Finite Group Databasebeta

← #34444673

Subgroups — #34444673

Order 20224.

476 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774059C211yesmaximal, characteristic, complemented
2#23765564C211yesmaximal, characteristic, complemented
3#23765557C211yesmaximal, characteristic, complemented
4#23765549C211yesmaximal, characteristic, complemented
5#23765539C211yesmaximal, characteristic, complemented
6#23765511C211yesmaximal, characteristic
7#23765506C211yesmaximal, characteristic, complemented
8#23765494C211yesmaximal, characteristic
9#23765489C211yesmaximal, characteristic, complemented
10#23765459C211yesmaximal, characteristic, complemented
11#23765459C211yesmaximal, characteristic, complemented
12#23765450C211yesmaximal, characteristic, complemented
13#23765445C211yesmaximal, characteristic, complemented
14#23765159C211yesmaximal, characteristic, complemented
15C79×#3175C211yesmaximal, characteristic
16#19099175—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579107C2211yescharacteristic
19C2×#14579105C2211yescharacteristic
20C2×#14579105C2211yescharacteristic
21C2×#14579101C2211yescharacteristic
22C22×C79⋊C16C2211yescharacteristic
23#19098965C2211yescharacteristic
24#19098943C2211yescharacteristic
25#19098932—21no—
26#19098821—21no—
27#19098804C2211yescharacteristic
28#19098804C2211yescharacteristic
29#19098671—21no—
30C4×Dic316C2211yescharacteristic
31C4×Dic316C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098390C2211yescharacteristic
34#19098388C2211yescharacteristic
35#19098388C2211yescharacteristic
36C4⋊(C79⋊C16)C2211yescharacteristic
37C4⋊(C79⋊C16)C2211yescharacteristic
38C4⋊(C79⋊C16)C2211yescharacteristic
39C4⋊(C79⋊C16)C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098367C2211yescharacteristic
42#19098366C2211yescharacteristic
43#19098366C2211yescharacteristic
44#19098366C2211yescharacteristic
45#19098366C2211yescharacteristic
46#19098364C2211yescharacteristic
47#19098364C2211yescharacteristic
48#542×C79—21no—
49#500×C79C2211yescharacteristic
50#495×C79C2211yescharacteristic
51#489×C79—21no—
52#468×C79—21no—
53#467×C79C2211yescharacteristic
54#466×C79C2211yescharacteristic
55#440×C79—21no—
56C4×Q16×C79C2211yescharacteristic
57C4×D8×C79C2211yescharacteristic
58#405×C79C2211yescharacteristic
59C2×Dic316—21no—
60C2×Dic316C2311yescharacteristic
61C2×Dic316C2311yescharacteristic
62#14579161—21no—
63#14579154—41no—
64#14579154—41no—
65#14579153—21no—
66#14579151—41no—
67#14579151—41no—
68#14579140—21no—
69#14579140—21no—
70C4×Dic158—21no—
71C4×Dic158—21no—
72#14579107—41no—
73#14579107—41no—
74#14579105—41no—
75#14579105—41no—
76#14579101—41no—
77C2×C79⋊C16—41no—
78C2×C79⋊C16C2311yescharacteristic
79C2×C79⋊C16C2311yescharacteristic
80C2×C79⋊C16C2311yescharacteristic
81C2×C79⋊C16C2311yescharacteristic
82#14579095C2311yescharacteristic
