Finite Group Databasebeta

← #29569099

Subgroups — #29569099

Order 15616.

500 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21452183C211yesmaximal, characteristic
2#21447378C211yesmaximal, characteristic, complemented
3#21447224C211yesmaximal, characteristic, complemented
4#21447222C211yesmaximal, characteristic, complemented
5#21443641C211yesmaximal, characteristic, complemented
6#21443639C211yesmaximal, characteristic, complemented
7C61×#1270C211yesmaximal, characteristic, complemented
8C2×#14519761—21no—
9C2×#14519749—21no—
10C2×#14519699—21no—
11C2×#14519698—21no—
12#16839684—21no—
13#16839678—21no—
14#16839671—21no—
15#16839664C2211yescharacteristic
16#16839660C2211yescharacteristic
17#16839637—21no—
18#16839632C2211yescharacteristic
19#16839525—21no—
20#16839525—21no—
21#16839525—21no—
22#16839524—21no—
23#16839517C2211yescharacteristic
24C61×#390C2211yescharacteristic
25C61×#336C2211yescharacteristic
26C61×#336C2211yescharacteristic
27C22×D244D411yescharacteristic
28C22×C22⋊D61—21no—
29C22×C22⋊D61—21no—
30C23×Dic61—21no—
31C22×#92418—21no—
32#14519761—42no—
33#14519761—42no—
34#14519749—42no—
35#14519749—42no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61D411yescharacteristic
41C2×C4⋊Dic61D411yescharacteristic
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—21no—
44C2×C22⋊Dic61—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—21no—
48C2×#11758198—21no—
49#14519699—42no—
50#14519698—42no—
51#14519684—21no—
52#14519684—21no—
53C22×D4×C61D411yescharacteristic
54C22×C8×C61—21no—
55C22×C8×C61—21no—
56C2×C4⋊C4×C61D411yescharacteristic
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60C24×C61—21no—
61C23×D61—21no—
62C2×D4×C61—21no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61D812yes—
66C22×C4×C61—21no—
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C2×C22⋊D61—42no—
71C2×C22⋊D61—42no—
72C2×C22⋊D61—42no—
73C2×C22⋊D61—81no—
74C2×C22⋊D61—81no—
75C2×C22⋊D61—81no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—42no—
80C22×Dic61—42no—
81C22×Dic61—41no—
82C22×Dic61—81no—
83C2×D244—21no—
84C2×D244—21no—
85C2×D244—41no—
86C2×D244—41no—
87C2×C8×C61—41no—
88C2×C8×C61—41no—
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C4⋊C4×C61—41no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C42×C61—41no—
99C22⋊Dic61—42no—
100#11758198—42no—
101#11758198—41no—
102#11758198—41no—
103#11758198—41no—
104#11758198—41no—
105#11758198—81no—
106C4⋊Dic61—21no—
107C4⋊Dic61—42no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4⋊Dic61—42no—
111C4⋊Dic61—41no—
112C4⋊Dic61—41no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115C4⋊Dic61SD1612yes—
116#11758196—41no—
117#11758196—41no—
118C2×#92418—41no—
119C2×#92418—41no—
120C2×#92418—41no—
121C23×C61—22no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61#12011yescharacteristic
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136D4×C61—42no—
137D4×C61—81no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×SD1611yescharacteristic
149C2×C4×C61#13511yescharacteristic
150C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C22⋊D61—84no—
154C22⋊D61—84no—
155C2×Dic61—22no—
156C2×Dic61—41no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—84no—
165C2×Dic61—81no—
166C2×Dic61—81no—
167C2×Dic61—81no—
168D244—42no—
169D244—81no—
170C8×C61—81no—
171C8×C61—81no—
172#92418—81no—
173#7795—611nomaximal, Sylow
174C22×C61—42no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61#44711yescharacteristic
182C22×C61#44911yescharacteristic
183C22×C61#45011yescharacteristic
184C22×C61#45411yescharacteristic
185C22×C61#45511yescharacteristic
186C22×C61#45511yescharacteristic
187C22×C61#45811yescharacteristic
188C2×D61—81no—
189C2×D61—81no—
190C2×D61—81no—
191C2×D61—81no—
192C2×D61—81no—
193C2×D61—81no—
194C2×D61—81no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—81no—
202Dic61—42no—
203Dic61—81no—
204Dic61—81no—
205Dic61—161no—
206#2682—611no—
207#2099—611no—
208#2071—611no—
209#2071—611no—
210#1271—611no—
211#1270—611no—
212#1270D6111yescharacteristic, complemented
