Finite Group Databasebeta

← #34192064

Subgroups — #34192064

Order 20224.

574 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098755C211yesmaximal, characteristic
2#23764207C211yesmaximal, characteristic, complemented
3#23764193C211yesmaximal, characteristic, complemented
4#23763401C211yesmaximal, characteristic, complemented
5C2×#19098313C211yesmaximal, characteristic, complemented
6#23761284C211yesmaximal, characteristic, complemented
7C2×#331×C79C211yesmaximal, characteristic, complemented
8C22×#11929609—21no—
9C2×C4×D316C2211yescharacteristic
10#19098755—22no—
11#19098755—22no—
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579083—21no—
14C2×#14579082—21no—
15C2×#14579082—21no—
16C2×#14579082—21no—
17C2×(C2×D79)⋊C8—21no—
18C2×(C2×D79)⋊C8—21no—
19C2×C4×C79⋊C8C2211yescharacteristic
20#19098433C2211yescharacteristic
21#19098326—21no—
22#19098313—22no—
23C2×#127×C79C2211yescharacteristic
24C2×C4⋊C8×C79C2211yescharacteristic
25C2×C4⋊C8×C79C2211yescharacteristic
26#331×C79—22no—
27C22×D316D411yescharacteristic
28C22×C4×D79—21no—
29C2×#11929609—24no—
30C2×#11929609—42no—
31C2×#11929609—42no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8D411yescharacteristic
34C22×C79⋊C8D411yescharacteristic
35C4×D316—21no—
36C4×D316—21no—
37C4×D316—21no—
38C4×D316—21no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79—21no—
41C2×C4⋊Dic79—21no—
42C2×C4⋊Dic79D411yescharacteristic
43C2×(C2×D79)⋊C4—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)D412yes—
46#14579083—42no—
47#14579082—24no—
48#14579082—42no—
49#14579082—42no—
50(C2×D79)⋊C8—42no—
51(C2×D79)⋊C8—42no—
52C4×C79⋊C8—21no—
53C4×C79⋊C8—21no—
54C22×D4×C79—21no—
55C22×D4×C79—21no—
56C22×D4×C79—21no—
57C22×C8×C79—21no—
58C22×C8×C79—21no—
59#127×C79—21no—
60#127×C79—21no—
61#127×C79—21no—
62#127×C79D412yes—
63C2×C42×C79C2311yescharacteristic
64C4⋊C8×C79—21no—
65C4⋊C8×C79—21no—
66C4⋊C8×C79—21no—
67C4⋊C8×C79—21no—
68C23×D79—21no—
69C2×D316—21no—
70C2×D316—21no—
71C2×D316—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316SD1612yes—
75C22×Dic79—21no—
76C22×Dic79—41no—
77C2×C4×D79—41no—
78C2×C4×D79—41no—
79C2×C4×D79—41no—
80C2×C4×D79—41no—
81C2×C4×D79—41no—
82C2×C4×D79—41no—
83#11929609—48no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8D812yescomplemented
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98C4⋊Dic79—41no—
99C4⋊Dic79—81no—
100(C2×D79)⋊C4—41no—
101(C2×D79)⋊C4—41no—
102C24×C79—41no—
103C2×D4×C79—22no—
104C2×D4×C79—22no—
105C2×D4×C79—22no—
106C2×D4×C79—41no—
107C2×D4×C79—41no—
108C2×D4×C79—41no—
109C2×D4×C79—41no—
110C2×D4×C79—41no—
111C2×D4×C79—41no—
112C2×D4×C79—41no—
113C2×D4×C79—41no—
114C2×D4×C79—41no—
115C2×D4×C79—41no—
116C2×D4×C79—41no—
117C2×D4×C79—81no—
118C2×D4×C79—81no—
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D411yescharacteristic
121C22×C4×C79#4111yescharacteristic
122C2×C8×C79—41no—
123C2×C8×C79—41no—
124C2×C8×C79—41no—
125C2×C8×C79—41no—
126C2×C8×C79—41no—
127C2×C8×C79—41no—
128C42×C79C2×D411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79#4111yescharacteristic
132C23×C79—42no—
133C23×C79—81no—
134C23×C79—81no—
135C23×C79—81no—
136C23×C79—81no—
137C23×C79—81no—
138C23×C79—81no—
139C23×C79#12111yescharacteristic
140C22×D79—22no—
141C22×D79—41no—
142C22×D79—41no—
143C22×D79—41no—
144C22×D79—41no—
145C22×D79—41no—
146C22×D79—41no—
147D4×C79—42no—
148D4×C79—42no—
149D4×C79—42no—
150D4×C79—42no—
151D4×C79—82no—
152D4×C79—82no—
153D4×C79—81no—
154D4×C79—81no—
155D4×C79—81no—
