Finite Group Databasebeta

← #34184094

Subgroups — #34184094

Order 20224.

508 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772435C211yesmaximal, characteristic, complemented
2#23763225C211yesmaximal, characteristic, complemented
3#23762011C211yesmaximal, characteristic, complemented
4#23761895C211yesmaximal, characteristic, complemented
5#23761298C211yesmaximal, characteristic, complemented
6#23761257C211yesmaximal, characteristic, complemented
7C79×#3109C211yesmaximal, characteristic
8#19098641—21no—
9#19098635—22no—
10D4×C79⋊C8—21no—
11#19098620C2211yescharacteristic
12#19098616C2211yescharacteristic
13C2×#14579082—21no—
14#19098527—21no—
15C2×C22⋊(C79⋊C8)—21no—
16C2×#14579072C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098325—21no—
19#19098303—22no—
20#19098297—21no—
21#19098297—21no—
22#532×C79C2211yescharacteristic
23C2×C4×D4×C79C2211yescharacteristic
24#481×C79—21no—
25#476×C79C411yescharacteristic, complemented
26#476×C79C411yescharacteristic, complemented
27#459×C79C411yescharacteristic, complemented
28#459×C79C411yescharacteristic, complemented
29#447×C79—21no—
30#442×C79—21no—
31#442×C79—21no—
32#432×C79C2211yescharacteristic
33C2×#11929608—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—22no—
36C2×C4⋊Dic79—21no—
37C2×C4×Dic79—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40#14579082—24no—
41C22⋊(C79⋊C8)—22no—
42C22⋊(C79⋊C8)—22no—
43C22⋊(C79⋊C8)—21no—
44C22⋊(C79⋊C8)D411yescharacteristic
45C22⋊(C79⋊C8)D411yescharacteristic
46#14579072—21no—
47#14579072—21no—
48#14579072D412yes—
49C4×C79⋊C8—21no—
50C79⋊#95—21no—
51C2×#41×C79—21no—
52C22×D4×C79D411yescharacteristic
53C23×C4×C79—21no—
54C2×D8×C79—21no—
55C2×D8×C79—21no—
56C4⋊Q8×C79C2×C411yescharacteristic
57#127×C79C2×C411yescharacteristic
58#124×C79—21no—
59#121×C79—21no—
60#121×C79—21no—
61C4×D4×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79C2×C411yescharacteristic
66C4×D4×C79C2×C411yescharacteristic
67C4×D4×C79D411yescharacteristic
68C4×D4×C79D411yescharacteristic
69C2×C4⋊C4×C79D411yescharacteristic
70C2×C22⋊C4×C79—21no—
71C2×C42×C79C2311yescharacteristic
72#106×C79—21no—
73#106×C79—21no—
74C4⋊C8×C79C2×C411yescharacteristic
75C4⋊C8×C79C2×C411yescharacteristic
76#102×C79—21no—
77#102×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80C22⋊C8×C79—21no—
81#97×C79—21no—
82C22×Dic79—41no—
83#11929608—82no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—82no—
91C4⋊Dic79—22no—
92C4⋊Dic79—41no—
93C4×Dic79—81no—
94C24×C79—21no—
95#41×C79—81no—
96#41×C79—81no—
97C2×Q8×C79—21no—
98C2×D4×C79—22no—
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79—41no—
104C2×D4×C79—41no—
105C2×D4×C79C22⋊C411yescharacteristic
106C2×D4×C79C22⋊C411yescharacteristic
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—21no—
112C22×C4×C79—41no—
113C22×C4×C79—82no—
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79#4111yescharacteristic
117D8×C79—42no—
118D8×C79—42no—
119C2×C8×C79—21no—
120C2×C8×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—21no—
124C4⋊C4×C79C22⋊C411yescharacteristic
125C4⋊C4×C79C22⋊C411yescharacteristic
126C22⋊C4×C79—21no—
127C22⋊C4×C79—21no—
128C22⋊C4×C79—21no—
129C22⋊C4×C79—21no—
130C22⋊C4×C79—41no—
