Finite Group Databasebeta

← #33008214

Subgroups — #33008214

Order 18688.

2278 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D73C211yesmaximal, characteristic
2C2×#17988449C214yesmaximal
3C2×#17988444C211yesmaximal, characteristic
4C2×C4×#11924005C211yesmaximal, characteristic
5#22608197C214yesmaximal, complemented
6#22608161C211yesmaximal, characteristic, complemented
7#22608160C211yesmaximal, characteristic, complemented
8#22608160C211yesmaximal, characteristic, complemented
9C73×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D73C2212yes—
11C23×C4×D73C2211yescharacteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C2×C4⋊C4×D73—24no—
16C2×C4⋊C4×D73—24no—
17C2×C4⋊C4×D73C418yescomplemented
18C22×#11924005C2212yes—
19C22×#11924005C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21#17988449—216no—
22#17988444—24no—
23C4×#11924005—24no—
24C2×#14556498C2214yes—
25C2×#14556498C2214yes—
26C2×#14556498C2214yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C22×C4⋊C4×C73C2211yescharacteristic
30C22×C4⋊C4×C73C2211yescharacteristic
31C22×C42×C73C2211yescharacteristic
32C2×#95×C73C2214yes—
33C24×D73C2311yescharacteristic
34C23×Dic73C2312yes—
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C23×Dic73C2311yescharacteristic
38C22×C4×D73—28no—
39C22×C4×D73—24no—
40C22×C4×D73—24no—
41C22×C4×D73—28no—
42C22×C4×D73—22no—
43C22×C4×D73—22no—
44C22×C4×D73—22no—
45C22×C4×D73—22no—
46C22×C4×D73C2×C418yes—
47C22×C4×D73C2×C414yes—
48C22×C4×D73D414yes—
49C22×C4×D73D414yes—
50C2×C4⋊Dic73—22no—
51C2×C4⋊Dic73—22no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—22no—
55C2×C4⋊Dic73C2×C414yes—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007C2×C418yes—
59C4⋊C4×D73—232no—
60C2×#11924005—24no—
61C2×#11924005—24no—
62C2×#11924005—24no—
63C2×#11924005—28no—
64C2×#11924005D418yes—
65C2×C4×Dic73—22no—
66C2×C4×Dic73—22no—
67C2×C4×Dic73—22no—
68#14556498—24no—
69#14556498—24no—
70#14556498—28no—
71#14556498—28no—
72#14556498—28no—
73C23×C4×C73C2312yes—
74C23×C4×C73C2312yes—
75C23×C4×C73C2312yes—
76C23×C4×C73C2311yescharacteristic
77C2×C4⋊C4×C73—22no—
78C2×C4⋊C4×C73—22no—
79C2×C4⋊C4×C73—22no—
80C2×C4⋊C4×C73—22no—
81C2×C4⋊C4×C73—22no—
82C2×C4⋊C4×C73C2×C414yes—
83C2×C42×C73—22no—
84C2×C42×C73—22no—
85C2×C42×C73—22no—
86#95×C73—28no—
87C23×D73—21no—
88C23×D73—21no—
89C23×D73—21no—
90C23×D73—21no—
91C23×D73—22no—
92C23×D73—22no—
93C23×D73C22×C412yes—
94C23×D73C2×D412yes—
95C23×D73C2×D412yes—
96C23×D73C2×D414yes—
97C23×D73#4114yes—
98C22×Dic73—21no—
99C22×Dic73—21no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—24no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—24no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—24no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—24no—
117C22×Dic73C22×C414yes—
118C22×Dic73C22×C412yes—
119C22×Dic73C2×D414yes—
120C22×Dic73C2×D412yes—
121C22×Dic73C2×D412yes—
122C2×C4×D73—28no—
123C2×C4×D73—28no—
124C2×C4×D73—28no—
125C2×C4×D73—28no—
126C2×C4×D73—28no—
127C2×C4×D73—28no—
128C2×C4×D73—216no—
129C2×C4×D73—232no—
130C2×C4×D73—416no—
131C2×C4×D73—48no—
132C2×C4×D73—48no—
133C2×C4×D73C22⋊C4116yes—
134C4⋊Dic73—28no—
135C4⋊Dic73—44no—
136#11924007—216no—
