Finite Group Databasebeta

← #32973104

Subgroups — #32973104

Order 18688.

632 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988311C211yesmaximal, characteristic
2#22608176C211yesmaximal, characteristic
3C2×#342×C73C211yesmaximal, characteristic
4C22×C4⋊Dic73C411yescharacteristic, complemented
5C22×C22⋊Dic73—21no—
6C2×C22⋊#10596831—21no—
7C2×#14556498—21no—
8C2×#14556498C411yescharacteristic, complemented
9#17988311—21no—
10#17988311C412yescomplemented
11C22×C22⋊C4×C73—21no—
12C22×C22⋊C4×C73C2211yescharacteristic
13#342×C73—21no—
14#342×C73C412yescomplemented
15C23×Dic73—21no—
16C23×Dic73—21no—
17C22×#10596831—41no—
18C2×C4⋊Dic73—41no—
19C2×C4⋊Dic73—42no—
20C2×C22⋊Dic73—42no—
21C2×C22⋊Dic73—42no—
22C2×C22⋊Dic73—41no—
23C2×C22⋊Dic73—41no—
24C22⋊#10596831—22no—
25C22⋊#10596831—41no—
26#14556498—41no—
27#14556498—42no—
28C25×C73D411yescharacteristic
29C23×C4×C73—41no—
30C23×C4×C73C2×C411yescharacteristic
31C23×C4×C73D411yescharacteristic
32C2×C22⋊C4×C73—22no—
33C2×C22⋊C4×C73—22no—
34C2×C22⋊C4×C73—22no—
35C2×C22⋊C4×C73—41no—
36C2×C22⋊C4×C73—41no—
37C2×C22⋊C4×C73—41no—
38C2×C22⋊C4×C73—41no—
39C2×C22⋊C4×C73—41no—
40C2×C22⋊C4×C73C2×C411yescharacteristic
41C2×C22⋊C4×C73C2×C411yescharacteristic
42C2×C22⋊C4×C73D411yescharacteristic
43C2×C22⋊C4×C73Q811yescharacteristic
44C22×Dic73—22no—
45C22×Dic73—22no—
46C22×Dic73—41no—
47C22×Dic73—41no—
48C22×Dic73—41no—
49C22×Dic73—41no—
50C22×Dic73—41no—
51C22×Dic73—41no—
52C22×Dic73—41no—
53C22×Dic73—41no—
54C22×Dic73—41no—
55C22×Dic73—41no—
56C22×Dic73—41no—
57C22×Dic73—41no—
58C2×#10596831—42no—
59C2×#10596831—81no—
60C2×#10596831—81no—
61C4⋊Dic73—42no—
62C4⋊Dic73—42no—
63C22⋊Dic73—44no—
64C22⋊Dic73—44no—
65C24×C73—21no—
66C24×C73—21no—
67C24×C73—21no—
68C24×C73—21no—
69C24×C73—41no—
70C24×C73—41no—
71C24×C73—41no—
72C24×C73—41no—
73C24×C73C22⋊C411yescharacteristic
74C24×C73C22⋊C411yescharacteristic
75C24×C73C4⋊C411yescharacteristic
76C24×C73D811yescharacteristic
77C24×C73SD1611yescharacteristic
78C24×C73SD1611yescharacteristic
79C24×C73Q1611yescharacteristic
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—41no—
85C22×C4×C73—41no—
86C22×C4×C73—41no—
87C22×C4×C73—41no—
88C22×C4×C73—42no—
89C22×C4×C73—42no—
90C22×C4×C73—42no—
91C22×C4×C73—81no—
92C22×C4×C73—81no—
93C22×C4×C73—81no—
94C22×C4×C73—81no—
95C22×C4×C73C4211yescharacteristic
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73C4⋊C411yescharacteristic
98C22×C4×C73C4⋊C411yescharacteristic
99C22⋊C4×C73—24no—
100C22⋊C4×C73—41no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—42no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—42no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73—21no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73—41no—
134C23×C73—41no—
135C23×C73—41no—
136C23×C73—41no—
137C23×C73—41no—
138C23×C73—41no—
139C23×C73—81no—
140C23×C73—81no—
141C23×C73—81no—
142C23×C73—81no—
143C23×C73—81no—
144C23×C73—81no—
145C23×C73#9511yescharacteristic
146C23×C73#9911yescharacteristic
147C23×C73#9911yescharacteristic
148C23×C73#10211yescharacteristic
149C23×C73#10211yescharacteristic
150C23×C73#10311yescharacteristic
