Finite Group Databasebeta

← #33007975

Subgroups — #33007975

Order 18688.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D73C211yesmaximal, characteristic
2C2×#17988453C211yesmaximal, characteristic, complemented
3C2×#17988452C211yesmaximal, characteristic, complemented
4C2×#17988444C211yesmaximal, characteristic
5C2×C4×#11924005C211yesmaximal, characteristic
6#22608186C214yesmaximal, complemented
7#22608160C211yesmaximal, characteristic, complemented
8#22608152C214yesmaximal
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×D73—24no—
18C22×#11924005C2212yes—
19C22×#11924005C2212yes—
20C22×C4×Dic73C2211yescharacteristic
21#17988453—24no—
22#17988452—24no—
23#17988444—24no—
24C4×#11924005C418yescomplemented
25C2×#14556498C2214yes—
26C2×#14556498C2214yes—
27C2×#14556498C2212yes—
28C2×#14556498C2214yes—
29C2×#14556498C2212yes—
30C22×C4⋊C4×C73C2211yescharacteristic
31C22×C4⋊C4×C73C2211yescharacteristic
32C22×C42×C73C2211yescharacteristic
33C2×#95×C73C2214yes—
34C24×D73C2311yescharacteristic
35C23×Dic73C2312yes—
36C23×Dic73C2312yes—
37C23×Dic73C2312yes—
38C23×Dic73C2311yescharacteristic
39C22×C4×D73—24no—
40C22×C4×D73—24no—
41C22×C4×D73—28no—
42C22×C4×D73—22no—
43C22×C4×D73—22no—
44C22×C4×D73—24no—
45C22×C4×D73—22no—
46C22×C4×D73—22no—
47C22×C4×D73—44no—
48C22×C4×D73C2×C414yes—
49C22×C4×D73D414yes—
50C22×C4×D73Q814yes—
51C2×C4⋊Dic73—22no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73—22no—
55C2×C4⋊Dic73—22no—
56C2×C4⋊Dic73D414yes—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007—24no—
60C4⋊C4×D73—232no—
61C2×#11924005—24no—
62C2×#11924005—24no—
63C2×#11924005—24no—
64C2×#11924005—28no—
65C2×#11924005C2×C418yes—
66C2×C4×Dic73—22no—
67C2×C4×Dic73—22no—
68C2×C4×Dic73C2×C414yescomplemented
69#14556498—24no—
70#14556498—28no—
71#14556498—28no—
72#14556498—44no—
73#14556498C2×C418yescomplemented
74C23×C4×C73C2312yes—
75C23×C4×C73C2312yes—
76C23×C4×C73C2312yes—
77C23×C4×C73C2311yescharacteristic
78C2×C4⋊C4×C73—22no—
79C2×C4⋊C4×C73—22no—
80C2×C4⋊C4×C73—22no—
81C2×C4⋊C4×C73—22no—
82C2×C4⋊C4×C73—22no—
83C2×C4⋊C4×C73D414yes—
84C2×C42×C73—22no—
85C2×C42×C73—22no—
86C2×C42×C73C2×C414yescomplemented
87#95×C73—44no—
88C23×D73—21no—
89C23×D73—21no—
90C23×D73—21no—
91C23×D73—21no—
92C23×D73—22no—
93C23×D73—22no—
94C23×D73C22×C412yes—
95C23×D73C2×D412yes—
96C23×D73C2×Q812yes—
97C23×D73#4114yes—
98C23×D73#4114yes—
99C22×Dic73—21no—
100C22×Dic73—21no—
101C22×Dic73—21no—
102C22×Dic73—21no—
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—24no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73—24no—
117C22×Dic73—24no—
118C22×Dic73C22×C414yes—
119C22×Dic73C22×C412yes—
120C22×Dic73C2×D414yes—
121C22×Dic73C2×D412yes—
122C22×Dic73C2×Q812yes—
123C2×C4×D73—28no—
124C2×C4×D73—28no—
125C2×C4×D73—28no—
126C2×C4×D73—28no—
127C2×C4×D73—28no—
128C2×C4×D73—28no—
129C2×C4×D73—216no—
130C2×C4×D73—416no—
131C2×C4×D73—416no—
132C2×C4×D73—48no—
133C2×C4×D73—48no—
134C2×C4×D73C4⋊C4116yes—
135C4⋊Dic73—28no—
136C4⋊Dic73—44no—
137#11924007—216no—
138#11924005—216no—
139#11924005—48no—
140C4×Dic73—28no—
141C24×C73C2411yescharacteristic
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—24no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—24no—
