Finite Group Databasebeta

← #29594427

Subgroups — #29594427

Order 15616.

240 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C16×D244C211yesmaximal, characteristic, complemented
2#21445205C211yesmaximal, characteristic, complemented
3#21445171C211yesmaximal, characteristic
4#21443891C211yesmaximal, characteristic, complemented
5#21443874C211yesmaximal, characteristic, complemented
6#21443865C211yesmaximal, characteristic, complemented
7C61×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D61—21no—
9#16839964—21no—
10#16839954—21no—
11C8×D244C2211yescharacteristic
12C4⋊#11758185C2211yescharacteristic
13#16839541—21no—
14#16839515C2211yescharacteristic
15#16839515C2211yescharacteristic
16#16839508—21no—
17#16839506—21no—
18C61×#500C2211yescharacteristic
19C61×#493C2211yescharacteristic
20C61×#357—21no—
21C61×#357—21no—
22C4×C16×C61C2211yescharacteristic
23C2×#11758201—41no—
24C2×#11758199—41no—
25C2×C8×D61—21no—
26C4×D244D411yescharacteristic
27C2×#11758185—21no—
28C16×D61—42no—
29C4⋊#92418—21no—
30C4⋊#92418—21no—
31C4⋊#92418D411yescharacteristic
32#14519701—41no—
33#14519698—41no—
34#14519689—21no—
35C2×D8×C61—21no—
36C2×D8×C61—21no—
37C4⋊Q8×C61—21no—
38#127×C61—21no—
39C2×C16×C61D411yescharacteristic
40C2×C16×C61D411yescharacteristic
41#107×C61—21no—
42#107×C61—21no—
43C4×C8×C61C2311yescharacteristic
44C2×Q8×C61—41no—
45C2×D4×C61—41no—
46C2×D4×C61—41no—
47C2×D4×C61—41no—
48C2×D244D811yescharacteristic
49C2×C4×D61—21no—
50D8×C61—41no—
51D8×C61—41no—
52D8×C61—81no—
53C2×C8×C61C2×D411yescharacteristic
54C2×C8×C61#4111yescharacteristic
55C4⋊C4×C61—41no—
56C4⋊C4×C61—41no—
57C42×C61C2×D411yescharacteristic
58#11758201—82no—
59#11758199—82no—
60#11758198—21no—
61C4⋊Dic61D811yescharacteristic
62C2×#92418—21no—
63C2×#92418—41no—
64C2×#92418—41no—
65C8×D61—41no—
66C8×D61—41no—
67C16×C61—21no—
68C16×C61—21no—
69#11758185—41no—
70C23×C61—81no—
71C22×D61—21no—
72Q8×C61—81no—
73Q8×C61—81no—
74D4×C61—81no—
75D4×C61—81no—
76D4×C61—81no—
77D4×C61—81no—
78D4×C61—161no—
79C2×C4×C61—81no—
80C2×C4×C61#12111yescharacteristic
81C2×C4×C61C2×D811yescharacteristic
82C2×C4×C61#13511yescharacteristic
83C2×Dic61—21no—
84D244—21no—
85D244D1612yes—
86C4×D61—41no—
87C4×D61—41no—
88C8×C61—21no—
89C8×C61#13511yescharacteristic
90C8×C61AGL(1,ℤ/8)11yescharacteristic
91#92418—41no—
92#92418—81no—
93#92418—81no—
94#12467—611nomaximal, Sylow
95C22×C61—161no—
96C22×C61—161no—
97C22×C61—161no—
98C22×C61#46611yescharacteristic
99C2×D61—22no—
100C2×D61—41no—
101C2×D61—41no—
102C4×C61—21no—
103C4×C61—161no—
104C4×C61#46311yescharacteristic
105C4×C61#47111yescharacteristic
106C4×C61C2×D1611yescharacteristic
107C4×C61#50811yescharacteristic
108Dic61—41no—
109#2240D6111yescharacteristic, complemented
110D4×C16—611no—
111#1673—611no—
112#1667—611no—
113#1371—611no—
114#1364—611no—
115#1361—611no—