83#14579095C2311yescharacteristic
84#14579093—21no—
85#14579086—21no—
86#14579086—21no—
87#14579086—21no—
88#14579086—21no—
89C2×Q16×C79C2311yescharacteristic
90C2×SD16×C79—21no—
91C2×D8×C79C2311yescharacteristic
92C2×M16×C79D411yescharacteristic
93C22×C8×C79C2311yescharacteristic
94C4⋊Q8×C79—21no—
95#126×C79—41no—
96#124×C79—21no—
97#124×C79—41no—
98#123×C79—41no—
99C22⋊Q8×C79—21no—
100C22⋊Q8×C79—41no—
101#121×C79—21no—
102C4×Q8×C79—21no—
103C4×D4×C79—21no—
104#117×C79D411yescharacteristic
105#107×C79—21no—
106#107×C79C2311yescharacteristic
107#107×C79C2311yescharacteristic
108#103×C79—21no—
109#103×C79—21no—
110#102×C79—21no—
111#102×C79—21no—
112#97×C79D411yescharacteristic
113#97×C79D411yescharacteristic
114C4×C8×C79C2311yescharacteristic
115C4×C8×C79C2311yescharacteristic
116C2×Dic158—21no—
117C2×Dic158—21no—
118Dic316—21no—
119Dic316—21no—
120Dic316—21no—
121Dic316—21no—
122Dic316—42no—
123C4⋊Dic79—21no—
124C4⋊Dic79—21no—
125C4⋊Dic79—41no—
126C4⋊Dic79—41no—
127Dic79⋊C4—41no—
128Dic79⋊C4—41no—
129Dic79⋊C4—41no—
130Dic79⋊C4—41no—
131Dic79⋊C4—41no—
132Dic79⋊C4—41no—
133C22⋊Dic79—41no—
134C22⋊Dic79—41no—
135C4×Dic79—41no—
136C4×Dic79—41no—
137C79⋊C16—21no—
138C79⋊C16—21no—
139C79⋊C16—21no—
140C79⋊C16—21no—
141C2×Q8×C79—21no—
142C2×Q8×C79—41no—
143C2×D4×C79—21no—
144C2×D4×C79—41no—
145C22×C4×C79—41no—
146C22×C4×C79C2×D411yescharacteristic
147Q16×C79—21no—
148Q16×C79—21no—
149SD16×C79—42no—
150D8×C79—21no—
151D8×C79—21no—
152M16×C79—41no—
153C2×C8×C79—21no—
154C2×C8×C79—21no—
155C2×C8×C79C2×D411yescharacteristic
156C2×C8×C79C2×D411yescharacteristic
157C2×C8×C79C2×D411yescharacteristic
158C2×C8×C79C2411yescharacteristic
159C4⋊C4×C79—21no—
160C4⋊C4×C79—21no—
161C4⋊C4×C79—41no—
162C4⋊C4×C79—41no—
163C4⋊C4×C79—41no—
164C4⋊C4×C79D811yescharacteristic
165C4⋊C4×C79D811yescharacteristic
166C22⋊C4×C79—41no—
167C22⋊C4×C79—41no—
168C22⋊C4×C79—41no—
169C22⋊C4×C79—41no—
170C22⋊C4×C79—41no—
171C22⋊C4×C79D811yescharacteristic
172C22⋊C4×C79D811yescharacteristic
173C42×C79—41no—
174C42×C79C2×D411yescharacteristic
175C42×C79C2×D411yescharacteristic
176C23×C79—41no—
177C23×C79C2×D811yescharacteristic
178Q8×C79—41no—
179Q8×C79—41no—
180Q8×C79—161no—
181D4×C79—41no—
182D4×C79—41no—
183D4×C79—161no—
184C2×C4×C79—21no—
185C2×C4×C79—21no—
186C2×C4×C79—41no—
187C2×C4×C79—41no—
188C2×C4×C79—41no—
189C2×C4×C79—161no—
190C2×C4×C79C2×D811yescharacteristic
191C2×C4×C79C2×D811yescharacteristic
192C2×C4×C79C2×D811yescharacteristic
193C2×C4×C79C2×D811yescharacteristic
194C2×C4×C79C2×D811yescharacteristic
195C2×C4×C79C22×D411yescharacteristic
196C2×Dic79—41no—
197C2×Dic79—41no—
198C2×Dic79—41no—
199C2×Dic79—41no—
200Dic158—41no—
201Dic158—41no—
202Dic158—41no—
203Dic158—41no—
204C8×C79—21no—
205C8×C79—21no—
206C8×C79—21no—
207C8×C79H(5,𝔽2)11yescharacteristic
208C8×C792−1+411yescharacteristic
209C22×C79—21no—
210C22×C79—21no—
