Finite Group Databasebeta

← #29600692

Subgroups — #29600692

Order 15616.

284 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 55 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#21458952C211yesmaximal, characteristic
2#21448139C211yesmaximal, characteristic, complemented
3#21448136C211yesmaximal, characteristic, complemented
4#21447928C211yesmaximal, characteristic, complemented
5#21444005C211yesmaximal, characteristic, complemented
6#21443958C211yesmaximal, characteristic, complemented
7C61×#1418C211yesmaximal, characteristic, complemented
8#135×D61—21no—
9#16840204—21no—
10#16840201—21no—
11#16840199—21no—
12#16840196—21no—
13#16840194C2211yescharacteristic
14#16840192—21no—
15C2×Dic488C2211yescharacteristic, complemented
16C2×#14519716C2211yescharacteristic
17#16840004—21no—
18Dic976—21no—
19#16839602—21no—
20#16839601—21no—
21#16839600—21no—
22#16839596C2211yescharacteristic, complemented
23#16839565—21no—
24C61×#508C2211yescharacteristic
25C61×M64C2211yescharacteristic, complemented
26C61×#367C2211yescharacteristic, complemented
27#41×D61—41no—
28#14519800—21no—
29#14519799—21no—
30#14519798—41no—
31#14519797—41no—
32#14519796—21no—
33C2×Dic244—21no—
34Q16×D61—21no—
35SD16×D61—41no—
36D8×D61—21no—
37C2×C8×D61D411yescharacteristic
38#14519757—41no—
39Dic488—21no—
40Dic488—21no—
41Dic488D411yescharacteristic
42#14519723—21no—
43#14519722—21no—
44#14519721—21no—
45#14519720—21no—
46D976D411yescharacteristic
47#14519718—21no—
48#14519716—21no—
49#14519716—21no—
50C2×#11758185D411yescharacteristic
51#14519700—41no—
52#14519683—21no—
53#135×C61—21no—
54Q32×C61D411yescharacteristic
55SD32×C61—21no—
56D16×C61D411yescharacteristic
57C2×C16×C61C2311yescharacteristic
58#107×C61—21no—
59C32×C61—21no—
60#41×C61—41no—
61#11758226—41no—
62Q8×D61—41no—
63#11758224—41no—
64#11758224—81no—
65D4×D61—41no—
66D4×D61—81no—
67#11758222—41no—
68#11758222—81no—
69C2×C4×D61—81no—
70C2×C4×D61D811yescharacteristic
71C2×Dic122—41no—
72Q16×C61—21no—
73SD16×C61—41no—
74D8×C61—21no—
75C2×C8×C61C2×D411yescharacteristic
76C4⋊C4×C61—41no—
77#11758202—41no—
78#11758201—41no—
79#11758200—41no—
80#11758199—41no—
81#11758198—81no—
82C4⋊Dic61—81no—
83C2×#92418D811yescharacteristic
84Dic244—21no—
85Dic244—41no—
86Dic244—41no—
87D488—21no—
88#11758190—41no—
89C8×D61—21no—
90C8×D61—21no—
91C16×C61C2×D411yescharacteristic
92C16×C61C2×D411yescharacteristic
93#11758185—21no—
94C22×D61—81no—
95C22×D61D1611yescharacteristic
96Q8×C61—41no—
97D4×C61—41no—
98D4×C61—81no—
99C2×C4×C61—81no—
100C2×C4×C61—81no—
101C2×C4×C61C2×D811yescharacteristic
102C22⋊D61—81no—
103C22⋊D61—81no—
104C22⋊D61—81no—
105C2×Dic61—81no—
106C2×Dic61—81no—
107C2×Dic61D1611yescharacteristic
108D244—41no—
109D244—81no—
110C4×D61—21no—
111C4×D61—21no—
112C4×D61—81no—
113C4×D61—81no—
114C4×D61—81no—
115Dic122—41no—
116Dic122—81no—
117Dic122—81no—
118Dic122—81no—
119C8×C61#12011yescharacteristic
120C8×C61AGL(1,ℤ/8)11yescharacteristic
121#92418—21no—
122#13668—611nomaximal, Sylow
123C22×C61—81no—
124C22×C61C2×D1611yescharacteristic
125C2×D61—21no—
126C2×D61—41no—
127C2×D61—81no—
128C2×D61—162no—
129C4×C61—81no—
130C4×C61—161no—
131C4×C61#44711yescharacteristic
132C4×C61#50911yescharacteristic
133Dic61—21no—
134Dic61—81no—
135Dic61—161no—
136#3414—611no—
137#2264—611no—
138#2264—611no—
139#2219—611no—
140#1418—611no—
141#1418D6111yescharacteristic, complemented
