Finite Group Databasebeta

← #34182909

Subgroups — #34182909

Order 20224.

540 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772517C211yesmaximal, characteristic, complemented
2#23764195C211yesmaximal, characteristic, complemented
3#23761791C211yesmaximal, characteristic, complemented
4#23761782C211yesmaximal, characteristic, complemented
5#23761251C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7C4×AGL(1,ℤ/8)×C79C211yesmaximal, characteristic
8#19098694—21no—
9#19098641—21no—
10#19098635—22no—
11#19098620—21no—
12#19098616C2211yescharacteristic
13#19098616C2211yescharacteristic
14#19098587—21no—
15C2×C4⋊(C79⋊C8)C2211yescharacteristic
16C2×C22⋊(C79⋊C8)—22no—
17C2×C4×C79⋊C8C2211yescharacteristic
18Q8⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)—21no—
20D4⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C79C411yescharacteristic
23C4×#41×C79C2211yescharacteristic
24C2×C4×D4×C79C2211yescharacteristic
25#442×C79—21no—
26#440×C79—21no—
27C4×SD16×C79—21no—
28C4×D8×C79—21no—
29#428×C79C411yescharacteristic
30#418×C79C411yescharacteristic
31#417×C79C411yescharacteristic
32C4×M16×C79C2211yescharacteristic
33C2×#11929608—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8D412yes—
36C4⋊(C79⋊C8)—22no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)D411yescharacteristic
41C4⋊(C79⋊C8)D411yescharacteristic
42C22⋊(C79⋊C8)—22no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—44no—
46#14579072—22no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8—21no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—21no—
51C2×#41×C79C2×C411yescharacteristic
52C22×D4×C79C2×C411yescharacteristic
53C23×C4×C79—21no—
54AGL(1,ℤ/8)×C79—22no—
55AGL(1,ℤ/8)×C79—22no—
56C2×SD16×C79—21no—
57C2×D8×C79—21no—
58C2×M16×C79C2×C411yescharacteristic
59C2×M16×C79C2×C411yescharacteristic
60C4×Q8×C79D411yescharacteristic
61C4×Q8×C79D411yescharacteristic
62C4×D4×C79—21no—
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79—21no—
68C4×D4×C79D411yescharacteristic
69C4×D4×C79D411yescharacteristic
70#117×C79—21no—
71#117×C79C2×C411yescharacteristic
72C2×C4⋊C4×C79C2×C411yescharacteristic
73C2×C22⋊C4×C79—21no—
74C2×C42×C79—21no—
75C2×C42×C79C2311yescharacteristic
76#107×C79—21no—
77#106×C79—21no—
78#103×C79—21no—
79#102×C79—21no—
80#102×C79—21no—
81#102×C79—21no—
82#97×C79—21no—
83C4×C8×C79—21no—
84#11929608—82no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—42no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C24×C79—21no—
94#41×C79—21no—
95#41×C79—21no—
96#41×C79—21no—
97#41×C79—21no—
98C2×Q8×C79C22⋊C411yescharacteristic
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—21no—
104C2×D4×C79—21no—
105C2×D4×C79C22⋊C411yescharacteristic
106C2×D4×C79M1612yescomplemented
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—42no—
112C22×C4×C79—42no—
113C22×C4×C79—41no—
114C22×C4×C79—41no—
115C22×C4×C79C22×C411yescharacteristic
116C22×C4×C79C2×D411yescharacteristic
117C22×C4×C79#4111yescharacteristic
118SD16×C79—24no—
119D8×C79—24no—
120M16×C79—21no—
121M16×C79—21no—
122M16×C79—21no—
123M16×C79M1612yescomplemented
124C2×C8×C79—21no—
125C2×C8×C79—21no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79—21no—
129C4⋊C4×C79—21no—
130C4⋊C4×C79C22⋊C411yescharacteristic
131C4⋊C4×C79C22⋊C411yescharacteristic
132C22⋊C4×C79—21no—
133C22⋊C4×C79—21no—
134C22⋊C4×C79—21no—
