Finite Group Databasebeta

← C4×#17988507 #33007795

Subgroups — C4×#17988507 #33007795

Order 18688.

1942 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic146C211yesmaximal, characteristic
2C2×#17988507C211yesmaximal, characteristic
3C2×C4×#11924007C212yesmaximal
4#22608201C211yesmaximal, characteristic
5#22608195C211yesmaximal, characteristic
6#22608194C212yesmaximal
7#22608184C211yesmaximal, characteristic
8#22608182C212yesmaximal
9C4×#14556498C211yesmaximal, characteristic
10C4×#14556498C212yesmaximal
11C2×C4×C4⋊C4×C73C211yesmaximal, characteristic
12C23×Dic146C2211yescharacteristic
13C2×C4×Dic146—28no—
14C2×C4×Dic146—24no—
15C22×C4⋊Dic73C2211yescharacteristic
16C22×#11924007C2212yes—
17C22×#11924007C2212yes—
18C22×#11924007C2212yes—
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2212yes—
21#17988507C418yesdirect product factor
22C4×#11924007—28no—
23C2×#14556498C2212yes—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C22×C4⋊C4×C73C2211yescharacteristic
33C22×C4⋊C4×C73C2211yescharacteristic
34C22×C42×C73C2211yescharacteristic
35C22×C42×C73C2211yescharacteristic
36C22×C42×C73C2211yescharacteristic
37C4×C4⋊C4×C73—24no—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2211yescharacteristic
40C22×Dic146—24no—
41C22×Dic146—24no—
42C22×Dic146—22no—
43C22×Dic146—22no—
44C22×Dic146C2×C414yes—
45C23×Dic73C2312yes—
46C23×Dic73C2312yes—
47C23×Dic73C2312yes—
48C23×Dic73C2312yes—
49C4×Dic146—232no—
50C2×C4⋊Dic73—24no—
51C2×C4⋊Dic73—22no—
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—24no—
60C2×#11924007C2×C418yes—
61C2×C4×Dic73—24no—
62C2×C4×Dic73—24no—
63C2×C4×Dic73—24no—
64C2×C4×Dic73—24no—
65C2×C4×Dic73D418yes—
66C2×C4×Dic73Q818yescomplemented
67#14556498—24no—
68#14556498—24no—
69#14556498—24no—
70#14556498—24no—
71#14556498—24no—
72#14556498—22no—
73#14556498—24no—
74#14556498C2×C414yes—
75#14556498C2×C418yes—
76C23×C4×C73C2311yescharacteristic
77C23×C4×C73C2311yescharacteristic
78C23×C4×C73C2311yescharacteristic
79C23×C4×C73C2311yescharacteristic
80C23×C4×C73C2311yescharacteristic
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2311yescharacteristic
83C2×C4⋊C4×C73—22no—
84C2×C4⋊C4×C73—22no—
85C2×C4⋊C4×C73—22no—
86C2×C4⋊C4×C73—22no—
87C2×C4⋊C4×C73—22no—
88C2×C4⋊C4×C73C2×C414yes—
89C2×C42×C73—22no—
90C2×C42×C73—24no—
91C2×C42×C73—22no—
92C2×C42×C73—22no—
93C2×C42×C73—24no—
94C2×C42×C73D414yes—
95C2×C42×C73Q814yescomplemented
96#95×C73—22no—
97#95×C73—22no—
98C2×Dic146—28no—
99C2×Dic146—216no—
100C2×Dic146—28no—
101C2×Dic146—28no—
102C2×Dic146—216no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—22no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73C22×C414yes—
124C22×Dic73C22×C414yes—
125C22×Dic73C2×D414yes—
126C22×Dic73C2×D414yes—
127C22×Dic73C2×Q814yes—
128C22×Dic73C2×Q814yes—
129C22×Dic73#4114yes—
130C22×Dic73#4114yes—
131C4⋊Dic73—28no—
132#11924007—216no—
133#11924007—216no—
134#11924007—216no—
135C4×Dic73—216no—
136C4×Dic73—216no—
137C24×C73C2411yescharacteristic
138C22×C4×C73—21no—
139C22×C4×C73—22no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73—21no—
