Finite Group Databasebeta

← #33022787

Subgroups — #33022787

Order 18688.

1266 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988525C211yesmaximal, characteristic
2C2×#17988482C211yesmaximal, characteristic
3C2×C4⋊C4×Dic73C211yesmaximal, characteristic
4#22608192C212yesmaximal
5#22608191C212yesmaximal
6#22608185C211yesmaximal, characteristic
7#22608184C211yesmaximal, characteristic
8#22608159C212yesmaximal
9#22608159C211yesmaximal, characteristic
10#22608159C212yesmaximal
11C2×#400×C73C211yesmaximal, characteristic
12C22×C4⋊Dic73C2211yescharacteristic
13C22×#11924007C2212yes—
14C22×#11924007C2211yescharacteristic
15C22×#11924007C2211yescharacteristic
16C22×#11924007C2211yescharacteristic
17C22×C4×Dic73C2212yes—
18C22×C4×Dic73C2211yescharacteristic
19#17988525—24no—
20#17988482—24no—
21C4⋊C4×Dic73—24no—
22C2×#14556498C2212yes—
23C2×#14556498C2211yescharacteristic
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C22×C4⋊C4×C73C2211yescharacteristic
34C22×C4⋊C4×C73C2211yescharacteristic
35#400×C73—24no—
36C2×#95×C73C2211yescharacteristic
37C2×#95×C73C2212yes—
38C2×#95×C73C2212yes—
39C23×Dic73C2312yes—
40C23×Dic73C2311yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2311yescharacteristic
43C23×Dic73C2312yes—
44C23×Dic73C2311yescharacteristic
45C2×C4⋊Dic73—24no—
46C2×C4⋊Dic73—22no—
47C2×#11924007—22no—
48C2×#11924007—22no—
49C2×#11924007—22no—
50C2×#11924007—24no—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007—22no—
55C2×#11924007—22no—
56C2×#11924007—22no—
57C2×#11924007Q814yes—
58C2×C4×Dic73—24no—
59C2×C4×Dic73—24no—
60C2×C4×Dic73—24no—
61C2×C4×Dic73—24no—
62C2×C4×Dic73D414yes—
63#14556498—24no—
64#14556498—22no—
65#14556498—22no—
66#14556498—24no—
67#14556498—24no—
68#14556498—24no—
69#14556498—24no—
70#14556498—22no—
71#14556498—42no—
72#14556498—42no—
73#14556498—42no—
74C23×C4×C73C2311yescharacteristic
75C23×C4×C73C2312yes—
76C23×C4×C73C2312yes—
77C23×C4×C73C2311yescharacteristic
78C23×C4×C73C2311yescharacteristic
79C2×C4⋊C4×C73—22no—
80C2×C4⋊C4×C73—22no—
81C2×C4⋊C4×C73—22no—
82C2×C4⋊C4×C73—22no—
83C2×C4⋊C4×C73—22no—
84C2×C4⋊C4×C73D414yes—
85#95×C73—24no—
86#95×C73—22no—
87#95×C73—24no—
88C22×Dic73—21no—
89C22×Dic73—22no—
90C22×Dic73—21no—
91C22×Dic73—22no—
92C22×Dic73—22no—
93C22×Dic73—22no—
94C22×Dic73—21no—
95C22×Dic73—21no—
96C22×Dic73—22no—
97C22×Dic73—21no—
98C22×Dic73—22no—
99C22×Dic73—21no—
100C22×Dic73—22no—
101C22×Dic73—21no—
102C22×Dic73—21no—
103C22×Dic73—22no—
104C22×Dic73—22no—
105C22×Dic73—21no—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—22no—
116C22×Dic73C2×D412yes—
117C22×Dic73C2×D412yes—
118C22×Dic73C2×Q812yes—
119C22×Dic73C2×Q812yes—
120C22×Dic73#4114yes—
121C22×Dic73#4112yes—
122C22×Dic73#4114yes—
123C22×Dic73#4114yes—
124C4⋊Dic73—28no—
125#11924007—28no—
126#11924007—44no—
127#11924007—44no—
128#11924007—48no—
129C4×Dic73—28no—
130C4×Dic73—48no—
131C24×C73C2411yescharacteristic