135C22⋊C4×C79—41no—
136C22⋊C4×C79—41no—
137C42×C79—21no—
138C42×C79—21no—
139C42×C79—21no—
140C42×C79C2×D411yescharacteristic
141C42×C79C2×D411yescharacteristic
142C42×C79C2×D411yescharacteristic
143C42×C79#4111yescharacteristic
144C23×C79—21no—
145C23×C79—22no—
146C23×C79—42no—
147C23×C79—42no—
148C23×C79—41no—
149C23×C79—41no—
150C23×C79C4×D411yescharacteristic
151Q8×C79—21no—
152Q8×C79—21no—
153D4×C79—21no—
154D4×C79—22no—
155D4×C79—21no—
156D4×C79—22no—
157D4×C79—41no—
158D4×C79—41no—
159D4×C79—41no—
160D4×C79—41no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—21no—
165C2×C4×C79—21no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—42no—
172C2×C4×C79—41no—
173C2×C4×C79—81no—
174C2×C4×C79—82no—
175C2×C4×C79—81no—
176C2×C4×C79—81no—
177C2×C4×C79C2×C22⋊C411yescharacteristic
178C2×C4×C79C4×D411yescharacteristic
179C2×C4×C79#12011yescharacteristic
180C2×C4×C79#12111yescharacteristic
181C2×C4×C79#12111yescharacteristic
182C2×C4×C79#12311yescharacteristic
183C2×C4×C79C2×M1611yescharacteristic
184C2×C4×C79C2×M1611yescharacteristic
185C2×C4×C79#13111yescharacteristic
186C2×C4×C79#13111yescharacteristic
187C8×C79—22no—
188C8×C79—41no—
189C79⋊C8—42no—
190C79⋊C8—42no—
191C79⋊C8—41no—
192C79⋊C8—41no—
193C79⋊C8—81no—
194C22×C79—21no—
195C22×C79—21no—
196C22×C79—44no—
197C22×C79—41no—
198C22×C79—42no—
199C22×C79—41no—
200C22×C79—41no—
201C22×C79—82no—
202C22×C79—81no—
203C22×C79—81no—
204C22×C79#38611yescharacteristic
205C22×C79#43511yescharacteristic
206C22×C79#43611yescharacteristic
207C4×C79—21no—
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—41no—
211C4×C79—41no—
212C4×C79—41no—
213C4×C79—81no—
214C4×C79#40711yescharacteristic
215C4×C79#40811yescharacteristic
216C4×C79#43511yescharacteristic
217C4×C79#43611yescharacteristic
218C4×C79#45311yescharacteristic
219C4×C79#45411yescharacteristic
220#8956—791nomaximal, Sylow
221C2×C79—21no—
222C2×C79—21no—
223C2×C79—81no—
224C2×C79—82no—
225C2×C79#187011yescharacteristic
226C2×C79#188111yescharacteristic
227C2×C79#188311yescharacteristic
228#2989—791no—
229C4×AGL(1,ℤ/8)D7911yescharacteristic
230#1954—791no—
231#1478—791no—
232#1476—791no—
233#1283—791no—
234#1278—791no—
235C79#895611yescharacteristic, complemented, Sylow
236C2×AGL(1,ℤ/8)Dic7911yescharacteristic
237C4×#41C2×D7911yescharacteristic
238C2×C4×D4C2×D7911yescharacteristic
239#446—1581no—
240#442—21no—
241#440—21no—
242C4×SD16—21no—
243C4×D8—21no—
244#436—1581no—
245#435—1582no—
246#432—1581no—
247#431—791no—
248#431—791no—
249#428Dic7911yescharacteristic
250#423—1581no—
251C2×C4⋊C8—791no—
252#418Dic7911yescharacteristic
253#417Dic7911yescharacteristic
254C2×C22⋊C8—1582no—
255C4×M16C2×D7911yescharacteristic
256C2×C4×C8—791no—
257Q8⋊C8—1581no—
258D4⋊C8—1581no—
259D4⋊C8—1581no—
260D4⋊C8—1581no—
261C2×#41C2×Dic7911yescharacteristic
262C22×D4C2×Dic7911yescharacteristic
263C23×C4—21no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)—22no—
266C2×SD16—21no—
267C2×D8—21no—
268C2×M16—1582no—
269C2×M16C2×Dic7911yescharacteristic
270C2×M16C2×Dic7911yescharacteristic
271C22×C8—792no—
272C22×C8—1581no—
273C4×Q8C22⋊D7911yescharacteristic
274C4×Q8C22⋊D7911yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4C22⋊D7911yescharacteristic
282C4×D4C22⋊D7911yescharacteristic
283#117—21no—
284#117C2×Dic7911yescharacteristic
