Finite Group Databasebeta

← #29580647

Subgroups — #29580647

Order 15616.

822 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 822 conjugacy classes of subgroups — too many to draw. Only its 79 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#21455071C212yesmaximal, complemented
2#21453843C211yesmaximal, characteristic, complemented
3#21444347C212yesmaximal, complemented
4#21443727C211yesmaximal, characteristic, complemented
5C61×#1782C211yesmaximal, characteristic, complemented
6C2×#14519802—22no—
7C2×#14519751C2212yes—
8#16839907—24no—
9#16839901—22no—
10#16839896—22no—
11#16839763—22no—
12#16839726C2212yes—
13#16839689—22no—
14#16839672—22no—
15#16839671—22no—
16#16839495—24no—
17C22×#41×C61C2211yescharacteristic
18C2×#99×C61—22no—
19C61×#381C2211yescharacteristic
20C61×#381C2211yescharacteristic
21C22×D244—22no—
22C22×D244D412yes—
23C22×C22⋊D61—22no—
24#14519802—48no—
25C2×#11758201—24no—
26C2×#11758199—24no—
27C2×#11758194—22no—
28#14519751—22no—
29#14519751—22no—
30#14519751—22no—
31#14519751—22no—
32C2×C4⋊Dic61D412yes—
33C2×#11758198—22no—
34C2×#11758198—24no—
35#14519701—24no—
36#14519698—24no—
37C22⋊#92418—22no—
38C22⋊#92418D414yescomplemented
39#14519684—22no—
40C2×#41×C61—22no—
41C2×#41×C61—22no—
42C2×#41×C61—44no—
43C22×Q8×C61D411yescharacteristic, complemented
44C22×D4×C61—21no—
45C22×D4×C61D411yescharacteristic, complemented
46C23×C4×C61—21no—
47C23×C4×C61C2311yescharacteristic
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61D412yes—
50#99×C61—44no—
51#95×C61—22no—
52#95×C61—22no—
53C24×C61—21no—
54C24×C61C2×D411yescharacteristic
55C23×D61—22no—
56#41×C61—28no—
57#41×C61—44no—
58#41×C61—44no—
59#41×C61—44no—
60C2×Q8×C61—22no—
61C2×Q8×C61—42no—
62C2×D4×C61—24no—
63C2×D4×C61—22no—
64C2×D4×C61—42no—
65C2×D4×C61—42no—
66C2×D4×C61—41no—
67C2×D4×C61—41no—
68C22×C4×C61—21no—
69C22×C4×C61—22no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61—41no—
75C22×C4×C61C2×D412yes—
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61#4111yescharacteristic
79C2×C22⋊D61—42no—
80C2×C22⋊D61—42no—
81C2×C22⋊D61—84no—
82C22×Dic61—22no—
83C2×D244—22no—
84C2×D244—22no—
85C2×D244—44no—
86C2×D244—84no—
87C2×D244—82no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90#11758201—48no—
91#11758199—48no—
92#11758198—42no—
93#11758198—42no—
94#11758198—84no—
95C4⋊Dic61—22no—
96C4⋊Dic61—22no—
97#11758194—44no—
98C2×#92418—24no—
99C23×C61—22no—
100C23×C61—22no—
101C23×C61—42no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61—82no—
105C23×C61#12011yescharacteristic
106C23×C61#12111yescharacteristic
107C23×C61#12711yescharacteristic
108C22×D61—42no—
109C22×D61—42no—
110C22×D61—44no—
111C22×D61—42no—
112C22×D61—44no—
113Q8×C61—24no—
114Q8×C61—42no—
115D4×C61—24no—
116D4×C61—42no—
117D4×C61—44no—
118D4×C61—44no—
119C2×C4×C61—24no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—48no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—44no—
128C2×C4×C61—42no—
129C2×C4×C61—81no—
130C2×C4×C61—82no—
131C2×C4×C61#12012yes—
132C2×C4×C61#12112yes—
133C22⋊D61—84no—
134C22⋊D61—84no—
135C2×Dic61—44no—
136C2×Dic61—42no—
137D244—44no—
138D244—82no—
139D244—88no—
140#92418—44no—
141#9980—611nomaximal, Sylow
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—22no—
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—42no—
148C22×C61—82no—
149C22×C61—82no—
150C22×C61—82no—
151C22×C61—82no—
152C22×C61#39211yescharacteristic
153C22×C61B(4,𝔽2)12yes—
154C2×D61—84no—
155C2×D61—84no—
156C2×D61—82no—
157C2×D61—82no—
158C2×D61—82no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—42no—
162C4×C61—82no—