211C22×C79—21no—
212C22×C79—81no—
213C22×C79—81no—
214C22×C79—81no—
215C22×C79C22×D811yescharacteristic
216C4×C79—21no—
217C4×C79—21no—
218C4×C79—21no—
219C4×C79—21no—
220C4×C79—21no—
221C4×C79—81no—
222C4×C79—81no—
223C4×C79—81no—
224C4×C79#53711yescharacteristic
225C4×C79#57511yescharacteristic
226Dic79—81no—
227Dic79—81no—
228Dic79—81no—
229Dic79—81no—
230#33824—791nomaximal, Sylow
231C2×C79—41no—
232C2×C79—161no—
233C2×C79#324511yescharacteristic
234C2×C79#341611yescharacteristic
235C2×C79#341711yescharacteristic
236#3177—791no—
237#3175D7911yescharacteristic
238#2229—791no—
239#2228—791no—
240#2227—791no—
241#2225—791no—
242#2220—791no—
243#2219—791no—
244#2217—791no—
245#2216—791no—
246#2211—791no—
247#2211—791no—
248#2209—791no—
249#2208—791no—
250#2151—791no—
251C79#3382411yescharacteristic, complemented, Sylow
252#542—21no—
253#541—1581no—
254C2×Q32—791no—
255C2×Q32—791no—
256C2×SD32—791no—
257C2×SD32—791no—
258C2×M32—791no—
259C22×C16—791no—
260#500C2×D7911yescharacteristic
261#499—791no—
262#495C2×D7911yescharacteristic
263#494—791no—
264#491—1581no—
265#489—21no—
266#470—1581no—
267#468—21no—
268#467—791no—
269#467—791no—
270#467C2×D7911yescharacteristic
271#466C2×D7911yescharacteristic
272#441—1581no—
273#440—21no—
274C4×Q16—791no—
275C4×Q16—791no—
276C4×Q16C2×D7911yescharacteristic
277C4×D8C2×D7911yescharacteristic
278#405C2×D7911yescharacteristic
279#367—791no—
280#367—791no—
281#366—791no—
282#366—791no—
283C4⋊C16—791no—
284C4⋊C16—791no—
285C4⋊C16—791no—
286C4⋊C16—791no—
287#358—791no—
288#358—791no—
289#358—791no—
290#358—791no—
291#358—791no—
292#358—791no—
293#357—791no—
294#357—791no—
295C2×Q16—791no—
296C2×Q16—791no—
297C2×Q16—1581no—
298C2×Q16C22×D7911yescharacteristic
299C2×SD16—21no—
300C2×D8C22×D7911yescharacteristic
301C2×M16C22⋊D7911yescharacteristic
302C22×C8C22×D7911yescharacteristic
303C4⋊Q8—21no—
304C4⋊Q8—1581no—
305#126—41no—
306#126—3161no—
307#126—3161no—
308#125—3161no—
309#125—3161no—
310#125—1581no—
311#124—21no—
312#124—41no—
313#123—41no—
314C22⋊Q8—21no—
315C22⋊Q8—41no—
316C22⋊Q8—1581no—
317C22⋊Q8—1581no—
318#121—21no—
319C4×Q8—21no—
320C4×Q8—1581no—
321C4×Q8—1581no—
322C4×D4—21no—
323#117C22⋊D7911yescharacteristic
324Q32—3161no—
325Q32—3161no—
326SD32—3161no—
327SD32—3161no—
328M32—3161no—
329C2×C16—3161no—
330C2×C16—791no—
331C2×C16—791no—
332C2×C16—791no—
333C2×C16—791no—
334#107—21no—
335#107—791no—
336#107—791no—
337#107C22×D7911yescharacteristic
338#107C22×D7911yescharacteristic
339#106—1581no—
340#103—21no—
341#103—21no—
342#103—1581no—
343#103—1581no—
344#103—1581no—
345#103—1581no—
346#102—21no—
347#102—21no—
348#97C22⋊D7911yescharacteristic
349#97C22⋊D7911yescharacteristic
350C4×C8C22×D7911yescharacteristic
351C4×C8C22×D7911yescharacteristic
352C2×Q8—21no—
353C2×Q8—41no—
354C2×Q8—1581no—
355C2×Q8—1581no—
356C2×D4—21no—
