescharacteristic
258C2×C4×C7#12311yescharacteristic
259C2×C4×C7#12311yescharacteristic
260C2×C4×C7#12311yescharacteristic
261C2×C4×C7#12311yescharacteristic
262C2×C4×C7#12311yescharacteristic
263C2×C4×C7#12311yescharacteristic
264C2×C4×C7#12311yescharacteristic
265C2×C4×C7#12411yescharacteristic
266C2×C4×C7#12411yescharacteristic
267C2×C4×C7#12611yescharacteristic
268C2×C4×C7#12611yescharacteristic
269C2×C4×C7#12611yescharacteristic
270C2×C4×C7#12611yescharacteristic
271C2×C4×C7#12611yescharacteristic
272C2×C4×C7#12611yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×C4×C7C2×#4111yescharacteristic
277C2×C4×C7C2×#4111yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C22⋊D7—42no—
280C22⋊D7—42no—
281C22⋊D7—42no—
282C22⋊D7—42no—
283C2×Dic7—22no—
284C2×Dic7—41no—
285C2×Dic7—41no—
286D28—22no—
287D28—22no—
288D28—41no—
289D28—41no—
290C4×D7—42no—
291C4×D7—42no—
292C4×D7—42no—
293C4×D7—42no—
294Dic14—22no—
295Dic14—22no—
296Dic14—41no—
297Dic14—41no—
298C7×C8—42no—
299C7⋊C8—24no—
300C7⋊C8—42no—
301C7⋊C8—42no—
302C2×#41—72no—
303C2×#41—72no—
304C2×#41—141no—
305C2×#41—141no—
306C2×#41—141no—
307C2×#41—141no—
308C2×#41—141no—
309C2×#41—141no—
310C2×#41—141no—
311C2×#41—141no—
312C2×#41—141no—
313C2×#41—141no—
314C22×Q8—71no—
315C22×D4—71no—
316C22×D4—141no—
317C23×C4—141no—
318C23×C4C22×D711yescharacteristic
319C2×M16—21no—
320C2×M16—21no—
321C2×M16—21no—
322C2×M16—21no—
323C2×M16—21no—
324C2×M16—72no—
325C2×M16—72no—
326C2×M16—72no—
327C2×M16—72no—
328C2×M16—141no—
329C2×M16—141no—
330C2×M16—141no—
331C2×M16—141no—
332C2×M16—141no—
333C2×M16—141no—
334C2×M16—141no—
335C2×M16—141no—
336C2×M16—141no—
337C2×M16—141no—
338C2×M16—141no—
339C2×M16—141no—
340C2×M16—141no—
341C2×M16—141no—
342C2×M16C22×D712yes—
343C22×C8—21no—
344C22×C8—72no—
345C22×C8—141no—
346C22×C8—141no—
347#117—21no—
348#117—21no—
349#117—21no—
350#117—21no—
351#117—21no—
352#117—21no—
353#117C22×D712yes—
354#117C22×D712yes—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C42—21no—
360C2×C42—21no—
361#108—148no—
362#103—144no—
363#103—144no—
364#102—144no—
365#102—144no—
366#101—144no—
367#100—144no—
368C22×C7—41no—
369C22×C7—41no—
370C22×C7—41no—
371C22×C7—41no—
372C22×C7—41no—
373C22×C7—41no—
374C22×C7—41no—
375C22×C7#39711yescharacteristic
376C22×C7#39711yescharacteristic
377C22×C7#39711yescharacteristic
378C22×C7#39711yescharacteristic
379C22×C7C2×#12311yescharacteristic
380C22×C7C2×#12411yescharacteristic
381C22×C7C2×#12611yescharacteristic
382C2×D7—42no—
383C2×D7—82no—
384C2×D7—82no—
385C2×D7—82no—
386C4×C7—82no—
387C4×C7—82no—
388C4×C7#39711yescharacteristic
389C4×C7#39711yescharacteristic
390C4×C7#39711yescharacteristic
391C4×C7#39711yescharacteristic
392C4×C7C2×#12111yescharacteristic
393C4×C7C2×#12311yescharacteristic
394C4×C7C2×#12311yescharacteristic
395C4×C7C2×#12611yescharacteristic
396Dic7—42no—
397C24—141no—
398C24#115911yescharacteristic
399#41—142no—
400#41—142no—
401#41—142no—
402#41—142no—
403#41—142no—
404#41—142no—
405#41—142no—
406#41—142no—
407#41—281no—
408#41—281no—
409#41—281no—
410#41—281no—
411#41—281no—
412#41—281no—
413#41—281no—
414#41—281no—
415C2×Q8—72no—
416C2×Q8—141no—
417C2×Q8—141no—
418C2×Q8—141no—
419C2×Q8—141no—
420C2×Q8—141no—
421C2×D4—72no—
422C2×D4—142no—
423C2×D4—142no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—141no—
427C2×D4—141no—
428C2×D4—141no—
429C2×D4—281no—
430C2×D4—281no—
431C2×D4—281no—
432C2×D4—281no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—141no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—281no—
439C22×C4—281no—
440C22×C4—281no—
441C22×C4—281no—
442C22×C4#115711yescharacteristic
443C22×C4#115711yescharacteristic
444C22×C4#115711yescharacteristic
445C22×C4#115711yescharacteristic
446C22×C4#115711yescharacteristic
447C22×C4#115711yescharacteristic
448C22×C4D4×D711yescharacteristic
449C22×C4D4×D711yescharacteristic
