Finite Group Databasebeta

← #29835383

Subgroups — #29835383

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D61C211yesmaximal, characteristic, complemented
2#21456244C211yesmaximal, characteristic
3#21448058C211yesmaximal, characteristic, complemented
4#21448056C211yesmaximal, characteristic, complemented
5#21448055C211yesmaximal, characteristic, complemented
6#21448045C212yesmaximal, complemented
7#21448041C212yesmaximal, complemented
8#21447530C212yesmaximal, complemented
9#21447528C212yesmaximal, complemented
10#21447412C211yesmaximal, characteristic, complemented
11C61×#2243C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13C2×#14519798—21no—
14C2×#14519796C2211yescharacteristic
15#16840311—21no—
16C4⋊Q8×D61—21no—
17C2×#14519716C2212yes—
18#16840167C2211yescharacteristic
19#16840166—21no—
20#16840166C2212yes—
21#16840165—21no—
22#16840146C2211yescharacteristic
23#16840144—21no—
24#16840140C2211yescharacteristic
25#16840137C2211yescharacteristic
26#16840133—21no—
27#107×D61—21no—
28#107×D61C2212yes—
29C4×C8×D61C2211yescharacteristic
30#16839567C2212yes—
31#16839566C2212yes—
32#16839565C2212yes—
33#16839564C2212yes—
34#16839563C2212yes—
35#16839562C2212yes—
36#16839538C2211yescharacteristic
37C16×Dic61C2212yes—
38C61×#500C2211yescharacteristic
39C61×#495C2211yescharacteristic
40C61×#358C2212yes—
41C61×#357C2212yes—
42C61×#346C2211yescharacteristic
43C2×#11758226—21no—
44C2×#11758224—21no—
45C2×Q8×D61—41no—
46#14519800—24no—
47#14519798—44no—
48#14519796—24no—
49C2×Dic244C2311yescharacteristic
50C2×#11758202—21no—
51C2×#11758190—21no—
52C2×#11758200—21no—
53C2×#11758201—21no—
54C2×D488C2311yescharacteristic
55C2×#11758199—21no—
56C2×C8×D61D411yescharacteristic
57C2×C8×D61C2311yescharacteristic
58#14519777—21no—
59#14519776—22no—
60#14519776—41no—
61#14519775—21no—
62#14519775—41no—
63#14519773—21no—
64#14519765—21no—
65#14519750—41no—
66#14519750—41no—
67#14519737—41no—
68#14519737—41no—
69C4⋊C4×D61—22no—
70C4⋊C4×D61—41no—
71C42×D61D411yescharacteristic
72#14519716—24no—
73C2×#11758185C2312yes—
74#14519711—21no—
75#14519711C2312yes—
76#14519710—22no—
77#14519710—22no—
78C4×#92418D411yescharacteristic
79C8×Dic61D411yescharacteristic
80C8×Dic61C2311yescharacteristic
81C2×Q16×C61C2311yescharacteristic
82C2×SD16×C61—21no—
83C2×D8×C61C2311yescharacteristic
84C4⋊Q8×C61—21no—
85#124×C61—21no—
86C2×C16×C61C2312yes—
87#107×C61—21no—
88#107×C61C2312yes—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—21no—
91C2×Q8×C61—41no—
92C2×D4×C61—21no—
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C22×Dic61—41no—
97#11758226—42no—
98#11758226—42no—
99Q8×D61—84no—
100#11758224—42no—
101#11758224—42no—
102C2×D244—21no—
103C2×D244—41no—
104C2×C4×D61—42no—
105C2×C4×D61—41no—
106C2×C4×D61D811yescharacteristic
107C2×C4×D61D811yescharacteristic
108C2×C4×D61C2×D411yescharacteristic
109C2×Dic122—21no—
110C2×Dic122—41no—
111C2×Dic122—42no—
112Q16×C61—22no—
113SD16×C61—42no—
114D8×C61—22no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2411yescharacteristic
117C4⋊C4×C61—22no—
118C4⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C42×C61C2×D411yescharacteristic
122#11758202—42no—