213C2×C61—82no—
214C2×C61#160711yescharacteristic
215C2×C61#161211yescharacteristic
216C2×C61#161611yescharacteristic
217C2×C61#162511yescharacteristic
218C2×C61#163911yescharacteristic
219C2×C61#164211yescharacteristic
220C2×C61#166211yescharacteristic
221D61—161no—
222C2×#123—1221no—
223C2×#121—1221no—
224C2×#102—1221no—
225C2×#102—1221no—
226#396—1221no—
227#394—1221no—
228#392—1221no—
229#390—611no—
230#390C2×D6111yescharacteristic
231#389—611no—
232#385—1221no—
233#384—611no—
234#340—1221no—
235#340—1221no—
236#340—1221no—
237#340—1221no—
238#336—611no—
239#336C2×D6111yescharacteristic
240#336C2×D6111yescharacteristic
241C61#779511yescharacteristic, complemented, Sylow
242C22×D4—611no—
243C22×D4—1221no—
244C22×D4—1221no—
245C22×D4D24411yescharacteristic
246C23×C4—1221no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—1221no—
250#123—2442no—
251#123—2442no—
252#121—2442no—
253#121—2442no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4D24411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1221no—
270C2×C42C22×D6111yescharacteristic
271#102—2442no—
272#102—2442no—
273#95—1221no—
274#95—1221no—
275C24—21no—
276C24—1221no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—1221no—
281C2×D4—1221no—
282C2×D4—4881no—
283C2×D4—4881no—
284C2×D4—4881no—
285C2×D4—2441no—
286C2×D4—2441no—
287C2×D4—2442no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4D48812yes—
291C22×C4—21no—
292C22×C4—1221no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—4881no—
296C22×C4—2442no—
297C22×C4—2441no—
298C22×C4—2442no—
299C22×C4C2×D24411yescharacteristic
300C22×C4D4×D6111yescharacteristic
301C22×C4D4×D6111yescharacteristic
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—2441no—
309C2×C8—2441no—
310C2×C8—2441no—
311C4⋊C4—41no—
312C4⋊C4—612no—
313C4⋊C4—1221no—
314C4⋊C4—2442no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2442no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—4881no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C42—41no—
336C23—22no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—2441no—
344C23—2441no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23#1451974711yescharacteristic
351D4—42no—
352D4—81no—
353D4—4881no—
354D4—4884no—
355D4—4884no—
356D4—2442no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—1222no—
367C2×C4—4881no—
368C2×C4—4881no—
369C2×C4—4881no—
370C2×C4—4884no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2441no—
377C2×C4—2441no—
378C2×C4—2441no—
379C2×C4C2×D48811yescharacteristic
380C2×C4SD16×D6111yescharacteristic
381C2×C4#1451979811yescharacteristic
382C2×C4#1451980111yescharacteristic
383C2×C4#1451980111yescharacteristic
384C2×C4#1451980511yescharacteristic
385C8—81no—
386C8—81no—
387C8—4881no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—4881no—
396C22—4881no—
397C22—4881no—
398C22—4881no—
399C22—4881no—
400C22—4881no—
401C22—4881no—
402C22#1683989811yescharacteristic
403C22#1683990811yescharacteristic
404C22#1683991411yescharacteristic
405C22#1683993611yescharacteristic
406C22#1683994111yescharacteristic
407C22#1683994211yescharacteristic
408C22#1683995311yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—81no—
416C4—9761no—
417C4—4881no—
418C4—4881no—
419C4—2442no—
420C2—82no—
421C2—9761no—
422C2#2144480711yescharacteristic
423C2#2144483411yescharacteristic
424C2#2144485711yescharacteristic
425C2#2144491011yescharacteristic
426C2#2144500011yescharacteristic
427C2#2144502011yescharacteristic
428C2#2144514411yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1270C2
socleC23×C61#120
prime radical#21452183C2
hypercenter#1270D61
hypocenterC61#7795
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #21447378C2semidirect
3. #21447224C2semidirect
4. #21447222C2semidirect
5. #21443641C2semidirect
6. #21443639C2semidirect
7. C61×#1270C2semidirect
212. #1270D61semidirect
241. C61#7795semidirect

Maximal subgroups

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

subgroupindex
#214521832
#214473782
#214472242
#214472222
#214436412
#214436392
C61×#12702
#779561