Finite Group Databasebeta

← #34184504

Subgroups — #34184504

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
2C2×#19098641C211yesmaximal, characteristic, complemented
3#23764205C211yesmaximal, characteristic, complemented
4C2×#19098306C211yesmaximal, characteristic, complemented
5#23761283C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#1802C211yesmaximal, characteristic, complemented
8C22×#11929609—21no—
9C22×#11929608—21no—
10C2×#14579158C2211yescharacteristic
11#19098758—22no—
12#19098758C414yescomplemented
13#19098641—22no—
14#19098641—22no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C4⋊(C79⋊C8)C2211yescharacteristic
17C2×#14579081—21no—
18C2×#14579081—21no—
19C2×#14579081—21no—
20C2×C22⋊(C79⋊C8)—21no—
21C2×C4×C79⋊C8C2211yescharacteristic
22#19098306—22no—
23C2×C4×D4×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×#102×C79—21no—
26#384×C79C2211yescharacteristic
27#340×C79—21no—
28C22×D316—21no—
29C22×D316—21no—
30C22×D316—21no—
31C2×#11929609—24no—
32C2×#11929609—42no—
33C2×#11929609—42no—
34C2×#11929608—24no—
35C2×#11929608—42no—
36C2×#11929608—42no—
37C22×C79⋊C8—21no—
38C22×C79⋊C8—21no—
39C22×C79⋊C8D411yescharacteristic
40C22×C79⋊C8D411yescharacteristic
41#14579158—21no—
42#14579158—21no—
43#14579158—21no—
44#14579158C2×C412yes—
45C4⋊(C79⋊C8)—21no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)C2×C412yes—
48C4⋊(C79⋊C8)D412yes—
49#14579081—24no—
50#14579081—42no—
51#14579081—42no—
52C22⋊(C79⋊C8)—24no—
53C4×C79⋊C8—21no—
54C4×C79⋊C8—21no—
55C22×D4×C79D411yescharacteristic
56C23×C4×C79—21no—
57C22×C8×C79—21no—
58C4×D4×C79—21no—
59C4×D4×C79—21no—
60C4×D4×C79C2×C412yes—
61C4×D4×C79D412yes—
62C2×C4⋊C4×C79—21no—
63C2×C4⋊C4×C79—21no—
64C2×C4⋊C4×C79—21no—
65C2×C4⋊C4×C79D411yescharacteristic
66C2×C22⋊C4×C79—21no—
67C2×C42×C79C2311yescharacteristic
68C4⋊C8×C79—21no—
69C4⋊C8×C79—21no—
70#102×C79—42no—
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—
88#11929609—48no—
89#11929608—48no—
90C2×C79⋊C8—22no—
91C2×C79⋊C8—21no—
92C2×C79⋊C8—21no—
93C2×C79⋊C8—22no—
94C2×C79⋊C8—21no—
95C2×C79⋊C8—21no—
96C2×C79⋊C8—21no—
97C2×C79⋊C8—41no—
98C2×C79⋊C8—41no—
99C2×C79⋊C8—41no—
100C2×C79⋊C8—41no—
101C2×C79⋊C8D812yescomplemented
102C24×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79—21no—
106C2×D4×C79—21no—
107C2×D4×C79—21no—
108C2×D4×C79C22⋊C412yes—
109C22×C4×C79—21no—
110C22×C4×C79—22no—
111C22×C4×C79—22no—
112C22×C4×C79—41no—
113C22×C4×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79—41no—
116C22×C4×C79—41no—
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120C2×C8×C79—41no—
121C2×C8×C79—41no—
122C2×C8×C79—41no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79—41no—
127C4⋊C4×C79—41no—
128C4⋊C4×C79—81no—
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×C79—81no—
165C2×C4×C79—81no—
166C2×C4×C79—81no—
167C2×C4×C79C2×C22⋊C411yescharacteristic
168C2×C4×C79C4×D411yescharacteristic
169C2×C4×C79C4×D411yescharacteristic
170C2×C4×C79#12011yescharacteristic
171C2×C4×C79#12111yescharacteristic
172C2×C4×C79#12311yescharacteristic
173C2×C4×C79C2×D811yescharacteristic
174C2×C4×C79#13511yescharacteristic
175C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C79#13711yescharacteristic
181D316—42no—
182D316—42no—
183D316—42no—
184D316—42no—
185D316—82no—
186D316—82no—
187D316—81no—
188D316—81no—
189D316—81no—
190D316—81no—
191C8×C79—81no—
192C79⋊C8—22no—
193C79⋊C8—42no—
194C79⋊C8—42no—
195C79⋊C8—41no—
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—161no—
221C4×C79#41711yescharacteristic
222C4×C79#41811yescharacteristic
223C4×C79C4×D811yescharacteristic
224C4×C79#44211yescharacteristic
225C4×C79#44711yescharacteristic
226C4×C79#44911yescharacteristic
227C4×C79#46811yescharacteristic
228C4×C79#46911yescharacteristic
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×#436—791no—
241#1957—791no—
242#1802D7911yescharacteristic, complemented
243C2×#331—791no—
244#1296—791no—
245#1277—791no—
246C79#922911yescharacteristic, complemented, Sylow
247C22×D8—1581no—
248C22×M16—1581no—
249C2×#127—791no—
250C2×C4×D4C2×D7911yescharacteristic
251#459—794no—
252#459—1582no—
253#436—1582no—
