Finite Group Databasebeta

← #29572591

Subgroups — #29572591

Order 15616.

1074 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 119 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#21459583C211yesmaximal, characteristic
2#21446282C212yesmaximal, complemented
3#21444178C212yesmaximal, complemented
4#21444157C211yesmaximal, characteristic, complemented
5C61×#1870C211yesmaximal, characteristic, complemented
6C2×#14519822C412yes—
7#16840416—22no—
8#16840407—24no—
9D4×D244—22no—
10#16840343—21no—
11#16840324—21no—
12#16840324C412yes—
13#16840312—21no—
14C2×#14519751—21no—
15C2×#14519751C2212yes—
16C2×#14519693—22no—
17#16839726C2212yescomplemented
18#16839530—22no—
19#16839488—21no—
20#16839486—21no—
21#16839485—24no—
22C22×M16×C61C2211yescharacteristic, complemented
23C2×C4×D4×C61C2211yescharacteristic
24C2×C22⋊C8×C61—21no—
25C2×C22⋊C8×C61C2211yescharacteristic, complemented
26C61×#336—21no—
27#14519822—44no—
28#14519822—44no—
29C2×#11758222—22no—
30C2×#11758224—24no—
31C22×D244—21no—
32C22×D244D412yes—
33C22×C22⋊D61—22no—
34C22×C22⋊D61C2×C414yes—
35C2×D4×D61—24no—
36C2×#11758194—22no—
37#14519777—21no—
38#14519774—21no—
39#14519765—22no—
40#14519761—24no—
41#14519761—24no—
42#14519757—24no—
43#14519752—24no—
44#14519751—22no—
45#14519751—22no—
46#14519751—22no—
47#14519751—22no—
48#14519751—22no—
49#14519751—22no—
50#14519751—22no—
51#14519751—22no—
52#14519747—24no—
53#14519747—24no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61D412yes—
58C2×#11758198—22no—
59C2×#11758198D414yes—
60#14519693—44no—
61C22⋊#92418—22no—
62C22⋊#92418—22no—
63C22×D4×C61C2×C411yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C2×M16×C61—22no—
67C2×M16×C61—42no—
68C22×C8×C61—21no—
69C22×C8×C61—21no—
70C4×D4×C61—22no—
71C4×D4×C61—22no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61C2×C411yescharacteristic
74C2×C22⋊C4×C61D411yescharacteristic
75C2×C42×C61D411yescharacteristic
76C22⋊C8×C61—22no—
77C22⋊C8×C61—42no—
78C24×C61C22⋊C411yescharacteristic
79C24×C61C22×C411yescharacteristic
80C23×D61—22no—
81C23×D61C22⋊C414yes—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—22no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61—41no—
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22⋊C411yescharacteristic
93C22×C4×C61C22⋊C411yescharacteristic
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D412yes—
98C2×C22⋊D61—24no—
99C2×C22⋊D61—24no—
100C2×C22⋊D61—24no—
101C2×C22⋊D61—44no—
102C2×C22⋊D61—44no—
103C2×C22⋊D61—44no—
104C2×C22⋊D61—44no—
105C22×Dic61—22no—
106C22×Dic61C22⋊C414yes—
107#11758224—416no—
108D4×D61—416no—
109#11758222—44no—
110#11758222—44no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244—22no—
116C2×D244—42no—
117C2×D244—42no—
118C2×D244—81no—
119C2×D244—81no—
120C2×C4×D61—24no—
121C2×Dic122—22no—
122M16×C61—24no—
123M16×C61—42no—
124C2×C8×C61—22no—
125C2×C8×C61—41no—
126C2×C8×C61—41no—
127C2×C8×C61—41no—
128C2×C8×C61—41no—
129C2×C8×C61—41no—
130C4⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C42×C61—21no—
134C42×C61—21no—
135#11758198—24no—
136#11758198—24no—
137#11758198—24no—
138#11758198—24no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143#11758194—42no—
