Finite Group Databasebeta

← C4×#17988523 #33007813

Subgroups — C4×#17988523 #33007813

Order 18688.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988523C211yesmaximal, characteristic
2C2×C4×#11924007C212yesmaximal, complemented
3C2×C4×C4⋊Dic73C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Dic73C212yesmaximal
5#22608200C211yesmaximal, characteristic, complemented
6#22608197C212yesmaximal, complemented
7#22608195C211yesmaximal, characteristic, complemented
8#22608176C211yesmaximal, characteristic
9#22608174C212yesmaximal
10C4×#14556498C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊C4×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×C22⋊Dic73C2212yes—
17C22×C22⋊Dic73C2212yes—
18C22×C4×Dic73C2212yes—
19C22×C4×Dic73C2212yes—
20#17988523C418yesdirect product factor
21C4×#11924007—28no—
22C4×C4⋊Dic73—24no—
23C4×C22⋊Dic73—28no—
24C2×#14556498C2212yes—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2211yescharacteristic
32C24×C4×C73C2211yescharacteristic
33C22×C22⋊C4×C73C2211yescharacteristic
34C22×C22⋊C4×C73C2211yescharacteristic
35C22×C42×C73C2211yescharacteristic
36C22×C42×C73C2211yescharacteristic
37C4×C22⋊C4×C73—24no—
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2211yescharacteristic
40C23×Dic73C2312yes—
41C23×Dic73C2312yes—
42C23×Dic73C2312yes—
43C23×Dic73C2312yes—
44C2×C4⋊Dic73—24no—
45C2×C4⋊Dic73—22no—
46C2×C4⋊Dic73—22no—
47C2×C4⋊Dic73—22no—
48C2×C4⋊Dic73C2×C414yescomplemented
49C2×#11924007—24no—
50C2×#11924007—24no—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007C2×C418yescomplemented
55C2×C22⋊Dic73—24no—
56C2×C22⋊Dic73—24no—
57C2×C22⋊Dic73—24no—
58C2×C22⋊Dic73—24no—
59C2×C22⋊Dic73—24no—
60C2×C22⋊Dic73C2×C418yes—
61C2×C4×Dic73—24no—
62C2×C4×Dic73—24no—
63C2×C4×Dic73—24no—
64C2×C4×Dic73—24no—
65C2×C4×Dic73—24no—
66C2×C4×Dic73D418yes—
67#14556498—24no—
68#14556498—24no—
69#14556498—24no—
70#14556498—24no—
71#14556498—24no—
72#14556498—24no—
73#14556498—22no—
74#14556498C2×C414yescomplemented
75C25×C73C2311yescharacteristic
76C23×C4×C73—22no—
77C23×C4×C73—24no—
78C23×C4×C73—24no—
79C23×C4×C73—22no—
80C23×C4×C73Q814yes—
81C23×C4×C73C2311yescharacteristic
82C23×C4×C73C2311yescharacteristic
83C23×C4×C73C2311yescharacteristic
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C23×C4×C73C2311yescharacteristic
87C2×C22⋊C4×C73—22no—
88C2×C22⋊C4×C73—22no—
89C2×C22⋊C4×C73—22no—
90C2×C22⋊C4×C73—22no—
91C2×C22⋊C4×C73—22no—
92C2×C22⋊C4×C73C2×C414yes—
93C2×C42×C73—24no—
94C2×C42×C73—22no—
95C2×C42×C73—24no—
96C2×C42×C73D414yes—
97#95×C73—22no—
98#95×C73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
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×Dic73C22×C414yes—
122C22×Dic73C22×C414yes—
123C22×Dic73C2×D414yes—
124C22×Dic73C2×D414yes—
125C22×Dic73#4114yes—
126C22×Dic73#4114yes—
127C4⋊Dic73—28no—
128C4⋊Dic73—28no—
129#11924007—216no—
130#11924007—216no—
131C22⋊Dic73—216no—
132C22⋊Dic73—216no—
133C4×Dic73—216no—
134C4×Dic73—48no—
135C24×C73—21no—
136C24×C73—22no—
137C24×C73—21no—
138C24×C73—21no—
139C24×C73—22no—
140C24×C73—21no—
141C24×C73—22no—
142C24×C73—22no—
143C24×C73C22×C412yes—
144C24×C73C2×Q812yes—
145C24×C73#4112yes—
146C24×C73C2411yescharacteristic
147C22×C4×C73—22no—
148C22×C4×C73—22no—
149C22×C4×C73—21no—
150C22×C4×C73—21no—
151C22×C4×C73—21no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—21no—