156C22×C4×C73—24no—
157C22×C4×C73C22×C414yes—
158C22×C4×C73C22×C412yes—
159C22×C4×C73C2×D412yes—
160C22×C4×C73C2×D412yes—
161C22×C4×C73C2×D414yes—
162C22×C4×C73C2×D412yes—
163C22×C4×C73C2×Q812yes—
164C22×C4×C73#4112yes—
165C22×C4×C73#4112yes—
166C4⋊C4×C73—28no—
167C4⋊C4×C73—44no—
168C42×C73—28no—
169C23×C73C23×C411yescharacteristic
170C23×C73C22×D411yescharacteristic
171C23×C73C22×D411yescharacteristic
172C23×C73C22×D411yescharacteristic
173C23×C73C22×Q811yescharacteristic
174C23×C73C2×#4112yes—
175C23×C73C2×#4112yes—
176C23×C73C2×#4111yescharacteristic
177C23×C73C2×#4111yescharacteristic
178C23×C73H(5,𝔽2)12yes—
179C23×C732−1+412yes—
180C22×D73—22no—
181C22×D73—22no—
182C22×D73—24no—
183C22×D73—22no—
184C22×D73—22no—
185C22×D73—22no—
186C22×D73—22no—
187C22×D73—22no—
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×D73—24no—
198C22×D73C2×C4⋊C414yes—
199C22×D73#11718yes—
200C22×D73#12414yes—
201C22×D73#12414yes—
202C22×D73#12518yes—
203C2×C4×C73—24no—
204C2×C4×C73—24no—
205C2×C4×C73—24no—
206C2×C4×C73—24no—
207C2×C4×C73—28no—
208C2×C4×C73—42no—
209C2×C4×C73—42no—
210C2×C4×C73—42no—
211C2×C4×C73—44no—
212C2×C4×C73—42no—
213C2×C4×C73—42no—
214C2×C4×C73—42no—
215C2×C4×C73—42no—
216C2×C4×C73—42no—
217C2×C4×C73—42no—
218C2×C4×C73—44no—
219C2×C4×C73—44no—
220C2×C4×C73C2×C4⋊C414yes—
221C2×C4×C73C4×D414yes—
222C2×C4×C73C4×D414yes—
223C2×C4×C73#12014yes—
224C2×C4×C73C22⋊Q814yes—
225C2×C4×C73C22⋊Q814yes—
226C2×C4×C73#12314yes—
227C2×Dic73—22no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—22no—
232C2×Dic73—22no—
233C2×Dic73—24no—
234C2×Dic73—24no—
235C2×Dic73—24no—
236C2×Dic73—24no—
237C2×Dic73—28no—
238C2×Dic73—42no—
239C2×Dic73—42no—
240C2×Dic73—44no—
241C2×Dic73—42no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—42no—
246C2×Dic73—42no—
247C2×Dic73—42no—
248C2×Dic73—44no—
249C2×Dic73—44no—
250C2×Dic73C2×C4⋊C414yes—
251C4×D73—232no—
252C4×D73—432no—
253C22×C73C22×C4⋊C411yescharacteristic
254C22×C73C2×#11712yes—
255C22×C73C2×C4×D411yescharacteristic
256C22×C73C2×C4×D411yescharacteristic
257C22×C73#52012yes—
258C22×C73C2×#12011yescharacteristic
259C22×C73C2×C22⋊Q811yescharacteristic
260C22×C73C2×C22⋊Q811yescharacteristic
261C22×C73C2×#12311yescharacteristic
262C22×C73C2×#12411yescharacteristic
263C22×C73C2×#12411yescharacteristic
264C22×C73C2×#12512yes—
265C22×C73#53511yescharacteristic
266C22×C73#53611yescharacteristic
267C22×C73#54412yes—
268C22×C73#54612yes—
269C22×C73#54612yes—
270C22×C73#54812yes—
271C22×C73#54812yes—
272C22×C73#55212yes—
273C22×C73#55212yes—
274C22×C73#55312yes—
275C22×C73#55312yes—
276C2×D73—24no—
277C2×D73—28no—
278C2×D73—28no—
279C2×D73—24no—
280C2×D73—24no—
281C2×D73—28no—
282C2×D73—28no—
283C2×D73—24no—
284C2×D73—24no—
285C2×D73—24no—
286C2×D73#38218yescomplemented
287C4×C73—44no—
288C4×C73—44no—
289C4×C73—82no—
290C4×C73#38518yes—
291Dic73—24no—
292Dic73—44no—
293Dic73—44no—
294Dic73—82no—
295#13881—731nomaximal, Sylow
296C2×C73C2×#38211yescharacteristic
297C2×C73C2×#38511yescharacteristic
298C2×C73#231711yescharacteristic
299C2×C73#235212yes—
300C2×C73#235212yes—
301C2×C73#243511yescharacteristic
302C2×C73#243611yescharacteristic
303C2×C73#244212yes—
304C2×C73#244212yes—