131C22⋊C4×C79—41no—
132C42×C79C22×C411yescharacteristic
133C42×C79C2×D411yescharacteristic
134C42×C79C2×D411yescharacteristic
135C42×C79#4111yescharacteristic
136C23×C79—21no—
137C23×C79—21no—
138C23×C79—42no—
139C23×C79—41no—
140C23×C79—41no—
141C23×C79—41no—
142C23×C79—81no—
143C23×C79#12111yescharacteristic
144Q8×C79—81no—
145D4×C79—21no—
146D4×C79—21no—
147D4×C79—21no—
148D4×C79—42no—
149D4×C79—41no—
150D4×C79—41no—
151D4×C79—81no—
152D4×C79—81no—
153D4×C79—81no—
154D4×C79—81no—
155D4×C79#10412yes—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—41no—
163C2×C4×C79—42no—
164C2×C4×C79—42no—
165C2×C4×C79—41no—
166C2×C4×C79—81no—
167C2×C4×C79—81no—
168C2×C4×C79—82no—
169C2×C4×C79—81no—
170C2×C4×C79C2×C22⋊C411yescharacteristic
171C2×C4×C79C4×D411yescharacteristic
172C2×C4×C79C4×D411yescharacteristic
173C2×C4×C79#12011yescharacteristic
174C2×C4×C79#12111yescharacteristic
175C2×C4×C79#12311yescharacteristic
176C2×Dic79—42no—
177C2×Dic79—81no—
178C2×Dic79—81no—
179C8×C79—41no—
180C8×C79—41no—
181C79⋊C8—42no—
182C79⋊C8—41no—
183C79⋊C8—42no—
184C79⋊C8—41no—
185C22×C79—21no—
186C22×C79—21no—
187C22×C79—22no—
188C22×C79—41no—
189C22×C79—41no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79—41no—
193C22×C79—82no—
194C22×C79—82no—
195C22×C79—81no—
196C22×C79—81no—
197C22×C79—81no—
198C22×C79#38611yescharacteristic
199C22×C79#47111yescharacteristic
200C22×C79#47211yescharacteristic
201C4×C79—21no—
202C4×C79—21no—
203C4×C79—21no—
204C4×C79—21no—
205C4×C79—21no—
206C4×C79—41no—
207C4×C79—41no—
208C4×C79—81no—
209C4×C79C2×#10411yescharacteristic
210C4×C79#42111yescharacteristic
211Dic79—82no—
212#9163—791nomaximal, Sylow
213C2×C79—21no—
214C2×C79—21no—
215C2×C79—42no—
216C2×C79—81no—
217C2×C79—161no—
218C2×C79#187111yescharacteristic
219C2×C79#190411yescharacteristic
220C2×C79#190711yescharacteristic
221#3109D7911yescharacteristic
222#2974—791no—
223#1762—791no—
224#1530—791no—
225#1502—791no—
226#1302—791no—
227#1285—791no—
228C79#916311yescharacteristic, complemented, Sylow
229#532C2×D7911yescharacteristic
230C2×C4×D4C2×D7911yescharacteristic
231#481—21no—
232#476Dic7911yescharacteristic, complemented
233#476Dic7911yescharacteristic, complemented
234#459Dic7911yescharacteristic, complemented
235#459Dic7911yescharacteristic, complemented
236#447—21no—
237#442—21no—
238#442—21no—
239#436—1581no—
240#435—1582no—
241C8×D4—1581no—
242#432—791no—
243#432C2×D7911yescharacteristic
244#431—791no—
245C2×#102—1581no—
246#407—1581no—
247C2×C22⋊C8—1581no—
248C2×#97—791no—
249C4×C4⋊C4—791no—
250#340—1581no—
251#329—1582no—
252#327—1581no—
253#327—1581no—
254C2×#41—21no—
255C22×D4C22⋊D7911yescharacteristic
256C23×C4—21no—
257C2×D8—21no—
258C2×D8—21no—
259C2×M16—1582no—
260C22×C8—1581no—
261C22×C8—1582no—
262C4⋊Q8C2×Dic7911yescharacteristic
263#127C2×Dic7911yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C2×Dic7911yescharacteristic
272C4×D4C2×Dic7911yescharacteristic
273C4×D4C22⋊D7911yescharacteristic
274C4×D4C22⋊D7911yescharacteristic
275C2×C4⋊C4—1581no—
276C2×C4⋊C4C22⋊D7911yescharacteristic
277C2×C22⋊C4—21no—
278C2×C42—1581no—
279C2×C42C22×D7911yescharacteristic
280#106—21no—
281#106—21no—
282C4⋊C8—1581no—
283C4⋊C8—1581no—
