Finite Group Databasebeta

← #34187259

Subgroups — #34187259

Order 20224.

700 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771701C211yesmaximal, characteristic
2#23763162C211yesmaximal, characteristic, complemented
3C2×#19098299C212yesmaximal, complemented
4C2×#19098296C212yesmaximal
5C79×C4⋊#95C211yesmaximal, characteristic, complemented
6C2×#14579149C411yescharacteristic, complemented
7C2×#14579140C2211yescharacteristic
8C2×#14579135C2211yescharacteristic
9C2×C22⋊(C79⋊C8)C2212yes—
10C2×#14579072—21no—
11#19098469C411yescharacteristic, complemented
12#19098465C411yescharacteristic, complemented
13#19098454—21no—
14#19098454—21no—
15#19098442—21no—
16#19098442—21no—
17#19098429C411yescharacteristic, complemented
18#19098318—21no—
19#19098299—24no—
20#19098296—24no—
21C22×C4⋊C4×C79—21no—
22C22×C42×C79C2211yescharacteristic
23C2×#95×C79—21no—
24C2×#95×C79C2212yes—
25C22×Dic158C2×C411yescharacteristic
26C22×D316C2×C411yescharacteristic
27C22×C22⋊D79—21no—
28C22×C79⋊C8—22no—
29#14579149—42no—
30#14579140—22no—
31#14579140—22no—
32#14579135—22no—
33#14579135—22no—
34C2×C4⋊Dic79C2×C411yescharacteristic
35C2×C4⋊Dic79C2×C411yescharacteristic
36C2×Dic79⋊C4—21no—
37C2×C22⋊Dic79—21no—
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×(C2×D79)⋊C4—21no—
41C2×(C2×D79)⋊C4—21no—
42C2×(C2×D79)⋊C4—21no—
43C22⋊(C79⋊C8)—22no—
44C22⋊(C79⋊C8)D414yes—
45#14579072—42no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—21no—
49C79⋊#95—21no—
50C23×C4×C79—22no—
51C23×C4×C79D411yescharacteristic
52C23×C4×C79D411yescharacteristic
53C23×C4×C79C2311yescharacteristic
54C2×C4⋊C4×C79—24no—
55C2×C4⋊C4×C79—42no—
56C2×C4⋊C4×C79—42no—
57C2×C42×C79—42no—
58C2×C42×C79C2×C411yescharacteristic
59C2×C42×C79C2×C411yescharacteristic
60C2×C42×C79D411yescharacteristic
61C2×C42×C79D411yescharacteristic
62#95×C79—22no—
63#95×C79—22no—
64#95×C79—42no—
65C23×D79—21no—
66C2×Dic158—22no—
67C2×Dic158—41no—
68C2×Dic158—41no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×C22⋊D79—41no—
73C2×C22⋊D79—41no—
74C2×C22⋊D79—41no—
75C2×C22⋊D79—41no—
76C2×C22⋊D79—41no—
77C2×C22⋊D79—41no—
78C22×Dic79—21no—
79C22×Dic79—21no—
80C22×Dic79—21no—
81C2×C79⋊C8—24no—
82C2×C79⋊C8—42no—
83C2×C79⋊C8—42no—
84C4⋊Dic79—22no—
85C4⋊Dic79—22no—
86Dic79⋊C4—42no—
87C22⋊Dic79—41no—
88C22⋊Dic79—41no—
89(C2×D79)⋊C4—42no—
90(C2×D79)⋊C4—42no—
91(C2×D79)⋊C4—42no—
92(C2×D79)⋊C4—42no—
93(C2×D79)⋊C4—42no—
94C24×C79C2×D411yescharacteristic
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—22no—
98C22×C4×C79—22no—
99C22×C4×C79—21no—
100C22×C4×C79—21no—
101C22×C4×C79—24no—
102C22×C4×C79—41no—
103C22×C4×C79—41no—
104C22×C4×C79—41no—
105C22×C4×C79—41no—
106C22×C4×C79—42no—
107C22×C4×C79—42no—
108C22×C4×C79—42no—
109C22×C4×C79—42no—
110C22×C4×C79—42no—
111C22×C4×C79—42no—
112C22×C4×C79C22⋊C411yescharacteristic
113C22×C4×C79C22⋊C411yescharacteristic
114C22×C4×C79C22⋊C411yescharacteristic
115C22×C4×C79C22⋊C411yescharacteristic
116C22×C4×C79C22×C411yescharacteristic
117C22×C4×C79C2×D411yescharacteristic
118C22×C4×C79C2×D412yes—
119C22×C4×C79#4112yes—
120C4⋊C4×C79—48no—
121C42×C79—22no—
122C42×C79—22no—
123C42×C79—41no—
124C42×C79—41no—
125C23×C79—21no—
126C23×C79—21no—
127C23×C79—21no—
128C23×C79—21no—
129C23×C79C2×C22⋊C411yescharacteristic
130C23×C79#12411yescharacteristic
131C23×C79#12711yescharacteristic
132C23×C79AGL(1,ℤ/8)11yescharacteristic