142#1400—611no—
143C2×C61—21no—
144C2×C61—161no—
145C2×C61#218411yescharacteristic
146D61—41no—
147D61—161no—
148C2×#135—1221no—
149#510—1221no—
150#510—1221no—
151#509—1221no—
152#509—1221no—
153#508—611no—
154#508—1221no—
155#508C2×D6111yescharacteristic
156C2×Q32—611no—
157C2×M32—611no—
158#467—1221no—
159Q64—1221no—
160Q64—1221no—
161SD64—1221no—
162SD64—1221no—
163M64—611no—
164M64C2×D6111yescharacteristic, complemented
165#367C2×D6111yescharacteristic, complemented
166#358—1221no—
167C61#1366811yescharacteristic, complemented, Sylow
168C2×#41—2441no—
169#135—21no—
170#135—1221no—
171#135—1221no—
172#135—1221no—
173#135—2441no—
174#135—2441no—
175C2×Q16—1221no—
176C2×Q16—1221no—
177C2×SD16—2441no—
178C2×D8—1221no—
179C22×C8—611no—
180C22⋊Q8—2441no—
181Q32—611no—
182Q32—1221no—
183Q32—1221no—
184Q32—1221no—
185Q32D24411yescharacteristic
186SD32—21no—
187SD32—1221no—
188SD32—1221no—
189SD32—1221no—
190D16—611no—
191D16—1221no—
192D16D24411yescharacteristic
193M32—1221no—
194M32—1221no—
195C2×C16—611no—
196C2×C16C22×D6111yescharacteristic
197#107—21no—
198#103—2441no—
199C32—21no—
200C32—1221no—
201#41—41no—
202#41—4881no—
203#41—4881no—
204#41—2441no—
205#41—2441no—
206#41—2441no—
207C2×Q8—2441no—
208C2×Q8—2441no—
209C2×D4—4881no—
210C2×D4—2441no—
211C22×C4—611no—
212C22×C4—4881no—
213Q16—21no—
214Q16—1221no—
215Q16—2441no—
216Q16—2441no—
217Q16—2441no—
218SD16—41no—
219SD16—2441no—
220SD16—2441no—
221SD16—2441no—
222D8—21no—
223D8—1221no—
224D8—2441no—
225C2×C8—611no—
226C2×C8—1221no—
227C2×C8—1221no—
228C2×C8D4×D6111yescharacteristic
229C4⋊C4—41no—
230C4⋊C4—4881no—
231C22⋊C4—4881no—
232C16—1221no—
233C16C2×D24411yescharacteristic
234C16D4×D6111yescharacteristic
235C23—611no—
236C23—4881no—
237Q8—41no—
238Q8—4881no—
239Q8—4881no—
240Q8—4881no—
241Q8—2441no—
242D4—41no—
243D4—81no—
244D4—4881no—
245D4—4881no—
246D4—4881no—
247D4—4881no—
248D4—2441no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—611no—
252C2×C4—1221no—
253C2×C4—1221no—
254C2×C4—4881no—
255C2×C4—4881no—
256C2×C4—4881no—
257C2×C4—4881no—
258C2×C4—4881no—
259C2×C4D8×D6111yescharacteristic
260C8—1221no—
261C8#1451974711yescharacteristic
262C8#1451980411yescharacteristic
263C22—81no—
264C22—1221no—
265C22—9762no—
266C22—4881no—
267C22—2441no—
268C22D16×D6111yescharacteristic
269C4—81no—
270C4—161no—
271C4—1221no—
272C4—9761no—
273C4—4881no—
274C4#1683989711yescharacteristic
275C4#1684019911yescharacteristic
276C2—21no—
277C2—161no—
278C2—9761no—
279C2—2441no—
280C2#2144773211yescharacteristic
subgroupG/H
centerC2#21447732
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
FittingC61×#1418C2
socleC2×C61#2184
prime radical#21458952C2
hypercenter#1418D61
hypocenterC61#13668
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1418

Complements

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

NH ≅ G/Ntype
2. #21448139C2semidirect
3. #21448136C2semidirect
4. #21447928C2semidirect
5. #21444005C2semidirect
6. #21443958C2semidirect
7. C61×#1418C2semidirect
15. C2×Dic488C22semidirect
22. #16839596C22semidirect
25. C61×M64C22semidirect
26. C61×#367C22semidirect
141. #1418D61semidirect
164. M64C2×D61semidirect
165. #367C2×D61semidirect
167. C61#13668semidirect

Maximal subgroups

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

subgroupindex
#214589522
#214481392
#214481362
#214479282
#214440052
#214439582
C61×#14182
#1366861