144#11758194—42no—
145C2×#92418—42no—
146C23×C61—21no—
147C23×C61—21no—
148C23×C61—21no—
149C23×C61—21no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—41no—
154C23×C61#10012yescomplemented
155C23×C61C2×C22⋊C412yes—
156C23×C61C2×C22⋊C411yescharacteristic
157C22×D61—24no—
158C22×D61—44no—
159C22×D61—44no—
160C22×D61—44no—
161C22×D61—44no—
162C22×D61—82no—
163C22×D61—82no—
164D4×C61—28no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—81no—
180C2×C4×C61#10012yescomplemented
181C2×C4×C61#12011yescharacteristic
182C2×C4×C61#12012yes—
183C2×C4×C61#12011yescharacteristic
184C22⋊D61—48no—
185C22⋊D61—48no—
186C22⋊D61—44no—
187C22⋊D61—44no—
188C2×Dic61—24no—
189C2×Dic61—44no—
190C2×Dic61—44no—
191D244—44no—
192D244—42no—
193D244—42no—
194D244—82no—
195D244—82no—
196D244—82no—
197C4×D61—48no—
198Dic122—44no—
199C8×C61—42no—
200C8×C61—81no—
201#92418—82no—
202#8536—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—42no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—81no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61#37911yescharacteristic
216C22×C61C2×#10011yescharacteristic
217C22×C61C2×#10011yescharacteristic
218C2×D61—44no—
219C2×D61—44no—
220C2×D61—44no—
221C2×D61—164no—
222C2×D61—164no—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—21no—
226C4×C61—41no—
227C4×C61—41no—
228C4×C61B(4,𝔽2)11yescharacteristic
229C4×C61#45811yescharacteristic
230Dic61—44no—
231#3496—611no—
232#1900—612no—
233#1870D6111yescharacteristic, complemented
234#1471—612no—
235#1465—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—42no—
239C2×C61—42no—
240C2×C61#178311yescharacteristic
241C2×C61#179011yescharacteristic
242C2×C61#179111yescharacteristic
243D61—84no—
244C2×H(5,𝔽2)—612no—
245C22×M16C2×D6111yescharacteristic, complemented
246#547—1222no—
247#546—1224no—
248D42—1222no—
249#537—1221no—
250#534—612no—
251#534—1221no—
252#532—1221no—
253C2×#121—612no—
254C2×#121—1221no—
255C2×C4×D4C2×D6111yescharacteristic
256C2×#100—1222no—
257#407—612no—
258C2×C22⋊C8—21no—
259C2×C22⋊C8C2×D6111yescharacteristic, complemented
260#343—1222no—
261#336—21no—
262#324—1221no—
263#323—1224no—
264#323—1221no—
265C61#853611yescharacteristic, complemented, Sylow
266H(5,𝔽2)—2444no—
267H(5,𝔽2)—2444no—
268C2×#41—1224no—
269C2×#41—1222no—
270C22×D4—614no—
271C22×D4—612no—
272C22×D4—1221no—
273C22×D4—1224no—
274C22×D4—1222no—
275C22×D4C4×D6111yescharacteristic
276C23×C4C22⋊D6111yescharacteristic
277C23×C4C22×D6111yescharacteristic
278C2×M16—22no—
279C2×M16—42no—
280C2×M16—1222no—
281C22×C8—21no—
282C22×C8—21no—
283C4⋊Q8—1221no—
284#127—1221no—
285#124—1222no—
286#123—1224no—
287#123—1224no—
288C22⋊Q8—1224no—
289#121—1224no—
290#121—1222no—
291#121—1222no—
292#121—1222no—
293#121—1222no—
294#121—1222no—
295#121—1222no—
296#121—1222no—
297#121—1222no—
298#120—1224no—
299#120—1224no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—1222no—
303C4×D4—1222no—
304C2×C4⋊C4—612no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4D24411yescharacteristic
307C2×C22⋊C4—614no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4C4×D6111yescharacteristic
310C2×C22⋊C4D24411yescharacteristic
311C2×C42C22⋊D6111yescharacteristic
