Finite Group Databasebeta

← #32978727

Subgroups — #32978727

Order 18688.

698 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609911C211yesmaximal, characteristic, complemented
2C2×#17988321C212yesmaximal
3#22608158C211yesmaximal, characteristic, complemented
4#22607809C212yesmaximal, complemented
5C4×#121×C73C211yesmaximal, characteristic
6C2×C4⋊#10596831—21no—
7C2×C22⋊#10596831C2212yes—
8C2×#14556498—21no—
9C2×#14556498—21no—
10C2×#14556498C2212yes—
11#17988342—21no—
12#17988321—24no—
13C2×#121×C73C2211yescharacteristic
14C2×C4×D4×C73—21no—
15C2×C4×D4×C73C411yescharacteristic, complemented
16C22×C42×C73C2211yescharacteristic
17#390×C73C411yescharacteristic, complemented
18#389×C73C411yescharacteristic, complemented
19#388×C73—21no—
20#386×C73—21no—
21#380×C73C2211yescharacteristic
22C4×C4⋊C4×C73C411yescharacteristic, complemented
23C4×C22⋊C4×C73—21no—
24C23×Dic73—22no—
25C22×#10596831—22no—
26C4⋊#10596831—42no—
27C22⋊#10596831—22no—
28C22⋊#10596831—22no—
29#14556498—24no—
30#14556498—22no—
31#14556498—22no—
32#14556498—42no—
33C22×D4×C73—21no—
34C22×D4×C73C2×C411yescharacteristic
35C23×C4×C73—21no—
36C23×C4×C73D411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73C2311yescharacteristic
39#121×C73—21no—
40#121×C73—21no—
41#121×C73D412yes—
42#121×C73Q812yes—
43C4×D4×C73—42no—
44C4×D4×C73—44no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C42×C73—21no—
55C2×C42×C73—42no—
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73D411yescharacteristic
59C2×C42×C73D411yescharacteristic
60#95×C73—21no—
61#95×C73—21no—
62#95×C73—21no—
63C22×Dic73—24no—
64C22×Dic73—42no—
65C22×Dic73—42no—
66C22×Dic73—42no—
67C22×Dic73—42no—
68C22×Dic73—42no—
69C22×Dic73—42no—
70C2×#10596831—42no—
71C2×#10596831—42no—
72C2×#10596831—42no—
73C24×C73—21no—
74C24×C73C2×D411yescharacteristic
75C2×D4×C73—22no—
76C2×D4×C73—22no—
77C2×D4×C73—21no—
78C2×D4×C73—41no—
79C2×D4×C73—41no—
80C2×D4×C73—42no—
81C2×D4×C73—42no—
82C2×D4×C73C4⋊C412yes—
83C22×C4×C73—21no—
84C22×C4×C73—22no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—22no—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—41no—
91C22×C4×C73—41no—
92C22×C4×C73—41no—
93C22×C4×C73—41no—
94C22×C4×C73—42no—
95C22×C4×C73—42no—
96C22×C4×C73—42no—
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22⋊C411yescharacteristic
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22×C411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×D411yescharacteristic
104C22×C4×C73C2×Q811yescharacteristic
105C22×C4×C73#4111yescharacteristic
106C22×C4×C73#4111yescharacteristic
107C4⋊C4×C73—21no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—42no—
111C4⋊C4×C73C4⋊C412yes—
112C22⋊C4×C73—22no—
113C22⋊C4×C73—22no—
114C22⋊C4×C73—42no—
115C22⋊C4×C73—42no—
116C22⋊C4×C73—42no—
117C22⋊C4×C73—42no—
118C42×C73—22no—
119C42×C73—22no—
120C42×C73—41no—
121C42×C73—41no—
122C42×C73—42no—
123C23×C73—21no—
124C23×C73—21no—
125C23×C73—21no—
126C23×C73—21no—
127C23×C73—22no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73—41no—
134C23×C73C2×C22⋊C411yescharacteristic
135C23×C73#12111yescharacteristic
