Finite Group Databasebeta

← #29592358

Subgroups — #29592358

Order 15616.

364 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 59 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#21447048C211yesmaximal, characteristic
2#21444821C211yesmaximal, characteristic
3#21444787C211yesmaximal, characteristic
4#21443849C211yesmaximal, characteristic, complemented
5#21443825C211yesmaximal, characteristic, complemented
6#21443763C211yesmaximal, characteristic, complemented
7C61×#2789C211yesmaximal, characteristic, complemented
8#16840343C2211yescharacteristic
9#16840075—21no—
10#16839901—21no—
11C2×#14519705—21no—
12#16839761—21no—
13#16839531C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C61×#401—21no—
19C61×#400—21no—
20C61×#398C2211yescharacteristic
21C61×#395C2211yescharacteristic
22C61×#391—21no—
23C61×#387—21no—
24C61×#387—21no—
25C61×#387—21no—
26C61×#382C2211yescharacteristic
27C2×#11758222—21no—
28C2×#11758201—41no—
29C2×#11758194—21no—
30#14519757—41no—
31#14519752—41no—
32#14519751—21no—
33#14519751—21no—
34#14519710—41no—
35#14519705—42no—
36#14519701—41no—
37#14519700—41no—
38#14519697—41no—
39#14519697—41no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C22×Q8×C61D411yescharacteristic
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61C2311yescharacteristic
49C2×C42×C61—21no—
50C2×C42×C61D411yescharacteristic
51#95×C61—21no—
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62#95×C61D411yescharacteristic
63C2×Q8×C61—42no—
64C2×Q8×C61—41no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C2×C22⋊D61—41no—
75#11758222—41no—
76#11758222—41no—
77#11758222—41no—
78#11758222—41no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—42no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C42×C61—21no—
92C42×C61—21no—
93C42×C61—81no—
94#11758201—82no—
95#11758198—41no—
96#11758198—41no—
97C4⋊Dic61—21no—
98C4⋊Dic61—21no—
99#11758194—41no—
100#11758194—41no—
101C2×#92418—41no—
102C2×#92418—41no—
103C2×#92418—41no—
104C23×C61#12011yescharacteristic
105C22×D61—41no—
106Q8×C61—42no—
107Q8×C61—81no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61#13511yescharacteristic
132C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
133C22⋊D61—81no—
134C22⋊D61—81no—
135C2×Dic61—41no—
136D244—41no—
137D244—41no—
138D244—81no—
139D244—81no—
140C4×D61—81no—
141C4×D61—81no—
142Dic122—41no—
143Dic122—41no—
144#92418—81no—
145#92418—81no—
146#92418—81no—
147#12143—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61#44911yescharacteristic
151C22×C61#45411yescharacteristic
152C22×C61B(4,𝔽2)11yescharacteristic
153C2×D61—81no—
154C2×D61—81no—
155C2×D61—81no—
156C4×C61—21no—
157C4×C61—21no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—42no—
161C4×C61—81no—
162C4×C61—81no—
163C4×C61—81no—
164C4×C61—81no—
165C4×C61—81no—
166Dic61—81no—
167#2789D6111yescharacteristic, complemented
168#2040—611no—
169#1610—611no—
170#1604—611no—
171#1354—611no—
172#1346—611no—
173#1324—611no—
174C2×C61—21no—
175C2×C61—21no—
176C2×C61#159711yescharacteristic
177C2×C61#219711yescharacteristic
178C2×C61#220311yescharacteristic
179D61—161no—
180#537—611no—
181#480—1221no—
182#448—1221no—
183C2×#104—1221no—
184#417—1221no—
185#401—21no—
186#400—21no—
187#398C2×D6111yescharacteristic
188#395C2×D6111yescharacteristic
189#391—21no—
190#387—21no—
191#387—21no—
192#387—21no—
193#382C2×D6111yescharacteristic
194#343—611no—
195#327—1221no—
196#327—1221no—
197#324—611no—
198#324—611no—
199C61#1214311yescharacteristic, complemented, Sylow
200C2×#41—1221no—
201C22×Q8C22⋊D6111yescharacteristic
202C2×SD16—2441no—
203C2×M16—1221no—
204C22⋊Q8—2441no—
205#121—1221no—
206#121—1221no—
207#121—2441no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D6111yescharacteristic
212C2×C4⋊C4C22⋊D6111yescharacteristic
213C2×C4⋊C4C22×D6111yescharacteristic
214C2×C42—21no—
215C2×C42C22⋊D6111yescharacteristic
216#107—2441no—
217#104—2442no—
218#103—2441no—
219#103—2441no—
220#102—2441no—
221#102—2441no—
222C22⋊C8—1221no—
223C22⋊C8—1221no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C22⋊D6111yescharacteristic
235#95C22⋊D6111yescharacteristic
236#41—2441no—
237#41—2441no—
238#41—2441no—
239#41—2441no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—1221no—
243C2×D4—1221no—
244C2×D4—1221no—
245C2×D4—1221no—
246C2×D4—2441no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—2441no—
254C22×C4C2×C22⋊D6111yescharacteristic
255C22×C4C2×C22⋊D6111yescharacteristic
256C22×C4C2×C22⋊D6111yescharacteristic
257SD16—4882no—
258M16—2441no—
259M16—2441no—
260C2×C8—2441no—
261C2×C8—2441no—
262C2×C8—2441no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—1221no—
271C4⋊C4—1221no—
272C22⋊C4—2441no—
273C22⋊C4—2441no—
274C42—21no—
275C42—21no—
276C42—81no—
277C23—2441no—
278C23#1451974611yescharacteristic
279Q8—42no—
280Q8—81no—
281Q8—2441no—
282Q8—2441no—
283D4—4881no—
284D4—4881no—
285D4—4881no—
286D4—4881no—
287D4—2441no—
288D4—2441no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—4881no—
313C2×C4—4881no—
314C2×C4—2441no—
315C2×C4#1451979711yescharacteristic
316C2×C4#1451980211yescharacteristic
317C8—4881no—
318C8—4881no—
319C8—4881no—
320C22—21no—
321C22—21no—
322C22—4881no—
323C22—4881no—
324C22—4881no—
325C22#1683990711yescharacteristic
326C22#1683993511yescharacteristic
327C22#1683994911yescharacteristic
328C4—21no—
329C4—21no—
330C4—41no—
331C4—41no—
332C4—42no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—4881no—
339C2—21no—
340C2—21no—
341C2—9761no—
342C2#2144474811yescharacteristic
343C2#2144780911yescharacteristic
344C2#2144784011yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2789C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2789D61
hypocenterC61#12143
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2789

Complements

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

NH ≅ G/Ntype
4. #21443849C2semidirect
5. #21443825C2semidirect
6. #21443763C2semidirect
7. C61×#2789C2semidirect
167. #2789D61semidirect
199. C61#12143semidirect

Maximal subgroups

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

subgroupindex
#214470482
#214448212
#214447872
#214438492
#214438252
#214437632
C61×#27892
#1214361