132C22×C4×C73—22no—
133C22×C4×C73—22no—
134C22×C4×C73—21no—
135C22×C4×C73—22no—
136C22×C4×C73—21no—
137C22×C4×C73—22no—
138C22×C4×C73—21no—
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73—21no—
142C22×C4×C73—21no—
143C22×C4×C73—22no—
144C22×C4×C73—22no—
145C22×C4×C73—21no—
146C22×C4×C73—22no—
147C22×C4×C73—21no—
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—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73C2×D412yes—
159C22×C4×C73C2×D414yes—
160C22×C4×C73C2×D412yes—
161C22×C4×C73C2×Q812yes—
162C22×C4×C73#4112yes—
163C4⋊C4×C73—28no—
164C4⋊C4×C73—44no—
165C23×C73C22×D411yescharacteristic
166C23×C73C22×D411yescharacteristic
167C23×C73C22×Q811yescharacteristic
168C23×C73C2×#4111yescharacteristic
169C23×C73C2×#4112yes—
170C23×C73C2×#4112yes—
171C23×C73C2×#4112yes—
172C23×C73C2×#4111yescharacteristic
173C23×C73H(5,𝔽2)11yescharacteristic
174C23×C732−1+412yes—
175C23×C732−1+411yescharacteristic
176C2×C4×C73—22no—
177C2×C4×C73—22no—
178C2×C4×C73—24no—
179C2×C4×C73—24no—
180C2×C4×C73—24no—
181C2×C4×C73—22no—
182C2×C4×C73—24no—
183C2×C4×C73—24no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—41no—
187C2×C4×C73—42no—
188C2×C4×C73—42no—
189C2×C4×C73—41no—
190C2×C4×C73—41no—
191C2×C4×C73—42no—
192C2×C4×C73—42no—
193C2×C4×C73—41no—
194C2×C4×C73—41no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73—42no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—42no—
201C2×C4×C73—42no—
202C2×C4×C73—42no—
203C2×C4×C73—42no—
204C2×C4×C73—42no—
205C2×C4×C73—42no—
206C2×C4×C73#12114yes—
207C2×Dic73—22no—
208C2×Dic73—22no—
209C2×Dic73—24no—
210C2×Dic73—22no—
211C2×Dic73—22no—
212C2×Dic73—22no—
213C2×Dic73—24no—
214C2×Dic73—24no—
215C2×Dic73—22no—
216C2×Dic73—41no—
217C2×Dic73—42no—
218C2×Dic73—42no—
219C2×Dic73—41no—
220C2×Dic73—42no—
221C2×Dic73—42no—
222C2×Dic73—41no—
223C2×Dic73—42no—
224C2×Dic73—42no—
225C2×Dic73—41no—
226C2×Dic73—41no—
227C2×Dic73—41no—
228C2×Dic73—41no—
229C2×Dic73—42no—
230C2×Dic73—42no—
231C2×Dic73—42no—
232C2×Dic73—42no—
233C2×Dic73—42no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—42no—
237C2×Dic73—42no—
238C2×Dic73C22⋊Q814yes—
239C2×Dic73#12314yes—
240C2×Dic73#12414yes—
241C2×Dic73#12518yes—
242C2×Dic73#12618yes—
243C22×C73C2×#12111yescharacteristic
244C22×C73C2×C22⋊Q811yescharacteristic
245C22×C73C2×#12311yescharacteristic
246C22×C73#52511yescharacteristic
247C22×C73C2×#12411yescharacteristic
248C22×C73C2×#12512yes—
249C22×C73C2×#12612yes—
250C22×C73#52912yes—
251C22×C73#53212yes—
252C22×C73#53611yescharacteristic
253C22×C73#53711yescharacteristic
254C22×C73#54112yes—
255C22×C73#54212yes—
256C22×C73#54712yes—
257C22×C73D4×Q811yescharacteristic
258C22×C73#55011yescharacteristic
259C22×C73#55212yes—
260C22×C73#55312yes—
261C22×C73#55411yescharacteristic
262C22×C73#55612yes—
263C22×C73#55612yes—
264C22×C73#55611yescharacteristic
265C22×C73#55712yes—
266C4×C73—24no—
267C4×C73—42no—
