Finite Group Databasebeta

← #34187400

Subgroups — #34187400

Order 20224.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23771701C211yesmaximal, characteristic
2#23763380C211yesmaximal, characteristic, complemented
3C2×#19098299C212yesmaximal, complemented
4C2×#19098296C212yesmaximal
5C4×#95×C79C211yesmaximal, characteristic, complemented
6C2×#14579149C411yescharacteristic, complemented
7C2×#14579140C2211yescharacteristic
8C2×#14579135C2211yescharacteristic
9C2×C4⋊(C79⋊C8)—21no—
10C2×C22⋊(C79⋊C8)C2212yes—
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×C42×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—
43C4⋊(C79⋊C8)—42no—
44C22⋊(C79⋊C8)—22no—
45C22⋊(C79⋊C8)—22no—
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×C42×C79—24no—
55C2×C42×C79—42no—
56C2×C42×C79—42no—
57C2×C42×C79—42no—
58C2×C42×C79C2×C411yescharacteristic
59C2×C42×C79C2×C411yescharacteristic
60C2×C42×C79D411yescharacteristic
61C2×C42×C79D411yescharacteristic
62#95×C79—24no—
63#95×C79—22no—
64#95×C79D414yes—
65C23×D79—21no—
66C2×Dic158—22no—
67C2×Dic158—41no—
68C2×Dic158—41no—
69C2×D316—22no—
70C2×D316—41no—
71C2×D316—41no—
72C2×C22⋊D79—42no—
73C2×C22⋊D79—42no—
74C2×C22⋊D79—41no—
75C2×C22⋊D79—41no—
76C22×Dic79—21no—
77C22×Dic79—21no—
78C22×Dic79—21no—
79C2×C79⋊C8—42no—
80C2×C79⋊C8—42no—
81C2×C79⋊C8—42no—
82C4⋊Dic79—22no—
83C4⋊Dic79—22no—
84Dic79⋊C4—41no—
85Dic79⋊C4—41no—
86C22⋊Dic79—42no—
87(C2×D79)⋊C4—42no—
88(C2×D79)⋊C4—42no—
89(C2×D79)⋊C4—42no—
90(C2×D79)⋊C4—42no—
91(C2×D79)⋊C4—41no—
92(C2×D79)⋊C4—41no—
93C24×C79C2×D411yescharacteristic
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—22no—
97C22×C4×C79—22no—
98C22×C4×C79—24no—
99C22×C4×C79—24no—
100C22×C4×C79—24no—
101C22×C4×C79—22no—
102C22×C4×C79—22no—
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×C79C22⋊C411yescharacteristic
111C22×C4×C79C22⋊C411yescharacteristic
112C22×C4×C79C22⋊C411yescharacteristic
113C22×C4×C79C22⋊C411yescharacteristic
114C22×C4×C79C22×C411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79C2×D412yes—
117C22×C4×C79#4112yes—
118C42×C79—22no—
119C42×C79—22no—
120C42×C79—41no—
121C42×C79—41no—
122C42×C79—48no—
123C23×C79—21no—
124C23×C79—21no—
125C23×C79—21no—
126C23×C79—21no—
127C23×C79C2×C22⋊C411yescharacteristic
128C23×C79#12112yes—
129C23×C79#13512yes—
130C23×C79AGL(1,ℤ/8)11yescharacteristic
131C23×C79#13711yescharacteristic
132C22×D79—41no—
133C22×D79—41no—
134C22×D79—41no—
135C22×D79—41no—
136C22×D79—41no—
137C22×D79—41no—
138C22×D79—41no—
139C2×C4×C79—24no—
140C2×C4×C79—24no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
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—28no—
152C2×C4×C79—21no—
153C2×C4×C79—21no—
154C2×C4×C79—21no—
155C2×C4×C79—42no—
156C2×C4×C79—42no—
157C2×C4×C79—44no—
158C2×C4×C79—44no—
159C2×C4×C79—44no—
160C2×C4×C79—44no—
161C2×C4×C79—44no—
162C2×C4×C79—44no—
163C2×C4×C79—44no—
164C2×C4×C79#9912yes—
