Finite Group Databasebeta

← #29593840

Subgroups — #29593840

Order 15616.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#21457541C211yesmaximal, characteristic
2#21448048C211yesmaximal, characteristic, complemented
3#21447963C211yesmaximal, characteristic, complemented
4#21443867C211yesmaximal, characteristic, complemented
5#21443779C211yesmaximal, characteristic, complemented
6D4⋊#11758185C211yesmaximal, characteristic, complemented
7C61×#1675C211yesmaximal, characteristic, complemented
8C2×#14519799—21no—
9#16840417—21no—
10#16840142C2211yescharacteristic
11#16840003—21no—
12#16839968—21no—
13#16839926—22no—
14C4×D488C2211yescharacteristic
15#16839589—21no—
16C4⋊#11758185C2211yescharacteristic
17#16839565—22no—
18#16839562—21no—
19C4×#11758185C2211yescharacteristic
20C61×#500C2211yescharacteristic
21C61×#476—21no—
22C8×D4×C61C2211yescharacteristic
23C61×#335C2211yescharacteristic
24C61×#333—21no—
25C2×#11758222—22no—
26#14519799—42no—
27#14519799—42no—
28C2×Dic244—21no—
29C2×Dic244D412yes—
30C2×#11758190—22no—
31C2×D488D411yescharacteristic
32#14519777—21no—
33#14519759—42no—
34#14519756—22no—
35#14519748—41no—
36C4×D244—21no—
37C4⋊C4×D61—41no—
38C2×#11758185—21no—
39C2×#11758185—21no—
40#14519711D411yescharacteristic
41#14519702—22no—
42#14519699—21no—
43C22×C8×C61—21no—
44C4⋊Q8×C61—21no—
45C4×D4×C61D411yescharacteristic
46#107×C61—21no—
47#107×C61—21no—
48#106×C61—41no—
49C4⋊C8×C61—21no—
50C4⋊C8×C61D411yescharacteristic
51#102×C61—41no—
52C22⋊C8×C61—21no—
53C4×C8×C61C2311yescharacteristic
54C2×Q8×C61—41no—
55C2×D4×C61D811yescharacteristic
56C22×C4×C61—21no—
57C2×C22⋊D61—42no—
58#11758222—44no—
59#11758222—82no—
60C2×D244—21no—
61C2×C4×D61—42no—
62C2×C4×D61—41no—
63C2×Dic122—22no—
64C2×C8×C61—21no—
65C2×C8×C61—41no—
66C2×C8×C61—41no—
67C2×C8×C61—41no—
68C2×C8×C61C2×D411yescharacteristic
69C2×C8×C61C2×D411yescharacteristic
70C4⋊C4×C61—41no—
71C4⋊C4×C61—41no—
72C4⋊C4×C61D811yescharacteristic
73C22⋊C4×C61—21no—
74C42×C61C2×D411yescharacteristic
75C22⋊Dic61—42no—
76#11758198—41no—
77C4⋊Dic61—21no—
78C4⋊Dic61—41no—
79C4⋊Dic61—41no—
80#11758196—42no—
81Dic244—42no—
82Dic244—41no—
83Dic244—41no—
84D488—21no—
85D488SD1612yes—
86#11758190—44no—
87#11758185—41no—
88#11758185—41no—
89C23×C61—21no—
90C22×D61—41no—
91Q8×C61—161no—
92D4×C61—21no—
93D4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—41no—
96C2×C4×C61—41no—
97C2×C4×C61—81no—
98C2×C4×C61#12011yescharacteristic
99C2×C4×C61C2×D811yescharacteristic
100C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
101C22⋊D61—84no—
102C2×Dic61—41no—
103C2×Dic61—42no—
104D244—22no—
105D244—41no—
106C4×D61—84no—
107C4×D61—81no—
108C4×D61—81no—
109Dic122—42no—
110Dic122—42no—
111C8×C61—21no—
112C8×C61—41no—
113C8×C61—81no—
114C8×C61C2×SD1611yescharacteristic
115C8×C61AGL(1,ℤ/8)11yescharacteristic
116#12375—611nomaximal, Sylow
117C22×C61—41no—
118C22×C61—41no—
119C22×C61—41no—
120C22×C61#44711yescharacteristic
121C2×D61—42no—
122C2×D61—81no—
123C2×D61—81no—
124C4×C61—21no—
125C4×C61—41no—
126C4×C61—161no—
127C4×C61#45011yescharacteristic
128C4×C61#45511yescharacteristic
129C4×C61#50811yescharacteristic
130C4×C61#51011yescharacteristic
131Dic61—82no—
132Dic61—81no—
133#3283—611no—
134#2241—611no—
135#2226—611no—
136#1675D6111yescharacteristic, complemented
137#1362—611no—
138#1331—611no—
139D4⋊C16—611no—
140C2×C61—81no—
