Finite Group Databasebeta

← #34445848

Subgroups — #34445848

Order 20224.

464 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775769C211yesmaximal, characteristic
2#23765676C211yesmaximal, characteristic, complemented
3#23765608C211yesmaximal, characteristic, complemented
4#23765605C211yesmaximal, characteristic, complemented
5#23765602C211yesmaximal, characteristic, complemented
6#23765600C211yesmaximal, characteristic, complemented
7#23765564C211yesmaximal, characteristic, complemented
8#23765561C211yesmaximal, characteristic, complemented
9#23765506C211yesmaximal, characteristic, complemented
10#23765506C211yesmaximal, characteristic, complemented
11C4×#14579107C211yesmaximal, characteristic, complemented
12#23765232C211yesmaximal, characteristic
13#23765189C211yesmaximal, characteristic, complemented
14#23765186C211yesmaximal, characteristic, complemented
15C79×#3336C211yesmaximal, characteristic
16#19099253—21no—
17C2×#14579107C2211yescharacteristic
18C2×#14579101C2211yescharacteristic
19C2×#14579101C2211yescharacteristic
20#19098965C2211yescharacteristic
21#19098905—21no—
22#19098894—21no—
23#19098871—21no—
24#19098804C2211yescharacteristic
25#19098804C2211yescharacteristic
26C4×Dic316C2211yescharacteristic
27#19098601C2211yescharacteristic
28C2×#14579095C2211yescharacteristic
29#19098390C2211yescharacteristic
30#19098390C2211yescharacteristic
31#19098390C2211yescharacteristic
32#19098390C2211yescharacteristic
33#19098388C2211yescharacteristic
34#19098388C2211yescharacteristic
35#19098388C2211yescharacteristic
36#19098388C2211yescharacteristic
37#19098388C2211yescharacteristic
38C4⋊(C79⋊C16)C2211yescharacteristic
39#19098367C2211yescharacteristic
40#19098367C2211yescharacteristic
41#19098367C2211yescharacteristic
42#19098366C2211yescharacteristic
43#19098366C2211yescharacteristic
44#19098366C2211yescharacteristic
45C22⋊(C79⋊C16)C2211yescharacteristic
46C22⋊(C79⋊C16)C2211yescharacteristic
47C4×C79⋊C16C2211yescharacteristic
48#552×C79—21no—
49#499×C79C2211yescharacteristic
50#485×C79—21no—
51#483×C79—21no—
52#479×C79—21no—
53#467×C79C2211yescharacteristic
54#467×C79C2211yescharacteristic
55C4×Q16×C79C2211yescharacteristic
56C8×D4×C79C2211yescharacteristic
57#427×C79C2211yescharacteristic
58C2×#107×C79C2211yescharacteristic
59C2×Dic316C2311yescharacteristic
60#14579154—41no—
61#14579153—21no—
62#14579151—41no—
63#14579142—41no—
64#14579140—21no—
65#14579140—21no—
66C4×Dic158—21no—
67#14579120—21no—
68#14579117—41no—
69C2×C4⋊Dic79—21no—
70#14579107—22no—
71#14579101—22no—
72#14579101—22no—
73C2×C79⋊C16C2311yescharacteristic
74C2×C79⋊C16C2311yescharacteristic
75C2×C79⋊C16C2311yescharacteristic
76C2×C79⋊C16C2311yescharacteristic
77#14579095—21no—
78#14579095C2311yescharacteristic
79#14579095C2311yescharacteristic
80#14579095C2311yescharacteristic
81#14579093—21no—
82#14579093—21no—
83#14579086—21no—
84#14579086—21no—
85#14579086—21no—
86C2×Q16×C79C2311yescharacteristic
87C22×C8×C79C2311yescharacteristic
88C22×C8×C79C2311yescharacteristic
89#126×C79—41no—
90#125×C79—21no—
91#125×C79—41no—
92#123×C79—41no—
93C22⋊Q8×C79—21no—
94C22⋊Q8×C79—21no—
95C4×Q8×C79—21no—
96C4×D4×C79D411yescharacteristic
97#117×C79—21no—
98#117×C79—41no—
99C2×C4⋊C4×C79—21no—
100#107×C79—21no—
101#107×C79C2311yescharacteristic
102#107×C79C2311yescharacteristic
103#107×C79C2311yescharacteristic
104#106×C79—21no—
105#106×C79—21no—
106C4⋊C8×C79D411yescharacteristic
107#103×C79—21no—
108#103×C79—21no—
109#103×C79—21no—
110C22⋊C8×C79D411yescharacteristic
111C22⋊C8×C79D411yescharacteristic
112C4×C8×C79C2311yescharacteristic
113C2×Dic158—21no—
114C22×Dic79—41no—
115Dic316—21no—
116Dic316—21no—
