Finite Group Databasebeta

← #32985116

Subgroups — #32985116

Order 18688.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609911C211yesmaximal, characteristic, complemented
2C2×#17988324C212yesmaximal, complemented
3C2×#17988321C212yesmaximal
4C4×#14556498C211yesmaximal, characteristic, complemented
5C73×#2850C211yesmaximal, characteristic
6C22×C4×Dic73—21no—
7C2×C4⋊#10596831—21no—
8C2×C22⋊#10596831C2212yes—
9C2×#14556498—21no—
10C2×#14556498C2212yes—
11#17988342—21no—
12#17988324—24no—
13#17988321—24no—
14C2×#124×C73C411yescharacteristic, complemented
15C2×C22⋊Q8×C73C2211yescharacteristic
16C2×#121×C73C2211yescharacteristic
17C22×C42×C73C2211yescharacteristic
18#391×C73C411yescharacteristic, complemented
19#390×C73C411yescharacteristic, complemented
20#388×C73—21no—
21#388×C73—21no—
22#386×C73—21no—
23#386×C73—21no—
24#382×C73C411yescharacteristic, complemented
25C23×Dic73—22no—
26C22×#10596831—22no—
27C2×C4×Dic73—24no—
28C2×C4×Dic73—42no—
29C2×C4×Dic73—42no—
30C4⋊#10596831—42no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556498—24no—
34#14556498—22no—
35#14556498D414yes—
36C22×Q8×C73C2×C411yescharacteristic
37C22×D4×C73—21no—
38C22×D4×C73C2×C411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C23×C4×C73C2311yescharacteristic
42#124×C73—42no—
43C22⋊Q8×C73—22no—
44C22⋊Q8×C73—22no—
45#121×C73—22no—
46#121×C73—22no—
47C2×C4⋊C4×C73—21no—
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C42×C73—42no—
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73C2×C411yescharacteristic
59C2×C42×C73D411yescharacteristic
60C2×C42×C73D411yescharacteristic
61#95×C73—21no—
62#95×C73—21no—
63#95×C73—21no—
64#95×C73—21no—
65C22×Dic73—24no—
66C22×Dic73—24no—
67C22×Dic73—24no—
68C22×Dic73—42no—
69C22×Dic73—42no—
70C22×Dic73—42no—
71C22×Dic73—42no—
72C2×#10596831—42no—
73C2×#10596831—42no—
74C2×#10596831—42no—
75C4×Dic73—48no—
76C24×C73—21no—
77C24×C73C2×D411yescharacteristic
78C2×Q8×C73—22no—
79C2×Q8×C73—41no—
80C2×Q8×C73—41no—
81C2×D4×C73—22no—
82C2×D4×C73—41no—
83C2×D4×C73—41no—
84C2×D4×C73—42no—
85C2×D4×C73—42no—
86C2×D4×C73—41no—
87C2×D4×C73—41no—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—22no—
93C22×C4×C73—22no—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73—21no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22⋊C411yescharacteristic
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C22×C411yescharacteristic
105C22×C4×C73C2×D412yes—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73#4112yes—
108C4⋊C4×C73—22no—
109C4⋊C4×C73—22no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—41no—
112C22⋊C4×C73—42no—
113C22⋊C4×C73—42no—
114C22⋊C4×C73—42no—
115C22⋊C4×C73—42no—
116C22⋊C4×C73—42no—
117C22⋊C4×C73—41no—
118C22⋊C4×C73—41no—
119C42×C73—22no—
120C42×C73—22no—
121C42×C73—41no—
122C42×C73—41no—
123C23×C73—21no—
124C23×C73—21no—
125C23×C73—21no—
126C23×C73—21no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73—41no—
134C23×C73C2×C22⋊C411yescharacteristic
135C23×C73#12112yes—
136C23×C73#13512yes—
137C23×C73AGL(1,ℤ/8)11yescharacteristic
138C23×C73#13711yescharacteristic
139Q8×C73—82no—
140D4×C73—82no—
141D4×C73—82no—
142D4×C73—82no—
143C2×C4×C73—24no—
144C2×C4×C73—24no—
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—21no—
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—42no—
159C2×C4×C73—42no—
160C2×C4×C73—44no—
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—41no—
170C2×C4×C73#9912yes—
171C2×C4×C73#10412yes—
172C2×C4×C73#10412yes—
173C2×C4×C73C4×D412yes—
174C2×C4×C73#12411yescharacteristic
175C2×C4×C73#12711yescharacteristic
176C2×Dic73—28no—
177C2×Dic73—44no—
178C2×Dic73—44no—
179C2×Dic73—44no—
180C2×Dic73—44no—
181C2×Dic73—44no—
182C2×Dic73—44no—
183#10596831—82no—
184C22×C73—22no—
185C22×C73—22no—
186C22×C73—22no—
187C22×C73—22no—
188C22×C73—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—21no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73—81no—
195C22×C73—81no—
196C22×C73—81no—
197C22×C73—81no—
198C22×C73—81no—
199C22×C73#39011yescharacteristic
200C22×C73C2×#9911yescharacteristic
201C22×C73#41011yescharacteristic
202C22×C73#41611yescharacteristic
203C22×C73#41711yescharacteristic
204C22×C73C2×#10411yescharacteristic
205C22×C73C2×#10411yescharacteristic
206C22×C73#46312yes—
207C22×C73#46412yes—
208C4×C73—21no—
209C4×C73—22no—
210C4×C73—21no—
211C4×C73—22no—
212C4×C73—22no—
213C4×C73—22no—
214C4×C73—41no—
215C4×C73—81no—
216C4×C73—81no—
217C4×C73—81no—
218Dic73—48no—
