Finite Group Databasebeta

← #34204242

Subgroups — #34204242

Order 20224.

600 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×#19098350C212yesmaximal
2#23763392C211yesmaximal, characteristic, complemented
3#23763388C211yesmaximal, characteristic, complemented
4#23761518C212yesmaximal, complemented
5C79×#2321C211yesmaximal, characteristic
6#19098587—21no—
7C2×#14579078C2212yes—
8C2×#14579076C2212yes—
9#19098423—21no—
10#19098350—24no—
11#19098332—21no—
12#19098330—21no—
13C2×#120×C79C2211yescharacteristic
14C2×C4×D4×C79C2211yescharacteristic
15C22×C22⋊C4×C79C411yescharacteristic, complemented
16#390×C79—21no—
17#388×C79—21no—
18#388×C79—21no—
19#386×C79—21no—
20#386×C79C2211yescharacteristic
21#385×C79C411yescharacteristic, complemented
22#379×C79C411yescharacteristic, complemented
23C4×C22⋊C4×C79C411yescharacteristic, complemented
24C2×#11929608—22no—
25C2×C4⋊Dic79—41no—
26C2×C22⋊Dic79—22no—
27C4⋊(C79⋊C8)—41no—
28C4⋊(C79⋊C8)—41no—
29#14579078—22no—
30#14579078—22no—
31#14579076—22no—
32#14579076—22no—
33C79⋊#95—41no—
34C25×C79C2×C411yescharacteristic
35C22×D4×C79—21no—
36C22×D4×C79C2311yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79C2×C411yescharacteristic
39C23×C4×C79D411yescharacteristic
40#120×C79—21no—
41#120×C79—21no—
42#120×C79D412yes—
43#120×C79Q812yes—
44C4×D4×C79—41no—
45C4×D4×C79—41no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79D411yescharacteristic
48C2×C22⋊C4×C79—21no—
49C2×C22⋊C4×C79—21no—
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79—21no—
54C2×C22⋊C4×C79—41no—
55C2×C22⋊C4×C79—41no—
56C2×C22⋊C4×C79C2×C411yescharacteristic
57C2×C22⋊C4×C79C2×C411yescharacteristic
58C2×C22⋊C4×C79C2×C411yescharacteristic
59C2×C22⋊C4×C79D411yescharacteristic
60C2×C42×C79—21no—
61C2×C42×C79D411yescharacteristic
62#95×C79—21no—
63#95×C79—21no—
64#95×C79—21no—
65C22×Dic79—42no—
66#11929608—42no—
67#11929608—42no—
68C2×C79⋊C8—42no—
69C4⋊Dic79—82no—
70C22⋊Dic79—42no—
71C22⋊Dic79—42no—
72C24×C79—21no—
73C24×C79—21no—
74C24×C79—41no—
75C24×C79—41no—
76C24×C79—81no—
77C24×C79—81no—
78C24×C79C4⋊C412yescomplemented
79C24×C79C22×C411yescharacteristic
80C24×C79C2×D411yescharacteristic
81C2×D4×C79—22no—
82C2×D4×C79—22no—
83C2×D4×C79—41no—
84C2×D4×C79—41no—
85C2×D4×C79—81no—
86C2×D4×C79—81no—
87C2×D4×C79C2×D411yescharacteristic
88C2×D4×C79C2×Q811yescharacteristic
89C2×D4×C79#4111yescharacteristic
90C2×D4×C79#4111yescharacteristic
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—21no—
98C22×C4×C79—41no—
99C22×C4×C79—81no—
100C22×C4×C79—81no—
101C22×C4×C79—81no—
102C22×C4×C79C22⋊C411yescharacteristic
103C22×C4×C79C22⋊C411yescharacteristic
104C22×C4×C79C22⋊C411yescharacteristic
105C22×C4×C79C22⋊C411yescharacteristic
106C22×C4×C79C2×D411yescharacteristic
107C4⋊C4×C79—41no—
108C4⋊C4×C79—81no—
109C22⋊C4×C79—21no—
110C22⋊C4×C79—22no—
111C22⋊C4×C79—22no—
112C22⋊C4×C79—41no—
113C22⋊C4×C79—41no—
114C22⋊C4×C79—42no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C22⋊C4×C79—81no—
119C22⋊C4×C79—81no—
120C22⋊C4×C79—81no—
121C22⋊C4×C79C4⋊C412yescomplemented
122C42×C79—21no—
123C42×C79—21no—
124C42×C79—81no—
125C23×C79—21no—
126C23×C79—21no—
127C23×C79—21no—
128C23×C79—21no—
129C23×C79—22no—
130C23×C79—41no—
131C23×C79—41no—