450C22×C4D4×D711yescharacteristic
451C22×C4D4×D711yescharacteristic
452C22×C4#115911yescharacteristic
453C22×C4#116111yescharacteristic
454C22×C4#116111yescharacteristic
455C22×C4C23×D711yescharacteristic
456M16—22no—
457M16—22no—
458M16—41no—
459M16—41no—
460M16—142no—
461M16—142no—
462M16—142no—
463M16—142no—
464M16—142no—
465M16—142no—
466M16—142no—
467M16—142no—
468M16—281no—
469M16—281no—
470M16—281no—
471M16—281no—
472C2×C8—22no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—74no—
476C2×C8—142no—
477C2×C8—142no—
478C2×C8—142no—
479C2×C8—142no—
480C2×C8—281no—
481C2×C8—281no—
482C2×C8—281no—
483C2×C8—281no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—41no—
496C2×C7—21no—
497C2×C7—21no—
498C2×C7—21no—
499C2×C7—21no—
500C2×C7—21no—
501C2×C7—21no—
502C2×C7—21no—
503C2×C7C2×#39711yescharacteristic
504D7—82no—
505C23—81no—
506C23—142no—
507C23—281no—
508C23—281no—
509C23—562no—
510C23#644611yescharacteristic
511C23#644611yescharacteristic
512C23#644811yescharacteristic
513C23#644811yescharacteristic
514C23#646111yescharacteristic
515C23#646111yescharacteristic
516C23C2×#115911yescharacteristic
517Q8—142no—
518Q8—142no—
519Q8—281no—
520Q8—281no—
521D4—142no—
522D4—142no—
523D4—282no—
524D4—282no—
525D4—282no—
526D4—282no—
527D4—281no—
528D4—281no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—142no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—282no—
537C2×C4—282no—
538C2×C4—282no—
539C2×C4—282no—
540C2×C4#643911yescharacteristic
541C2×C4#643911yescharacteristic
542C2×C4#643911yescharacteristic
543C2×C4#643911yescharacteristic
544C2×C4#644611yescharacteristic
545C2×C4#644611yescharacteristic
546C2×C4#644711yescharacteristic
547C2×C4#644711yescharacteristic
548C2×C4#644711yescharacteristic
549C2×C4#644711yescharacteristic
550C2×C4#644811yescharacteristic
551C2×C4#644811yescharacteristic
552C2×C4#645711yescharacteristic
553C2×C4#645711yescharacteristic
554C2×C4#645711yescharacteristic
555C2×C4#645711yescharacteristic
556C2×C4#645711yescharacteristic
557C2×C4#645711yescharacteristic
558C2×C4#645711yescharacteristic
559C2×C4#645711yescharacteristic
560C2×C4#646111yescharacteristic
561C2×C4#646111yescharacteristic
562C2×C4C2×#115711yescharacteristic
563C2×C4C2×#115711yescharacteristic
564C2×C4C2×#115711yescharacteristic
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×D4×D711yescharacteristic
567C2×C4C2×#116111yescharacteristic
568C8—42no—
569C8—144no—
570C8—282no—
571C8—282no—
572C7#2365211yescharacteristic, complemented, Sylow
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—282no—
581C22—562no—
582C22—562no—
583C22—562no—
584C22#8944711yescharacteristic
585C22#8944711yescharacteristic
586C22#8944711yescharacteristic
587C22#8944711yescharacteristic
588C22C2×#644611yescharacteristic
589C22C2×#644811yescharacteristic
590C22C2×#646111yescharacteristic
591C4—82no—
592C4—82no—
593C4—282no—
594C4#8944711yescharacteristic
595C4#8944711yescharacteristic
596C4#8944711yescharacteristic
597C4#8944711yescharacteristic
598C4C2×#643911yescharacteristic
599C4C2×#644711yescharacteristic
600C4C2×#645711yescharacteristic
601C4C2×#645711yescharacteristic
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—562no—
610C2C2×#8944711yescharacteristic
subgroupG/H
centerC4#89447
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×H(3,ℤ/4)C2
socleC2×C7C2×#397
prime radicalC22×#1157C22
hypercenterC2×H(3,ℤ/4)D7
hypocenterC7#23652
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
2. C2×#89896C2semidirect
5. C2×C7×H(3,ℤ/4)C2semidirect
6. C2×#89328C2semidirect
7. C2×#89328C2semidirect
8. #11726463C2semidirect
9. #11726460C2semidirect
110. C2×H(3,ℤ/4)D7semidirect
572. C7#23652semidirect

Maximal subgroups

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

subgroupindex
C2×#899042
C2×#898962
#117396582
C2×C7×H(3,ℤ/4)2
C2×#893282
#117264632
#117264602
#236527