157C22×C4×C73—21no—
158C22×C4×C73—21no—
159C22×C4×C73—21no—
160C22×C4×C73—22no—
161C22×C4×C73—21no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—216no—
166C22×C4×C73—28no—
167C22×C4×C73—44no—
168C22×C4×C73—48no—
169C22×C4×C73—44no—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C22×C412yes—
172C22×C4×C73C2×D414yes—
173C22×C4×C73C2×D412yes—
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×Q812yes—
177C22×C4×C73C2×Q812yes—
178C22×C4×C73#4114yes—
179C22×C4×C73#4112yes—
180C22×C4×C73#4112yes—
181C22×C4×C73#4112yes—
182C22⋊C4×C73—28no—
183C22⋊C4×C73—28no—
184C42×C73—28no—
185C42×C73—44no—
186C23×C73—22no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—24no—
191C23×C73—22no—
192C23×C73—22no—
193C23×C73—24no—
194C23×C73—22no—
195C23×C73—24no—
196C23×C73—24no—
197C23×C73—22no—
198C23×C73—41no—
199C23×C73—42no—
200C23×C73—41no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—41no—
206C23×C73—41no—
207C23×C73—42no—
208C23×C73C4×Q814yes—
209C23×C73C23×C411yescharacteristic
210C23×C73C22×D412yes—
211C23×C73C22×D411yescharacteristic
212C23×C73C22×Q811yescharacteristic
213C23×C73C2×#4112yes—
214C23×C73C2×#4112yes—
215C23×C73C2×#4112yes—
216C23×C73C2×#4111yescharacteristic
217C23×C73C2×#4111yescharacteristic
218C23×C73C2×#4111yescharacteristic
219C23×C73C2×#4111yescharacteristic
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—24no—
232C2×C4×C73—22no—
233C2×C4×C73—41no—
234C2×C4×C73—42no—
235C2×C4×C73—41no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—41no—
240C2×C4×C73—41no—
241C2×C4×C73—41no—
242C2×C4×C73—41no—
243C2×C4×C73—42no—
244C2×C4×C73—416no—
245C2×C4×C73C4×D414yes—
246C2×C4×C73C22⋊Q818yes—
247C2×C4×C73C22⋊Q814yes—
248C2×C4×C73#12314yes—
249C2×C4×C73#12318yes—
250C2×C4×C73#12714yes—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—24no—
256C2×Dic73—24no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—24no—
262C2×Dic73—24no—
263C2×Dic73—24no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—42no—
267C2×Dic73—42no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—42no—
273C2×Dic73—42no—
274C2×Dic73—42no—
275C2×Dic73—42no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73C4×D418yes—
279C22×C73—28no—
280C22×C73—24no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—42no—
284C22×C73—44no—
285C22×C73—42no—
286C22×C73—44no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—44no—
290C22×C73C2×C4×D412yes—
291C22×C73C2×C4×D411yescharacteristic
292C22×C73C2×C4×Q811yescharacteristic
293C22×C73C4×#4112yes—
294C22×C73C4×#4111yescharacteristic
295C22×C73C2×C22⋊Q812yes—
296C22×C73C2×C22⋊Q811yescharacteristic
297C22×C73C2×#12311yescharacteristic
298C22×C73C2×#12312yes—
299C22×C73#52511yescharacteristic
300C22×C73#52512yes—
301C22×C73#52512yes—
302C22×C73#52511yescharacteristic
303C22×C73#52911yescharacteristic
304C22×C73#52912yes—
305C22×C73#52912yes—
306C22×C73#52912yes—
307C22×C73#52912yes—
308C22×C73C2×#12711yescharacteristic
309C22×C73#53211yescharacteristic
310C22×C73#53212yes—
311C22×C73#53312yes—
312C22×C73#53311yescharacteristic
313C4×C73—24no—
314C4×C73—42no—
315C4×C73—42no—
316C4×C73—42no—
317C4×C73—42no—
318C4×C73#39918yescomplemented
319Dic73—28no—
320Dic73—44no—
321Dic73—44no—
322Dic73—44no—
323C4×#399—731nomaximal, Sylow
324C2×C73—44no—