145C22×C4×C73—21no—
146C22×C4×C73—21no—
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—22no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—21no—
153C22×C4×C73—21no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73C22×C412yes—
161C22×C4×C73C22×C412yes—
162C22×C4×C73C22×C412yes—
163C22×C4×C73C2×D414yes—
164C22×C4×C73C2×D412yes—
165C22×C4×C73C2×D412yes—
166C22×C4×C73C2×D412yes—
167C22×C4×C73C2×Q814yes—
168C22×C4×C73C2×Q812yes—
169C22×C4×C73C2×Q812yes—
170C22×C4×C73C2×Q812yes—
171C22×C4×C73#4112yes—
172C22×C4×C73#4112yes—
173C22×C4×C73#4112yes—
174C22×C4×C73#4112yes—
175C4⋊C4×C73—28no—
176C4⋊C4×C73—28no—
177C42×C73—28no—
178C42×C73—28no—
179C42×C73—44no—
180C23×C73C23×C411yescharacteristic
181C23×C73C22×D412yes—
182C23×C73C22×D411yescharacteristic
183C23×C73C22×Q812yes—
184C23×C73C22×Q811yescharacteristic
185C23×C73C2×#4112yes—
186C23×C73C2×#4112yes—
187C23×C73C2×#4111yescharacteristic
188C23×C73C2×#4111yescharacteristic
189C23×C73C2×#4111yescharacteristic
190C23×C73C2×#4111yescharacteristic
191C2×C4×C73—22no—
192C2×C4×C73—24no—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—22no—
200C2×C4×C73—22no—
201C2×C4×C73—22no—
202C2×C4×C73—24no—
203C2×C4×C73—22no—
204C2×C4×C73—22no—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—24no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—41no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—41no—
218C2×C4×C73—41no—
219C2×C4×C73—42no—
220C2×C4×C73—42no—
221C2×C4×C73C4×D414yes—
222C2×C4×C73C4×Q814yescomplemented
223C2×C4×C73#12014yes—
224C2×C4×C73C22⋊Q818yes—
225C2×C4×C73C22⋊Q814yes—
226C2×C4×C73C4⋊Q814yes—
227C2×C4×C73C4⋊Q818yes—
228C2×Dic73—24no—
229C2×Dic73—24no—
230C2×Dic73—24no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—24no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—24no—
242C2×Dic73—24no—
243C2×Dic73—24no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—42no—
253C2×Dic73—42no—
254C2×Dic73C4×D418yes—
255C2×Dic73C4×Q818yescomplemented
256Dic146—232no—
257C22×C73C2×C4×D412yes—
258C22×C73C2×C4×D411yescharacteristic
259C22×C73C2×C4×Q812yes—
260C22×C73C2×C4×Q811yescharacteristic
261C22×C73C4×#4111yescharacteristic
262C22×C73C2×#12011yescharacteristic
263C22×C73C2×C22⋊Q812yes—
264C22×C73C2×C22⋊Q811yescharacteristic
265C22×C73#52511yescharacteristic
266C22×C73#52512yes—
267C22×C73#52512yes—
268C22×C73#52511yescharacteristic
269C22×C73#52911yescharacteristic
270C22×C73#52912yes—
271C22×C73C2×C4⋊Q811yescharacteristic
272C22×C73C2×C4⋊Q812yes—
273C22×C73#53211yescharacteristic
274C22×C73#53212yes—
275C22×C73#53312yes—
276C22×C73#53312yes—
277C22×C73#53312yes—
278C22×C73#53312yes—
279C22×C73#53311yescharacteristic
280C4×C73—24no—
281C4×C73—24no—
282C4×C73—42no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73—42no—
286C4×C73#39518yescomplemented
287Dic73—28no—
288Dic73—28no—
289Dic73—44no—
290Dic73—44no—
291C4×#395—731nomaximal, Sylow
292C2×C73C4×#12011yescharacteristic
293C2×C73C4×C22⋊Q812yes—
294C2×C73C4×C22⋊Q811yescharacteristic
295C2×C73C4×C4⋊Q811yescharacteristic
296C2×C73C4×C4⋊Q812yes—