136C23×C73C22⋊Q811yescharacteristic
137C23×C73#13512yes—
138C23×C73AGL(1,ℤ/8)12yes—
139D4×C73—42no—
140D4×C73—42no—
141D4×C73—44no—
142D4×C73—44no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—21no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—22no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—44no—
163C2×C4×C73—44no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73—41no—
172C2×C4×C73—84no—
173C2×C4×C73#9912yes—
174C2×C4×C73#10412yes—
175C2×C4×C73C2×C4⋊C411yescharacteristic
176C2×C4×C73#11711yescharacteristic
177C2×C4×C73#12111yescharacteristic
178C2×C4×C73C22⋊Q811yescharacteristic
179C2×Dic73—44no—
180C2×Dic73—44no—
181C2×Dic73—44no—
182C2×Dic73—82no—
183C2×Dic73—82no—
184C2×Dic73—82no—
185C2×Dic73—82no—
186#10596831—82no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—22no—
192C22×C73—22no—
193C22×C73—22no—
194C22×C73—22no—
195C22×C73—42no—
196C22×C73—42no—
197C22×C73—42no—
198C22×C73—81no—
199C22×C73—81no—
200C22×C73—81no—
201C22×C73—81no—
202C22×C73#38011yescharacteristic
203C22×C73C2×#9911yescharacteristic
204C22×C73#41011yescharacteristic
205C22×C73#41611yescharacteristic
206C22×C73#41811yescharacteristic
207C22×C73C2×#10411yescharacteristic
208C22×C73#42111yescharacteristic
209C22×C73#46312yes—
210C22×C73#47812yes—
211C4×C73—21no—
212C4×C73—21no—
213C4×C73—21no—
214C4×C73—21no—
215C4×C73—22no—
216C4×C73—22no—
217C4×C73—42no—
218C4×C73—41no—
219C4×C73—41no—
220C4×C73—81no—
221C4×C73—81no—
222Dic73—84no—
223#8575—731nomaximal, Sylow
224C2×C73—22no—
225C2×C73—22no—
226C2×C73—82no—
227C2×C73#150111yescharacteristic
228C2×C73#150511yescharacteristic
229C2×C73#151011yescharacteristic
230C2×C73#152111yescharacteristic
231C2×C73#179911yescharacteristic
232C2×C73#180411yescharacteristic
233C2×C73#180611yescharacteristic
234C4×#121D7311yescharacteristic
235#1797—731no—
236C2×#327—732no—
237#1434—731no—
238#1294—732no—
239C73#857511yescharacteristic, complemented, Sylow
240C2×#121C2×D7311yescharacteristic
241C2×C4×D4—21no—
242C2×C4×D4Dic7311yescharacteristic, complemented
243C22×C42C2×D7311yescharacteristic
244C2×C4⋊C8—1461no—
245C2×C22⋊C8—732no—
246#390Dic7311yescharacteristic, complemented
247#389Dic7311yescharacteristic, complemented
248#388—21no—
249#386—21no—
250#380C2×D7311yescharacteristic
251C4×C4⋊C4Dic7311yescharacteristic, complemented
252C4×C22⋊C4—21no—
253C2×#95—732no—
254C2×#95—1461no—
255C2×#95—1461no—
256#336—1461no—
257#327—1464no—
258C22×D4—21no—
259C22×D4C2×Dic7311yescharacteristic
260C23×C4—21no—
261C23×C4—1462no—
262C23×C4C22⋊D7311yescharacteristic
263C23×C4C22⋊D7311yescharacteristic
264C23×C4C22×D7311yescharacteristic
265C22×C8—1462no—
266#121—21no—
267#121—21no—
268#121Dic14612yes—
269#121D29212yes—
270C4×D4—42no—
271C4×D4—44no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×Dic7311yescharacteristic
274C2×C4⋊C4C2×Dic7311yescharacteristic
275C2×C4⋊C4C2×Dic7311yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282C2×C42—42no—
283C2×C42C2×Dic7311yescharacteristic
284C2×C42C2×Dic7311yescharacteristic
285C2×C42C22⋊D7311yescharacteristic
286C2×C42C22⋊D7311yescharacteristic
