Finite Group Databasebeta

← #32980770

Subgroups — #32980770

Order 18688.

664 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988864C211yesmaximal, characteristic
2#22612831C211yesmaximal, characteristic
3#22610855C211yesmaximal, characteristic, complemented
4C2×#17988316C211yesmaximal, characteristic
5#22607814C211yesmaximal, characteristic, complemented
6#22607771C211yesmaximal, characteristic, complemented
7C73×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9C2×#14556553—21no—
10C2×C4⋊C4×D73—21no—
11#17988864—22no—
12#17988864C414yescomplemented
13C2×#14556524—21no—
14C2×C4⋊#10596831C2211yescharacteristic
15C2×C4⋊#10596831C2211yescharacteristic
16C2×#14556515—21no—
17C2×#14556515—21no—
18C2×#14556511—21no—
19#17988483—21no—
20#17988458—22no—
21#17988444—21no—
22C4×C4⋊Dic73C2211yescharacteristic
23#17988316—22no—
24C2×C4⋊Q8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26#384×C73C2211yescharacteristic
27#340×C73—22no—
28C22×D292D411yescharacteristic
29C22×C4×D73—22no—
30C22×C4×D73—21no—
31C22×#10596831—21no—
32C22×#10596831—21no—
33C4×D292—21no—
34C4×D292—21no—
35C4×D292C2×C412yes—
36C4×D292D412yes—
37#14556553—42no—
38#14556553—42no—
39C2×C4⋊Dic73D411yescharacteristic
40C2×C4⋊Dic73D411yescharacteristic
41C2×C4⋊Dic73D411yescharacteristic
42C2×#11924007—22no—
43C4⋊C4×D73—42no—
44C4⋊C4×D73—42no—
45C2×#11924005—22no—
46C2×#11924005—21no—
47C2×C4×Dic73—21no—
48#14556524—24no—
49C4⋊#10596831—21no—
50C4⋊#10596831—21no—
51C4⋊#10596831—21no—
52C4⋊#10596831C2×C412yescomplemented
53#14556515—42no—
54#14556515—42no—
55#14556511—24no—
56#14556498—22no—
57#14556498—21no—
58C22×Q8×C73—21no—
59C22×C8×C73—21no—
60C4⋊Q8×C73—21no—
61C4⋊Q8×C73—21no—
62C4⋊Q8×C73—21no—
63C4⋊Q8×C73C2×C412yescomplemented
64C2×C4⋊C4×C73—21no—
65C2×C4⋊C4×C73—21no—
66C2×C4⋊C4×C73—21no—
67C2×C4⋊C4×C73—21no—
68C2×C4⋊C4×C73D412yes—
69C2×C42×C73C2311yescharacteristic
70C4×C8×C73—21no—
71C4×C8×C73—21no—
72C23×D73—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292C22⋊C412yes—
77C2×D292SD1612yes—
78C2×D292Q1612yes—
79C22×Dic73—21no—
80C22×Dic73—21no—
81C22×Dic73—21no—
82C2×C4×D73—22no—
83C2×C4×D73—22no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×C4×D73—44no—
89C2×C4×D73—44no—
90C2×C4×D73—82no—
91C2×#10596831—22no—
92C2×#10596831—41no—
93C2×#10596831—41no—
94C2×#10596831—41no—
95C2×#10596831—41no—
96C2×#10596831—41no—
97C4⋊Dic73—21no—
98C4⋊Dic73—21no—
99C4⋊Dic73—21no—
100C4⋊Dic73—21no—
101C4⋊Dic73C22⋊C412yes—
102C4⋊Dic73D812yes—
103#11924007—44no—
104#11924005—22no—
105#11924005—22no—
106#11924005—44no—
107C4×Dic73—41no—
108C4×Dic73—41no—
109C2×Q8×C73—22no—
110C2×Q8×C73—41no—
111C2×Q8×C73—41no—
112C2×Q8×C73—41no—
113C2×Q8×C73—41no—
114C2×Q8×C73—41no—
115C22×C4×C73—22no—
116C22×C4×C73—41no—
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C22×C4×C73C2×D411yescharacteristic
120C2×C8×C73—22no—
121C2×C8×C73—41no—
122C2×C8×C73—41no—
123C4⋊C4×C73—22no—
124C4⋊C4×C73—22no—
125C4⋊C4×C73—42no—
126C4⋊C4×C73—41no—
127C4⋊C4×C73—41no—
128C4⋊C4×C73—41no—
129C4⋊C4×C73—41no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73—41no—
132C42×C73C22×C411yescharacteristic
133C42×C73C2×D411yescharacteristic
134C42×C73#4111yescharacteristic
135C42×C73#4111yescharacteristic
136C23×C73#12011yescharacteristic
137C22×D73—22no—
138C22×D73—22no—
139C22×D73—22no—
140C22×D73—41no—
141C22×D73—41no—
142C22×D73—41no—
143C22×D73—41no—
144Q8×C73—42no—
145Q8×C73—42no—
146Q8×C73—81no—
147Q8×C73—81no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—42no—
154C2×C4×C73—81no—