132C23×C79—41no—
133C23×C79—41no—
134C23×C79—41no—
135C23×C79—41no—
136C23×C79—41no—
137C23×C79—41no—
138C23×C79—42no—
139C23×C79—42no—
140C23×C79—41no—
141C23×C79—41no—
142C23×C79—82no—
143C23×C79—82no—
144C23×C79—82no—
145C23×C79—82no—
146C23×C79—84no—
147C23×C79C2×C22⋊C411yescharacteristic
148C23×C79C2×C4⋊C411yescharacteristic
149C23×C79#11711yescharacteristic
150C23×C79#12111yescharacteristic
151C23×C79C22⋊Q811yescharacteristic
152D4×C79—41no—
153D4×C79—41no—
154D4×C79—41no—
155D4×C79—41no—
156D4×C79—82no—
157D4×C79—82no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—22no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—41no—
165C2×C4×C79—42no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—41no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79—81no—
178C2×C4×C79—81no—
179C2×C4×C79—81no—
180C2×C4×C79—82no—
181C2×C4×C79#9911yescharacteristic
182C2×C4×C79#9911yescharacteristic
183C2×C4×C79#12111yescharacteristic
184C2×C4×C79C22⋊Q811yescharacteristic
185C2×Dic79—82no—
186C2×Dic79—82no—
187C2×Dic79—82no—
188C79⋊C8—82no—
189C22×C79—21no—
190C22×C79—21no—
191C22×C79—21no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—21no—
195C22×C79—41no—
196C22×C79—41no—
197C22×C79—41no—
198C22×C79—41no—
199C22×C79—42no—
200C22×C79—42no—
201C22×C79—42no—
202C22×C79—41no—
203C22×C79—41no—
204C22×C79—41no—
205C22×C79—41no—
206C22×C79—42no—
207C22×C79—81no—
208C22×C79—81no—
209C22×C79—81no—
210C22×C79—84no—
211C22×C79—84no—
212C22×C79—81no—
213C22×C79—81no—
214C22×C79#38011yescharacteristic
215C22×C79C2×#9911yescharacteristic
216C22×C79#41011yescharacteristic
217C4×C79—21no—
218C4×C79—21no—
219C4×C79—21no—
220C4×C79—21no—
221C4×C79—42no—
222C4×C79—41no—
223C4×C79—41no—
224C4×C79—41no—
225C4×C79—81no—
226C4×C79—81no—
227Dic79—162no—
228#12795—791nomaximal, Sylow
229C2×C79—21no—
230C2×C79—21no—
231C2×C79—41no—
232C2×C79—41no—
233C2×C79—41no—
234C2×C79—41no—
235C2×C79—82no—
236C2×C79#179911yescharacteristic
237C2×C79#212111yescharacteristic
238C2×C79#212211yescharacteristic
239#2321D7911yescharacteristic
240C2×AΣL(1,𝔽16)—792no—
241#1800—791no—
242#1799—791no—
243#1390—792no—
244C79#1279511yescharacteristic, complemented, Sylow
245C2×#120C2×D7911yescharacteristic
246C2×C4×D4C2×D7911yescharacteristic
247C22×C22⋊C4Dic7911yescharacteristic, complemented
248#423—1581no—
249C2×#100—792no—
250C2×#99—792no—
251#390—21no—
252#388—21no—
253#388—21no—
254#386—21no—
255#386C2×D7911yescharacteristic
256#385Dic7911yescharacteristic, complemented
257#380—1581no—
258#379Dic7911yescharacteristic, complemented
259C4×C22⋊C4Dic7911yescharacteristic, complemented
260AΣL(1,𝔽16)—1584no—
261#343—1581no—
262#342—1581no—
263C25C2×Dic7911yescharacteristic
264C22×D4—21no—
265C22×D4C22×D7911yescharacteristic
266C23×C4C2×Dic7911yescharacteristic
267C23×C4C2×Dic7911yescharacteristic
268C23×C4C22⋊D7911yescharacteristic
269C2×M16—1582no—
270#120—21no—
271#120—21no—
272#120Dic15812yes—
273#120D31612yes—
274C4×D4—41no—
275C4×D4—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—3161no—
278C2×C4⋊C4C22⋊D7911yescharacteristic
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—1582no—
288C2×C22⋊C4C2×Dic7911yescharacteristic
289C2×C22⋊C4C2×Dic7911yescharacteristic
