Finite Group Databasebeta

← #29593829

Subgroups — #29593829

Order 15616.

360 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 67 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#21457577C211yesmaximal, characteristic
2#21448042C211yesmaximal, characteristic, complemented
3#21447972C211yesmaximal, characteristic, complemented
4#21445219C211yesmaximal, characteristic, complemented
5#21443782C211yesmaximal, characteristic, complemented
6#21443773C211yesmaximal, characteristic, complemented
7C61×#1361C211yesmaximal, characteristic, complemented
8C2×#14519796—21no—
9#16840432—21no—
10#16840167C2211yescharacteristic
11#16840053—21no—
12#16839995—21no—
13#16839959—21no—
14#16839904—22no—
15C4×Dic244C2211yescharacteristic
16C8×D244C2211yescharacteristic
17#16839588—21no—
18#16839567—21no—
19#16839563—22no—
20C4×#11758185C2211yescharacteristic
21#16839510—21no—
22C61×#493C2211yescharacteristic
23C61×C4⋊C16C2211yescharacteristic
24C61×#335C2211yescharacteristic
25C2×#11758224—22no—
26#14519796—42no—
27#14519796—42no—
28C2×Dic244D411yescharacteristic
29C2×#11758200—22no—
30C2×C8×D61—21no—
31#14519777—21no—
32#14519773—41no—
33#14519749—42no—
34#14519748—22no—
35C4×Dic122—21no—
36C4×D244D411yescharacteristic
37D4×Dic61—41no—
38C2×#11758185—21no—
39#14519711—21no—
40#14519711—21no—
41#14519711D411yescharacteristic
42#14519709—41no—
43C4⋊#92418D411yescharacteristic
44#14519702—21no—
45#14519699—41no—
46#14519698—22no—
47#14519689—21no—
48C2×D8×C61—21no—
49C2×D8×C61D412yes—
50#127×C61—21no—
51C2×C16×C61—21no—
52C4⋊C8×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×D4×C61—22no—
55C2×D4×C61—41no—
56C2×D4×C61—41no—
57C2×C22⋊D61—42no—
58C2×C22⋊D61—42no—
59C22×Dic61—42no—
60#11758224—44no—
61#11758224—82no—
62C2×D244D811yescharacteristic
63C2×C4×D61—21no—
64C2×Dic122—21no—
65C2×Dic122—41no—
66D8×C61—42no—
67D8×C61—41no—
68D8×C61—41no—
69C2×C8×C61—41no—
70C2×C8×C61C2×D411yescharacteristic
71C2×C8×C61C2×D411yescharacteristic
72C42×C61C2×D411yescharacteristic
73C22⋊Dic61—42no—
74#11758200—44no—
75#11758198—21no—
76C4⋊Dic61—21no—
77C4⋊Dic61—41no—
78C4⋊Dic61—41no—
79C4⋊Dic61D811yescharacteristic
80#11758196—41no—
81C4×Dic61—41no—
82C2×#92418—21no—
83Dic244—21no—
84Dic244SD1612yes—
85C8×D61—41no—
86C8×D61—41no—
87C16×C61—41no—
88#11758185—41no—
89C23×C61—42no—
90C22×D61—21no—
91D4×C61—42no—
92D4×C61—42no—
93D4×C61—81no—
94D4×C61—81no—
95D4×C61—161no—
96C2×C4×C61#12011yescharacteristic
97C2×C4×C61C2×D811yescharacteristic
98C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
99C22⋊D61—84no—
100C22⋊D61—162no—
101C2×Dic61—21no—
102C2×Dic61—41no—
103C2×Dic61—41no—
104C2×Dic61—84no—
105C2×Dic61—81no—
106D244—21no—
107D244—21no—
108C4×D61—41no—
109C4×D61—41no—
110Dic122—22no—
111Dic122—41no—
112Dic122—161no—
113C8×C61—21no—
114C8×C61—81no—
115C8×C61C2×SD1611yescharacteristic
116C8×C61AGL(1,ℤ/8)11yescharacteristic
117#92418—41no—
118#12373—611nomaximal, Sylow
119C22×C61—82no—
120C22×C61—82no—
121C22×C61—82no—
122C22×C61#44711yescharacteristic
123C2×D61—41no—
124C2×D61—41no—
125C2×D61—41no—
126C4×C61—21no—
127C4×C61#45011yescharacteristic
128C4×C61#45511yescharacteristic
129C4×C61#50811yescharacteristic
130C4×C61#51011yescharacteristic
131Dic61—42no—
132Dic61—41no—
133Dic61—81no—
134Dic61—161no—
135#3287—611no—
136#2240—611no—
137#2227—611no—
138#1675—611no—
139#1361D6111yescharacteristic, complemented