123#11758201—42no—
124#11758200—42no—
125#11758199—42no—
126#11758198—41no—
127#11758198—41no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130#11758196—42no—
131#11758196—42no—
132#11758196—41no—
133#11758196—41no—
134C4×Dic61D811yescharacteristic
135C4×Dic61D811yescharacteristic
136C4×Dic61C2×D411yescharacteristic
137C2×#92418C2×D411yescharacteristic
138C2×#92418C2×D411yescharacteristic
139Dic244—22no—
140D488—22no—
141#11758190—42no—
142C8×D61SD1614yes—
143C8×D61#4114yes—
144C16×C61—22no—
145#11758185—22no—
146C23×C61—41no—
147C22×D61—41no—
148C22×D61C2×D811yescharacteristic
149Q8×C61—41no—
150Q8×C61—41no—
151Q8×C61—82no—
152D4×C61—41no—
153D4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—82no—
160C22⋊D61—82no—
161C22⋊D61—82no—
162C2×Dic61—42no—
163C2×Dic61—41no—
164C2×Dic61—82no—
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167C2×Dic61C2×D811yescharacteristic
168D244—41no—
169D244—41no—
170D244—82no—
171C4×D61—22no—
172C4×D61—22no—
173C4×D61—84no—
174C4×D61—82no—
175C4×D61#12412yes—
176C4×D61C2×SD1612yes—
177Dic122—41no—
178Dic122—41no—
179Dic122—82no—
180Dic122—84no—
181C8×C61C2×SD1612yes—
182C8×C61C2×#4112yes—
183#92418#12412yes—
184#92418C2×SD1612yes—
185#33707—611nomaximal, Sylow
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61C22×D811yescharacteristic
190C2×D61—21no—
191C2×D61—21no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C2×D61#48612yes—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—81no—
199C4×C61—82no—
200C4×C61C2×#12411yescharacteristic
201C4×C61C22×SD1611yescharacteristic
202Dic61—81no—
203Dic61—82no—
204Dic61#48612yes—
205Dic61#50912yes—
206Dic61#51012yes—
207C2×#500—611no—
208#3147—611no—
209#2243—611no—
210#2243—611no—
211#2243—611no—
212#2243D6111yescharacteristic, complemented
213#2241—612no—
214#2240—612no—
215#2139—612no—
216#2139—612no—
217C4×M32—611no—
218C2×C61—161no—
219C2×C61C2×#48611yescharacteristic
220C2×C61C2×#50911yescharacteristic
221C2×C61C2×#51011yescharacteristic
222D61—22no—
223D61—161no—
224C2×#135—611no—
225C2×#135—611no—
226C2×#135—1221no—
227#532—1221no—
228C2×C4⋊Q8—1221no—
229C2×M32—612no—
230#500—612no—
231#500—611no—
232#500—1221no—
233#500—1221no—
234#500C2×D6111yescharacteristic
235#495—611no—
236#495—1221no—
237#495C2×D6111yescharacteristic
238#494—611no—
239#493—611no—
240#492—1221no—
241C2×#107—612no—
242C2×#107—1221no—
243C2×C4×C8—611no—
244#358—612no—
245#358—612no—
246#358—612no—
247#358C2×D6112yes—
248#357—612no—
249#357—612no—
250#357—612no—
251#357C2×D6112yes—
252#346—611no—
253#346C2×D6111yescharacteristic
254C4×C16—612no—
255C61#3370711yescharacteristic, complemented, Sylow
256C2×#41—1221no—
257C2×#41—1221no—
258C22×Q8—2441no—
259#135—1224no—
260#135—1224no—
261#135—2444no—
262C2×Q16—611no—
263C2×Q16—1221no—
264C2×Q16C22×D6111yescharacteristic
265C2×SD16—21no—
266C2×SD16—1221no—
267C2×SD16—1221no—
268C2×SD16—1221no—
269C2×D8—611no—
270C2×D8—1221no—
271C2×D8C22×D6111yescharacteristic
272C22×C8—611no—
273C22×C8—611no—
274C4⋊Q8—21no—
275C4⋊Q8—1222no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278C4⋊Q8—2441no—
279C4⋊Q8—2441no—
280#127—1221no—
281#124—21no—
282#124—1221no—
283#121—2441no—