156D4×C79—81no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79#12011yescharacteristic
160C2×C4×C79#12111yescharacteristic
161C2×C4×C79#12111yescharacteristic
162C2×C4×C79#12311yescharacteristic
163C2×C4×C79#12311yescharacteristic
164C2×C4×C79#12411yescharacteristic
165C2×C4×C79C2×D811yescharacteristic
166C2×C4×C79C2×SD1611yescharacteristic
167C2×C4×C79#13511yescharacteristic
168C2×C4×C79#13511yescharacteristic
169C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176C2×Dic79—81no—
177C2×Dic79—81no—
178C2×Dic79—81no—
179D316—22no—
180D316—22no—
181D316—41no—
182D316—41no—
183C4×D79—81no—
184C4×D79—81no—
185C4×D79—81no—
186C4×D79—81no—
187C8×C79—81no—
188C8×C79—81no—
189C79⋊C8—22no—
190C79⋊C8—42no—
191C79⋊C8—41no—
192C22×C79—82no—
193C22×C79—82no—
194C22×C79—82no—
195C22×C79—161no—
196C22×C79—161no—
197C22×C79—161no—
198C22×C79—161no—
199C22×C79#39411yescharacteristic
200C22×C79#45911yescharacteristic
201C22×C79#46311yescharacteristic
202C22×C79#46511yescharacteristic
203C22×C79#46911yescharacteristic
204C22×C79#47111yescharacteristic
205C22×C79#47211yescharacteristic
206C2×D79—42no—
207C2×D79—42no—
208C2×D79—42no—
209C2×D79—81no—
210C2×D79—81no—
211C2×D79—81no—
212C2×D79—81no—
213C4×C79—21no—
214C4×C79—21no—
215C4×C79#44911yescharacteristic
216C4×C79#45011yescharacteristic
217C4×C79#45911yescharacteristic
218C4×C79#46311yescharacteristic
219C4×C79#46611yescharacteristic
220C4×C79#46911yescharacteristic
221C4×C79#48111yescharacteristic
222C4×C79#48211yescharacteristic
223Dic79—81no—
224Dic79—161no—
225#10473—791nomaximal, Sylow
226C2×C79—162no—
227C2×C79#166511yescharacteristic
228C2×C79#167711yescharacteristic
229C2×C79#168711yescharacteristic
230C2×C79#170111yescharacteristic
231C2×C79#203411yescharacteristic
232C2×C79#205111yescharacteristic
233C2×C79#205411yescharacteristic
234D79—82no—
235C2×#459—791no—
236#1957—791no—
237#1954—791no—
238#1802—791no—
239C2×#334—791no—
240C2×#331D7911yescharacteristic, complemented
241#1296—791no—
242C79#1047311yescharacteristic, complemented, Sylow
243C22×D8—1581no—
244C2×#127C2×D7911yescharacteristic
245C2×C4×D4—791no—
246#459—1582no—
247#459—1582no—
248C2×C4⋊C8—791no—
249C2×C4⋊C8C2×D7911yescharacteristic
250C2×C4⋊C8C2×D7911yescharacteristic
251C2×#102—1581no—
252C2×#102—1581no—
253C2×#102—1581no—
254C2×#102—1581no—
255C2×C22⋊C8—1581no—
256C2×C22⋊C8—1581no—
257C2×C4×C8—791no—
258#384—791no—
259#340—1581no—
260#334—1582no—
261#331—22no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—791no—
266C23×C4—1581no—
267C2×D8—3162no—
268C2×D8—3162no—
269C2×D8—1584no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8—791no—
273C22×C8—791no—
274C22×C8—1581no—
275#127—21no—
276#127—21no—
277#127—21no—
278#127C22⋊D7912yes—
279C4×D4—1581no—
280C4×D4—1581no—
281C4×D4—1581no—
282C4×D4—1581no—
283C2×C4⋊C4—791no—
284C2×C4⋊C4—1581no—
285C2×C4⋊C4—1581no—
286C2×C4⋊C4—1581no—
287C2×C22⋊C4—1581no—
288C2×C42C22×D7911yescharacteristic
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—792no—
294C4⋊C8—1581no—
295#102—3162no—
296#102—3162no—
297#102—3162no—
298#102—1584no—
299C22⋊C8—3162no—
300C22⋊C8—3162no—
301C4×C8—1581no—
302C4×C8—1581no—
303C24—41no—
304C24—1581no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—81no—
320C2×D4—81no—
321C2×D4—792no—
322C2×D4—1581no—
323C2×D4—1581no—
324C2×D4—1581no—
325C2×D4—1581no—
326C2×D4—1581no—
327C22×C4—3161no—
328C22×C4—3161no—
329C22×C4—3161no—
330C22×C4—3161no—
331C22×C4—3161no—
332C22×C4—3161no—