219#9713—731nomaximal, Sylow
220C2×C73—22no—
221C2×C73—22no—
222C2×C73—161no—
223C2×C73#150011yescharacteristic
224C2×C73#150111yescharacteristic
225C2×C73#150411yescharacteristic
226C2×C73#150911yescharacteristic
227C2×C73#196211yescharacteristic
228C2×C73#196411yescharacteristic
229C2×C73#198311yescharacteristic
230#2850D7311yescharacteristic
231#1797—731no—
232C2×#328—732no—
233C2×#327—732no—
234C4×#95—731no—
235C73#971311yescharacteristic, complemented, Sylow
236C2×#124Dic7311yescharacteristic, complemented
237C2×C22⋊Q8C2×D7311yescharacteristic
238C2×#121C2×D7311yescharacteristic
239C22×C42—1461no—
240C22×C42C2×D7311yescharacteristic
241C2×C4⋊C8—1461no—
242C2×C22⋊C8—732no—
243#391Dic7311yescharacteristic, complemented
244#390Dic7311yescharacteristic, complemented
245#388—21no—
246#388—21no—
247#386—21no—
248#386—21no—
249#382Dic7311yescharacteristic, complemented
250C2×#95—732no—
251C2×#95—1461no—
252#336—1461no—
253#328—1464no—
254#327—1464no—
255C22×Q8C2×Dic7311yescharacteristic
256C22×D4—21no—
257C22×D4C2×Dic7311yescharacteristic
258C23×C4—1462no—
259C23×C4C22⋊D7311yescharacteristic
260C23×C4C22⋊D7311yescharacteristic
261C23×C4C22×D7311yescharacteristic
262C22×C8—1462no—
263#124—42no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266#121—22no—
267#121—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C2×Dic7311yescharacteristic
270C2×C4⋊C4C2×Dic7311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—42no—
278C2×C42—2922no—
279C2×C42—2922no—
280C2×C42—1464no—
281C2×C42C2×Dic7311yescharacteristic
282C2×C42C2×Dic7311yescharacteristic
283C2×C42C22⋊D7311yescharacteristic
284C2×C42C22⋊D7311yescharacteristic
285C4⋊C8—2922no—
286C22⋊C8—1462no—
287C22⋊C8—1462no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—734no—
293#95—1464no—
294#95—1462no—
295C24—21no—
296C24C2×C22⋊D7311yescharacteristic
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—2922no—
320C22×C4—2922no—
321C22×C4—2922no—
322C22×C4—2922no—
323C22×C4—1464no—
324C22×C4—1464no—
325C22×C4—1464no—
326C22×C4C22⋊Dic7311yescharacteristic
327C22×C4C22⋊Dic7311yescharacteristic
328C22×C4C22⋊Dic7311yescharacteristic
329C22×C4C22⋊Dic7311yescharacteristic
330C22×C4C22×Dic7311yescharacteristic
331C22×C4D4×D7312yes—
332C22×C4C2×C22⋊D7311yescharacteristic
333C22×C4#1192402712yes—
334C2×C8—2922no—
335C2×C8—2922no—
336C2×C8—2922no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—2928no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C2×C22⋊Dic7311yescharacteristic
365C23#1455656212yes—
366C23#1455661112yes—
367C23#1455661611yescharacteristic
368C23#1455662111yescharacteristic
369Q8—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
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—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—2924no—
401C2×C4—2924no—
402C2×C4—2924no—
403C2×C4—2924no—
404C2×C4—2924no—
405C2×C4—2924no—
406C2×C4—1468no—
407C2×C4#1455650612yes—
408C2×C4#1455651812yes—
409C2×C4#1455651812yes—
410C2×C4D4×Dic7312yes—
411C2×C4#1455657811yescharacteristic
412C2×C4#1455658711yescharacteristic
413C8—5842no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#1798849011yescharacteristic
430C22C2×#1455650611yescharacteristic
431C22#1798856111yescharacteristic
432C22#1798858311yescharacteristic
433C22#1798858811yescharacteristic
434C22C2×#1455651811yescharacteristic
435C22C2×#1455651811yescharacteristic
436C22#1798879912yes—
437C22#1798880612yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—22no—
443C4—21no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—2928no—
449C2—22no—
450C2—22no—
451C2—161no—
452C2#2260841711yescharacteristic
453C2#2260842211yescharacteristic
454C2#2260843511yescharacteristic
455C2#2260845611yescharacteristic
456C2#2261076011yescharacteristic
457C2#2261076811yescharacteristic
458C2#2261086411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2850C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#121×C73C22
cosocleC23×C4×C73C23
hypercenter#2850D73
hypocenterC73#9713
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. #22609911C2semidirect
2. C2×#17988324C2semidirect
4. C4×#14556498C2semidirect
14. C2×#124×C73C4semidirect
18. #391×C73C4semidirect
19. #390×C73C4semidirect
24. #382×C73C4semidirect
235. C73#9713semidirect
236. C2×#124Dic73semidirect
243. #391Dic73semidirect
244. #390Dic73semidirect
249. #382Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099112
C2×#179883242
C2×#179883212
C4×#145564982
C73×#28502
#971373