137#11924005—216no—
138#11924005—48no—
139C4×Dic73—44no—
140C24×C73C2411yescharacteristic
141C22×C4×C73—24no—
142C22×C4×C73—22no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—22no—
146C22×C4×C73—22no—
147C22×C4×C73—22no—
148C22×C4×C73—24no—
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—24no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—24no—
156C22×C4×C73C22×C414yes—
157C22×C4×C73C22×C412yes—
158C22×C4×C73C2×D414yes—
159C22×C4×C73C2×D412yes—
160C22×C4×C73C2×D412yes—
161C22×C4×C73#4112yes—
162C22×C4×C73#4112yes—
163C22×C4×C73#4112yes—
164C22×C4×C73#4112yes—
165C4⋊C4×C73—28no—
166C4⋊C4×C73—44no—
167C42×C73—44no—
168C23×C73C23×C411yescharacteristic
169C23×C73C22×D411yescharacteristic
170C23×C73C22×D411yescharacteristic
171C23×C73C22×D412yes—
172C23×C73C2×#4112yes—
173C23×C73C2×#4111yescharacteristic
174C23×C73C2×#4111yescharacteristic
175C23×C73C2×#4111yescharacteristic
176C23×C73C2×#4111yescharacteristic
177C23×C732−1+412yes—
178C23×C732−1+412yes—
179C22×D73—22no—
180C22×D73—22no—
181C22×D73—24no—
182C22×D73—22no—
183C22×D73—22no—
184C22×D73—22no—
185C22×D73—22no—
186C22×D73—22no—
187C22×D73—24no—
188C22×D73—22no—
189C22×D73—24no—
190C22×D73—24no—
191C22×D73—24no—
192C22×D73—24no—
193C22×D73—24no—
194C22×D73—24no—
195C22×D73—24no—
196C22×D73—24no—
197C22×D73C2×C22⋊C414yes—
198C22×D73C4×D418yes—
199C22×D73#12118yes—
200C22×D73#12414yes—
201C22×D73#12714yes—
202C2×C4×C73—24no—
203C2×C4×C73—28no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—44no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—44no—
218C2×C4×C73—44no—
219C2×C4×C73C2×C22⋊C414yes—
220C2×C4×C73#11714yes—
221C2×C4×C73#11714yes—
222C2×C4×C73#12314yes—
223C2×C4×C73#12314yes—
224C2×C4×C73#12314yes—
225C2×C4×C73#12314yes—
226C2×Dic73—22no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—24no—
233C2×Dic73—28no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—44no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—42no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—44no—
248C2×Dic73—44no—
249C2×Dic73C2×C22⋊C414yes—
250C4×D73—232no—
251C4×D73—432no—
252C22×C73C22×C22⋊C411yescharacteristic
253C22×C73C2×#11711yescharacteristic
254C22×C73C2×#11711yescharacteristic
255C22×C73C2×C4×D412yes—
256C22×C73#51912yes—
257C22×C73C2×#12112yes—
258C22×C73C2×#12311yescharacteristic
259C22×C73C2×#12311yescharacteristic
260C22×C73C2×#12311yescharacteristic
261C22×C73C2×#12311yescharacteristic
262C22×C73C2×#12411yescharacteristic
263C22×C73C2×#12711yescharacteristic
264C22×C73#53611yescharacteristic
265C22×C73#53612yes—
266C22×C73#53611yescharacteristic
267C22×C73#54712yes—
268C22×C73#54712yes—
269C22×C73#54712yes—
270C22×C73#54712yes—
271C22×C73#55212yes—
272C22×C73#55212yes—
273C22×C73#55212yes—
274C22×C73#55212yes—
275C2×D73—24no—
276C2×D73—28no—
277C2×D73—28no—
278C2×D73—28no—
279C2×D73—28no—
280C2×D73—24no—
281C2×D73—24no—
282C2×D73—24no—
283C2×D73—24no—
284C2×D73—24no—
285C2×D73#39018yes—
286C4×C73—44no—
287C4×C73—44no—
288C4×C73—82no—
289C4×C73#38118yescomplemented
290Dic73—24no—
291Dic73—44no—
292Dic73—44no—
293Dic73—82no—
294#13904—731nomaximal, Sylow
295C2×C73C2×#38111yescharacteristic