305C2×C73#250311yescharacteristic
306C2×C73#250411yescharacteristic
307D73—28no—
308C23×C4⋊C4—731no—
309C2×#382—731no—
310C2×#382—731no—
311C2×#380—731no—
312C2×C4×C22⋊C4—731no—
313#1442—734no—
314C4⋊#95—731no—
315C4⋊#95D7311yescharacteristic, complemented
316C22⋊#95—734no—
317C73#1388111yescharacteristic, complemented, Sylow
318C24×C4—732no—
319C24×C4—731no—
320C22×C4⋊C4—731no—
321C22×C4⋊C4—732no—
322C22×C4⋊C4—731no—
323C22×C4⋊C4—1464no—
324C22×C4⋊C4—1464no—
325C22×C4⋊C4—1464no—
326C22×C4⋊C4C2×D7311yescharacteristic
327C22×C4⋊C4C2×D7311yescharacteristic
328C22×C22⋊C4—732no—
329C22×C22⋊C4—732no—
330C22×C42—731no—
331C22×C42C2×D7311yescharacteristic
332#382—1464no—
333#382—1464no—
334#380—1464no—
335C4×C22⋊C4—738no—
336C2×#95—734no—
337C2×#95—732no—
338C2×#95—734no—
339C2×#95—734no—
340C2×#95—732no—
341C2×#95C2×D7314yes—
342C25—731no—
343C23×C4—2924no—
344C23×C4—731no—
345C23×C4—734no—
346C23×C4—734no—
347C23×C4—734no—
348C23×C4—732no—
349C23×C4—732no—
350C23×C4—732no—
351C23×C4—1468no—
352C23×C4—1464no—
353C23×C4—1464no—
354C23×C4—1464no—
355C23×C4—1462no—
356C23×C4—1462no—
357C23×C4—1462no—
358C23×C4—1462no—
359C23×C4C22×D7312yes—
360C23×C4C22×D7312yes—
361C23×C4C22×D7312yes—
362C23×C4C22×D7311yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—734no—
369C2×C4⋊C4—14632no—
370C2×C4⋊C4—1462no—
371C2×C4⋊C4—1462no—
372C2×C4⋊C4—1462no—
373C2×C4⋊C4—1464no—
374C2×C4⋊C4—1464no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4—1464no—
377C2×C4⋊C4—1462no—
378C2×C4⋊C4D29214yes—
379C2×C22⋊C4—738no—
380C2×C22⋊C4—1464no—
381C2×C22⋊C4—1468no—
382C2×C22⋊C4—1464no—
383C2×C22⋊C4—1464no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—734no—
387C2×C42—1462no—
388C2×C42—1462no—
389C2×C42C4×D7314yescomplemented
390#95—44no—
391#95—2924no—
392#95—738no—
393#95—1464no—
394#95—1468no—
395#95—1468no—
396C24—732no—
397C24—732no—
398C24—732no—
399C24—734no—
400C24—734no—
401C24—1461no—
402C24—1461no—
403C24—1461no—
404C24—1461no—
405C24—1462no—
406C24—1462no—
407C24C23×D7311yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—29216no—
424C22×C4—2928no—
425C22×C4—2928no—
426C22×C4—29216no—
427C22×C4—732no—
428C22×C4—732no—
429C22×C4—732no—
430C22×C4—7316no—
431C22×C4—734no—
432C22×C4—734no—
433C22×C4—14616no—
434C22×C4—1462no—
435C22×C4—1462no—
436C22×C4—1462no—
437C22×C4—1464no—
438C22×C4—1464no—
439C22×C4—1462no—
440C22×C4—1462no—
441C22×C4—1462no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1464no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1464no—
448C22×C4—1462no—
449C22×C4—1461no—
450C22×C4—1461no—
451C22×C4—1461no—
452C22×C4—1461no—
453C22×C4—1468no—
454C22×C4—1468no—
455C22×C4—1468no—
456C22×C4—1468no—
457C22×C4—1468no—
458C22×C4—1468no—
459C22×C4C2×C4×D7314yes—
460C22×C4C2×C4×D7312yes—
461C22×C4D4×D7312yes—
462C22×C4D4×D7312yes—
463C22×C4C2×D29214yes—
464C22×C4C2×D29212yes—
465C22×C4Q8×D7312yes—
466C22×C4#1192402712yes—
467C22×C4#1192402812yes—
468C4⋊C4—28no—
469C4⋊C4—44no—
470C4⋊C4—2924no—
471C4⋊C4—14616no—
472C4⋊C4—1468no—
473C22⋊C4—2928no—
474C22⋊C4—14616no—
475C42—28no—
476C42—1468no—
477C23—734no—
478C23—738no—
479C23—734no—
480C23—734no—
481C23—738no—
482C23—1462no—
483C23—1462no—
484C23—1464no—
485C23—1464no—
486C23—1464no—