268C4×C73—44no—
269C4×C73—81no—
270C4×C73—82no—
271Dic73—44no—
272Dic73—42no—
273Dic73—42no—
274Dic73—81no—
275Dic73—82no—
276Dic73#40018yescomplemented
277#15156—731nomaximal, Sylow
278C2×C73C2×#40011yescharacteristic
279C2×C73#250612yes—
280C2×C73#250912yes—
281C2×C73#251212yes—
282C2×C73#251412yes—
283C2×C73#253811yescharacteristic
284C2×C73#254711yescharacteristic
285C2×C73#256211yescharacteristic
286C2×C73#258111yescharacteristic
287C2×C73#262511yescharacteristic
288C2×C73#285411yescharacteristic
289C2×#400—731no—
290C2×#400D7311yescharacteristic
291C2×#389—731no—
292C2×C4×C4⋊C4—731no—
293#1443—732no—
294#1443—732no—
295#1441—731no—
296#1441—731no—
297C4⋊#95—732no—
298C4⋊#95—731no—
299C4⋊#95—732no—
300C73#1515611yescharacteristic, complemented, Sylow
301C22×C4⋊C4—731no—
302C22×C4⋊C4—731no—
303C22×C4⋊C4—732no—
304C22×C4⋊C4—731no—
305C22×C4⋊C4—731no—
306C22×C4⋊C4C2×D7311yescharacteristic
307C22×C4⋊C4C2×D7311yescharacteristic
308C22×C42—731no—
309C22×C42—732no—
310#400—24no—
311#400—1464no—
312#389—1464no—
313C4×C4⋊C4—1464no—
314C2×#95—732no—
315C2×#95—732no—
316C2×#95—732no—
317C2×#95—731no—
318C2×#95—732no—
319C2×#95—732no—
320C2×#95—732no—
321C2×#95—732no—
322C2×#95—732no—
323C2×#95—731no—
324C2×#95—731no—
325C2×#95C2×D7311yescharacteristic
326C2×#95C2×D7312yes—
327C2×#95C2×D7312yes—
328C23×C4—731no—
329C23×C4—731no—
330C23×C4—732no—
331C23×C4—731no—
332C23×C4—731no—
333C23×C4—732no—
334C23×C4C22×D7311yescharacteristic
335C23×C4C22×D7312yes—
336C23×C4C22×D7312yes—
337C23×C4C22×D7311yescharacteristic
338C23×C4C22×D7311yescharacteristic
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—734no—
345C2×C4⋊C4—1464no—
346C2×C4⋊C4—1464no—
347C2×C4⋊C4—1464no—
348C2×C4⋊C4—1464no—
349C2×C4⋊C4—1462no—
350C2×C4⋊C4—1462no—
351C2×C4⋊C4—1462no—
352C2×C4⋊C4—1462no—
353C2×C4⋊C4—1462no—
354C2×C4⋊C4—1462no—
355C2×C4⋊C4—1464no—
356C2×C4⋊C4—1462no—
357C2×C4⋊C4C22⋊D7314yes—
358C2×C42—734no—
359C2×C42—1464no—
360C2×C42—1464no—
361C2×C42—1464no—
362C2×C42—1464no—
363#95—24no—
364#95—22no—
365#95—24no—
366#95—2922no—
367#95—2922no—
368#95—2922no—
369#95—1464no—
370#95—1464no—
371#95—1464no—
372#95—1462no—
373#95—1462no—
374#95—1462no—
375#95—1464no—
376#95—1464no—
377C24C23×D7311yescharacteristic
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—732no—
405C22×C4—732no—
406C22×C4—734no—
407C22×C4—734no—
408C22×C4—732no—
409C22×C4—732no—
410C22×C4—732no—
411C22×C4—734no—
412C22×C4—1461no—
413C22×C4—1462no—
414C22×C4—1461no—
415C22×C4—1462no—
416C22×C4—1462no—
417C22×C4—1462no—
418C22×C4—1462no—
419C22×C4—1462no—
420C22×C4—1462no—
421C22×C4—1461no—
422C22×C4—1462no—
423C22×C4—1462no—
424C22×C4—1462no—
425C22×C4—1461no—
426C22×C4—1461no—
427C22×C4—1462no—
428C22×C4—1462no—
429C22×C4—1462no—
430C22×C4—1461no—
431C22×C4—1462no—
432C22×C4—1461no—
433C22×C4—1461no—
434C22×C4—1462no—
435C22×C4—1461no—
436C22×C4—1462no—
437C22×C4—1462no—
438C22×C4—1461no—