287C4⋊C8—2922no—
288C22⋊C8—1462no—
289C22⋊C8—1462no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—2922no—
294#95—1464no—
295#95—1462no—
296#95—1462no—
297C24—21no—
298C24C2×C22⋊D7311yescharacteristic
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4C4⋊Dic7312yes—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—2922no—
322C22×C4—2922no—
323C22×C4—2922no—
324C22×C4—2922no—
325C22×C4—2922no—
326C22×C4—2922no—
327C22×C4—1464no—
328C22×C4C22⋊Dic7311yescharacteristic
329C22×C4C22⋊Dic7311yescharacteristic
330C22×C4C22⋊Dic7311yescharacteristic
331C22×C4C22⋊Dic7311yescharacteristic
332C22×C4C22×Dic7311yescharacteristic
333C22×C4C2×C22⋊D7311yescharacteristic
334C22×C4C2×D29211yescharacteristic
335C22×C4C2×Dic14611yescharacteristic
336C22×C4#1192402811yescharacteristic
337C22×C4#1192402811yescharacteristic
338C2×C8—2922no—
339C2×C8—2922no—
340C2×C8—2922no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4C4⋊Dic7312yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—42no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23C2×C22⋊Dic7311yescharacteristic
369C23#1455656511yescharacteristic
370C23#1455657011yescharacteristic
371C23#1455661112yes—
372C23#1455661712yes—
373D4—42no—
374D4—42no—
375D4—44no—
376D4—44no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—2924no—
408C2×C4—2924no—
409C2×C4—2924no—
410C2×C4—5842no—
411C2×C4—5842no—
412C2×C4—5842no—
413C2×C4—5842no—
414C2×C4#1455650612yes—
415C2×C4#1455651812yes—
416C2×C4C2×C4⋊Dic7311yescharacteristic
417C2×C4#1455655011yescharacteristic
418C2×C4#1455656511yescharacteristic
419C2×C4#1455657011yescharacteristic
420C8—5842no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—21no—
428C22—21no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22#1798844711yescharacteristic
437C22C2×#1455650611yescharacteristic
438C22#1798856111yescharacteristic
439C22#1798858311yescharacteristic
440C22#1798859211yescharacteristic
441C22C2×#1455651811yescharacteristic
442C22#1798860411yescharacteristic
443C22#1798880212yes—
444C22#1798888712yes—
445C4—22no—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—42no—
452C4—41no—
453C4—41no—
454C4—81no—
455C4—81no—
456C4—5844no—
457C2—22no—
458C2—22no—
459C2—82no—
460C2#2260842211yescharacteristic
461C2#2260843911yescharacteristic
462C2#2260846011yescharacteristic
463C2#2260850811yescharacteristic
464C2#2260991911yescharacteristic
465C2#2260994111yescharacteristic
466C2#2260995111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC4×#121×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#121×C73C22
cosocleC23×C4×C73C23
hypercenterC4×#121D73
hypocenterC73#8575
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #22609911C2semidirect
3. #22608158C2semidirect
4. #22607809C2semidirect
15. C2×C4×D4×C73C4semidirect
17. #390×C73C4semidirect
18. #389×C73C4semidirect
22. C4×C4⋊C4×C73C4semidirect
239. C73#8575semidirect
242. C2×C4×D4Dic73semidirect
246. #390Dic73semidirect
247. #389Dic73semidirect
251. C4×C4⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099112
C2×#179883212
#226081582
#226078092
C4×#121×C732
#857573