254#436—1582no—
255C2×C4⋊C8—791no—
256C2×C4⋊C8—791no—
257C2×C4⋊C8C2×D7911yescharacteristic
258C2×#102—21no—
259C2×#102—1581no—
260C2×#102—1581no—
261C2×#102—1581no—
262C2×C22⋊C8—1581no—
263C2×C4×C8—791no—
264#384C2×D7911yescharacteristic
265#340—21no—
266#331—1582no—
267C22×D4—1581no—
268C22×D4—1581no—
269C22×D4—1581no—
270C22×D4C22⋊D7911yescharacteristic
271C23×C4—21no—
272C2×D8—3162no—
273C2×D8—3162no—
274C2×D8—1584no—
275C2×M16—3162no—
276C2×M16—3162no—
277C2×M16—1584no—
278C22×C8—21no—
279C22×C8—791no—
280C22×C8—791no—
281C22×C8—1581no—
282C22×C8—1581no—
283#127—792no—
284#127—1581no—
285#127—1581no—
286#127—1581no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C4×D7912yes—
290C4×D4D31612yes—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4C22⋊D7911yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D7911yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—792no—
300C4⋊C8—792no—
301C4⋊C8—1581no—
302C4⋊C8—1581no—
303#102—42no—
304#102—3162no—
305#102—3162no—
306#102—1584no—
307C22⋊C8—1584no—
308C4×C8—1581no—
309C4×C8—1581no—
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—41no—
342C22×C4D4×D7911yescharacteristic
343C22×C4C2×C22⋊D7911yescharacteristic
344C22×C4#1192962311yescharacteristic
345D8—3168no—
346M16—3168no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—3161no—
351C2×C8—3161no—
352C2×C8—3161no—
353C2×C8—3161no—
354C2×C8—792no—
355C2×C8—1582no—
356C2×C8—1581no—
357C2×C8—1582no—
358C2×C8—1581no—
359C2×C8—1581no—
360C2×C8—1581no—
361C2×C8—1581no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—81no—
368C4⋊C4(C2×D79)⋊C412yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42C2×C4×D7911yescharacteristic
372C42D4×D7911yescharacteristic
373C42C2×D31611yescharacteristic
374C42#1192962311yescharacteristic
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#1457913311yescharacteristic
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—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4C2×(C2×D79)⋊C411yescharacteristic
417C2×C4(C22⋊D79)⋊C411yescharacteristic
418C2×C4(C22⋊D79)⋊C411yescharacteristic
419C2×C4#1457913111yescharacteristic
420C2×C4#1457913311yescharacteristic
421C2×C4#1457914511yescharacteristic
422C2×C4D8×D7911yescharacteristic
423C2×C4#1457918211yescharacteristic
424C2×C4#1457918511yescharacteristic
425C2×C4#1457918611yescharacteristic
426C2×C4#1457918611yescharacteristic
427C2×C4#1457918711yescharacteristic
428C2×C4#1457918811yescharacteristic
429C2×C4#1457919211yescharacteristic
430C8—81no—
431C8—3162no—
432C8—3161no—
433C8—3162no—
434C8—1582no—
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#1909844011yescharacteristic
450C22#1909875611yescharacteristic
451C22#1909878011yescharacteristic
452C22#1909881111yescharacteristic
453C22#1909881411yescharacteristic
454C22#1909882811yescharacteristic
455C22#1909883111yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—161no—
460C4#1909856211yescharacteristic
461C4#1909856711yescharacteristic
462C4#1909864311yescharacteristic
463C4#1909867211yescharacteristic
464C4#1909869811yescharacteristic
465C4#1909870911yescharacteristic
466C4#1909881111yescharacteristic
467C4#1909881411yescharacteristic
468C2—82no—
469C2—12642no—
470C2#2376286711yescharacteristic
471C2#2376287711yescharacteristic
472C2#2376293511yescharacteristic
473C2#2376296311yescharacteristic
474C2#2376374611yescharacteristic
475C2#2376383611yescharacteristic
476C2#2376390611yescharacteristic
subgroupG/H
centerC23#14579133
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1802C2
socleC23×C79#121
prime radicalC2×#19098758C2
cosocleC2×C42×C79C23
hypercenter#1802D79
hypocenterC79#9229
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#19098641C2semidirect
3. #23764205C2semidirect
4. C2×#19098306C2semidirect
5. #23761283C2semidirect
6. #23761234C2semidirect
7. C79×#1802C2semidirect
12. #19098758C4semidirect
101. C2×C79⋊C8D8semidirect
242. #1802D79semidirect
246. C79#9229semidirect

Maximal subgroups

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

subgroupindex
C2×#190987582
C2×#190986412
#237642052
C2×#190983062
#237612832
#237612342
C79×#18022
#922979