140#1332—611no—
141D4⋊C16—611no—
142C2×C61—162no—
143C2×C61#163911yescharacteristic
144C2×C61#218911yescharacteristic
145C2×C61#219211yescharacteristic
146D61—81no—
147C2×#135—1221no—
148#550—1221no—
149#500—611no—
150#493C2×D6111yescharacteristic
151#476—1221no—
152#466—1221no—
153#460—1221no—
154#449—1222no—
155C4×Q16—611no—
156C8×D4—611no—
157#366—1221no—
158C4⋊C16C2×D6111yescharacteristic
159#358—1221no—
160#357—1222no—
161C4×C16—611no—
162#335C2×D6111yescharacteristic
163#333—1221no—
164C61#1237311yescharacteristic, complemented, Sylow
165C2×#41—1222no—
166#135—2442no—
167#135—2442no—
168C2×Q16—611no—
169C2×SD16—1222no—
170C2×D8—21no—
171C2×D8D24412yes—
172C22×C8—1221no—
173C4⋊Q8—1221no—
174#127—21no—
175#127—2441no—
176#121—1222no—
177#121—2442no—
178C4×Q8—1221no—
179C4×D4—611no—
180C4×D4—2441no—
181C2×C16—21no—
182C2×C16—1221no—
183#107—611no—
184#107—1221no—
185#107—1221no—
186#106—2441no—
187C4⋊C8—21no—
188C4⋊C8—611no—
189#103—1221no—
190#102—1222no—
191#102—2441no—
192C22⋊C8—1221no—
193C4×C8C22×D6111yescharacteristic
194#41—4882no—
195#41—2444no—
196C2×Q8—1221no—
197C2×Q8—2441no—
198C2×D4—22no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—611no—
202C2×D4—2442no—
203C2×D4—2442no—
204C22×C4—1221no—
205C22×C4—2442no—
206Q16—612no—
207Q16—1221no—
208SD16—2444no—
209D8—42no—
210D8—41no—
211D8—41no—
212C2×C8—41no—
213C2×C8—1221no—
214C2×C8—2441no—
215C2×C8—2441no—
216C2×C8C2×D24411yescharacteristic
217C2×C8D4×D6111yescharacteristic
218C4⋊C4—611no—
219C4⋊C4—1221no—
220C4⋊C4—2441no—
221C4⋊C4—2441no—
222C4⋊C4—2441no—
223C22⋊C4—1221no—
224C22⋊C4—2442no—
225C42—2441no—
226C42D4×D6111yescharacteristic
227C16—41no—
228C16—2441no—
229C23—42no—
230C23—1221no—
231Q8—1222no—
232Q8—9761no—
233Q8—2441no—
234D4—42no—
235D4—42no—
236D4—81no—
237D4—81no—
238D4—161no—
239D4—1221no—
240D4—1221no—
241D4—9762no—
242D4—4884no—
243C2×C4—1221no—
244C2×C4—4881no—
245C2×C4—4884no—
246C2×C4—2441no—
247C2×C4—2441no—
248C2×C4—2441no—
249C2×C4—2441no—
250C2×C4#1451974711yescharacteristic
251C2×C4D8×D6111yescharacteristic
252C2×C4#1451980411yescharacteristic
253C8—21no—
254C8—81no—
255C8—2441no—
256C8SD16×D6111yescharacteristic
257C8#1451980111yescharacteristic
258C22—82no—
259C22—82no—
260C22—82no—
261C22—2441no—
262C22—2441no—
263C22—2441no—
264C22#1683989711yescharacteristic
265C4—21no—
266C4—9761no—
267C4—4881no—
268C4—2441no—
269C4—2442no—
270C4#1683991411yescharacteristic
271C4#1683994211yescharacteristic
272C4#1684019111yescharacteristic
273C4#1684020111yescharacteristic
274C2—162no—
275C2—4881no—
276C2#2144499911yescharacteristic
277C2#2144776011yescharacteristic
278C2#2144777911yescharacteristic
subgroupG/H
centerC22#16839897
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1361C2
socleC22×C61#447
prime radical#21457577C2
hypercenter#1361D61
hypocenterC61#12373
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1361

Complements

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

NH ≅ G/Ntype
2. #21448042C2semidirect
3. #21447972C2semidirect
4. #21445219C2semidirect
5. #21443782C2semidirect
6. #21443773C2semidirect
7. C61×#1361C2semidirect
139. #1361D61semidirect
164. C61#12373semidirect

Maximal subgroups

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

subgroupindex
#214575772
#214480422
#214479722
#214452192
#214437822
#214437732
C61×#13612
#1237361