297C2×C73C2×#39511yescharacteristic
298C2×C73#238811yescharacteristic
299C2×C73#239611yescharacteristic
300C2×C73#239612yes—
301C2×C73#239911yescharacteristic
302C2×C73#239912yes—
303C22×C4×Q8—731no—
304C2×#395—731no—
305C2×C4×C4⋊C4—732no—
306C2×C4×C4⋊C4D7311yescharacteristic
307#1446—731no—
308#1444—731no—
309#1444—732no—
310#1441—732no—
311#1441—731no—
312C4×#95—732no—
313C4×#95—731no—
314C73C4×#39511yescharacteristic, complemented, Sylow
315C23×Q8—731no—
316C2×C4×Q8—1468no—
317C2×C4×Q8—1464no—
318C22×C4⋊C4—732no—
319C22×C4⋊C4—732no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—732no—
322C22×C4⋊C4C2×D7311yescharacteristic
323C22×C4⋊C4C2×D7311yescharacteristic
324C22×C42—732no—
325C22×C42—732no—
326C22×C42C2×D7311yescharacteristic
327C22×C42C2×D7311yescharacteristic
328C22×C42C2×D7311yescharacteristic
329#395—738no—
330C4×C4⋊C4—24no—
331C4×C4⋊C4—1468no—
332C2×#95—732no—
333C2×#95—731no—
334C2×#95—732no—
335C2×#95—732no—
336C2×#95—732no—
337C2×#95—732no—
338C2×#95—731no—
339C2×#95—732no—
340C2×#95—732no—
341C2×#95C2×D7311yescharacteristic
342C2×#95C2×D7311yescharacteristic
343C22×Q8—734no—
344C22×Q8—1464no—
345C22×Q8—1464no—
346C22×Q8—1462no—
347C22×Q8—1462no—
348C23×C4—732no—
349C23×C4—732no—
350C23×C4—732no—
351C23×C4—732no—
352C23×C4C22×D7311yescharacteristic
353C23×C4C22×D7311yescharacteristic
354C23×C4C22×D7311yescharacteristic
355C23×C4C22×D7311yescharacteristic
356C23×C4C22×D7311yescharacteristic
357C23×C4C22×D7311yescharacteristic
358C23×C4C22×D7311yescharacteristic
359C4×Q8—14632no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—738no—
366C2×C4⋊C4—1464no—
367C2×C4⋊C4—1464no—
368C2×C4⋊C4—1464no—
369C2×C4⋊C4—1464no—
370C2×C4⋊C4—1464no—
371C2×C4⋊C4—1464no—
372C2×C4⋊C4—1464no—
373C2×C4⋊C4—1464no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1464no—
376C2×C4⋊C4C4×D7314yes—
377C2×C42—24no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—24no—
381C2×C42—22no—
382C2×C42—738no—
383C2×C42—738no—
384C2×C42—1464no—
385C2×C42—1464no—
386C2×C42—1464no—
387C2×C42—1464no—
388C2×C42Dic14614yescomplemented
389C2×C42C22⋊D7314yes—
390#95—22no—
391#95—22no—
392#95—734no—
393#95—738no—
394#95—1464no—
395#95—1464no—
396#95—1464no—
397#95—1464no—
398#95—1464no—
399#95—1462no—
400#95—1464no—
401C24C23×D7311yescharacteristic
402C2×Q8—14616no—
403C2×Q8—1468no—
404C2×Q8—1468no—
405C2×Q8—14616no—
406C2×Q8—1468no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—734no—
430C22×C4—734no—
431C22×C4—734no—
432C22×C4—734no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—734no—
436C22×C4—734no—
437C22×C4—1462no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1462no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4C2×C4×D7312yes—
458C22×C4C2×C4×D7312yes—
459C22×C4C2×C4×D7312yes—
460C22×C4D4×D7314yes—
461C22×C4C2×C22⋊D7312yes—
462C22×C4C2×C22⋊D7312yes—
463C22×C4C2×C22⋊D7312yes—
464C22×C4Q8×D7314yes—
465C22×C4C2×Dic14612yes—
466C22×C4C2×Dic14612yes—
467C22×C4C2×Dic14612yes—
468C22×C4#1192402812yes—
469C22×C4#1192402812yes—
470C22×C4#1192402812yes—
