Finite Group Databasebeta

← #34184506

Subgroups — #34184506

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098758C211yesmaximal, characteristic
2#23764206C211yesmaximal, characteristic, complemented
3#23763400C211yesmaximal, characteristic, complemented
4C2×#19098306C211yesmaximal, characteristic, complemented
5#23761284C211yesmaximal, characteristic, complemented
6#23761235C211yesmaximal, characteristic, complemented
7C2×#436×C79C211yesmaximal, characteristic, complemented
8C22×#11929609—21no—
9C2×#14579158C2211yescharacteristic
10#19098758—22no—
11#19098758C414yescomplemented
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×C4⋊(C79⋊C8)C2211yescharacteristic
14C2×#14579083—21no—
15C2×#14579083—21no—
16C2×#14579082—21no—
17C2×#14579081—21no—
18#19098433C2211yescharacteristic
19#19098325—21no—
20#19098306—22no—
21C22×M16×C79—21no—
22C2×C4×D4×C79C2211yescharacteristic
23#436×C79—22no—
24#436×C79—22no—
25C2×C4⋊C8×C79C2211yescharacteristic
26C2×C22⋊C8×C79—21no—
27C2×C4×C8×C79C2211yescharacteristic
28C22×D316—21no—
29C22×D316—21no—
30C22×D316—21no—
31C2×#11929609—24no—
32C2×#11929609—42no—
33C2×#11929609—42no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36#14579158—21no—
37#14579158—21no—
38#14579158—21no—
39#14579158C2×C412yes—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79—21no—
43C4⋊(C79⋊C8)—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)C2×C412yes—
47#14579083—42no—
48#14579083—42no—
49#14579082—42no—
50#14579081—24no—
51C22×D4×C79D411yescharacteristic
52C23×C4×C79—21no—
53C2×M16×C79—24no—
54C2×M16×C79—42no—
55C2×M16×C79—42no—
56C22×C8×C79—21no—
57C22×C8×C79D411yescharacteristic
58C22×C8×C79D411yescharacteristic
59C4×D4×C79—21no—
60C4×D4×C79—21no—
61C4×D4×C79C2×C412yes—
62C4×D4×C79D412yes—
63C2×C4⋊C4×C79D411yescharacteristic
64C2×C22⋊C4×C79—21no—
65C2×C42×C79C2311yescharacteristic
66C4⋊C8×C79—21no—
67C4⋊C8×C79D412yes—
68C22⋊C8×C79—24no—
69C4×C8×C79—21no—
70C4×C8×C79—21no—
71C23×D79—41no—
72C2×D316—22no—
73C2×D316—22no—
74C2×D316—22no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—41no—
85C2×D316—41no—
86C2×D316—81no—
87C2×D316—81no—
88C22×Dic79—41no—
89#11929609—48no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—41no—
99C4⋊Dic79—41no—
100C4⋊Dic79—81no—
101C24×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—21no—
107C2×D4×C79C22⋊C412yes—
108C22×C4×C79—21no—
109C22×C4×C79—22no—
110C22×C4×C79—22no—
111C22×C4×C79—41no—
112C22×C4×C79—41no—
113C22×C4×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79#4111yescharacteristic
118M16×C79—48no—
119C2×C8×C79—21no—
120C2×C8×C79—21no—
121C2×C8×C79—22no—
122C2×C8×C79—21no—
123C2×C8×C79—21no—
124C2×C8×C79—21no—
125C2×C8×C79—41no—
126C2×C8×C79—41no—
127C2×C8×C79D812yescomplemented
128C4⋊C4×C79—21no—
129C4⋊C4×C79C22⋊C412yes—
130C22⋊C4×C79—22no—
131C22⋊C4×C79—22no—
132C42×C79C22×C411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79C2×D411yescharacteristic
135C42×C79#4111yescharacteristic
136C23×C79—22no—
137C23×C79—41no—
138C23×C79—41no—
139C23×C79—41no—
140C23×C79—41no—
141C23×C79—41no—
142C23×C79—41no—
143C23×C79#12111yescharacteristic
144C22×D79—42no—
145C22×D79—81no—
146C22×D79—81no—
147C22×D79—81no—
148C22×D79—81no—
149C22×D79—81no—
150C22×D79—81no—
151D4×C79—22no—
152D4×C79—22no—
153D4×C79—41no—
154D4×C79—41no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—22no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79—42no—
163C2×C4×C79—42no—
164C2×C4×C79C2×C22⋊C411yescharacteristic
165C2×C4×C79C4×D411yescharacteristic