117C4⋊Dic79—21no—
118C4⋊Dic79—21no—
119C4⋊Dic79—21no—
120C4⋊Dic79—41no—
121Dic79⋊C4—41no—
122Dic79⋊C4—41no—
123Dic79⋊C4—41no—
124Dic79⋊C4—41no—
125Dic79⋊C4—41no—
126C22⋊Dic79—41no—
127C22⋊Dic79—41no—
128C22⋊Dic79—41no—
129C4×Dic79—41no—
130C4×Dic79—41no—
131C79⋊C16—21no—
132C79⋊C16—21no—
133C79⋊C16—21no—
134C79⋊C16#4112yes—
135C2×Q8×C79—21no—
136C2×D4×C79D811yescharacteristic
137C22×C4×C79—41no—
138C22×C4×C79C2×D411yescharacteristic
139C22×C4×C79C2×D411yescharacteristic
140Q16×C79—21no—
141Q16×C79—21no—
142C2×C8×C79—21no—
143C2×C8×C79—21no—
144C2×C8×C79—21no—
145C2×C8×C79—21no—
146C2×C8×C79C2×D411yescharacteristic
147C2×C8×C79C2×D411yescharacteristic
148C2×C8×C79C2×D411yescharacteristic
149C2×C8×C79C2411yescharacteristic
150C4⋊C4×C79—21no—
151C4⋊C4×C79—21no—
152C4⋊C4×C79—21no—
153C4⋊C4×C79—41no—
154C4⋊C4×C79—41no—
155C4⋊C4×C79—41no—
156C4⋊C4×C79—41no—
157C4⋊C4×C79—41no—
158C4⋊C4×C79—41no—
159C4⋊C4×C79D811yescharacteristic
160C22⋊C4×C79—41no—
161C22⋊C4×C79—41no—
162C22⋊C4×C79—41no—
163C22⋊C4×C79D811yescharacteristic
164C22⋊C4×C79D811yescharacteristic
165C42×C79—41no—
166C42×C79—41no—
167C42×C79C2×D411yescharacteristic
168C23×C79C2×D811yescharacteristic
169C23×C79C2×D811yescharacteristic
170Q8×C79—41no—
171Q8×C79—41no—
172D4×C79—41no—
173C2×C4×C79—21no—
174C2×C4×C79—21no—
175C2×C4×C79—21no—
176C2×C4×C79—21no—
177C2×C4×C79—41no—
178C2×C4×C79—41no—
179C2×C4×C79—41no—
180C2×C4×C79—41no—
181C2×C4×C79—82no—
182C2×C4×C79C2×D811yescharacteristic
183C2×C4×C79C2×D811yescharacteristic
184C2×C4×C79C2×D811yescharacteristic
185C2×C4×C79C2×D811yescharacteristic
186C2×C4×C79C22×D411yescharacteristic
187C2×Dic79—41no—
188C2×Dic79—41no—
189C2×Dic79—41no—
190C2×Dic79—41no—
191C2×Dic79—82no—
192Dic158—41no—
193Dic158—41no—
194C8×C79—21no—
195C8×C79—21no—
196C8×C79—21no—
197C8×C79C2×#4111yescharacteristic
198C8×C792−1+411yescharacteristic
199C22×C79—21no—
200C22×C79—21no—
201C22×C79—21no—
202C22×C79—21no—
203C22×C79—21no—
204C22×C79—21no—
205C22×C79C22×D811yescharacteristic
206C4×C79—21no—
207C4×C79—21no—
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—81no—
211C4×C79—81no—
212C4×C79—81no—
213C4×C79—81no—
214C4×C79#54711yescharacteristic
215C4×C79#57811yescharacteristic
216Dic79—81no—
217Dic79—81no—
218Dic79—81no—
219Dic79—81no—
220#33919—791nomaximal, Sylow
221C2×C79—41no—
222C2×C79—41no—
223C2×C79#333411yescharacteristic
224C2×C79#341411yescharacteristic
225C2×C79#341711yescharacteristic
226#3336—791no—
227#3336D7911yescharacteristic
228#2253—791no—
229#2237—791no—
230#2237—791no—
231#2236—791no—
232#2236—791no—
233#2229—791no—
234#2229—791no—
235#2219—791no—
236#2219—791no—
237C4×Q32—791no—
238#2169—791no—
239#2159—791no—
240#2159—791no—
241C79#3391911yescharacteristic, complemented, Sylow
242#552—21no—
243#552—1581no—
244C2×Q32—791no—
245C2×M32—791no—
246C2×M32—791no—
247#499—791no—
248#499C2×D7911yescharacteristic
249#485—21no—
250#485—1581no—
251#483—21no—
252#483—1581no—
253#479—21no—
254#479—1581no—
255#467—791no—
256#467—791no—
257#467C2×D7911yescharacteristic
258#467C2×D7911yescharacteristic
259C4×Q16—791no—
260C4×Q16C2×D7911yescharacteristic
261C8×D4C2×D7911yescharacteristic
262#427—791no—
263#427C2×D7911yescharacteristic
264C2×#107—791no—
265C2×#107C2×D7911yescharacteristic
266#367—791no—
267#367—791no—
268#367—791no—
269#367—791no—
270#366—791no—
271#366—791no—
272#366—791no—
273#366—791no—
274#366—791no—
275C4⋊C16—791no—
276#358—791no—
277#358—791no—
278#358—791no—
279#358—791no—
280#358—791no—
281#358—791no—
282C22⋊C16—791no—
283C22⋊C16—791no—
284C4×C16—791no—
285C2×Q16—791no—
286C2×Q16C22×D7911yescharacteristic
287C22×C8C22×D7911yescharacteristic
288C22×C8C22×D7911yescharacteristic