155C2×C4×C73—81no—
156C2×C4×C73C2×C22⋊C411yescharacteristic
157C2×C4×C73C4×D411yescharacteristic
158C2×C4×C73C4×D411yescharacteristic
159C2×C4×C73#12111yescharacteristic
160C2×C4×C73#12111yescharacteristic
161C2×C4×C73#12311yescharacteristic
162C2×C4×C73C2×D811yescharacteristic
163C2×C4×C73C2×SD1611yescharacteristic
164C2×C4×C73C2×Q1611yescharacteristic
165C2×C4×C73#13511yescharacteristic
166C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C73#13711yescharacteristic
170C2×Dic73—22no—
171C2×Dic73—22no—
172C2×Dic73—41no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175C2×Dic73—41no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178C2×Dic73—41no—
179D292—22no—
180D292—22no—
181D292—22no—
182D292#10314yescomplemented
183C4×D73—42no—
184C4×D73—42no—
185C4×D73—42no—
186C4×D73—42no—
187C4×D73—88no—
188C8×C73—42no—
189#10596831—42no—
190#10596831—81no—
191C22×C73#38611yescharacteristic
192C22×C73#44711yescharacteristic
193C22×C73#44911yescharacteristic
194C22×C73#45011yescharacteristic
195C22×C73#45111yescharacteristic
196C22×C73#45511yescharacteristic
197C22×C73#45611yescharacteristic
198C2×D73—24no—
199C2×D73—42no—
200C2×D73—42no—
201C2×D73—42no—
202C2×D73—42no—
203C2×D73—42no—
204C2×D73—42no—
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—82no—
208C4×C73—82no—
209C4×C73C2×#10311yescharacteristic
210C4×C73#41711yescharacteristic
211C4×C73C4×D811yescharacteristic
212C4×C73#44211yescharacteristic
213C4×C73#45911yescharacteristic
214C4×C73#46311yescharacteristic
215C4×C73#48311yescharacteristic
216C4×C73#48411yescharacteristic
217Dic73—42no—
218Dic73—42no—
219Dic73—81no—
220#8981—731nomaximal, Sylow
221C2×C73#163911yescharacteristic
222C2×C73#164411yescharacteristic
223C2×C73#164911yescharacteristic
224C2×C73#165411yescharacteristic
225C2×C73#187211yescharacteristic
226C2×C73#187911yescharacteristic
227C2×C73#188211yescharacteristic
228D73—44no—
229C2×#474—731no—
230D4×C4⋊C4—731no—
231#1990D7311yescharacteristic, complemented
232#1981—731no—
233C2×D4⋊C8—731no—
234#1296—731no—
235#1279—731no—
236C73#898111yescharacteristic, complemented, Sylow
237C2×C4⋊Q8C2×D7311yescharacteristic
238C2×C4×D4—731no—
239C2×C4×D4—1461no—
240C22×C4⋊C4—1461no—
241#474—734no—
242#474—1462no—
243C2×#107—1461no—
244C2×C4⋊C8—731no—
245C2×C4⋊C8—731no—
246C2×#103—1461no—
247C2×#103—1461no—
248C2×#102—1461no—
249C2×C4×C8C2×D7311yescharacteristic
250#389—1461no—
251#385—1462no—
252#384C2×D7311yescharacteristic
253#380—1461no—
254C4×C4⋊C4—731no—
255#340—22no—
256D4⋊C8—1462no—
257C22×Q8—21no—
258C22×D4—731no—
259C23×C4—1462no—
260C23×C4—1461no—
261C22×C8—21no—
262C22×C8—1461no—
263C22×C8—1461no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C4×D7312yescomplemented
268C4×D4—2922no—
269C4×D4—2922no—
270C4×D4—732no—
271C4×D4—732no—
272C4×D4—1461no—
273C4×D4—1461no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—2922no—
279C2×C4⋊C4—2922no—
280C2×C4⋊C4—731no—
281C2×C4⋊C4—731no—
282C2×C4⋊C4—731no—
283C2×C4⋊C4—1462no—
284C2×C4⋊C4C22⋊D7312yes—
285C2×C22⋊C4—1462no—
286C2×C22⋊C4—1461no—
287C2×C42—1461no—
288C2×C42C22×D7311yescharacteristic
289#107—1464no—
290C4⋊C8—732no—
291C4⋊C8—1461no—
292C4⋊C8—1461no—
293C4⋊C8—1461no—
294#103—2922no—
295#103—2922no—
296#102—1464no—
297C4×C8—21no—
298C4×C8—21no—
299#95—1462no—
300#95—1461no—
301C24—1461no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—732no—
309C2×D4—732no—
310C2×D4—732no—
311C2×D4—1461no—
312C2×D4—1461no—
313C2×D4—1461no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—2924no—
317C22×C4—2924no—