166C2×C4×C79C4×D411yescharacteristic
167C2×C4×C79#12011yescharacteristic
168C2×C4×C79#12111yescharacteristic
169C2×C4×C79#12311yescharacteristic
170C2×C4×C79C2×D811yescharacteristic
171C2×C4×C79#13511yescharacteristic
172C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C79#13711yescharacteristic
178C2×Dic79—81no—
179C2×Dic79—81no—
180C2×Dic79—81no—
181D316—42no—
182D316—42no—
183D316—42no—
184D316—42no—
185D316—82no—
186D316—82no—
187D316—81no—
188D316—81no—
189D316—81no—
190D316—81no—
191C8×C79—22no—
192C8×C79—42no—
193C8×C79—41no—
194C79⋊C8—42no—
195C79⋊C8—81no—
196C22×C79—42no—
197C22×C79—42no—
198C22×C79—42no—
199C22×C79—81no—
200C22×C79—81no—
201C22×C79—81no—
202C22×C79—81no—
203C22×C79#38611yescharacteristic
204C22×C79#45911yescharacteristic
205C22×C79#46311yescharacteristic
206C22×C79#46811yescharacteristic
207C22×C79#46911yescharacteristic
208C22×C79#47111yescharacteristic
209C22×C79#47211yescharacteristic
210C2×D79—82no—
211C2×D79—82no—
212C2×D79—82no—
213C2×D79—161no—
214C2×D79—161no—
215C2×D79—161no—
216C2×D79—161no—
217C4×C79—21no—
218C4×C79—21no—
219C4×C79—42no—
220C4×C79#41711yescharacteristic
221C4×C79#41811yescharacteristic
222C4×C79C4×D811yescharacteristic
223C4×C79#44211yescharacteristic
224C4×C79#44711yescharacteristic
225C4×C79#44911yescharacteristic
226C4×C79#46811yescharacteristic
227C4×C79#46911yescharacteristic
228Dic79—161no—
229#9229—791nomaximal, Sylow
230C2×C79—82no—
231C2×C79#168511yescharacteristic
232C2×C79#168711yescharacteristic
233C2×C79#169711yescharacteristic
234C2×C79#170111yescharacteristic
235C2×C79#187911yescharacteristic
236C2×C79#189511yescharacteristic
237C2×C79#190711yescharacteristic
238D79—162no—
239C2×#459—791no—
240C2×#436D7911yescharacteristic, complemented
241#1957—791no—
242#1802—791no—
243C2×#331—791no—
244#1296—791no—
245#1277—791no—
246C79#922911yescharacteristic, complemented, Sylow
247C22×D8—1581no—
248C22×M16—21no—
249C2×#127—791no—
250C2×C4×D4C2×D7911yescharacteristic
251#459—794no—
252#459—1582no—
253#436—22no—
254#436—22no—
255C2×C4⋊C8—791no—
256C2×C4⋊C8—791no—
257C2×C4⋊C8C2×D7911yescharacteristic
258C2×#102—1581no—
259C2×#102—1581no—
260C2×#102—1581no—
261C2×#102—1581no—
262C2×C22⋊C8—21no—
263C2×C4×C8C2×D7911yescharacteristic
264#384—791no—
265#340—1581no—
266#331—1582no—
267C22×D4—1581no—
268C22×D4—1581no—
269C22×D4—1581no—
270C22×D4D31611yescharacteristic
271C23×C4—21no—
272C2×D8—3162no—
273C2×D8—3162no—
274C2×D8—1584no—
275C2×M16—24no—
276C2×M16—42no—
277C2×M16—42no—
278C22×C8—21no—
279C22×C8—1581no—
280C22×C8—1581no—
281C22×C8C22⋊D7911yescharacteristic
282C22×C8C22⋊D7911yescharacteristic
283#127—792no—
284#127—1581no—
285#127—1581no—
286#127—1581no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C4×D7912yes—
290C4×D4C22⋊D7912yes—
291C2×C4⋊C4—1581no—
292C2×C4⋊C4—1581no—
293C2×C4⋊C4—1581no—
294C2×C4⋊C4D31611yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D7911yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—792no—
299C4⋊C8—1581no—
300C4⋊C8—1581no—
301C4⋊C8—1581no—
302C4⋊C8C22⋊D7912yes—
303#102—3162no—
304#102—3162no—
305#102—3162no—
306#102—1584no—
307C22⋊C8—24no—
308C4×C8—21no—
309C4×C8—21no—
310C24—21no—
311C24—3161no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—3161no—
318C2×D4—3161no—
319C2×D4—3161no—
320C2×D4—3161no—
321C2×D4—3161no—
322C2×D4—3161no—
323C2×D4—3161no—
324C2×D4—3161no—
325C2×D4—3161no—
326C2×D4—3161no—
327C2×D4—3161no—
328C2×D4—6321no—
329C2×D4—6321no—
330C2×D4—1582no—
331C2×D4—1582no—
332C2×D4—1582no—
