Finite Group Databasebeta

← #34015983

Subgroups — #34015983

Order 18688.

1210 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626786C212yesmaximal, complemented
2#22625195C212yesmaximal, complemented
3#22625186C212yesmaximal, complemented
4#22625182C212yesmaximal, complemented
5#22625180C212yesmaximal, complemented
6#22625166C212yesmaximal, complemented
7#22625160C212yesmaximal, complemented
8C2×#17989068C211yesmaximal, characteristic
9#17989617—28no—
10#17989607C2214yes—
11#17989558C2214yes—
12#17989555C2214yes—
13#17989433C2214yes—
14#17989433C2214yes—
15#17989432C2214yes—
16#17989430C2214yes—
17C22×#11924028C411yescharacteristic
18#17989068—28no—
19C2×#14556556C411yescharacteristic
20C2×#14556556C2211yescharacteristic
21C2×#14556553C411yescharacteristic
22C2×#14556553C2211yescharacteristic
23C2×C4×C22⋊D73C411yescharacteristic
24C2×C4×C22⋊D73C411yescharacteristic
25C2×C4×C22⋊D73C2211yescharacteristic
26C2×C4×C22⋊D73C2211yescharacteristic
27C2×#14556550C2211yescharacteristic
28C2×#14556549C411yescharacteristic
29C2×#14556549C411yescharacteristic
30C2×C4⋊C4×D73C2211yescharacteristic
31C2×C4⋊C4×D73C2211yescharacteristic
32C22×C4⋊C4×C73C411yescharacteristic
33C2×#11924038—24no—
34C2×#11924036—24no—
35#14556658—24no—
36#14556648—24no—
37#14556648—24no—
38C22⋊#10596833C2318yes—
39#14556646—24no—
40C2×#11924028—24no—
41C2×#11924028—24no—
42C2×#11924028—24no—
43C22×Dic146C2×C411yescharacteristic
44C22×D292C2×C411yescharacteristic
45C22×C22⋊D73C2×C411yescharacteristic
46C22×C22⋊D73C2×C411yescharacteristic
47C22×C4×D73C2×C411yescharacteristic
48C22×C4×D73C2×C411yescharacteristic
49C22×C4×D73C2×C411yescharacteristic
50C22×C4×D73C2×C411yescharacteristic
51C22×C4×D73C2311yescharacteristic
52C22×C4×D73C2311yescharacteristic
53#14556556—24no—
54#14556556—24no—
55#14556553—24no—
56#14556553D418yes—
57C4×C22⋊D73—24no—
58C4×C22⋊D73—24no—
59C4×C22⋊D73—24no—
60C4×C22⋊D73—24no—
61#14556550—24no—
62#14556549—24no—
63#14556549—24no—
64C2×C4⋊Dic73C2×C411yescharacteristic
65C2×C4⋊Dic73C2311yescharacteristic
66C2×#11924007C2×C411yescharacteristic
67C2×#11924007C2×C411yescharacteristic
68C2×#11924007C2311yescharacteristic
69C2×#11924007C2311yescharacteristic
70C4⋊C4×D73—24no—
71C4⋊C4×D73—24no—
72C2×C22⋊Dic73C2×C411yescharacteristic
73C2×C22⋊Dic73C2311yescharacteristic
74C2×#11924005C2×C411yescharacteristic
75C2×#11924005C2×C411yescharacteristic
76C2×#11924005C2×C411yescharacteristic
77C2×#11924005C2×C411yescharacteristic
78C2×C4×Dic73C2×C411yescharacteristic
79C2×C4×Dic73C2×C411yescharacteristic
80C2×C4×Dic73C2×C411yescharacteristic
81C2×C4×Dic73C2×C411yescharacteristic
82C2×C4×Dic73C2×C411yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84C23×C4×C73C2×C411yescharacteristic
85C23×C4×C73C2×C411yescharacteristic
86C23×C4×C73C2×C411yescharacteristic
87C2×C4⋊C4×C73—24no—
88C2×C4⋊C4×C73C2×C411yescharacteristic
89C2×C4⋊C4×C73C2×C411yescharacteristic
90C2×C4⋊C4×C73C2×C411yescharacteristic
91C2×C4⋊C4×C73C2×C411yescharacteristic
92#11924038—84no—