284C4⋊C8C2×Dic7911yescharacteristic
285C4⋊C8C2×Dic7911yescharacteristic
286#102—21no—
287#102—21no—
288#102—21no—
289#102—21no—
290#102—1584no—
291C22⋊C8—21no—
292C22⋊C8—791no—
293C22⋊C8—791no—
294C22⋊C8—1582no—
295C22⋊C8—1582no—
296C22⋊C8—1581no—
297#97—21no—
298#97—792no—
299#97—1581no—
300#97—1581no—
301C4×C8—1581no—
302#95—1581no—
303C24—21no—
304#41—81no—
305#41—81no—
306C2×Q8—21no—
307C2×D4—22no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4C22⋊Dic7911yescharacteristic
315C2×D4C22⋊Dic7911yescharacteristic
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—82no—
323C22×C4—3161no—
324C22×C4D4×D7911yescharacteristic
325C22×C4C2×C22⋊D7911yescharacteristic
326C22×C4#1192962311yescharacteristic
327D8—42no—
328D8—42no—
329M16—6322no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—3161no—
333C2×C8—3161no—
334C2×C8—6322no—
335C2×C8—1582no—
336C2×C8—1581no—
337C2×C8—1581no—
338C2×C8—1582no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—3161no—
343C4⋊C4—1582no—
344C4⋊C4C22⋊Dic7911yescharacteristic
345C4⋊C4C22⋊Dic7911yescharacteristic
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C42—6321no—
353C42C22×Dic7911yescharacteristic
354C42D4×D7911yescharacteristic
355C42C2×C22⋊D7911yescharacteristic
356C42#1192962311yescharacteristic
357C23—21no—
358C23—21no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—81no—
364C23#1457913311yescharacteristic
365Q8—81no—
366D4—21no—
367D4—21no—
368D4—21no—
369D4—42no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4#1457908812yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—3162no—
392C2×C4—6321no—
393C2×C4—6321no—
394C2×C4C2×C22⋊Dic7911yescharacteristic
395C2×C4D4×Dic7911yescharacteristic
396C2×C4D4×Dic7911yescharacteristic
397C2×C4#1457912911yescharacteristic
398C2×C4#1457913311yescharacteristic
399C2×C4#1457914211yescharacteristic
400C8—41no—
401C8—41no—
402C8—3162no—
403C8—3161no—
404C8—3162no—
405C8—3161no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—82no—
415C22—82no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22#1909844311yescharacteristic
420C22#1909882411yescharacteristic
421C22#1909883111yescharacteristic
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—41no—
428C4—41no—
429C4—81no—
430C4—6322no—
431C4C2×#1457908811yescharacteristic
432C4#1909858011yescharacteristic
433C2—21no—
434C2—21no—
435C2—42no—
436C2—81no—
437C2—161no—
438C2#2376370311yescharacteristic
439C2#2376388911yescharacteristic
440C2#2376391011yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3109C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3109D79
hypocenterC79#9163
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #23772435C2semidirect
2. #23763225C2semidirect
3. #23762011C2semidirect
4. #23761895C2semidirect
5. #23761298C2semidirect
6. #23761257C2semidirect
25. #476×C79C4semidirect
26. #476×C79C4semidirect
27. #459×C79C4semidirect
28. #459×C79C4semidirect
228. C79#9163semidirect
232. #476Dic79semidirect
233. #476Dic79semidirect
234. #459Dic79semidirect
235. #459Dic79semidirect

Maximal subgroups

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

subgroupindex
#237724352
#237632252
#237620112
#237618952
#237612982
#237612572
C79×#31092
#916379