333C2×D4(C2×D79)⋊C412yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—3161no—
342C22×C4C2×C22⋊D7911yescharacteristic
343C22×C4C2×D31611yescharacteristic
344C22×C4#1192962411yescharacteristic
345D8—3168no—
346M16—48no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—3161no—
356C2×C8—3161no—
357C2×C8—3161no—
358C2×C8—3161no—
359C2×C8—3161no—
360C2×C8—1582no—
361C2×C8#1192960912yescomplemented
362C4⋊C4—21no—
363C4⋊C4—3161no—
364C4⋊C4—3161no—
365C4⋊C4—3161no—
366C4⋊C4—3161no—
367C4⋊C4—6321no—
368C4⋊C4(C2×D79)⋊C412yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42C2×C4×D7911yescharacteristic
372C42C2×C22⋊D7911yescharacteristic
373C42C2×C22⋊D7911yescharacteristic
374C42#1192962411yescharacteristic
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—3162no—
383C23—6321no—
384C23—6321no—
385C23—6321no—
386C23—6321no—
387C23—6321no—
388C23—6321no—
389C23#1457913511yescharacteristic
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4—3162no—
395D4—3162no—
396D4—3162no—
397D4—3162no—
398D4—6322no—
399D4—6322no—
400D4—6321no—
401D4—6321no—
402D4—6321no—
403D4—6321no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—6321no—
414C2×C4—6321no—
415C2×C4—6321no—
416C2×C4C2×(C2×D79)⋊C411yescharacteristic
417C2×C4C4×C22⋊D7911yescharacteristic
418C2×C4C4×C22⋊D7911yescharacteristic
419C2×C4#1457913011yescharacteristic
420C2×C4#1457913511yescharacteristic
421C2×C4#1457914411yescharacteristic
422C2×C4C2×#1192960911yescharacteristic
423C2×C4#1457918111yescharacteristic
424C2×C4#1457918611yescharacteristic
425C2×C4#1457918711yescharacteristic
426C2×C4#1457918911yescharacteristic
427C2×C4#1457918911yescharacteristic
428C2×C4#1457918911yescharacteristic
429C2×C4#1457919411yescharacteristic
430C8—22no—
431C8—42no—
432C8—41no—
433C8—3162no—
434C8—6321no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—6322no—
443C22—6322no—
444C22—6322no—
445C22—12641no—
446C22—12641no—
447C22—12641no—
448C22—12641no—
449C22#1909844211yescharacteristic
450C22#1909875811yescharacteristic
451C22#1909877811yescharacteristic
452C22#1909880911yescharacteristic
453C22#1909881611yescharacteristic
454C22#1909882611yescharacteristic
455C22#1909883311yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—12641no—
460C4#1909856511yescharacteristic
461C4#1909856911yescharacteristic
462C4C4×#1192960911yescharacteristic
463C4#1909867311yescharacteristic
464C4#1909869711yescharacteristic
465C4#1909870811yescharacteristic
466C4#1909880911yescharacteristic
467C4#1909881611yescharacteristic
468C2—82no—
469C2—12642no—
470C2#2376286511yescharacteristic
471C2#2376287911yescharacteristic
472C2#2376293311yescharacteristic
473C2#2376296111yescharacteristic
474C2#2376374811yescharacteristic
475C2#2376383811yescharacteristic
476C2#2376390811yescharacteristic
subgroupG/H
centerC23#14579135
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC2×#436×C79C2
socleC23×C79#121
prime radicalC2×#19098758C2
cosocleC2×C42×C79C23
hypercenterC2×#436D79
hypocenterC79#9229
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
2. #23764206C2semidirect
3. #23763400C2semidirect
4. C2×#19098306C2semidirect
5. #23761284C2semidirect
6. #23761235C2semidirect
7. C2×#436×C79C2semidirect
11. #19098758C4semidirect
127. C2×C8×C79D8semidirect
240. C2×#436D79semidirect
246. C79#9229semidirect
361. C2×C8#11929609semidirect

Maximal subgroups

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

subgroupindex
C2×#190987582
#237642062
#237634002
C2×#190983062
#237612842
#237612352
C2×#436×C792
#922979