312#100—2444no—
313C22⋊C8—22no—
314C22⋊C8—42no—
315C22⋊C8—1222no—
316C22⋊C8—1222no—
317C24—614no—
318C24—1222no—
319C24#1175819811yescharacteristic
320C24C2×C4×D6111yescharacteristic
321#41—2444no—
322#41—24416no—
323#41—2444no—
324C2×Q8—1222no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1224no—
332C2×D4—1224no—
333C2×D4—1224no—
334C2×D4—1222no—
335C2×D4—1222no—
336C2×D4—4881no—
337C2×D4—4881no—
338C2×D4—2442no—
339C2×D4—2442no—
340C2×D4—24416no—
341C2×D4—2444no—
342C2×D4—2444no—
343C2×D4—2444no—
344C2×D4—2444no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—614no—
352C22×C4—1224no—
353C22×C4—1222no—
354C22×C4#1175819811yescharacteristic
355C22×C4#1175819811yescharacteristic
356C22×C4#1175819811yescharacteristic
357C22×C4C2×D24411yescharacteristic
358C22×C4D4×D6112yes—
359C22×C4D4×D6112yes—
360C22×C4C2×C22⋊D6111yescharacteristic
361M16—24no—
362M16—42no—
363M16—2442no—
364M16—2442no—
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—2442no—
372C4⋊C4—22no—
373C4⋊C4—1222no—
374C4⋊C4—1222no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—1224no—
380C22⋊C4—1224no—
381C22⋊C4—1224no—
382C22⋊C4—1224no—
383C42—21no—
384C42—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—1224no—
394C23—4882no—
395C23—4882no—
396C23—2444no—
397C23—2444no—
398C23—2444no—
399C23—2444no—
400C23#1451969312yescomplemented
401C23C22⋊C4×D6112yes—
402C23C2×#1175819811yescharacteristic
403Q8—2444no—
404D4—28no—
405D4—4882no—
406D4—4882no—
407D4—4882no—
408D4—2444no—
409D4—2448no—
410D4—2448no—
411D4—2444no—
412D4—2444no—
413D4—2442no—
414D4—2442no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—1224no—
431C2×C4—2448no—
432C2×C4—2444no—
433C2×C4—2444no—
434C2×C4#1451969312yescomplemented
435C2×C4#1451974511yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4#1451974712yes—
438C8—42no—
439C8—81no—
440C8—4882no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—9764no—
454C22—9764no—
455C22—2444no—
456C22—2444no—
457C22—2444no—
458C22#1683961711yescharacteristic
459C22C2×#1451969311yescharacteristic
460C22C2×#1451969311yescharacteristic
461C4—22no—
462C4—21no—
463C4—22no—
464C4—41no—
465C4—41no—
466C4—2444no—
467C4#1683994811yescharacteristic
468C4#1683995111yescharacteristic
469C2—21no—
470C2—21no—
471C2—42no—
472C2—42no—
473C2—4884no—
474C2#2144573911yescharacteristic
475C2#2144576611yescharacteristic
476C2#2144576911yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1870C2
socleC22×C61#379
prime radical#21459583C2
cosocleC23×C4×C61C23
hypercenter#1870D61
hypocenterC61#8536
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. #21446282C2semidirect
3. #21444178C2semidirect
4. #21444157C2semidirect
5. C61×#1870C2semidirect
17. #16839726C22semidirect
22. C22×M16×C61C22semidirect
25. C2×C22⋊C8×C61C22semidirect
154. C23×C61#100semidirect
180. C2×C4×C61#100semidirect
233. #1870D61semidirect
245. C22×M16C2×D61semidirect
259. C2×C22⋊C8C2×D61semidirect
265. C61#8536semidirect
400. C23#14519693semidirect
434. C2×C4#14519693semidirect

Maximal subgroups

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

subgroupindex
#214595832
#214462822
#214441782
#214441572
C61×#18702
#853661