165C2×C4×C79#10412yes—
166C2×C4×C79#10412yes—
167C2×C4×C79C4×D412yes—
168C2×C4×C79#12411yescharacteristic
169C2×C4×C79#12711yescharacteristic
170C22⋊D79—82no—
171C22⋊D79—82no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176C2×Dic79—41no—
177C2×Dic79—41no—
178C2×Dic79—41no—
179C2×Dic79—41no—
180C2×Dic79—41no—
181D316—82no—
182Dic158—82no—
183C79⋊C8—82no—
184C22×C79—22no—
185C22×C79—22no—
186C22×C79—22no—
187C22×C79—22no—
188C22×C79—21no—
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79#39011yescharacteristic
193C22×C79C2×#9911yescharacteristic
194C22×C79#41011yescharacteristic
195C22×C79#41611yescharacteristic
196C22×C79#41711yescharacteristic
197C22×C79C2×#10411yescharacteristic
198C22×C79C2×#10411yescharacteristic
199C22×C79#46312yes—
200C22×C79#46412yes—
201C2×D79—81no—
202C2×D79—81no—
203C2×D79—81no—
204C2×D79—81no—
205C2×D79—81no—
206C2×D79—81no—
207C2×D79—81no—
208C4×C79—22no—
209C4×C79—21no—
210C4×C79—21no—
211C4×C79—22no—
212C4×C79—22no—
213C4×C79—22no—
214C4×C79—48no—
215C4×C79—41no—
216Dic79—81no—
217Dic79—81no—
218Dic79—81no—
219#9713—791nomaximal, Sylow
220C2×C79—22no—
221C2×C79—22no—
222C2×C79#150011yescharacteristic
223C2×C79#150111yescharacteristic
224C2×C79#150411yescharacteristic
225C2×C79#150911yescharacteristic
226C2×C79#196211yescharacteristic
227C2×C79#196411yescharacteristic
228C2×C79#198311yescharacteristic
229D79—161no—
230#2850—791no—
231#1797—791no—
232C2×#328—792no—
233C2×#327—792no—
234C4×#95D7911yescharacteristic, complemented
235C79#971311yescharacteristic, complemented, Sylow
236C2×#124—791no—
237C2×C22⋊Q8—791no—
238C2×#121—791no—
239C22×C42—21no—
240C22×C42C2×D7911yescharacteristic
241C2×C4⋊C8—1581no—
242C2×C22⋊C8—792no—
243#391—791no—
244#390—791no—
245#388—1581no—
246#388—1581no—
247#386—1581no—
248#386—1581no—
249#382—791no—
250C2×#95—21no—
251C2×#95C2×D7912yes—
252#336—1581no—
253#328—1584no—
254#327—1584no—
255C22×Q8—791no—
256C22×D4—791no—
257C22×D4—1581no—
258C23×C4—22no—
259C23×C4C22⋊D7911yescharacteristic
260C23×C4C22⋊D7911yescharacteristic
261C23×C4C22×D7911yescharacteristic
262C22×C8—1582no—
263#124—3162no—
264C22⋊Q8—1582no—
265C22⋊Q8—1582no—
266#121—1582no—
267#121—1582no—
268C2×C4⋊C4—791no—
269C2×C4⋊C4—791no—
270C2×C4⋊C4—1581no—
271C2×C22⋊C4—1581no—
272C2×C22⋊C4—1581no—
273C2×C22⋊C4—1581no—
274C2×C22⋊C4—1581no—
275C2×C22⋊C4—1581no—
276C2×C22⋊C4—1581no—
277C2×C42—24no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C4×D7911yescharacteristic
282C2×C42C4×D7911yescharacteristic
283C2×C42D31611yescharacteristic
284C2×C42D31611yescharacteristic
285C4⋊C8—3162no—
286C22⋊C8—1582no—
287C22⋊C8—1582no—
288#95—24no—
289#95—22no—
290#95—1581no—
291#95—1581no—
292#95—1581no—
293#95—1581no—
294#95D31614yes—
295C24—1581no—
296C24C2×C22⋊D7911yescharacteristic
297C2×Q8—3161no—
298C2×Q8—3161no—
299C2×Q8—1582no—
300C2×D4—3161no—
301C2×D4—3161no—
302C2×D4—3162no—
303C2×D4—3162no—
304C2×D4—3161no—