116C2×C61—321no—
117C2×C61#168111yescharacteristic
118C2×C61#221511yescharacteristic
119C2×C61#222311yescharacteristic
120D61—42no—
121C22×C16—1221no—
122#500C2×D6111yescharacteristic
123#493C2×D6111yescharacteristic
124#461—1221no—
125#459—1221no—
126C8×D4—611no—
127C4⋊C16—611no—
128#357—21no—
129#357—21no—
130C22⋊C16—1221no—
131C4×C16C2×D6111yescharacteristic
132#335—611no—
133#335—611no—
134#332—1221no—
135#331—1221no—
136C61#1246711yescharacteristic, complemented, Sylow
137C2×SD16—2441no—
138C2×D8—21no—
139C2×D8—21no—
140C2×D8—2441no—
141C22×C8—1221no—
142C4⋊Q8—21no—
143#127—21no—
144C4×D4—611no—
145C2×C16—1221no—
146C2×C16—2442no—
147C2×C16C22⋊D6111yescharacteristic
148C2×C16C22⋊D6111yescharacteristic
149#107—21no—
150#107—21no—
151C4⋊C8—611no—
152C4⋊C8—1221no—
153C4⋊C8—1221no—
154#103—2441no—
155#102—2441no—
156C22⋊C8—1221no—
157C4×C8C22×D6111yescharacteristic
158C2×Q8—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—611no—
163C22×C4—1221no—
164SD16—4882no—
165D8—41no—
166D8—41no—
167D8—81no—
168D8—4882no—
169C2×C8—1221no—
170C2×C8—2441no—
171C2×C8—2441no—
172C2×C8—2441no—
173C2×C8—2441no—
174C2×C8#1175822411yescharacteristic
175C2×C8C2×C22⋊D6111yescharacteristic
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—611no—
179C22⋊C4—1221no—
180C42D4×D6111yescharacteristic
181C16—21no—
182C16—21no—
183C16—2441no—
184C23—81no—
185C23—1221no—
186Q8—81no—
187Q8—81no—
188D4—81no—
189D4—81no—
190D4—81no—
191D4—81no—
192D4—161no—
193D4—612no—
194D4—1221no—
195C2×C4—81no—
196C2×C4—1221no—
197C2×C4—2441no—
198C2×C4—2441no—
199C2×C4#1451974811yescharacteristic
200C2×C4D8×D6111yescharacteristic
201C2×C4#1451979611yescharacteristic
202C8—21no—
203C8—4881no—
204C8—4881no—
205C8—2441no—
206C8#1451979711yescharacteristic
207C8#1451980211yescharacteristic
208C22—161no—
209C22—161no—
210C22—161no—
211C22—1222no—
212C22—2441no—
213C22—2441no—
214C22#1683999511yescharacteristic
215C4—21no—
216C4—161no—
217C4—2441no—
218C4#1683997411yescharacteristic
219C4#1684002211yescharacteristic
220C4D16×D6111yescharacteristic
221C4#1684019111yescharacteristic
222C2—321no—
223C2—2442no—
224C2#2144525311yescharacteristic
225C2#2144790211yescharacteristic
226C2#2144794611yescharacteristic
subgroupG/H
centerC22#16839995
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2240C2
socleC22×C61#466
prime radical#21445171C2
hypercenter#2240D61
hypocenterC61#12467
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C16×D244C2semidirect
2. #21445205C2semidirect
4. #21443891C2semidirect
5. #21443874C2semidirect
6. #21443865C2semidirect
7. C61×#2240C2semidirect
109. #2240D61semidirect
136. C61#12467semidirect

Maximal subgroups

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

subgroupindex
C16×D2442
#214452052
#214451712
#214438912
#214438742
#214438652
C61×#22402
#1246761