325C2×C73C4×#12311yescharacteristic
326C2×C73C4×#12312yes—
327C2×C73C4×C22⋊Q812yes—
328C2×C73C4×C22⋊Q811yescharacteristic
329C2×C73C4×#12711yescharacteristic
330C2×C73C2×#39911yescharacteristic
331C2×C73#239511yescharacteristic
332C2×C73#239512yes—
333C2×C73#239611yescharacteristic
334C2×C73#239612yes—
335C2×C73#239711yescharacteristic
336C2×#399—731no—
337C2×C4×C4⋊C4—731no—
338C2×C4×C4⋊C4—732no—
339C2×C4×C22⋊C4—732no—
340C2×C4×C22⋊C4D7311yescharacteristic
341#1445—732no—
342#1445—731no—
343#1444—731no—
344#1439—732no—
345#1439—731no—
346C4×#95—731no—
347C73C4×#39911yescharacteristic, complemented, Sylow
348C24×C4C2×D7311yescharacteristic
349C22×C4⋊C4—732no—
350C22×C4⋊C4—731no—
351C22×C4⋊C4—732no—
352C22×C4⋊C4—731no—
353C22×C22⋊C4—732no—
354C22×C22⋊C4—732no—
355C22×C22⋊C4C2×D7311yescharacteristic
356C22×C22⋊C4C2×D7311yescharacteristic
357C22×C42—732no—
358C22×C42—732no—
359C22×C42C2×D7311yescharacteristic
360C22×C42C2×D7311yescharacteristic
361#399—738no—
362C4×C4⋊C4—1468no—
363C4×C4⋊C4—1464no—
364C4×C22⋊C4—24no—
365C4×C22⋊C4—1468no—
366C2×#95—731no—
367C2×#95—732no—
368C2×#95—731no—
369C2×#95—732no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95—732no—
373C2×#95—732no—
374C2×#95C2×D7311yescharacteristic
375C2×#95C2×D7311yescharacteristic
376C25C22×D7311yescharacteristic
377C23×C4—24no—
378C23×C4—22no—
379C23×C4—24no—
380C23×C4—22no—
381C23×C4—732no—
382C23×C4—732no—
383C23×C4—732no—
384C23×C4—732no—
385C23×C4Dic14614yes—
386C23×C4C22×D7311yescharacteristic
387C23×C4C22×D7311yescharacteristic
388C23×C4C22×D7311yescharacteristic
389C23×C4C22×D7311yescharacteristic
390C23×C4C22×D7311yescharacteristic
391C23×C4C22×D7311yescharacteristic
392C2×C4⋊C4—738no—
393C2×C4⋊C4—734no—
394C2×C4⋊C4—1464no—
395C2×C4⋊C4—1464no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1464no—
399C2×C4⋊C4—1462no—
400C2×C4⋊C4—1462no—
401C2×C4⋊C4—1464no—
402C2×C4⋊C4—1462no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—22no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—738no—
409C2×C22⋊C4—1464no—
410C2×C22⋊C4—1464no—
411C2×C22⋊C4—1464no—
412C2×C22⋊C4—1464no—
413C2×C22⋊C4—1464no—
414C2×C22⋊C4C4×D7314yes—
415C2×C42—24no—
416C2×C42—24no—
417C2×C42—22no—
418C2×C42—738no—
419C2×C42—1464no—
420C2×C42—1464no—
421C2×C42—1464no—
422C2×C42—1464no—
423C2×C42—1464no—
424C2×C42C22⋊D7314yes—
425#95—22no—
426#95—22no—
427#95—734no—
428#95—1464no—
429#95—1464no—
430#95—1464no—
431#95—1464no—
432#95—1464no—
433#95—1464no—
434#95—1462no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24—21no—
443C24C2×C4×D7312yes—
444C24C2×Dic14612yes—
445C24#1192402812yes—
446C24C23×D7311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—216no—
466C22×C4—28no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—734no—
471C22×C4—734no—
472C22×C4—734no—
473C22×C4—734no—
474C22×C4—734no—
475C22×C4—734no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1462no—
479C22×C4—1462no—
480C22×C4—1462no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1462no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4C2×C4×D7312yes—
499C22×C4C2×C4×D7312yes—
500C22×C4D4×D7314yes—
501C22×C4C2×C22⋊D7312yes—
502C22×C4C2×C22⋊D7312yes—
503C22×C4C2×C22⋊D7312yes—
504C22×C4C2×Dic14612yes—
505C22×C4C2×Dic14612yes—
506C22×C4#1192402714yes—
507C22×C4#1192402812yes—
508C22×C4#1192402812yes—
509C22×C4#1192402812yes—