93#11924036—84no—
94C2×#10596833—28no—
95C23×D73C22×C411yescharacteristic
96C23×D73C22×C411yescharacteristic
97#11924028—232no—
98C2×Dic146—22no—
99C2×Dic146—22no—
100C2×Dic146—22no—
101C2×D292—22no—
102C2×D292—22no—
103C2×D292C22⋊C414yes—
104C2×C22⋊D73—22no—
105C2×C22⋊D73—22no—
106C2×C22⋊D73—22no—
107C2×C22⋊D73—22no—
108C2×C22⋊D73—22no—
109C2×C22⋊D73M1614yescomplemented
110C22×Dic73C22×C411yescharacteristic
111C22×Dic73C22×C411yescharacteristic
112C22×Dic73C22×C411yescharacteristic
113C22×Dic73C22×C411yescharacteristic
114C22×Dic73C22×C411yescharacteristic
115C22×Dic73C2411yescharacteristic
116C2×C4×D73—22no—
117C2×C4×D73—22no—
118C2×C4×D73—22no—
119C2×C4×D73—22no—
120C2×C4×D73—22no—
121C2×C4×D73—22no—
122C2×C4×D73—22no—
123C2×C4×D73—81no—
124C2×C4×D73—81no—
125C2×C4×D73—81no—
126C2×C4×D73—81no—
127C2×C4×D73M1614yescomplemented
128C2×C4×D73C2×D414yes—
129C2×C4×D73C2×D414yes—
130C4⋊Dic73—22no—
131C4⋊Dic73—22no—
132#11924007—22no—
133#11924007—22no—
134#11924007—22no—
135#11924007—22no—
136C22⋊Dic73—22no—
137C22⋊Dic73—22no—
138#11924005—22no—
139#11924005—22no—
140#11924005C22⋊C414yes—
141#11924005C22⋊C414yes—
142C4×Dic73—22no—
143C4×Dic73—22no—
144C4×Dic73—22no—
145C4×Dic73—22no—
146C4×Dic73—22no—
147C4×Dic73C2×D414yes—
148C24×C73C22×C411yescharacteristic
149C22×C4×C73—22no—
150C22×C4×C73—22no—
151C22×C4×C73—22no—
152C22×C4×C73—22no—
153C22×C4×C73—22no—
154C22×C4×C73—81no—
155C22×C4×C73—81no—
156C22×C4×C73C22×C411yescharacteristic
157C22×C4×C73C22×C411yescharacteristic
158C22×C4×C73C22×C411yescharacteristic
159C22×C4×C73C22×C411yescharacteristic
160C22×C4×C73C22×C411yescharacteristic
161C22×C4×C73C22×C411yescharacteristic
162C4⋊C4×C73—22no—
163C4⋊C4×C73—22no—
164C4⋊C4×C73—22no—
165C4⋊C4×C73C22⋊C414yes—
166C23×C73—21no—
167C23×C73—21no—
168C23×C73—41no—
169C23×C73—41no—
170C23×C73C2×M1612yes—
171C23×C73C23×C411yescharacteristic
172C22×D73—21no—
173C22×D73—21no—
174C22×D73—41no—
175C22×D73—41no—
176C22×D73—41no—
177C22×D73—41no—
178C22×D73C2×C22⋊C412yes—
179C22×D73C2×C22⋊C412yes—
180C22×D73C2×M1612yes—
181C22×D73C2×M1612yes—
182C2×C4×C73—28no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—21no—
186C2×C4×C73—21no—
187C2×C4×C73—21no—
188C2×C4×C73—21no—
189C2×C4×C73—41no—
190C2×C4×C73—41no—
191C2×C4×C73—41no—
192C2×C4×C73—41no—
193C2×C4×C73—82no—
194C2×C4×C73—82no—
195C2×C4×C73C2×C22⋊C412yes—
196C2×C4×C73C2×C22⋊C412yes—
197C2×C4×C73C2×C22⋊C412yes—
198C2×C4×C73C2×M1612yes—
199C22⋊D73—28no—
200C22⋊D73—28no—
201C2×Dic73—21no—
202C2×Dic73—21no—
203C2×Dic73—21no—
204C2×Dic73—21no—
205C2×Dic73—21no—
206C2×Dic73—21no—
207C2×Dic73—41no—
208C2×Dic73—41no—
209C2×Dic73—41no—
210C2×Dic73—41no—
211C2×Dic73C2×C22⋊C412yes—
212C2×Dic73C2×M1612yes—
213C2×Dic73C2×M1612yes—
214C2×Dic73C22×D412yes—
215D292—28no—
216C4×D73—28no—
217C4×D73—28no—
218C4×D73—44no—
219C4×D73—44no—
220C4×D73—82no—
221C4×D73—82no—
222Dic146—28no—
223#10596833—48no—
224C22×C73—21no—
225C22×C73—21no—
226C22×C73—22no—
227C22×C73—42no—
228C22×C73—42no—
