Finite Group Databasebeta

← C4×#17988484 #33007178

Subgroups — C4×#17988484 #33007178

Order 18688.

1918 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
1C2×#17988484C212yesmaximal
2C2×C4×#11924007C211yesmaximal, characteristic
3C2×C4×#11924007C211yesmaximal, characteristic
4C2×C4×#11924007C211yesmaximal, characteristic
5C2×C4×C4⋊Dic73C211yesmaximal, characteristic
6C2×C42×Dic73C211yesmaximal, characteristic
7#22608165C212yesmaximal
8#22608162C212yesmaximal
9#22608159C212yesmaximal
10C4×#14556498C211yesmaximal, characteristic
11C4×#95×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2212yes—
13C22×#11924007C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×#11924007C2211yescharacteristic
17C22×C4×Dic73C2211yescharacteristic
18C22×C4×Dic73C2212yes—
19C22×C4×Dic73C2211yescharacteristic
20C22×C4×Dic73C2212yes—
21C22×C4×Dic73C2211yescharacteristic
22C22×C4×Dic73C2211yescharacteristic
23#17988484C4116yesdirect product factor
24C4×#11924007—24no—
25C4×#11924007—24no—
26C4×#11924007C418yescomplemented
27C4×C4⋊Dic73—24no—
28C2×#14556498C2212yes—
29C2×#14556498C2212yes—
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34C2×#14556498C2211yescharacteristic
35C42×Dic73—24no—
36C22×C42×C73C2211yescharacteristic
37C22×C42×C73C2211yescharacteristic
38C22×C42×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C2×#95×C73C2212yes—
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2312yes—
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2312yes—
46C23×Dic73C2311yescharacteristic
47C2×C4⋊Dic73—24no—
48C2×C4⋊Dic73—24no—
49C2×C4⋊Dic73C2×C418yes—
50C2×#11924007—24no—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007C2×C414yes—
56C2×#11924007C2×C418yes—
57C2×#11924007C2×C418yes—
58C2×#11924007C2×C414yes—
59C2×#11924007C2×C418yes—
60C2×C4×Dic73—24no—
61C2×C4×Dic73—24no—
62C2×C4×Dic73—24no—
63C2×C4×Dic73—24no—
64C2×C4×Dic73—22no—
65C2×C4×Dic73—22no—
66C2×C4×Dic73—22no—
67C2×C4×Dic73—22no—
68C2×C4×Dic73—22no—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—22no—
71C2×C4×Dic73—22no—
72C2×C4×Dic73C2×C414yes—
73C2×C4×Dic73C2×C418yes—
74C2×C4×Dic73C2×C414yes—
75C2×C4×Dic73D414yes—
76C2×C4×Dic73D414yes—
77C2×C4×Dic73Q814yes—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—24no—
82#14556498C2×C414yes—
83#14556498C2×C414yes—
84#14556498C2×C418yes—
85C23×C4×C73C2311yescharacteristic
86C23×C4×C73C2312yes—
87C23×C4×C73C2311yescharacteristic
88C23×C4×C73C2312yes—
89C23×C4×C73C2311yescharacteristic
90C2×C42×C73—22no—
91C2×C42×C73—22no—
92C2×C42×C73—22no—
93C2×C42×C73—22no—
94C2×C42×C73—22no—
95C2×C42×C73—22no—
96C2×C42×C73—22no—
97C2×C42×C73C2×C414yes—
98C2×C42×C73Q814yes—
99#95×C73—24no—
100#95×C73C2×C418yes—
101C22×Dic73—22no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—21no—
106C22×Dic73—22no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—21no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—21no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73C22×C412yes—
121C22×Dic73C22×C414yes—
122C22×Dic73C22×C412yes—
123C22×Dic73C22×C414yes—
124C22×Dic73C22×C412yes—
125C22×Dic73C22×C414yes—
126C22×Dic73C22×C412yes—
127C22×Dic73C22×C414yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×D414yes—
130C22×Dic73C2×D412yes—
131C22×Dic73C2×Q812yes—
132C22×Dic73C2×Q812yes—