296C2×C73C2×#39011yescharacteristic
297C2×C73#231011yescharacteristic
298C2×C73#234412yes—
299C2×C73#234412yes—
300C2×C73#248612yes—
301C2×C73#248612yes—
302C2×C73#250111yescharacteristic
303C2×C73#250111yescharacteristic
304C2×C73#250311yescharacteristic
305C2×C73#250311yescharacteristic
306D73—28no—
307C23×C4⋊C4—731no—
308C2×#381—734no—
309C2×#380—731no—
310C2×C4×C22⋊C4—731no—
311#1445—734no—
312C4⋊#95—731no—
313C4⋊#95—731no—
314C4⋊#95—731no—
315C4⋊#95D7311yescharacteristic, complemented
316C73#1390411yescharacteristic, complemented, Sylow
317C24×C4—731no—
318C24×C4—732no—
319C22×C4⋊C4—731no—
320C22×C4⋊C4—732no—
321C22×C4⋊C4—738no—
322C22×C4⋊C4—731no—
323C22×C4⋊C4—1464no—
324C22×C4⋊C4—1464no—
325C22×C4⋊C4C2×D7311yescharacteristic
326C22×C4⋊C4C2×D7311yescharacteristic
327C22×C22⋊C4—732no—
328C22×C22⋊C4—732no—
329C22×C42—731no—
330C22×C42C2×D7311yescharacteristic
331#381—14616no—
332#380—1464no—
333C4×C22⋊C4—1464no—
334C2×#95—734no—
335C2×#95—734no—
336C2×#95—734no—
337C2×#95—732no—
338C2×#95—732no—
339C2×#95C2×D7314yes—
340C25—731no—
341C23×C4—731no—
342C23×C4—734no—
343C23×C4—734no—
344C23×C4—734no—
345C23×C4—732no—
346C23×C4—732no—
347C23×C4—738no—
348C23×C4—732no—
349C23×C4—1464no—
350C23×C4—1464no—
351C23×C4—1468no—
352C23×C4—1468no—
353C23×C4—1462no—
354C23×C4—1462no—
355C23×C4—1462no—
356C23×C4—1462no—
357C23×C4C22×D7312yes—
358C23×C4C22×D7312yes—
359C23×C4C22×D7312yes—
360C23×C4C22×D7311yescharacteristic
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—738no—
367C2×C4⋊C4—734no—
368C2×C4⋊C4—14632no—
369C2×C4⋊C4—1462no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1464no—
373C2×C4⋊C4—1464no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4C4×D7314yes—
377C2×C22⋊C4—738no—
378C2×C22⋊C4—1464no—
379C2×C22⋊C4—1468no—
380C2×C22⋊C4—1464no—
381C2×C22⋊C4—1464no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—1462no—
386C2×C42—1462no—
387C2×C42—1462no—
388#95—28no—
389#95—1464no—
390#95—1464no—
391#95—1468no—
392#95—1468no—
393#95—1468no—
394C24—732no—
395C24—734no—
396C24—732no—
397C24—732no—
398C24—734no—
399C24—1461no—
400C24—1461no—
401C24—1461no—
402C24—1461no—
403C24—1462no—
404C24—1462no—
405C24C23×D7311yescharacteristic
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—2928no—
422C22×C4—29216no—
423C22×C4—2928no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—732no—
427C22×C4—7316no—
428C22×C4—734no—
429C22×C4—734no—
430C22×C4—14616no—
431C22×C4—14632no—
432C22×C4—1464no—
433C22×C4—1462no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1464no—
437C22×C4—1464no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1464no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1461no—
448C22×C4—1461no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1468no—
452C22×C4—1468no—
453C22×C4—1468no—
454C22×C4—1468no—
455C22×C4—1468no—
456C22×C4—1468no—
457C22×C4C2×C4×D7314yes—
458C22×C4C2×C4×D7312yes—
459C22×C4D4×D7314yes—
460C22×C4D4×D7312yes—
461C22×C4D4×D7312yes—
462C22×C4#1192402712yes—
463C22×C4#1192402812yes—
464C22×C4#1192402812yes—
465C22×C4#1192402912yes—
466C4⋊C4—28no—
467C4⋊C4—44no—
468C4⋊C4—2924no—
469C4⋊C4—14616no—
470C4⋊C4—1468no—
471C22⋊C4—2928no—
472C22⋊C4—14616no—