229C22×C73—41no—
230C22×C73—41no—
231C22×C73C22×C22⋊C411yescharacteristic
232C22×C73C22×M1611yescharacteristic
233C22×C73C22×M1611yescharacteristic
234C2×D73—22no—
235C2×D73—22no—
236C2×D73—22no—
237C2×D73—22no—
238C2×D73—22no—
239C2×D73—42no—
240C2×D73—42no—
241C2×D73—42no—
242C2×D73—42no—
243C2×D73#40714yes—
244C4×C73—22no—
245C4×C73—22no—
246C4×C73—22no—
247C4×C73—22no—
248C4×C73—41no—
249C4×C73—41no—
250Dic73—22no—
251Dic73—22no—
252Dic73—22no—
253Dic73—22no—
254Dic73—41no—
255Dic73—41no—
256#17368—731nomaximal, Sylow
257C2×C73—21no—
258C2×C73—21no—
259C2×C73—42no—
260C2×C73C2×#40711yescharacteristic
261C2×C73#288311yescharacteristic
262C2×C73#288911yescharacteristic
263D73—24no—
264D73—42no—
265C2×#520—731no—
266#2973—732no—
267#1467—732no—
268#1465—732no—
269#1464—732no—
270#1463—732no—
271#1459—732no—
272C2×#324—732no—
273C73#1736811yescharacteristic, complemented, Sylow
274C22×#41—731no—
275#520—1468no—
276C2×C4×Q8—731no—
277C2×C4×Q8—731no—
278C2×C4×D4—731no—
279C2×C4×D4—731no—
280C2×C4×D4—731no—
281C2×C4×D4—731no—
282C2×C4×D4—731no—
283C2×C4×D4—731no—
284C2×#117—731no—
285C2×#117—731no—
286C2×#117—731no—
287C22×C4⋊C4—731no—
288C22×C4⋊C4—731no—
289C22×C4⋊C4#6440811yescharacteristic
290#435—1468no—
291#432—734no—
292#407—734no—
293C2×C22⋊C8—734no—
294#324—734no—
295#324—734no—
296#324—734no—
297#323—734no—
298C2×#41—1464no—
299C2×#41—1464no—
300C2×#41—1464no—
301C22×Q8—731no—
302C22×D4—731no—
303C22×D4—731no—
304C22×D4—731no—
305C23×C4—731no—
306C23×C4—731no—
307C23×C4—731no—
308C23×C4—731no—
309C23×C4—731no—
310C23×C4—731no—
311C23×C4C2×#6440811yescharacteristic
312C23×C4C2×#6440811yescharacteristic
313C23×C4C2×#6440811yescharacteristic
314C2×M16—1464no—
315C22×C8—1464no—
316C4×Q8—1464no—
317C4×Q8—1464no—
318C4×D4—738no—
319C4×D4—1464no—
320C4×D4—1464no—
321C4×D4—1464no—
322C4×D4—1464no—
323C4×D4—1464no—
324#117—1464no—
325#117—1464no—
326#117—1464no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—731no—
329C2×C4⋊C4—731no—
330C2×C4⋊C4—731no—
331C2×C4⋊C4—731no—
332C2×C4⋊C4—731no—
333C2×C4⋊C4—731no—
334C2×C4⋊C4—1464no—
335C2×C4⋊C4—1464no—
336C2×C4⋊C4C2×#6440811yescharacteristic
337C2×C4⋊C4C2×#6440811yescharacteristic
338C2×C4⋊C4C2×#6440811yescharacteristic
339C2×C4⋊C4C2×#6440811yescharacteristic
340C2×C22⋊C4—731no—
341C2×C22⋊C4—731no—
342C2×C22⋊C4—731no—
343C2×C22⋊C4—731no—
344C2×C22⋊C4—731no—
345C2×C22⋊C4—731no—
346C2×C42—731no—
347C2×C42—731no—
348C2×C42—731no—
349C2×C42—731no—
350C2×C42—731no—
351C2×C42—731no—
352C4⋊C8—1464no—
353C22⋊C8—738no—
354C22⋊C8—1464no—
355C22⋊C8—1464no—
356#97—1464no—
357C24—731no—
358C24—731no—
359C24C22×#6440811yescharacteristic
360#41—14632no—
361C2×Q8—1462no—
362C2×Q8—1462no—
363C2×Q8—1462no—
364C2×D4—734no—
365C2×D4—734no—
366C2×D4—1462no—
367C2×D4—1462no—
368C2×D4—1462no—
369C2×D4—1462no—
370C2×D4—1462no—
371C2×D4—1462no—
372C2×D4—1462no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—5841no—
381C22×C4—5841no—
382C22×C4—5841no—
383C22×C4—5841no—
384C22×C4—731no—
385C22×C4—731no—