510C4⋊C4—14616no—
511C4⋊C4—14616no—
512C4⋊C4—1468no—
513C4⋊C4—1468no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—14616no—
517C22⋊C4—14616no—
518C42—28no—
519C42—44no—
520C42—2928no—
521C42—14616no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C4×Dic14614yes—
545C23C22×C4×D7311yescharacteristic
546C23C2×D4×D7312yes—
547C23C22×C22⋊D7311yescharacteristic
548C23C22×Dic14611yescharacteristic
549C23#41×D7312yes—
550C23#41×D7312yes—
551C23C2×#1192402712yes—
552C23C2×#1192402811yescharacteristic
553C23C2×#1192402811yescharacteristic
554C23C2×#1192402811yescharacteristic
555C23C2×#1192402811yescharacteristic
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—416no—
581C2×C4—2922no—
582C2×C4—2922no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2922no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2922no—
591C2×C4—2922no—
592C2×C4—2922no—
593C2×C4—738no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1464no—
597C2×C4—1464no—
598C2×C4—1464no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1464no—
603C2×C4—1464no—
604C2×C4—1464no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4C4×C22⋊D7314yes—
610C2×C4#1455656818yes—
611C2×C4#1455657014yes—
612C2×C4#1455657214yes—
613C2×C4#1455657318yes—
614C2×C4#1455658714yes—
615C22—28no—
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22C4×D4×D7312yes—
627C22C2×C4×C22⋊D7311yescharacteristic
628C22C2×C4×Dic14611yescharacteristic
629C22C4×#1192402712yes—
630C22C4×#1192402811yescharacteristic
631C22C2×#1455656812yes—
632C22C2×#1455657011yescharacteristic
633C22C2×#1455657211yescharacteristic
634C22C2×#1455657312yes—
635C22#1798909411yescharacteristic
636C22#1798909612yes—
637C22#1798909612yes—
638C22#1798909811yescharacteristic
639C22#1798911512yes—
640C22#1798911712yes—
641C22#1798911712yes—
642C22#1798911912yes—
643C22#1798912111yescharacteristic
644C22C2×#1455658711yescharacteristic
645C22#1798913311yescharacteristic
646C22#1798913412yes—
647C22#1798914012yes—
648C22#1798914411yescharacteristic
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—2924no—
655C4—2924no—
656C4—2924no—
657C4—1468no—
658C4#1798852318yesdirect product factor
659C2—44no—
660C2C4×#1455657211yescharacteristic
661C2C4×#1455657312yes—
662C2C4×#1455656812yes—
663C2C4×#1455657011yescharacteristic
664C2C4×#1455658711yescharacteristic
665C2C2×#1798852311yescharacteristic
666C2#2261319011yescharacteristic
667C2#2261319212yes—
668C2#2261319911yescharacteristic
669C2#2261320512yes—
670C2#2261321011yescharacteristic
subgroupG/H
centerC23×C4C22×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC2×C4×C22⋊C4×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×C4×C22⋊C4D73
hypocenterC73C4×#399
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11924007C2semidirect
3. C2×C4×C4⋊Dic73C2semidirect
5. #22608200C2semidirect
6. #22608197C2semidirect
7. #22608195C2semidirect
10. C4×#14556498C2semidirect
20. #17988523C4direct (N × H)
48. C2×C4⋊Dic73C2×C4semidirect
54. C2×#11924007C2×C4semidirect
74. #14556498C2×C4semidirect
318. C4×C73#399semidirect
347. C73C4×#399semidirect
658. C4#17988523direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179885232
C2×C4×#119240072
C2×C4×C4⋊Dic732
C2×C4×C22⋊Dic732
#226082002
#226081972
#226081952
#226081762
#226081742
C4×#145564982
C2×C4×C22⋊C4×C732
C4×#39973