284#121—2441no—
285C4×D4—2441no—
286C4×D4—2441no—
287C2×C4⋊C4—1222no—
288C2×C4⋊C4—2441no—
289C2×C42—611no—
290M32—1224no—
291C2×C16—612no—
292C2×C16C22×D6112yes—
293#107—21no—
294#107—612no—
295#107—1222no—
296#107—1222no—
297#107—1221no—
298#107C22×D6112yes—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8C22×D6111yescharacteristic
303#41—2442no—
304#41—2442no—
305#41—2442no—
306#41—2442no—
307C2×Q8—21no—
308C2×Q8—41no—
309C2×Q8—1221no—
310C2×Q8—4884no—
311C2×Q8—2442no—
312C2×Q8—2441no—
313C2×D4—21no—
314C2×D4—1221no—
315C2×D4—2441no—
316C2×D4—2441no—
317C2×D4—2441no—
318C2×D4—2441no—
319C22×C4—611no—
320C22×C4—611no—
321C22×C4—611no—
322C22×C4—2441no—
323C22×C4—2442no—
324C22×C4—2441no—
325Q16—22no—
326Q16—1222no—
327Q16—2442no—
328SD16—42no—
329SD16—2442no—
330SD16—2442no—
331SD16—2442no—
332D8—22no—
333D8—1222no—
334D8—2442no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—614no—
338C2×C8—614no—
339C2×C8D4×D6111yescharacteristic
340C2×C8C23×D6111yescharacteristic
341C4⋊C4—22no—
342C4⋊C4—41no—
343C4⋊C4—1222no—
344C4⋊C4—2442no—
345C4⋊C4—2442no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C42—611no—
354C42—611no—
355C42—611no—
356C42D4×D6111yescharacteristic
357C16—22no—
358C16—1222no—
359C23—41no—
360C23—611no—
361C23—2441no—
362Q8—41no—
363Q8—41no—
364Q8—82no—
365Q8—4882no—
366Q8—4884no—
367Q8—2441no—
368Q8—2441no—
369D4—41no—
370D4—41no—
371D4—4882no—
372D4—4882no—
373D4—4882no—
374D4—4882no—
375D4—2441no—
376D4—2441no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—611no—
380C2×C4—611no—
381C2×C4—611no—
382C2×C4—612no—
383C2×C4—612no—
384C2×C4—1222no—
385C2×C4—1222no—
386C2×C4—4882no—
387C2×C4—4884no—
388C2×C4—4882no—
389C2×C4—2441no—
390C2×C4—2442no—
391C2×C4D8×D6111yescharacteristic
392C2×C4D8×D6111yescharacteristic
393C2×C4C2×D4×D6111yescharacteristic
394C8—612no—
395C8—612no—
396C8SD16×D6112yes—
397C8#41×D6112yes—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—612no—
402C22—1221no—
403C22—1221no—
404C22—4881no—
405C22—4881no—
406C22—4881no—
407C22C2×D8×D6111yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—82no—
412C4—612no—
413C4—612no—
414C4—612no—
415C4—4881no—
416C4—4882no—
417C4#124×D6111yescharacteristic
418C4C2×SD16×D6111yescharacteristic
419C2—161no—
420C2—1222no—
421C2—9761no—
422C2#486×D6111yescharacteristic
423C2#509×D6111yescharacteristic
424C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2243C2
socleC22×C61C22×D8
prime radical#21456244C2
hypercenter#2243D61
hypocenterC61#33707
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2243

Complements

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

NH ≅ G/Ntype
1. #500×D61C2semidirect
3. #21448058C2semidirect
4. #21448056C2semidirect
5. #21448055C2semidirect
6. #21448045C2semidirect
7. #21448041C2semidirect
8. #21447530C2semidirect
9. #21447528C2semidirect
10. #21447412C2semidirect
11. C61×#2243C2semidirect
212. #2243D61semidirect
255. C61#33707semidirect

Maximal subgroups

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

subgroupindex
#500×D612
#214562442
#214480582
#214480562
#214480552
#214480452
#214480412
#214475302
#214475282
#214474122
C61×#22432
#3370761