163C4×C61—161no—
164Dic61—82no—
165#3006—612no—
166#2815—611no—
167#1782D6111yescharacteristic, complemented
168#1510—612no—
169#1306—611no—
170C2×C61—22no—
171C2×C61—41no—
172C2×C61—41no—
173C2×C61—82no—
174C2×C61#200812yes—
175C2×C61#202111yescharacteristic
176D61—162no—
177C22×#41C2×D6111yescharacteristic
178C2×AGL(1,ℤ/8)—1222no—
179C2×#121—612no—
180#449—1224no—
181#448—1222no—
182#447—1222no—
183#417—1222no—
184C2×#99—22no—
185#407—612no—
186#397—1222no—
187#392—1222no—
188#392—1222no—
189#381C2×D6111yescharacteristic
190#381C2×D6111yescharacteristic
191#327—1224no—
192C61#998011yescharacteristic, complemented, Sylow
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—44no—
196C22×Q8D24411yescharacteristic, complemented
197C22×D4—21no—
198C22×D4—612no—
199C22×D4—1222no—
200C22×D4—1222no—
201C22×D4D24411yescharacteristic, complemented
202C23×C4—21no—
203C23×C4C22×D6111yescharacteristic
204AGL(1,ℤ/8)—2448no—
205C2×SD16—1224no—
206C2×D8—1224no—
207C2×M16—1222no—
208#121—1222no—
209#121—1222no—
210#121—1222no—
211#121—1222no—
212C2×C4⋊C4—612no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1224no—
215C2×C22⋊C4—1222no—
216C2×C22⋊C4C22⋊D6112yes—
217#103—1224no—
218#102—1224no—
219#99—44no—
220C22⋊C8—614no—
221C22⋊C8—1222no—
222#95—22no—
223#95—22no—
224#95—1222no—
225C24—21no—
226C24—1222no—
227C24C2×C22⋊D6111yescharacteristic
228#41—28no—
229#41—44no—
230#41—44no—
231#41—44no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—1222no—
241C2×D4—1222no—
242C2×D4—4884no—
243C2×D4—4882no—
244C2×D4—4884no—
245C2×D4—2444no—
246C2×D4—2442no—
247C2×D4—2442no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—1222no—
256C22×C4C2×D24411yescharacteristic
257C22×C4#1175822211yescharacteristic
258C22×C4D4×D6112yes—
259C22×C4D4×D6112yes—
260SD16—2448no—
261D8—2448no—
262M16—2444no—
263C2×C8—1224no—
264C4⋊C4—1222no—
265C4⋊C4—1222no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—4884no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C23—22no—
272C23—22no—
273C23—42no—
274C23—41no—
275C23—41no—
276C23—82no—
277C23—2442no—
278C23—2442no—
279C23—2444no—
280C23—2444no—
281C23—2442no—
282C23#1451974511yescharacteristic
283C23#1451975111yescharacteristic
284C23#1451977311yescharacteristic
285Q8—24no—
286Q8—42no—
287D4—24no—
288D4—42no—
289D4—44no—
290D4—44no—
291D4—4882no—
292D4—4888no—
293D4—4884no—
294D4—4884no—
295D4—2444no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—48no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—2444no—
309C2×C4—2442no—
310C2×C4#1451974712yes—
311C2×C4#1451975012yes—
312C8—2444no—
313C22—22no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—4884no—
324C22—4884no—
325C22—4882no—
326C22—4882no—
327C22—4882no—
328C22#1683967111yescharacteristic
329C22#1683994812yes—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—82no—
334C4—161no—
335C4—4882no—
336C2—22no—
337C2—41no—
338C2—41no—
339C2—82no—
340C2—9762no—
341C2#2144687112yes—
342C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1782C2
socleC22×C61#392
hypercenter#1782D61
hypocenterC61#9980
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
1. #21455071C2semidirect
2. #21453843C2semidirect
3. #21444347C2semidirect
4. #21443727C2semidirect
5. C61×#1782C2semidirect
38. C22⋊#92418D4semidirect
43. C22×Q8×C61D4semidirect
45. C22×D4×C61D4semidirect
167. #1782D61semidirect
192. C61#9980semidirect
196. C22×Q8D244semidirect
201. C22×D4D244semidirect

Maximal subgroups

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

subgroupindex
#214550712
#214538432
#214443472
#214437272
C61×#17822
#998061