439C22×C4—1461no—
440C22×C4D4×D7312yes—
441C22×C4C2×C22⋊D7314yes—
442C22×C4C2×C22⋊D7312yes—
443C22×C4Q8×D7312yes—
444C22×C4#1192402712yes—
445C4⋊C4—28no—
446C4⋊C4—44no—
447C4⋊C4—2924no—
448C4⋊C4—2928no—
449C4⋊C4—2924no—
450C4⋊C4—1468no—
451C4⋊C4—1468no—
452C42—2928no—
453C42—1468no—
454C23C2×D4×D7311yescharacteristic
455C23C22×C22⋊D7311yescharacteristic
456C23C2×Q8×D7311yescharacteristic
457C23#41×D7312yes—
458C23#41×D7312yes—
459C23#41×D7312yes—
460C23#41×D7311yescharacteristic
461C23C2×#1192402711yescharacteristic
462C23#1455663711yescharacteristic
463C23#1455663812yes—
464C23#1455663911yescharacteristic
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—2922no—
496C2×C4—2921no—
497C2×C4—2922no—
498C2×C4—2922no—
499C2×C4—2921no—
500C2×C4—2922no—
501C2×C4—2922no—
502C2×C4—2921no—
503C2×C4—2921no—
504C2×C4—2922no—
505C2×C4—2922no—
506C2×C4—2922no—
507C2×C4—2922no—
508C2×C4—2922no—
509C2×C4—2922no—
510C2×C4—2922no—
511C2×C4—2922no—
512C2×C4—2921no—
513C2×C4—2921no—
514C2×C4—2921no—
515C2×C4—2922no—
516C2×C4—2922no—
517C2×C4—734no—
518C2×C4—734no—
519C2×C4—734no—
520C2×C4—738no—
521C2×C4—738no—
522C2×C4—1462no—
523C2×C4—1462no—
524C2×C4—1462no—
525C2×C4—1464no—
526C2×C4—1464no—
527C2×C4—1462no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1464no—
531C2×C4#1455656314yes—
532C22C2×#1455656311yescharacteristic
533C22C22⋊Q8×D7311yescharacteristic
534C22#123×D7311yescharacteristic
535C22#1798909511yescharacteristic
536C22#124×D7311yescharacteristic
537C22#125×D7312yes—
538C22#126×D7312yes—
539C22#1798911412yes—
540C22#1798913312yes—
541C22#1798915711yescharacteristic
542C22#1798916711yescharacteristic
543C22#1798919612yes—
544C22#1798920312yes—
545C22#1798923612yes—
546C22Q8×C22⋊D7311yescharacteristic
547C22#1798925611yescharacteristic
548C22#1798926912yes—
549C22#1798928812yes—
550C22#1798929511yescharacteristic
551C22#1798930412yes—
552C22#1798930612yes—
553C22#1798930811yescharacteristic
554C22#1798931412yes—
555C4—24no—
556C4—42no—
557C4—44no—
558C4—81no—
559C4—82no—
560C4—2924no—
561C4—2922no—
562C4—2922no—
563C4—5841no—
564C4—5842no—
565C4—738no—
566C2#400×D7311yescharacteristic
567C2#2261444612yes—
568C2#2261447512yes—
569C2#2261451612yes—
570C2#2261453412yes—
571C2#2261485011yescharacteristic
572C2#2261492411yescharacteristic
573C2#2261509311yescharacteristic
574C2#2261529411yescharacteristic
575C2#2261573211yescharacteristic
576C2#2261826011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutator, socle, prime radical, cosocleC24×C73C24
FittingC2×#400×C73C2
hypercenterC2×#400D73
hypocenterC73#15156
LCSG ▷ C24×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
276. Dic73#400semidirect
300. C73#15156semidirect

Maximal subgroups

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

subgroupindex
C2×#179885252
C2×#179884822
C2×C4⋊C4×Dic732
#226081922
#226081912
#226081852
#226081842
#226081592
C2×#400×C732
#1515673