151C23×C73#10311yescharacteristic
152C23×C73#10411yescharacteristic
153C23×C73#10411yescharacteristic
154C23×C73#10611yescharacteristic
155C23×C73#10711yescharacteristic
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—42no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—44no—
170C2×C4×C73—81no—
171C2×C4×C73—82no—
172C2×C4×C73—82no—
173C2×C4×C73—82no—
174C2×C4×C73—82no—
175C2×C4×C73—82no—
176C2×C4×C73—82no—
177C2×Dic73—42no—
178C2×Dic73—42no—
179C2×Dic73—42no—
180C2×Dic73—42no—
181C2×Dic73—42no—
182C2×Dic73—42no—
183C2×Dic73—81no—
184C2×Dic73—81no—
185C2×Dic73—81no—
186C2×Dic73—81no—
187C2×Dic73—81no—
188C2×Dic73—81no—
189C2×Dic73—81no—
190C2×Dic73—81no—
191#10596831—82no—
192C22×C73—21no—
193C22×C73—21no—
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—21no—
200C22×C73—21no—
201C22×C73—21no—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—21no—
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—41no—
209C22×C73—41no—
210C22×C73—41no—
211C22×C73—41no—
212C22×C73—41no—
213C22×C73—41no—
214C22×C73—41no—
215C22×C73—41no—
216C22×C73—41no—
217C22×C73—41no—
218C22×C73—41no—
219C22×C73—41no—
220C22×C73—41no—
221C22×C73—41no—
222C22×C73—41no—
223C22×C73—41no—
224C22×C73—81no—
225C22×C73—81no—
226C22×C73—81no—
227C22×C73—81no—
228C22×C73—81no—
229C22×C73—81no—
230C22×C73#32711yescharacteristic
231C22×C73#32711yescharacteristic
232C22×C73#32811yescharacteristic
233C22×C73#32811yescharacteristic
234C22×C73#33911yescharacteristic
235C22×C73#34011yescharacteristic
236C22×C73#34211yescharacteristic
237C22×C73AΣL(1,𝔽16)11yescharacteristic
238C22×C73#35211yescharacteristic
239C22×C73#35311yescharacteristic
240C22×C73#35411yescharacteristic
241C4×C73—41no—
242C4×C73—41no—
243C4×C73—41no—
244C4×C73—41no—
245C4×C73—84no—
246Dic73—82no—
247Dic73—82no—
248#7250—731nomaximal, Sylow
249C2×C73—21no—
250C2×C73—21no—
251C2×C73—21no—
252C2×C73—21no—
253C2×C73—41no—
254C2×C73—41no—
255C2×C73—41no—
256C2×C73—41no—
257C2×C73#129411yescharacteristic
258C2×C73#134311yescharacteristic
259C2×C73#134411yescharacteristic
260C2×C73#135111yescharacteristic
261C2×C73#135211yescharacteristic
262C2×C73#139011yescharacteristic
263C2×C73#139311yescharacteristic
264C2×#342D7311yescharacteristic
265C2×#323—731no—
266#1439—731no—
267C73#725011yescharacteristic, complemented, Sylow
268C22×C4⋊C4—731no—
269C22×C22⋊C4—21no—
270C22×C22⋊C4—1461no—
271C22×C22⋊C4C2×D7311yescharacteristic
272C2×C22⋊C8—1461no—
273C2×#95—731no—
274C2×#95—1461no—
275#342—21no—
276#342Dic7312yescomplemented
277#323—732no—
278#323—1461no—
279C25C22⋊D7311yescharacteristic
280C23×C4—41no—
281C23×C4—1461no—
282C23×C4—1461no—
283C23×C4C4×D7311yescharacteristic
284C23×C4D29211yescharacteristic
285C22×C8—2921no—
286C2×C4⋊C4—2921no—
287C2×C4⋊C4—2922no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—2921no—
297C2×C22⋊C4—2921no—
298C2×C22⋊C4—2922no—
299C2×C22⋊C4—2922no—
300C2×C22⋊C4Dic14611yescharacteristic
301C2×C22⋊C4C4×D7311yescharacteristic
302C2×C22⋊C4C2×Dic7311yescharacteristic
303C2×C22⋊C4C22⋊D7311yescharacteristic
304C22⋊C8—2921no—
305C22⋊C8—1462no—
306#95—2921no—
