Finite Group Databasebeta

← #32984972

Subgroups — #32984972

Order 18688.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊#10596831C211yesmaximal, characteristic, complemented
2C2×#17988324C212yesmaximal, complemented
3C2×#17988321C212yesmaximal
4#22608158C211yesmaximal, characteristic, complemented
5C73×#2850C211yesmaximal, characteristic
6C2×C22⋊#10596831C2212yes—
7C2×C4×#10596831—21no—
8C2×#14556498—21no—
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—
27C22⋊#10596831—22no—
28C22⋊#10596831D414yes—
29C4×#10596831—42no—
30#14556498—24no—
31#14556498—22no—
32#14556498—22no—
33#14556498—42no—
34C22×Q8×C73C2×C411yescharacteristic
35C22×D4×C73—21no—
36C22×D4×C73C2×C411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40#124×C73—42no—
41C22⋊Q8×C73—22no—
42C22⋊Q8×C73—22no—
43#121×C73—22no—
44#121×C73—22no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73—21no—
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—42no—
55C2×C42×C73C2×C411yescharacteristic
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C42×C73D411yescharacteristic
59#95×C73—21no—
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—24no—
71C2×#10596831—42no—
72C2×#10596831—42no—
73C24×C73—21no—
74C24×C73C2×D411yescharacteristic
75C2×Q8×C73—22no—
76C2×Q8×C73—41no—
77C2×Q8×C73—41no—
78C2×D4×C73—22no—
79C2×D4×C73—41no—
80C2×D4×C73—41no—
81C2×D4×C73—41no—
82C2×D4×C73—41no—
83C2×D4×C73—42no—
84C2×D4×C73—42no—
85C22×C4×C73—22no—
86C22×C4×C73—22no—
87C22×C4×C73—22no—
88C22×C4×C73—22no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—41no—
93C22×C4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22⋊C411yescharacteristic
100C22×C4×C73C22×C411yescharacteristic
101C22×C4×C73C2×D412yes—
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73#4112yes—
104C4⋊C4×C73—22no—
105C4⋊C4×C73—22no—
106C4⋊C4×C73—42no—
107C22⋊C4×C73—42no—
108C22⋊C4×C73—42no—
109C22⋊C4×C73—41no—
110C22⋊C4×C73—41no—
111C22⋊C4×C73—42no—
112C22⋊C4×C73—42no—
113C22⋊C4×C73—42no—
114C42×C73—22no—
115C42×C73—22no—
116C42×C73—41no—
117C42×C73—41no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73C2×C22⋊C411yescharacteristic
130C23×C73#12411yescharacteristic
131C23×C73#12711yescharacteristic
132C23×C73#13512yes—
133C23×C73#13512yes—
134Q8×C73—82no—
135D4×C73—82no—
136D4×C73—82no—
137D4×C73—82no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
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—21no—
152C2×C4×C73—42no—
153C2×C4×C73—42no—
154C2×C4×C73—44no—
155C2×C4×C73—44no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73#9912yes—
166C2×C4×C73#10412yes—
167C2×C4×C73C4×D412yes—
168C2×C4×C73#12112yes—
169C2×Dic73—44no—
170C2×Dic73—44no—
171C2×Dic73—44no—
172C2×Dic73—82no—
173C2×Dic73—82no—
174C2×Dic73—82no—
175C2×Dic73—82no—
176#10596831—44no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73—21no—
180C22×C73—21no—
181C22×C73—22no—
182C22×C73—22no—
183C22×C73—22no—
184C22×C73—22no—
185C22×C73—81no—
186C22×C73—81no—
187C22×C73—81no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73—81no—
192C22×C73#39011yescharacteristic
193C22×C73C2×#9911yescharacteristic
194C22×C73#41011yescharacteristic
195C22×C73#41611yescharacteristic
196C22×C73#41611yescharacteristic
197C22×C73C2×#10411yescharacteristic
198C22×C73#42111yescharacteristic
199C22×C73#48812yes—
200C22×C73#49512yes—
201C4×C73—22no—
202C4×C73—22no—
203C4×C73—22no—
204C4×C73—22no—
205C4×C73—41no—
206C4×C73—41no—
207C4×C73—81no—
208C4×C73—81no—
209C4×C73—81no—
210Dic73—84no—
211#9681—731nomaximal, Sylow
212C2×C73—22no—
213C2×C73—22no—
214C2×C73—161no—
215C2×C73#150011yescharacteristic
216C2×C73#150111yescharacteristic
217C2×C73#151611yescharacteristic
218C2×C73#152111yescharacteristic
219C2×C73#195811yescharacteristic
220C2×C73#196411yescharacteristic