357C2×D4—41no—
358C22×C4—41no—
359C22×C4C2×C22⋊D7911yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—3162no—
363Q16—1581no—
364Q16—1581no—
365Q16—1581no—
366Q16—1581no—
367SD16—42no—
368D8—21no—
369D8—21no—
370M16—41no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8C2×C22⋊D7911yescharacteristic
374C2×C8C2×C22⋊D7911yescharacteristic
375C2×C8C2×C22⋊D7911yescharacteristic
376C2×C8C23×D7911yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—3161no—
383C4⋊C4—3161no—
384C4⋊C4—3161no—
385C4⋊C4—3161no—
386C4⋊C4—3161no—
387C4⋊C4—3161no—
388C4⋊C4—3161no—
389C4⋊C4—3161no—
390C4⋊C4—1581no—
391C4⋊C4—1581no—
392C4⋊C4#1192960911yescharacteristic
393C4⋊C4#1192960911yescharacteristic
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—3161no—
400C22⋊C4—3161no—
401C22⋊C4#1192960911yescharacteristic
402C22⋊C4#1192960911yescharacteristic
403C42—41no—
404C42—3161no—
405C42—3161no—
406C42C2×C22⋊D7911yescharacteristic
407C42C2×C22⋊D7911yescharacteristic
408C16—1581no—
409C16—1581no—
410C16—1581no—
411C16—1581no—
412C23—41no—
413C23C2×#1192960911yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—161no—
417Q8—3161no—
418Q8—3161no—
419Q8—3161no—
420Q8—3161no—
421D4—41no—
422D4—41no—
423D4—161no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—161no—
430C2×C4—3161no—
431C2×C4—3161no—
432C2×C4—3161no—
433C2×C4—3161no—
434C2×C4C2×#1192960911yescharacteristic
435C2×C4C2×#1192960911yescharacteristic
436C2×C4C2×#1192960911yescharacteristic
437C2×C4C2×#1192960911yescharacteristic
438C2×C4C2×#1192960911yescharacteristic
439C2×C4C22×C22⋊D7911yescharacteristic
440C8—21no—
441C8—21no—
442C8—21no—
443C8#1457920711yescharacteristic
444C8#1457920811yescharacteristic
445C22—21no—
446C22—21no—
447C22—21no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22C22×#1192960911yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—6321no—
461C4—6321no—
462C4—6321no—
463C4—6321no—
464C4#1909914111yescharacteristic
465C4#1909936111yescharacteristic
466C2—41no—
467C2—161no—
468C2#2377470711yescharacteristic
469C2#2377654711yescharacteristic
470C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3175C2
socleC22×C79C22×D8
prime radical#466×C79C22
hypercenter#3175D79
hypocenterC79#33824
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #23774059C2semidirect
2. #23765564C2semidirect
3. #23765557C2semidirect
4. #23765549C2semidirect
5. #23765539C2semidirect
7. #23765506C2semidirect
9. #23765489C2semidirect
10. #23765459C2semidirect
11. #23765459C2semidirect
12. #23765450C2semidirect
13. #23765445C2semidirect
14. #23765159C2semidirect
251. C79#33824semidirect

Maximal subgroups

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

subgroupindex
#237740592
#237655642
#237655572
#237655492
#237655392
#237655112
#237655062
#237654942
#237654892
#237654592
#237654502
#237654452
#237651592
C79×#31752
#3382479