133C23×C79AGL(1,ℤ/8)11yescharacteristic
134C23×C79#13711yescharacteristic
135C23×C79#13711yescharacteristic
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C22×D79—41no—
140C22×D79—41no—
141C22×D79—41no—
142C22×D79—41no—
143C2×C4×C79—24no—
144C2×C4×C79—24no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—21no—
148C2×C4×C79—21no—
149C2×C4×C79—21no—
150C2×C4×C79—21no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—21no—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—42no—
160C2×C4×C79—42no—
161C2×C4×C79—42no—
162C2×C4×C79—42no—
163C2×C4×C79—44no—
164C2×C4×C79—44no—
165C2×C4×C79—44no—
166C2×C4×C79—82no—
167C2×C4×C79—82no—
168C2×C4×C79—82no—
169C2×C4×C79—82no—
170C2×C4×C79#9912yes—
171C2×C4×C79#10412yes—
172C2×C4×C79C4×D412yes—
173C2×C4×C79#12112yes—
174C22⋊D79—81no—
175C22⋊D79—81no—
176C22⋊D79—81no—
177C22⋊D79—81no—
178C2×Dic79—41no—
179C2×Dic79—41no—
180C2×Dic79—41no—
181C2×Dic79—41no—
182C2×Dic79—41no—
183C2×Dic79—41no—
184C2×Dic79—41no—
185C2×Dic79—41no—
186C2×Dic79—41no—
187D316—82no—
188Dic158—82no—
189C79⋊C8—44no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—21no—
195C22×C79—21no—
196C22×C79—21no—
197C22×C79—21no—
198C22×C79—21no—
199C22×C79—21no—
200C22×C79—21no—
201C22×C79—21no—
202C22×C79#39011yescharacteristic
203C22×C79C2×#9911yescharacteristic
204C22×C79#41011yescharacteristic
205C22×C79#41711yescharacteristic
206C22×C79#41711yescharacteristic
207C22×C79C2×#10411yescharacteristic
208C22×C79#42111yescharacteristic
209C22×C79#48911yescharacteristic
210C22×C79#48911yescharacteristic
211C22×C79#49611yescharacteristic
212C22×C79#49711yescharacteristic
213C2×D79—81no—
214C2×D79—81no—
215C2×D79—81no—
216C2×D79—81no—
217C2×D79—81no—
218C2×D79—81no—
219C2×D79—81no—
220C4×C79—22no—
221C4×C79—22no—
222C4×C79—22no—
223C4×C79—22no—
224C4×C79—41no—
225C4×C79—41no—
226C4×C79—84no—
227Dic79—81no—
228Dic79—81no—
229Dic79—81no—
230#9682—791nomaximal, Sylow
231C2×C79—21no—
232C2×C79—21no—
233C2×C79—21no—
234C2×C79—21no—
235C2×C79#150411yescharacteristic
236C2×C79#150911yescharacteristic
237C2×C79#151211yescharacteristic
238C2×C79#151311yescharacteristic
239C2×C79#195911yescharacteristic
240C2×C79#196411yescharacteristic
241C2×C79#198011yescharacteristic
242D79—161no—
243#2850—791no—
244#1749—791no—
245C2×#328—792no—
246C2×#327—792no—
247C4⋊#95D7911yescharacteristic, complemented
248C79#968211yescharacteristic, complemented, Sylow
249C2×#124—791no—
250C2×C22⋊Q8—791no—
251C2×#121—791no—
252C22×C4⋊C4—21no—
253C22×C42C2×D7911yescharacteristic
254C2×C22⋊C8—792no—
255C2×#97—1581no—
256#391—791no—
257#390—791no—
258#388—1581no—
259#388—1581no—
260#386—1581no—
261#386—1581no—
262#382—791no—
263C2×#95—21no—
264C2×#95C2×D7912yes—
265#336—1581no—
266#328—1584no—
267#327—1584no—
268C22×Q8—791no—
269C22×D4—791no—
270C22×D4—1581no—
271C23×C4—22no—
272C23×C4C22⋊D7911yescharacteristic
273C23×C4C22⋊D7911yescharacteristic
274C23×C4C22×D7911yescharacteristic
275C22×C8—1582no—
276#124—3162no—
277C22⋊Q8—1582no—
278C22⋊Q8—1582no—
279#121—1582no—
280#121—1582no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—791no—
285C2×C4⋊C4—791no—
286C2×C4⋊C4—1581no—
287C2×C22⋊C4—1581no—
288C2×C22⋊C4—1581no—
289C2×C22⋊C4—1581no—
290C2×C22⋊C4—1581no—
291C2×C22⋊C4—1581no—
292C2×C22⋊C4—1581no—
293C2×C42—42no—
294C2×C42C4×D7911yescharacteristic
295C2×C42C4×D7911yescharacteristic
296C2×C42D31611yescharacteristic