221C2×C73#198011yescharacteristic
222#2850D7311yescharacteristic
223C4×C22⋊C8—731no—
224C2×#328—732no—
225C2×#327—732no—
226#1434—731no—
227C73#968111yescharacteristic, complemented, Sylow
228C2×#124Dic7311yescharacteristic, complemented
229C2×C22⋊Q8C2×D7311yescharacteristic
230C2×#121C2×D7311yescharacteristic
231C22×C42C2×D7311yescharacteristic
232C2×C22⋊C8—732no—
233C2×C4×C8—1461no—
234#391Dic7311yescharacteristic, complemented
235#390Dic7311yescharacteristic, complemented
236#388—21no—
237#388—21no—
238#386—21no—
239#386—21no—
240#382Dic7311yescharacteristic, complemented
241C2×#95—732no—
242C2×#95—1461no—
243C2×#95—1461no—
244#336—1461no—
245#328—1464no—
246#327—1464no—
247C22×Q8C2×Dic7311yescharacteristic
248C22×D4—21no—
249C22×D4C2×Dic7311yescharacteristic
250C23×C4—1462no—
251C23×C4C22⋊D7311yescharacteristic
252C23×C4C22⋊D7311yescharacteristic
253C23×C4C22×D7311yescharacteristic
254C22×C8—1462no—
255#124—42no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258#121—22no—
259#121—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C2×Dic7311yescharacteristic
262C2×C4⋊C4C2×Dic7311yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—42no—
270C2×C42C2×Dic7311yescharacteristic
271C2×C42C2×Dic7311yescharacteristic
272C2×C42C22⋊D7311yescharacteristic
273C2×C42C22⋊D7311yescharacteristic
274C22⋊C8—734no—
275C22⋊C8—1462no—
276C4×C8—2922no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—2922no—
282#95—1464no—
283#95—1462no—
284#95—1462no—
285C24—21no—
286C24C2×C22⋊D7311yescharacteristic
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—2922no—
309C22×C4—2922no—
310C22×C4—2922no—
311C22×C4—2922no—
312C22×C4—2922no—
313C22×C4—2922no—
314C22×C4—1464no—
315C22×C4C22⋊Dic7311yescharacteristic
316C22×C4C22⋊Dic7311yescharacteristic
317C22×C4C22⋊Dic7311yescharacteristic
318C22×C4C22⋊Dic7311yescharacteristic
319C22×C4C22×Dic7311yescharacteristic
320C22×C4D4×D7312yes—
321C22×C4C2×C22⋊D7311yescharacteristic
322C22×C4#1192402712yes—
323C2×C8—2922no—
324C2×C8—2922no—
325C2×C8—1464no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C22⋊Dic7311yescharacteristic
352C23#1455657811yescharacteristic
353C23#1455658711yescharacteristic
354C23#1455661112yes—
355C23#1455661112yes—
356Q8—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—2924no—
388C2×C4—2924no—
389C2×C4—2924no—
390C2×C4—5842no—
391C2×C4—5842no—
392C2×C4—5842no—
393C2×C4—5842no—
394C2×C4#1455650612yes—
395C2×C4#1455651812yes—
396C2×C4D4×Dic7312yes—
397C2×C4#1455656212yes—
398C8—2924no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22#1798849011yescharacteristic
415C22C2×#1455650611yescharacteristic
416C22#1798856111yescharacteristic
417C22#1798858311yescharacteristic
418C22#1798858311yescharacteristic
419C22C2×#1455651811yescharacteristic
420C22#1798860411yescharacteristic
421C22#1798894112yes—
422C22#1798896912yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—5844no—
433C2—22no—
434C2—22no—
435C2—161no—
436C2#2260841711yescharacteristic
437C2#2260842211yescharacteristic
438C2#2260848711yescharacteristic
439C2#2260850811yescharacteristic
440C2#2261074311yescharacteristic
441C2#2261076811yescharacteristic
442C2#2261085111yescharacteristic
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#9681
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. C4×C22⋊#10596831C2semidirect
2. C2×#17988324C2semidirect
4. #22608158C2semidirect
14. C2×#124×C73C4semidirect
18. #391×C73C4semidirect
19. #390×C73C4semidirect
24. #382×C73C4semidirect
227. C73#9681semidirect
228. C2×#124Dic73semidirect
234. #391Dic73semidirect
235. #390Dic73semidirect
240. #382Dic73semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#105968312
C2×#179883242
C2×#179883212
#226081582
C73×#28502
#968173