290C2×C22⋊C4C2×Dic7911yescharacteristic
291C2×C22⋊C4C22⋊D7911yescharacteristic
292C2×C42—21no—
293C2×C42C22⋊D7911yescharacteristic
294C4⋊C8—3161no—
295C4⋊C8—3161no—
296#100—1582no—
297#100—1582no—
298#99—1582no—
299#99—1582no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—3161no—
304C24—21no—
305C24—21no—
306C24—41no—
307C24—41no—
308C24—81no—
309C24—81no—
310C24C4⋊Dic7912yescomplemented
311C24C22×Dic7911yescharacteristic
312C24C2×C22⋊D7911yescharacteristic
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—81no—
318C2×D4—81no—
319C2×D4C2×D31611yescharacteristic
320C2×D4C2×Dic15811yescharacteristic
321C2×D4#1192962411yescharacteristic
322C2×D4#1192962411yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—41no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—3162no—
335C22×C4C22⋊Dic7911yescharacteristic
336C22×C4C22⋊Dic7911yescharacteristic
337C22×C4C22⋊Dic7911yescharacteristic
338C22×C4C22⋊Dic7911yescharacteristic
339C22×C4C2×C22⋊D7911yescharacteristic
340M16—3162no—
341M16—3162no—
342C2×C8—3162no—
343C4⋊C4—41no—
344C4⋊C4—81no—
345C4⋊C4—6322no—
346C22⋊C4—21no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—81no—
356C22⋊C4—81no—
357C22⋊C4—81no—
358C22⋊C4—3162no—
359C22⋊C4—3162no—
360C22⋊C4C4⋊Dic7912yescomplemented
361C42—21no—
362C42—21no—
363C42—81no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—84no—
386C23C2×C22⋊Dic7911yescharacteristic
387C23C2×C4⋊Dic7911yescharacteristic
388C23#1457912011yescharacteristic
389C23#1457913511yescharacteristic
390C23#1457914011yescharacteristic
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—82no—
396D4—82no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—82no—
420C2×C4—6322no—
421C2×C4—6322no—
422C2×C4—6322no—
423C2×C4#1457907611yescharacteristic
424C2×C4#1457907611yescharacteristic
425C2×C4#1457913511yescharacteristic
426C2×C4#1457914011yescharacteristic
427C8—6322no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—42no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—84no—
450C22—84no—
451C22—81no—
452C22—81no—
453C22#1909842311yescharacteristic
454C22C2×#1457907611yescharacteristic
455C22#1909853711yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—81no—
465C4—81no—
466C4—12642no—
467C2—21no—
468C2—21no—
469C2—41no—
470C2—41no—
471C2—41no—
472C2—41no—
473C2—82no—
474C2#2376338811yescharacteristic
475C2#2376504611yescharacteristic
476C2#2376505011yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC24×C79C22×C4
FrattiniC22×D4C22×D79
FittingC79×#2321C2
socleC22×C79C2×#99
prime radicalC2×#120×C79C22
cosocleC22×D4×C79C23
hypercenter#2321D79
hypocenterC79#12795
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
2. #23763392C2semidirect
3. #23763388C2semidirect
4. #23761518C2semidirect
15. C22×C22⋊C4×C79C4semidirect
21. #385×C79C4semidirect
22. #379×C79C4semidirect
23. C4×C22⋊C4×C79C4semidirect
78. C24×C79C4⋊C4semidirect
121. C22⋊C4×C79C4⋊C4semidirect
244. C79#12795semidirect
247. C22×C22⋊C4Dic79semidirect
256. #385Dic79semidirect
258. #379Dic79semidirect
259. C4×C22⋊C4Dic79semidirect
310. C24C4⋊Dic79semidirect
360. C22⋊C4C4⋊Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190983502
#237633922
#237633882
#237615182
C79×#23212
#1279579