141C2×C61#163911yescharacteristic
142C2×C61#218911yescharacteristic
143C2×C61#219311yescharacteristic
144D61—82no—
145C2×#135—1221no—
146#547—1221no—
147#500C2×D6111yescharacteristic
148#494—611no—
149#476—21no—
150#467—1221no—
151#461—1221no—
152#452—1222no—
153C4×D8—611no—
154C8×D4C2×D6111yescharacteristic
155#367—1221no—
156C4⋊C16—611no—
157#358—1222no—
158#357—1221no—
159C4×C16—611no—
160#335C2×D6111yescharacteristic
161#333—21no—
162C61#1237511yescharacteristic, complemented, Sylow
163C2×#41—1222no—
164#135—2442no—
165#135—2442no—
166C2×Q16—612no—
167C2×Q16—1221no—
168C2×SD16—1222no—
169C2×D8—611no—
170C22×C8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—1221no—
173#123—2442no—
174C22⋊Q8—1222no—
175#121—2441no—
176C4×D4—1221no—
177C4×D4C22⋊D6111yescharacteristic
178C2×C4⋊C4—2441no—
179C2×C16—1221no—
180C2×C16—1221no—
181#107—21no—
182#107—21no—
183#107—611no—
184#106—41no—
185C4⋊C8—21no—
186C4⋊C8C22⋊D6111yescharacteristic
187#103—1222no—
188#102—41no—
189#102—1221no—
190C22⋊C8—21no—
191C4×C8C22×D6111yescharacteristic
192#41—4882no—
193#41—2444no—
194C2×Q8—41no—
195C2×Q8—1222no—
196C2×D4—1221no—
197C2×D4—2442no—
198C2×D4#1175819911yescharacteristic
199C22×C4—21no—
200C22×C4—2442no—
201C22×C4—2441no—
202Q16—2442no—
203Q16—2441no—
204Q16—2441no—
205SD16—2444no—
206D8—612no—
207D8—1221no—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8D4×D6111yescharacteristic
213C2×C8D4×D6111yescharacteristic
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—1221no—
217C4⋊C4—2442no—
218C4⋊C4—2441no—
219C4⋊C4—2441no—
220C4⋊C4#1175819911yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—2442no—
223C22⋊C4—2441no—
224C42C2×C22⋊D6111yescharacteristic
225C16—2441no—
226C16—2441no—
227C23—21no—
228C23—2441no—
229Q8—161no—
230Q8—2442no—
231Q8—2442no—
232D4—21no—
233D4—21no—
234D4—1222no—
235D4—4884no—
236D4—2441no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—4884no—
242C2×C4—4881no—
243C2×C4—4881no—
244C2×C4—2441no—
245C2×C4—2442no—
246C2×C4#1451974511yescharacteristic
247C2×C4C2×#1175819911yescharacteristic
248C2×C4#1451980211yescharacteristic
249C8—21no—
250C8—41no—
251C8—81no—
252C8SD16×D6111yescharacteristic
253C8#1451980411yescharacteristic
254C22—41no—
255C22—41no—
256C22—41no—
257C22—4881no—
258C22—4881no—
259C22—2442no—
260C22#1683989511yescharacteristic
261C4—21no—
262C4—41no—
263C4—161no—
264C4—4881no—
265C4—4882no—
266C4#1683991111yescharacteristic
267C4#1683993811yescharacteristic
268C4#1684019211yescharacteristic
269C4#1684020311yescharacteristic
270C2—81no—
271C2—4882no—
272C2#2144499611yescharacteristic
273C2#2144775711yescharacteristic
274C2#2144778111yescharacteristic
subgroupG/H
centerC22#16839895
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#1675C2
socleC22×C61#447
prime radical#21457541C2
hypercenter#1675D61
hypocenterC61#12375
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1675

Complements

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

NH ≅ G/Ntype
2. #21448048C2semidirect
3. #21447963C2semidirect
4. #21443867C2semidirect
5. #21443779C2semidirect
6. D4⋊#11758185C2semidirect
7. C61×#1675C2semidirect
136. #1675D61semidirect
162. C61#12375semidirect

Maximal subgroups

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

subgroupindex
#214575412
#214480482
#214479632
#214438672
#214437792
D4⋊#117581852
C61×#16752
#1237561