133C22×Dic73#4114yes—
134C22×Dic73#4114yes—
135C4⋊Dic73—216no—
136#11924007—28no—
137#11924007—216no—
138#11924007—216no—
139#11924007C42116yescomplemented
140C4×Dic73—216no—
141C4×Dic73—28no—
142C4×Dic73—28no—
143C4×Dic73—216no—
144C4×Dic73—44no—
145C4×Dic73C4⋊C4116yescomplemented
146C24×C73C2411yescharacteristic
147C22×C4×C73—22no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—21no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—21no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73C22×C414yes—
165C22×C4×C73C22×C412yes—
166C22×C4×C73C22×C414yes—
167C22×C4×C73C22×C412yes—
168C22×C4×C73C22×C414yes—
169C22×C4×C73C22×C412yes—
170C22×C4×C73C2×D412yes—
171C22×C4×C73C2×D412yes—
172C22×C4×C73C2×Q812yes—
173C22×C4×C73C2×Q814yes—
174C22×C4×C73C2×Q812yes—
175C22×C4×C73#4112yes—
176C22×C4×C73#4114yes—
177C22×C4×C73#4112yes—
178C42×C73—28no—
179C42×C73—28no—
180C42×C73—44no—
181C23×C73C23×C411yescharacteristic
182C23×C73C23×C412yes—
183C23×C73C22×D411yescharacteristic
184C23×C73C22×D411yescharacteristic
185C23×C73C22×Q811yescharacteristic
186C23×C73C22×Q811yescharacteristic
187C23×C73C2×#4112yes—
188C23×C73C2×#4112yes—
189C23×C73C2×#4111yescharacteristic
190C23×C73C2×#4111yescharacteristic
191C23×C73C2×#4112yes—
192C2×C4×C73—22no—
193C2×C4×C73—24no—
194C2×C4×C73—24no—
195C2×C4×C73—24no—
196C2×C4×C73—24no—
197C2×C4×C73—22no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—22no—
203C2×C4×C73—22no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—22no—
207C2×C4×C73—24no—
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×C73C2×C4214yes—
215C2×C4×C73C2×C4⋊C414yes—
216C2×C4×C73C4×D414yes—
217C2×C4×C73C4×Q818yes—
218C2×C4×C73C4×Q814yes—
219C2×C4×C73#12114yes—
220C2×C4×C73C22⋊Q814yes—
221C2×C4×C73#12514yes—
222C2×C4×C73C4⋊Q814yes—
223C2×Dic73—24no—
224C2×Dic73—22no—
225C2×Dic73—24no—
226C2×Dic73—24no—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—24no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—22no—
242C2×Dic73—24no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—42no—
249C2×Dic73C2×C4214yes—
250C2×Dic73C2×C4214yes—
251C2×Dic73C2×C4⋊C414yes—
252C2×Dic73C2×C4⋊C414yes—
253C2×Dic73#11714yes—
254C2×Dic73C4×D418yes—
255C2×Dic73C4×D418yes—
256C2×Dic73C4×Q818yes—
257C22×C73C22×C4211yescharacteristic
258C22×C73C22×C4⋊C411yescharacteristic
259C22×C73C2×#11711yescharacteristic
260C22×C73C2×C4×D412yes—
261C22×C73C2×C4×D412yes—
262C22×C73C2×C4×D411yescharacteristic
263C22×C73C2×C4×Q812yes—
264C22×C73C2×C4×Q812yes—
265C22×C73C2×C4×Q811yescharacteristic
266C22×C73C4×#4112yes—
267C22×C73C4×#4111yescharacteristic
268C22×C73C4×#4112yes—
269C22×C73C4×#4111yescharacteristic
270C22×C73C2×#12111yescharacteristic
271C22×C73C2×C22⋊Q811yescharacteristic
272C22×C73#52512yes—
273C22×C73C2×#12511yescharacteristic
274C22×C73#52912yes—
275C22×C73#52912yes—
276C22×C73C2×C4⋊Q811yescharacteristic
277C22×C73#53212yes—
278C22×C73#53312yes—
279C22×C73#53312yes—
280C4×C73—24no—
281C4×C73—28no—
282C4×C73—42no—
283C4×C73—44no—
284C4×C73#38918yescomplemented
285Dic73—24no—
286Dic73—24no—
287Dic73—28no—
288Dic73—42no—
289Dic73—44no—
290Dic73C4×C4⋊C418yescomplemented
291C4×#389—731nomaximal, Sylow
292C2×C73C2×C4×C4⋊C411yescharacteristic
293C2×C73C42×D411yescharacteristic