386C22×C4—734no—
387C22×C4—734no—
388C22×C4—731no—
389C22×C4—731no—
390C22×C4—731no—
391C22×C4—731no—
392C22×C4—734no—
393C22×C4—1462no—
394C22×C4—1462no—
395C22×C4—1462no—
396C22×C4—1462no—
397C22×C4—1462no—
398C22×C4—1462no—
399C22×C4—1462no—
400C22×C4C22×#6440811yescharacteristic
401C22×C4C22×#6440811yescharacteristic
402C22×C4C22×#6440811yescharacteristic
403C22×C4C22×#6440811yescharacteristic
404C22×C4C22×#6440811yescharacteristic
405C22×C4C22×#6440811yescharacteristic
406M16—5844no—
407C2×C8—5844no—
408C2×C8—1468no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—1462no—
413C4⋊C4—1462no—
414C4⋊C4—1462no—
415C4⋊C4—1462no—
416C4⋊C4—1462no—
417C4⋊C4—1462no—
418C4⋊C4C22⋊#6440814yes—
419C22⋊C4—734no—
420C22⋊C4—734no—
421C22⋊C4—1462no—
422C22⋊C4—1462no—
423C22⋊C4—1462no—
424C22⋊C4—1462no—
425C42—734no—
426C42—1462no—
427C42—1462no—
428C42—1462no—
429C42—1462no—
430C42—1462no—
431C23—21no—
432C23—21no—
433C23—41no—
434C23—41no—
435C23—2921no—
436C23—2921no—
437C23—2921no—
438C23—2921no—
439C23—732no—
440C23—732no—
441C23—732no—
442C23—732no—
443C23—1461no—
444C23—1461no—
445C23D73⋊M1612yes—
446C23C23×#6440811yescharacteristic
447Q8—1468no—
448D4—1468no—
449D4—1468no—
450D4—1468no—
451C2×C4—28no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—2924no—
465C2×C4—2924no—
466C2×C4—2921no—
467C2×C4—2921no—
468C2×C4—2921no—
469C2×C4—2921no—
470C2×C4—5842no—
471C2×C4—5842no—
472C2×C4—732no—
473C2×C4—732no—
474C2×C4—732no—
475C2×C4—732no—
476C2×C4—1468no—
477C2×C4—1468no—
478C2×C4—1461no—
479C2×C4—1461no—
480C2×C4—1461no—
481C2×C4—1461no—
482C2×C4—1461no—
483C2×C4—1461no—
484C2×C4C2×C22⋊#6440812yes—
485C2×C4C2×C22⋊#6440812yes—
486C2×C4C2×C22⋊#6440812yes—
487C2×C4D73⋊M1612yes—
488C8—2928no—
489C22—21no—
490C22—21no—
491C22—22no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—2922no—
497C22—2922no—
498C22—2922no—
499C22—2922no—
500C22—734no—
501C22—1462no—
502C22—1462no—
503C22—1462no—
504C22—1462no—
505C22—1462no—
506C22C22×C22⋊#6440811yescharacteristic
507C22C2×D73⋊M1611yescharacteristic
508C22C2×D73⋊M1611yescharacteristic
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—2921no—
516C4—2921no—
517C4—1462no—
518C4—1462no—
519C4—1462no—
520C4—1462no—
521C2—21no—
522C2—21no—
523C2—42no—
524C2—2922no—
525C2—1464no—
526C2D73⋊#40711yescharacteristic
527C2#2262653011yescharacteristic
528C2#2262654411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC22×#11924028C4
cosocleC22×Dic73C24
hypercenterC22×C4⋊C4#64408
hypocenterC73#17368
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #22626786C2semidirect
2. #22625195C2semidirect
3. #22625186C2semidirect
4. #22625182C2semidirect
5. #22625180C2semidirect
6. #22625166C2semidirect
7. #22625160C2semidirect
109. C2×C22⋊D73M16semidirect
127. C2×C4×D73M16semidirect
273. C73#17368semidirect

Maximal subgroups

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

subgroupindex
#226267862
#226251952
#226251862
#226251822
#226251802
#226251662
#226251602
C2×#179890682
#1736873