318C22×C4—2921no—
319C22×C4—2921no—
320C22×C4—2921no—
321C22×C4—2921no—
322C22×C4—5842no—
323C22×C4—1461no—
324C22×C4—1461no—
325C22×C4—1461no—
326C22×C4—1462no—
327C22×C4—1462no—
328C22×C4D4×D7311yescharacteristic
329C22×C4D4×D7311yescharacteristic
330C22×C4C2×C22⋊D7311yescharacteristic
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—2921no—
335C2×C8—2921no—
336C2×C8—2921no—
337C2×C8—2921no—
338C2×C8—2921no—
339C2×C8—1462no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—2924no—
350C4⋊C4—732no—
351C4⋊C4—732no—
352C4⋊C4—1461no—
353C4⋊C4—1461no—
354C4⋊C4—1461no—
355C4⋊C4—1461no—
356C22⋊C4—2924no—
357C22⋊C4—1462no—
358C22⋊C4—1462no—
359C42—2921no—
360C42—2921no—
361C42C2×C4×D7311yescharacteristic
362C42D4×D7311yescharacteristic
363C42#1192402711yescharacteristic
364C42#1192402811yescharacteristic
365C23—2921no—
366C23—2921no—
367C23—2921no—
368C23—2921no—
369C23—1462no—
370C23—1462no—
371C23—1462no—
372C23#1455655911yescharacteristic
373Q8—42no—
374Q8—42no—
375Q8—81no—
376Q8—81no—
377D4—734no—
378D4—1462no—
379D4—1462no—
380D4—1462no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—2921no—
390C2×C4—2921no—
391C2×C4—2921no—
392C2×C4—2921no—
393C2×C4—2922no—
394C2×C4—2922no—
395C2×C4—2922no—
396C2×C4—2922no—
397C2×C4—2921no—
398C2×C4—2921no—
399C2×C4—2921no—
400C2×C4—5848no—
401C2×C4—1462no—
402C2×C4—1462no—
403C2×C4C22⋊C4×D7311yescharacteristic
404C2×C4#1455655111yescharacteristic
405C2×C4C4×C22⋊D7311yescharacteristic
406C2×C4#1455656211yescharacteristic
407C2×C4#1455656411yescharacteristic
408C2×C4#1455657311yescharacteristic
409C2×C4D8×D7311yescharacteristic
410C2×C4SD16×D7311yescharacteristic
411C2×C4Q16×D7311yescharacteristic
412C2×C4#1455661211yescharacteristic
413C2×C4#1455661511yescharacteristic
414C2×C4#1455661811yescharacteristic
415C2×C4#1455661811yescharacteristic
416C2×C4#1455662311yescharacteristic
417C8—42no—
418C8—2922no—
419C8—5841no—
420C22—2922no—
421C22—2922no—
422C22—2922no—
423C22—2922no—
424C22—2922no—
425C22—2922no—
426C22—1464no—
427C22#1798846311yescharacteristic
428C22#1798871911yescharacteristic
429C22#1798873111yescharacteristic
430C22#1798873611yescharacteristic
431C22#1798874111yescharacteristic
432C22#1798876411yescharacteristic
433C22#1798876911yescharacteristic
434C4—21no—
435C4—21no—
436C4—82no—
437C4—82no—
438C4—2922no—
439C4—2922no—
440C4—5841no—
441C4#103×D7311yescharacteristic
442C4#1798858711yescharacteristic
443C4#1798866711yescharacteristic
444C4#1798869611yescharacteristic
445C4#1798878111yescharacteristic
446C4#1798880111yescharacteristic
447C4#1798891611yescharacteristic
448C4#1798892111yescharacteristic
449C2—2924no—
450C2#2260910711yescharacteristic
451C2#2260914211yescharacteristic
452C2#2260917711yescharacteristic
453C2#2260921211yescharacteristic
454C2#2261023511yescharacteristic
455C2#2261027611yescharacteristic
456C2#2261030311yescharacteristic
subgroupG/H
centerC23#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1990C2
socleC23×C73#120
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1990D73
hypocenterC73#8981
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
3. #22610855C2semidirect
5. #22607814C2semidirect
6. #22607771C2semidirect
7. C73×#1990C2semidirect
12. #17988864C4semidirect
52. C4⋊#10596831C2×C4semidirect
63. C4⋊Q8×C73C2×C4semidirect
182. D292#103semidirect
231. #1990D73semidirect
236. C73#8981semidirect
267. C4⋊Q8C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179888642
#226128312
#226108552
C2×#179883162
#226078142
#226077712
C73×#19902
#898173