294C2×C73C42×Q811yescharacteristic
295C2×C73C2×#38911yescharacteristic
296C2×C73#229311yescharacteristic
297C2×C73#229511yescharacteristic
298C2×C73#229611yescharacteristic
299C2×C73C4×#12112yes—
300C2×C73C4×C22⋊Q812yes—
301C2×C73C4×#12512yes—
302C2×C73C4×C4⋊Q812yes—
303C2×#389—732no—
304C2×C4×C4⋊C4—731no—
305C2×C4×C4⋊C4—731no—
306C2×C4×C4⋊C4—731no—
307C2×C4×C4⋊C4—731no—
308C2×C43—731no—
309#1436—732no—
310#1436—732no—
311C4⋊#95—732no—
312C4×#95—731no—
313C4×#95D7311yescharacteristic
314C73C4×#38911yescharacteristic, complemented, Sylow
315C22×C4⋊C4—731no—
316C22×C4⋊C4—731no—
317C22×C4⋊C4—732no—
318C22×C4⋊C4—732no—
319C22×C4⋊C4—732no—
320C22×C42—731no—
321C22×C42—732no—
322C22×C42—731no—
323C22×C42—731no—
324C22×C42—732no—
325C22×C42—731no—
326C22×C42C2×D7311yescharacteristic
327C22×C42C2×D7311yescharacteristic
328C22×C42C2×D7311yescharacteristic
329#389—7316no—
330C4×C4⋊C4—738no—
331C4×C4⋊C4—1464no—
332C4×C4⋊C4—1464no—
333C4×C4⋊C4—1464no—
334C2×#95—732no—
335C2×#95—731no—
336C2×#95—731no—
337C2×#95—732no—
338C2×#95—732no—
339C2×#95—732no—
340C2×#95—732no—
341C2×#95C2×D7312yes—
342C2×#95C2×D7312yes—
343C43—1464no—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—732no—
347C23×C4—731no—
348C23×C4—732no—
349C23×C4—731no—
350C23×C4C22×D7312yes—
351C23×C4C22×D7311yescharacteristic
352C23×C4C22×D7311yescharacteristic
353C23×C4C22×D7312yes—
354C23×C4C22×D7311yescharacteristic
355C2×C4⋊C4—734no—
356C2×C4⋊C4—738no—
357C2×C4⋊C4—734no—
358C2×C4⋊C4—738no—
359C2×C4⋊C4—738no—
360C2×C4⋊C4—738no—
361C2×C4⋊C4—1464no—
362C2×C4⋊C4—1464no—
363C2×C4⋊C4—1464no—
364C2×C4⋊C4—1464no—
365C2×C4⋊C4—1464no—
366C2×C4⋊C4—1464no—
367C2×C4⋊C4—1464no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—734no—
376C2×C42—734no—
377C2×C42—734no—
378C2×C42—734no—
379C2×C42—738no—
380C2×C42—734no—
381C2×C42—1462no—
382C2×C42—1462no—
383C2×C42—1462no—
384C2×C42—1464no—
385C2×C42—1464no—
386C2×C42—1464no—
387C2×C42—1464no—
388C2×C42—1462no—
389C2×C42—1462no—
390C2×C42—1462no—
391C2×C42—1464no—
392C2×C42—1462no—
393C2×C42Dic14614yes—
394C2×C42C4×D7314yes—
395#95—24no—
396#95—734no—
397#95—734no—
398#95—738no—
399#95—1464no—
400#95—1464no—
401#95—1464no—
402#95—1464no—
403#95C4×D7318yes—
404C24C23×D7311yescharacteristic
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—732no—
423C22×C4—734no—
424C22×C4—732no—
425C22×C4—732no—
426C22×C4—734no—
427C22×C4—732no—
428C22×C4—732no—
429C22×C4—734no—
430C22×C4—734no—
431C22×C4—732no—
432C22×C4—734no—
433C22×C4—732no—
434C22×C4—734no—
435C22×C4—732no—
436C22×C4—734no—
437C22×C4—1461no—
438C22×C4—1462no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1461no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1462no—
448C22×C4—1461no—
449C22×C4—1462no—
450C22×C4—1462no—
451C22×C4—1462no—
452C22×C4—1461no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4—1462no—
456C22×C4C2×C4×D7312yes—
457C22×C4C2×C4×D7314yes—
458C22×C4C2×C4×D7312yes—
459C22×C4C2×C4×D7314yes—
460C22×C4C2×C4×D7314yes—
461C22×C4C2×C4×D7312yes—
462C22×C4D4×D7312yes—
463C22×C4D4×D7312yes—
464C22×C4Q8×D7312yes—
465C22×C4C2×Dic14614yes—
466C22×C4C2×Dic14612yes—
467C22×C4#1192402712yes—
468C22×C4#1192402814yes—
469C22×C4#1192402812yes—
470C4⋊C4—7316no—
471C4⋊C4—1468no—
472C4⋊C4—14616no—
473C4⋊C4—14616no—
474C4⋊C4—14616no—