471C22×C4#1192402812yes—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—14616no—
475C4⋊C4—14616no—
476C4⋊C4—14616no—
477C4⋊C4—1468no—
478C42—28no—
479C42—28no—
480C42—44no—
481C42—14616no—
482C42—14616no—
483C23C22×C4×D7311yescharacteristic
484C23C2×D4×D7312yes—
485C23C22×C22⋊D7311yescharacteristic
486C23C2×Q8×D7312yes—
487C23C22×Dic14611yescharacteristic
488C23#41×D7312yes—
489C23#41×D7312yes—
490C23C2×#1192402811yescharacteristic
491C23C2×#1192402811yescharacteristic
492C23C2×#1192402811yescharacteristic
493C23C2×#1192402811yescharacteristic
494Q8—14632no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—2922no—
526C2×C4—2922no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—738no—
534C2×C4—738no—
535C2×C4—1464no—
536C2×C4—1464no—
537C2×C4—1464no—
538C2×C4—1464no—
539C2×C4—1464no—
540C2×C4—1464no—
541C2×C4—1464no—
542C2×C4—1464no—
543C2×C4—1464no—
544C2×C4—1464no—
545C2×C4—1464no—
546C2×C4—1464no—
547C2×C4—1464no—
548C2×C4—1464no—
549C2×C4—1464no—
550C2×C4—1464no—
551C2×C4—1464no—
552C2×C4—1464no—
553C2×C4C4×C22⋊D7314yes—
554C2×C4C4×Dic14614yescomplemented
555C2×C4#1455655914yes—
556C2×C4#1455656918yes—
557C2×C4#1455657014yes—
558C2×C4#1455658914yes—
559C2×C4#1455659018yes—
560C22C4×D4×D7312yes—
561C22C2×C4×C22⋊D7311yescharacteristic
562C22C4×Q8×D7312yes—
563C22C2×C4×Dic14611yescharacteristic
564C22C4×#1192402811yescharacteristic
565C22C2×#1455655911yescharacteristic
566C22C2×#1455656912yes—
567C22C2×#1455657011yescharacteristic
568C22#1798909411yescharacteristic
569C22#1798909612yes—
570C22#1798909612yes—
571C22#1798909811yescharacteristic
572C22#1798911512yes—
573C22#1798912111yescharacteristic
574C22C2×#1455658911yescharacteristic
575C22C2×#1455659012yes—
576C22#1798913311yescharacteristic
577C22#1798913412yes—
578C22#1798914012yes—
579C22#1798914112yes—
580C22#1798914112yes—
581C22#1798914112yes—
582C22#1798914411yescharacteristic
583C4—24no—
584C4—24no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—2924no—
590C4—2924no—
591C4—1468no—
592C4—1468no—
593C4#1798850718yesdirect product factor
594C2C4×#1455655911yescharacteristic
595C2C4×#1455656912yes—
596C2C4×#1455657011yescharacteristic
597C2C4×#1455658911yescharacteristic
598C2C4×#1455659012yes—
599C2C2×#1798850711yescharacteristic
600C2#2261314611yescharacteristic
601C2#2261319911yescharacteristic
602C2#2261320712yes—
603C2#2261322211yescharacteristic
604C2#2261322512yes—
subgroupG/H
centerC23×C4C22×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC2×C4×C4⋊C4×C73C2
socle, prime radical, cosocleC24×C73C24
hypercenterC2×C4×C4⋊C4D73
hypocenterC73C4×#395
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
21. #17988507C4direct (N × H)
66. C2×C4×Dic73Q8semidirect
95. C2×C42×C73Q8semidirect
222. C2×C4×C73C4×Q8semidirect
255. C2×Dic73C4×Q8semidirect
286. C4×C73#395semidirect
314. C73C4×#395semidirect
388. C2×C42Dic146semidirect
554. C2×C4C4×Dic146semidirect
593. C4#17988507direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Dic1462
C2×#179885072
C2×C4×#119240072
#226082012
#226081952
#226081942
#226081842
#226081822
C4×#145564982
C2×C4×C4⋊C4×C732
C4×#39573