487C23—1464no—
488C23—1464no—
489C23—1464no—
490C23—1464no—
491C23—1464no—
492C23—1464no—
493C23—1462no—
494C23—1462no—
495C23—1464no—
496C23—1462no—
497C23—1462no—
498C23—1462no—
499C23—1462no—
500C23C22×C4×D7311yescharacteristic
501C23C2×D4×D7311yescharacteristic
502C23C2×D4×D7311yescharacteristic
503C23C22×D29211yescharacteristic
504C23C2×Q8×D7311yescharacteristic
505C23#41×D7312yes—
506C23#41×D7312yes—
507C23C2×#1192402711yescharacteristic
508C23C2×#1192402811yescharacteristic
509C23#1455663612yes—
510C23#1455663912yes—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—29232no—
529C2×C4—2922no—
530C2×C4—2924no—
531C2×C4—2922no—
532C2×C4—2922no—
533C2×C4—2922no—
534C2×C4—2922no—
535C2×C4—2922no—
536C2×C4—2922no—
537C2×C4—2924no—
538C2×C4—2924no—
539C2×C4—2922no—
540C2×C4—2922no—
541C2×C4—734no—
542C2×C4—1468no—
543C2×C4—1464no—
544C2×C4—1464no—
545C2×C4—1464no—
546C2×C4—1464no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1462no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4—1462no—
553C2×C4—14632no—
554C2×C4C4⋊C4×D7314yes—
555C2×C4#1455655114yes—
556C2×C4C4×D29214yes—
557C2×C4#1455656114yes—
558C2×C4#1455656914yes—
559C2×C4#1455657114yes—
560C2×C4#1455657314yes—
561C22—738no—
562C22—1464no—
563C22—1464no—
564C22—1464no—
565C22—1468no—
566C22—1464no—
567C22—1468no—
568C22—1464no—
569C22—1468no—
570C22—1464no—
571C22—1468no—
572C22C2×C4⋊C4×D7311yescharacteristic
573C22#117×D7312yes—
574C22C2×#1455655111yescharacteristic
575C22C2×C4×D29211yescharacteristic
576C22#1798906812yes—
577C22C2×#1455656111yescharacteristic
578C22C2×#1455656911yescharacteristic
579C22C2×#1455657111yescharacteristic
580C22C2×#1455657311yescharacteristic
581C22#124×D7311yescharacteristic
582C22#124×D7311yescharacteristic
583C22#125×D7312yes—
584C22#1798915011yescharacteristic
585C22#1798916211yescharacteristic
586C22#1798921412yes—
587C22#1798923212yes—
588C22#1798923412yes—
589C22#1798924712yes—
590C22#1798924912yes—
591C22#1798927112yes—
592C22#1798927812yes—
593C22#1798928412yes—
594C22#1798928712yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—2924no—
599C4—2924no—
600C4—5842no—
601C4—1464no—
602C4#1798846118yes—
603C2—1468no—
604C2#382×D7311yescharacteristic
605C2C2×#1798846111yescharacteristic
606C2#2261255211yescharacteristic
607C2#2261285312yes—
608C2#2261286312yes—
609C2#2261358111yescharacteristic
610C2#2261358811yescharacteristic
611C2#2261365312yes—
612C2#2261365712yes—
613C2#2261441511yescharacteristic
614C2#2261442211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×C4⋊#95C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenterC4⋊#95D73
hypocenterC73#13881
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#17988453C2semidirect
3. C2×#17988452C2semidirect
6. #22608186C2semidirect
7. #22608160C2semidirect
9. C73×C4⋊#95C2semidirect
24. C4×#11924005C4semidirect
68. C2×C4×Dic73C2×C4semidirect
73. #14556498C2×C4semidirect
86. C2×C42×C73C2×C4semidirect
286. C2×D73#382semidirect
315. C4⋊#95D73semidirect
317. C73#13881semidirect
389. C2×C42C4×D73semidirect

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×#179884532
C2×#179884522
C2×#179884442
C2×C4×#119240052
#226081862
#226081602
#226081522
C73×C4⋊#952
#1388173