475C42—28no—
476C42—28no—
477C42—44no—
478C42—2924no—
479C42—7316no—
480C42—14616no—
481C42—14616no—
482C42—1468no—
483C42—1468no—
484C23C22×C4×D7311yescharacteristic
485C23C22×C4×D7312yes—
486C23C2×D4×D7311yescharacteristic
487C23C2×D4×D7311yescharacteristic
488C23C2×Q8×D7311yescharacteristic
489C23C22×Dic14611yescharacteristic
490C23#41×D7312yes—
491C23#41×D7312yes—
492C23C2×#1192402711yescharacteristic
493C23C2×#1192402812yes—
494C23C2×#1192402811yescharacteristic
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—2922no—
518C2×C4—2922no—
519C2×C4—2922no—
520C2×C4—2922no—
521C2×C4—2922no—
522C2×C4—2922no—
523C2×C4—734no—
524C2×C4—734no—
525C2×C4—734no—
526C2×C4—738no—
527C2×C4—738no—
528C2×C4—734no—
529C2×C4—734no—
530C2×C4—738no—
531C2×C4—1462no—
532C2×C4—1464no—
533C2×C4—1464no—
534C2×C4—1462no—
535C2×C4—1462no—
536C2×C4—1464no—
537C2×C4—1464no—
538C2×C4—1464no—
539C2×C4—1464no—
540C2×C4—1464no—
541C2×C4—1462no—
542C2×C4—1464no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—1462no—
546C2×C4—1464no—
547C2×C4—1464no—
548C2×C4—1464no—
549C2×C4—1464no—
550C2×C4—1464no—
551C2×C4C42×D7314yes—
552C2×C4C4⋊C4×D7314yes—
553C2×C4#1455655114yes—
554C2×C4C4×Dic14618yes—
555C2×C4C4×Dic14614yes—
556C2×C4#1455656414yes—
557C2×C4#1455656814yes—
558C2×C4#1455658114yes—
559C2×C4#1455659014yes—
560C22C2×C42×D7311yescharacteristic
561C22C2×C4⋊C4×D7311yescharacteristic
562C22#117×D7311yescharacteristic
563C22C4×D4×D7312yes—
564C22C4×D4×D7312yes—
565C22C2×#1455655111yescharacteristic
566C22C4×Q8×D7312yes—
567C22C2×C4×Dic14612yes—
568C22C2×C4×Dic14611yescharacteristic
569C22C4×#1192402712yes—
570C22#1798905511yescharacteristic
571C22C4×#1192402812yes—
572C22C4×#1192402811yescharacteristic
573C22C2×#1455656411yescharacteristic
574C22C2×#1455656811yescharacteristic
575C22#1798909612yes—
576C22C2×#1455658111yescharacteristic
577C22#1798911512yes—
578C22#1798911712yes—
579C22C2×#1455659011yescharacteristic
580C22#1798913412yes—
581C22#1798914012yes—
582C22#1798914112yes—
583C4—24no—
584C4—28no—
585C4—42no—
586C4—44no—
587C4—2922no—
588C4—2924no—
589C4—738no—
590C4—1464no—
591C4—1464no—
592C4—1468no—
593C4#1798848418yesdirect product factor
594C2C4×C4⋊C4×D7311yescharacteristic
595C2C4×#1455655111yescharacteristic
596C2C42×Dic14611yescharacteristic
597C2C2×#1798848411yescharacteristic
598C2#2261239411yescharacteristic
599C2#2261240311yescharacteristic
600C2#2261241111yescharacteristic
601C2C4×#1455656412yes—
602C2C4×#1455656812yes—
603C2C4×#1455658112yes—
604C2C4×#1455659012yes—
subgroupG/H
centerC23×C4C22×D73
commutatorC22×C73C22×C42
FrattiniC24C23×D73
FittingC4×#95×C73C2
socle, prime radical, cosocleC24×C73C24
hypercenterC4×#95D73
hypocenterC73C4×#389
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23×C4 ◁ C4×#95

Complements

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

NH ≅ G/Ntype
23. #17988484C4direct (N × H)
26. C4×#11924007C4semidirect
139. #11924007C42semidirect
145. C4×Dic73C4⋊C4semidirect
284. C4×C73#389semidirect
290. Dic73C4×C4⋊C4semidirect
314. C73C4×#389semidirect
593. C4#17988484direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179884842
C2×C4×#119240072
C2×C4×C4⋊Dic732
C2×C42×Dic732
#226081652
#226081622
#226081592
C4×#145564982
C4×#95×C732
C4×#38973