297C2×C42D31611yescharacteristic
298C22⋊C8—794no—
299C22⋊C8—1582no—
300#97—3162no—
301#95—22no—
302#95—22no—
303#95—42no—
304#95—1581no—
305#95—1581no—
306#95—1581no—
307#95—1581no—
308C24—1581no—
309C24C2×C22⋊D7911yescharacteristic
310C2×Q8—3161no—
311C2×Q8—3161no—
312C2×Q8—1582no—
313C2×D4—3161no—
314C2×D4—3161no—
315C2×D4—3161no—
316C2×D4—3161no—
317C2×D4—3161no—
318C2×D4—3161no—
319C2×D4—3161no—
320C2×D4—3161no—
321C2×D4—1582no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—1581no—
340C22×C4—1581no—
341C22×C4—1581no—
342C22×C4(C2×D79)⋊C411yescharacteristic
343C22×C4(C2×D79)⋊C411yescharacteristic
344C22×C4(C2×D79)⋊C411yescharacteristic
345C22×C4(C2×D79)⋊C411yescharacteristic
346C22×C4C2×C4×D7911yescharacteristic
347C22×C4D4×D7912yes—
348C22×C4C2×D31611yescharacteristic
349C22×C4#1192962512yes—
350C2×C8—3162no—
351C2×C8—3162no—
352C2×C8—1584no—
353C4⋊C4—48no—
354C4⋊C4—3162no—
355C4⋊C4—1582no—
356C4⋊C4—1582no—
357C22⋊C4—3162no—
358C22⋊C4—3162no—
359C22⋊C4—3161no—
360C22⋊C4—3161no—
361C22⋊C4—3162no—
362C22⋊C4—3162no—
363C22⋊C4—3162no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—3161no—
373C23—3161no—
374C23—3161no—
375C23—3161no—
376C23—3161no—
377C23—3161no—
378C23—3161no—
379C23C2×(C2×D79)⋊C411yescharacteristic
380C23#1457915011yescharacteristic
381C23#1457915711yescharacteristic
382C23#1457918711yescharacteristic
383C23#1457918711yescharacteristic
384C23#1457919211yescharacteristic
385C23#1457919211yescharacteristic
386Q8—6322no—
387D4—6322no—
388D4—6321no—
389D4—6321no—
390D4—6321no—
391D4—6321no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—3161no—
420C2×C4—3161no—
421C2×C4—3161no—
422C2×C4—3161no—
423C2×C4—3161no—
424C2×C4—3161no—
425C2×C4—3161no—
426C2×C4—3161no—
427C2×C4—3161no—
428C2×C4#1457907512yes—
429C2×C4#1457908912yes—
430C2×C4#1457912312yes—
431C2×C4#1457913612yes—
432C8—3164no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—6321no—
446C22—6321no—
447C22—6321no—
448C22—6321no—
449C22—6321no—
450C22—6321no—
451C22—6321no—
452C22#1909846411yescharacteristic
453C22C2×#1457907511yescharacteristic
454C22#1909853611yescharacteristic
455C22#1909856211yescharacteristic
456C22#1909856211yescharacteristic
457C22C2×#1457908911yescharacteristic
458C22#1909858111yescharacteristic
459C22#1909892511yescharacteristic
460C22#1909892511yescharacteristic
461C22#1909895011yescharacteristic
462C22#1909895511yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—84no—
470C4—6321no—
471C4—6321no—
472C4—6321no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—12641no—
478C2#2376190211yescharacteristic
479C2#2376192311yescharacteristic
480C2#2376193611yescharacteristic
481C2#2376194111yescharacteristic
482C2#2376421411yescharacteristic
483C2#2376423511yescharacteristic
484C2#2376431811yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×C4⋊#95C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenterC4⋊#95D79
hypocenterC79#9682
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. #23763162C2semidirect
3. C2×#19098299C2semidirect
5. C79×C4⋊#95C2semidirect
6. C2×#14579149C4semidirect
11. #19098469C4semidirect
12. #19098465C4semidirect
17. #19098429C4semidirect
247. C4⋊#95D79semidirect
248. C79#9682semidirect

Maximal subgroups

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

subgroupindex
#237717012
#237631622
C2×#190982992
C2×#190982962
C79×C4⋊#952
#968279