289#126—41no—
290#126—3161no—
291#125—21no—
292#125—41no—
293#125—3161no—
294#125—1581no—
295#123—41no—
296#123—3161no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—1581no—
300C22⋊Q8—1581no—
301C4×Q8—21no—
302C4×Q8—1581no—
303C4×D4C22⋊D7911yescharacteristic
304#117—21no—
305#117—41no—
306#117—3161no—
307#117—1581no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—1581no—
310Q32—1582no—
311M32—1582no—
312M32—1582no—
313C2×C16—791no—
314C2×C16—791no—
315C2×C16—791no—
316C2×C16—791no—
317#107—21no—
318#107—791no—
319#107—791no—
320#107—791no—
321#107—1581no—
322#107C22×D7911yescharacteristic
323#107C22×D7911yescharacteristic
324#107C22×D7911yescharacteristic
325#106—21no—
326#106—21no—
327#106—1581no—
328#106—1581no—
329C4⋊C8C22⋊D7911yescharacteristic
330#103—21no—
331#103—21no—
332#103—21no—
333#103—1581no—
334#103—1581no—
335#103—1581no—
336C22⋊C8C22⋊D7911yescharacteristic
337C22⋊C8C22⋊D7911yescharacteristic
338C4×C8C22×D7911yescharacteristic
339C2×Q8—21no—
340C2×Q8—1581no—
341C2×D4#1192960911yescharacteristic
342C22×C4—41no—
343C22×C4—3161no—
344C22×C4C2×C22⋊D7911yescharacteristic
345C22×C4C2×C22⋊D7911yescharacteristic
346Q16—21no—
347Q16—21no—
348Q16—1581no—
349Q16—1581no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8C2×C22⋊D7911yescharacteristic
355C2×C8C2×C22⋊D7911yescharacteristic
356C2×C8C2×C22⋊D7911yescharacteristic
357C2×C8C23×D7911yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—3161no—
368C4⋊C4—3161no—
369C4⋊C4—3161no—
370C4⋊C4—3161no—
371C4⋊C4—3161no—
372C4⋊C4—3161no—
373C4⋊C4—1581no—
374C4⋊C4—1581no—
375C4⋊C4—1581no—
376C4⋊C4#1192960911yescharacteristic
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—3161no—
381C22⋊C4—3161no—
382C22⋊C4—3161no—
383C22⋊C4#1192960911yescharacteristic
384C22⋊C4#1192960911yescharacteristic
385C42—41no—
386C42—41no—
387C42—3161no—
388C42—3161no—
389C42C2×C22⋊D7911yescharacteristic
390C16—792no—
391C16—1581no—
392C16—1581no—
393C16—1581no—
394C23C2×#1192960911yescharacteristic
395C23C2×#1192960911yescharacteristic
396Q8—41no—
397Q8—41no—
398Q8—3161no—
399Q8—3161no—
400D4—41no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—3161no—
411C2×C4—3161no—
412C2×C4—3161no—
413C2×C4—3161no—
414C2×C4—6322no—
415C2×C4C2×#1192960911yescharacteristic
416C2×C4C2×#1192960911yescharacteristic
417C2×C4C2×#1192960911yescharacteristic
418C2×C4C2×#1192960911yescharacteristic
419C2×C4C22×C22⋊D7911yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8#41×D7911yescharacteristic
424C8#1457920811yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22C22×#1192960911yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—6321no—
441C4—6321no—
442C4—6321no—
443C4—6321no—
444C4#1909921211yescharacteristic
445C4#1909937811yescharacteristic
446C2—41no—
447C2—41no—
448C2#2377574311yescharacteristic
449C2#2377653411yescharacteristic
450C2#2377655111yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#3336C2
socleC22×C79C22×D8
prime radicalC2×D4×C79D8
hypercenter#3336D79
hypocenterC79#33919
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #23765676C2semidirect
3. #23765608C2semidirect
4. #23765605C2semidirect
5. #23765602C2semidirect
6. #23765600C2semidirect
7. #23765564C2semidirect
8. #23765561C2semidirect
9. #23765506C2semidirect
10. #23765506C2semidirect
11. C4×#14579107C2semidirect
13. #23765189C2semidirect
14. #23765186C2semidirect
241. C79#33919semidirect

Maximal subgroups

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

subgroupindex
#237757692
#237656762
#237656082
#237656052
#237656022
#237656002
#237655642
#237655612
#237655062
C4×#145791072
#237652322
#237651892
#237651862
C79×#33362
#3391979