285C2×C4⋊C4C2×Dic7911yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C22×D7911yescharacteristic
289#107—21no—
290#106—21no—
291C4⋊C8—791no—
292C4⋊C8—791no—
293C4⋊C8—1582no—
294C4⋊C8—1581no—
295C4⋊C8—1581no—
296C4⋊C8—1581no—
297#103—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—3164no—
302C22⋊C8—1581no—
303C22⋊C8—1582no—
304C22⋊C8—1581no—
305#97—21no—
306#97—1582no—
307C4×C8—21no—
308C4×C8—1581no—
309C4×C8—1581no—
310C4×C8—1581no—
311C4×C8—1581no—
312C24—21no—
313#41—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317C2×Q8C22⋊Dic7911yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4C22⋊Dic7911yescharacteristic
325C2×D4#1192960812yescomplemented
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C22×Dic7911yescharacteristic
335C22×C4D4×D7911yescharacteristic
336C22×C4#1192962311yescharacteristic
337SD16—24no—
338D8—24no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16—6322no—
343M16#1192960812yescomplemented
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—3161no—
347C2×C8—3161no—
348C2×C8—3162no—
349C2×C8—3161no—
350C2×C8—1582no—
351C2×C8—1581no—
352C2×C8—1581no—
353C2×C8—1582no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4C22⋊Dic7911yescharacteristic
359C4⋊C4C22⋊Dic7911yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42D4×D7911yescharacteristic
369C42C2×C22⋊D7911yescharacteristic
370C42C2×C22⋊D7911yescharacteristic
371C42#1192962311yescharacteristic
372C23—22no—
373C23—21no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23D4×Dic7911yescharacteristic
379Q8—21no—
380Q8—21no—
381D4—22no—
382D4—21no—
383D4—22no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4C2×C22⋊Dic7911yescharacteristic
406C2×C4D4×Dic7911yescharacteristic
407C2×C4#1457912911yescharacteristic
408C2×C4#1457913311yescharacteristic
409C2×C4#1457913311yescharacteristic
410C2×C4#1457914211yescharacteristic
411C2×C4C2×#1192960811yescharacteristic
412C2×C4C2×#1192960811yescharacteristic
413C2×C4#1457916811yescharacteristic
414C2×C4#1457916811yescharacteristic
415C8—22no—
416C8—41no—
417C8—3162no—
418C8—3162no—
419C8—3161no—
420C8—3161no—
421C8—6321no—
422C22—21no—
423C22—21no—
424C22—44no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22#1909844311yescharacteristic
433C22#1909863511yescharacteristic
434C22#1909864111yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4#1909852711yescharacteristic
443C4#1909853011yescharacteristic
444C4#1909863511yescharacteristic
445C4#1909864111yescharacteristic
446C4#1909872911yescharacteristic
447C4#1909873411yescharacteristic
448C2—21no—
449C2—21no—
450C2—82no—
451C2—81no—
452C2#2376369811yescharacteristic
453C2#2376376411yescharacteristic
454C2#2376377811yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC4×AGL(1,ℤ/8)×C79C2
socleC22×C79#386
prime radicalC2×AGL(1,ℤ/8)×C79C4
cosocleC2×C42×C79C23
hypercenterC4×AGL(1,ℤ/8)D79
hypocenterC79#8956
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23772517C2semidirect
2. #23764195C2semidirect
3. #23761791C2semidirect
4. #23761782C2semidirect
5. #23761251C2semidirect
6. #23761238C2semidirect
106. C2×D4×C79M16semidirect
123. M16×C79M16semidirect
235. C79#8956semidirect
325. C2×D4#11929608semidirect
343. M16#11929608semidirect

Maximal subgroups

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

subgroupindex
#237725172
#237641952
#237617912
#237617822
#237612512
#237612382
C4×AGL(1,ℤ/8)×C792
#895679