473C42—44no—
474C42—2924no—
475C23—734no—
476C23—734no—
477C23—738no—
478C23—738no—
479C23—734no—
480C23—1462no—
481C23—1464no—
482C23—1462no—
483C23—1464no—
484C23—1464no—
485C23—1464no—
486C23—1464no—
487C23—1464no—
488C23—1464no—
489C23—1464no—
490C23—1464no—
491C23—1462no—
492C23—1462no—
493C23—1464no—
494C23—1462no—
495C23—1462no—
496C23—1462no—
497C23—1462no—
498C23C22×C4×D7311yescharacteristic
499C23C2×D4×D7311yescharacteristic
500C23C2×D4×D7312yes—
501C23C2×D4×D7311yescharacteristic
502C23#41×D7312yes—
503C23C2×#1192402711yescharacteristic
504C23C2×#1192402811yescharacteristic
505C23C2×#1192402811yescharacteristic
506C23C2×#1192402911yescharacteristic
507C23#1455663812yes—
508C23#1455663912yes—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—29232no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2924no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2924no—
537C2×C4—2924no—
538C2×C4—2922no—
539C2×C4—734no—
540C2×C4—1464no—
541C2×C4—1464no—
542C2×C4—1464no—
543C2×C4—1468no—
544C2×C4—1464no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—14632no—
552C2×C4C22⋊C4×D7314yes—
553C2×C4#1455654814yes—
554C2×C4#1455654914yes—
555C2×C4#1455657314yes—
556C2×C4#1455657314yes—
557C2×C4#1455657614yes—
558C2×C4#1455657614yes—
559C22—738no—
560C22—1464no—
561C22—1468no—
562C22—1468no—
563C22—1464no—
564C22—1464no—
565C22—1464no—
566C22—1464no—
567C22—1464no—
568C22—1468no—
569C22—1468no—
570C22C2×C22⋊C4×D7311yescharacteristic
571C22C2×#1455654811yescharacteristic
572C22C2×#1455654911yescharacteristic
573C22C4×D4×D7312yes—
574C22#1798906312yes—
575C22#121×D7312yes—
576C22C2×#1455657311yescharacteristic
577C22C2×#1455657311yescharacteristic
578C22C2×#1455657611yescharacteristic
579C22C2×#1455657611yescharacteristic
580C22#124×D7311yescharacteristic
581C22#127×D7311yescharacteristic
582C22#1798915711yescharacteristic
583C22#1798916012yes—
584C22#1798916211yescharacteristic
585C22#1798923712yes—
586C22#1798923812yes—
587C22#1798924012yes—
588C22#1798924212yes—
589C22#1798927112yes—
590C22#1798927512yes—
591C22#1798927612yes—
592C22#1798927812yes—
593C4—44no—
594C4—44no—
595C4—82no—
596C4—2924no—
597C4—2924no—
598C4—5842no—
599C4—1464no—
600C4#1798844918yescomplemented
601C2—1468no—
602C2C2×#1798844911yescharacteristic
603C2#390×D7311yescharacteristic
604C2#2261251411yescharacteristic
605C2#2261279612yes—
606C2#2261279912yes—
607C2#2261423512yes—
608C2#2261423712yes—
609C2#2261439911yescharacteristic
610C2#2261440111yescharacteristic
611C2#2261441311yescharacteristic
612C2#2261441511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C4⋊#95C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenterC4⋊#95D73
hypocenterC73#13904
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. #22608197C2semidirect
6. #22608161C2semidirect
7. #22608160C2semidirect
8. #22608160C2semidirect
9. C73×C4⋊#95C2semidirect
17. C2×C4⋊C4×D73C4semidirect
289. C4×C73#381semidirect
315. C4⋊#95D73semidirect
316. C73#13904semidirect
600. C4#17988449semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D732
C2×#179884492
C2×#179884442
C2×C4×#119240052
#226081972
#226081612
#226081602
C73×C4⋊#952
#1390473