333C22×C4—3161no—
334C22×C4—1581no—
335C22×C4D4×D7911yescharacteristic
336C22×C4D4×D7911yescharacteristic
337C22×C4#1192962411yescharacteristic
338D8—3168no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—3161no—
346C2×C8—3161no—
347C2×C8—792no—
348C2×C8—1582no—
349C2×C8—1581no—
350C2×C8—1581no—
351C2×C8—1581no—
352C2×C8—1581no—
353C2×C8—1581no—
354C4⋊C4—3161no—
355C4⋊C4—3161no—
356C4⋊C4—3161no—
357C4⋊C4—3161no—
358C4⋊C4—6321no—
359C4⋊C4—1581no—
360C4⋊C4—1581no—
361C22⋊C4—3161no—
362C22⋊C4—3161no—
363C42D4×D7911yescharacteristic
364C42C2×C22⋊D7911yescharacteristic
365C42#1192962311yescharacteristic
366C42#1192962411yescharacteristic
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—3161no—
375C23—3161no—
376C23—3161no—
377C23—3161no—
378C23—3161no—
379C23—3161no—
380C23—1582no—
381C23#1457913411yescharacteristic
382D4—42no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—82no—
387D4—82no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—3161no—
393D4—3161no—
394D4—1582no—
395D4—1582no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—3161no—
399C2×C4—3161no—
400C2×C4—3161no—
401C2×C4—6321no—
402C2×C4—6321no—
403C2×C4—6321no—
404C2×C4—6321no—
405C2×C4—6321no—
406C2×C4—6321no—
407C2×C4—6321no—
408C2×C4#1457912911yescharacteristic
409C2×C4#1457913311yescharacteristic
410C2×C4#1457913411yescharacteristic
411C2×C4#1457914311yescharacteristic
412C2×C4#1457914411yescharacteristic
413C2×C4#1457914711yescharacteristic
414C2×C4D8×D7911yescharacteristic
415C2×C4SD16×D7911yescharacteristic
416C2×C4#1457918011yescharacteristic
417C2×C4#1457918211yescharacteristic
418C2×C4#1457918511yescharacteristic
419C2×C4#1457918511yescharacteristic
420C2×C4#1457918811yescharacteristic
421C2×C4#1457918811yescharacteristic
422C8—81no—
423C8—81no—
424C8—3162no—
425C8—3161no—
426C8—1582no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—3162no—
435C22—3162no—
436C22—3162no—
437C22—6321no—
438C22—6321no—
439C22—6321no—
440C22—6321no—
441C22#1909847811yescharacteristic
442C22#1909875711yescharacteristic
443C22#1909877711yescharacteristic
444C22#1909879111yescharacteristic
445C22#1909881511yescharacteristic
446C22#1909882511yescharacteristic
447C22#1909883211yescharacteristic
448C4—21no—
449C4—21no—
450C4—6321no—
451C4—12641no—
452C4#1909870511yescharacteristic
453C4#1909871211yescharacteristic
454C4#1909875611yescharacteristic
455C4#1909878011yescharacteristic
456C4#1909879811yescharacteristic
457C4#1909881511yescharacteristic
458C4#1909888011yescharacteristic
459C4#1909888511yescharacteristic
460C2—162no—
461C2—6322no—
462C2#2376274011yescharacteristic
463C2#2376280811yescharacteristic
464C2#2376287811yescharacteristic
465C2#2376296011yescharacteristic
466C2#2376458711yescharacteristic
467C2#2376469211yescharacteristic
468C2#2376470911yescharacteristic
subgroupG/H
centerC23#14579134
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC2×#331×C79C2
socleC23×C79#121
prime radicalC2×#19098755C2
hypercenterC2×#331D79
hypocenterC79#10473
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. #23764207C2semidirect
3. #23764193C2semidirect
4. #23763401C2semidirect
5. C2×#19098313C2semidirect
6. #23761284C2semidirect
7. C2×#331×C79C2semidirect
92. C2×C79⋊C8D8semidirect
240. C2×#331D79semidirect
242. C79#10473semidirect

Maximal subgroups

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

subgroupindex
C2×#190987552
#237642072
#237641932
#237634012
C2×#190983132
#237612842
C2×#331×C792
#1047379