307#95—2922no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—41no—
313C24—41no—
314C24—41no—
315C24—41no—
316C24#1192400511yescharacteristic
317C24C22⋊Dic7311yescharacteristic
318C24#1192400711yescharacteristic
319C24#1192401311yescharacteristic
320C24#1192401511yescharacteristic
321C24#1192401611yescharacteristic
322C24#1192401811yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—81no—
335C22×C4—81no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—2921no—
339C22×C4—2921no—
340C22×C4—2921no—
341C22×C4—2921no—
342C22×C4—2921no—
343C22×C4—2921no—
344C22×C4—2921no—
345C22×C4—2921no—
346C22×C4—2921no—
347C22×C4—2921no—
348C22×C4—2921no—
349C22×C4—2921no—
350C22×C4—1462no—
351C22×C4—1462no—
352C22×C4C4×Dic7311yescharacteristic
353C22×C4#1192400511yescharacteristic
354C22×C4#1192400711yescharacteristic
355C22×C4C4⋊Dic7311yescharacteristic
356C2×C8—2922no—
357C2×C8—5841no—
358C2×C8—5841no—
359C4⋊C4—2922no—
360C4⋊C4—2922no—
361C22⋊C4—24no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—2924no—
370C22⋊C4—2924no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23#1455649811yescharacteristic
410C23#1455650511yescharacteristic
411C23#1455650511yescharacteristic
412C23#1455651111yescharacteristic
413C23#1455651211yescharacteristic
414C23#1455651411yescharacteristic
415C23#1455651511yescharacteristic
416C23#1455651911yescharacteristic
417C23#1455651911yescharacteristic
418C23#1455652211yescharacteristic
419C23#1455652411yescharacteristic
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—2922no—
442C2×C4—2922no—
443C2×C4—2922no—
444C2×C4—2922no—
445C2×C4—2922no—
446C2×C4—2922no—
447C2×C4—5841no—
448C2×C4—5841no—
449C2×C4—5841no—
450C2×C4—5841no—
451C2×C4—5841no—
452C2×C4—5841no—
453C2×C4—5841no—
454C2×C4—5841no—
455C8—5842no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22#1798832011yescharacteristic
495C22#1798832011yescharacteristic
496C22#1798832311yescharacteristic
497C22#1798832311yescharacteristic
498C22#1798834711yescharacteristic
499C22#1798834911yescharacteristic
500C22#1798835311yescharacteristic
501C22#1798837311yescharacteristic
502C22#1798837611yescharacteristic
503C22#1798837911yescharacteristic
504C22#1798838111yescharacteristic
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—84no—
510C4—5842no—
511C4—5842no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—21no—
516C2—41no—
517C2—41no—
518C2—41no—
519C2—41no—
520C2#2260781011yescharacteristic
521C2#2260792711yescharacteristic
522C2#2260793011yescharacteristic
523C2#2260795111yescharacteristic
524C2#2260795411yescharacteristic
525C2#2260804811yescharacteristic
526C2#2260805511yescharacteristic
subgroupG/H
centerC23#14556505
commutatorC22×C4×C73C42
FrattiniC22×C22⋊C4C2×D73
FittingC2×#342×C73C2
socleC23×C73#99
prime radicalC25×C73D4
cosocleC22×C22⋊C4×C73C22
hypercenterC2×#342D73
hypocenterC73#7250
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. C22×C4⋊Dic73C4semidirect
8. C2×#14556498C4semidirect
10. #17988311C4semidirect
14. #342×C73C4semidirect
267. C73#7250semidirect
276. #342Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179883112
#226081762
C2×#342×C732
#725073