305C2×D4—3161no—
306C2×D4—1582no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—1581no—
324C22×C4—1581no—
325C22×C4—1581no—
326C22×C4(C2×D79)⋊C411yescharacteristic
327C22×C4(C2×D79)⋊C411yescharacteristic
328C22×C4(C2×D79)⋊C411yescharacteristic
329C22×C4(C2×D79)⋊C411yescharacteristic
330C22×C4C2×C4×D7911yescharacteristic
331C22×C4C2×D31611yescharacteristic
332C22×C4C2×D31612yes—
333C22×C4#1192962412yes—
334C2×C8—3162no—
335C2×C8—3162no—
336C2×C8—3162no—
337C4⋊C4—3161no—
338C4⋊C4—3161no—
339C4⋊C4—1582no—
340C4⋊C4—1582no—
341C22⋊C4—3162no—
342C22⋊C4—3162no—
343C22⋊C4—3162no—
344C22⋊C4—3162no—
345C22⋊C4—3162no—
346C22⋊C4—3161no—
347C22⋊C4—3161no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—48no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—3161no—
358C23—3161no—
359C23—3161no—
360C23—3161no—
361C23—3161no—
362C23—3161no—
363C23—3161no—
364C23C2×(C2×D79)⋊C411yescharacteristic
365C23#1457913512yes—
366C23#1457918112yes—
367C23#1457918711yescharacteristic
368C23#1457919211yescharacteristic
369Q8—6322no—
370D4—6322no—
371D4—6322no—
372D4—6322no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—3161no—
399C2×C4—3161no—
400C2×C4—3161no—
401C2×C4—3161no—
402C2×C4—3161no—
403C2×C4—3161no—
404C2×C4—3161no—
405C2×C4—3161no—
406C2×C4—3161no—
407C2×C4#1457907512yes—
408C2×C4#1457908912yes—
409C2×C4#1457908912yes—
410C2×C4C4×D31612yes—
411C2×C4#1457914911yescharacteristic
412C2×C4#1457915811yescharacteristic
413C8—6322no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—6321no—
423C22—6321no—
424C22—6321no—
425C22—6321no—
426C22—6321no—
427C22—6321no—
428C22—6321no—
429C22#1909846511yescharacteristic
430C22C2×#1457907511yescharacteristic
431C22#1909853611yescharacteristic
432C22#1909855711yescharacteristic
433C22#1909856211yescharacteristic
434C22C2×#1457908911yescharacteristic
435C22C2×#1457908911yescharacteristic
436C22#1909877812yes—
437C22#1909878512yes—
438C4—21no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—48no—
445C4—41no—
446C4—6321no—
447C4—6321no—
448C4—6321no—
449C2—22no—
450C2—22no—
451C2—12641no—
452C2#2376188411yescharacteristic
453C2#2376188911yescharacteristic
454C2#2376190211yescharacteristic
455C2#2376192311yescharacteristic
456C2#2376422811yescharacteristic
457C2#2376423611yescharacteristic
458C2#2376433211yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC4×#95×C79C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenterC4×#95D79
hypocenterC79#9713
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. #23763380C2semidirect
3. C2×#19098299C2semidirect
5. C4×#95×C79C2semidirect
6. C2×#14579149C4semidirect
11. #19098469C4semidirect
12. #19098465C4semidirect
17. #19098429C4semidirect
234. C4×#95D79semidirect
235. C79#9713semidirect

Maximal subgroups

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

subgroupindex
#237717012
#237633802
C2×#190982992
C2×#190982962
C4×#95×C792
#971379