Finite Group Databasebeta

← #29567401

Subgroups — #29567401

Order 15616.

222 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 39 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#21446646C211yesmaximal, characteristic
2#21443770C211yesmaximal, characteristic, complemented
3C61×#1327C211yesmaximal, characteristic, complemented
4#16840312C411yescharacteristic
5C2×#14519703—21no—
6C2×#14519703—21no—
7C42×Dic61C411yescharacteristic
8#16839544—21no—
9C61×#423C2211yescharacteristic
10C61×#349—21no—
11C2×#11758222—21no—
12#14519777C811yescharacteristic
13#14519774C811yescharacteristic
14#14519765—21no—
15#14519751—41no—
16C4×D244—41no—
17C2×C4×Dic61—21no—
18C2×C4×Dic61—41no—
19#14519717—41no—
20#14519703—42no—
21#14519703—42no—
22C2×M16×C61D411yescharacteristic
23C2×M16×C61D411yescharacteristic
24C2×C42×C61C2×C411yescharacteristic
25M32×C61—41no—
26C4⋊C8×C61—41no—
27C22×C4×C61—21no—
28C22×C4×C61C22⋊C411yescharacteristic
29C2×C22⋊D61—41no—
30C22×Dic61—41no—
31#11758222—41no—
32#11758222—41no—
33#11758222—41no—
34#11758222—41no—
35C2×D244—21no—
36C2×D244—41no—
37C2×C4×D61—41no—
38C2×Dic122—21no—
39M16×C61—41no—
40M16×C61—41no—
41C2×C8×C61—21no—
42C2×C8×C61—21no—
43C42×C61—21no—
44C42×C61C2×C811yescharacteristic
45C42×C61M1611yescharacteristic
46#11758198—42no—
47C4⋊Dic61—41no—
48C4×Dic61—22no—
49C4×Dic61—41no—
50C4×Dic61—44no—
51C16×C61—81no—
52#11758185—81no—
53C23×C61#10011yescharacteristic
54C22×D61—41no—
55C2×C4×C61—21no—
56C2×C4×C61—21no—
57C2×C4×C61—21no—
58C2×C4×C61—21no—
59C2×C4×C61—42no—
60C2×C4×C61C22⋊C811yescharacteristic
61C2×C4×C61#9911yescharacteristic
62C22⋊D61—81no—
63C22⋊D61—81no—
64C2×Dic61—42no—
65C2×Dic61—44no—
66C2×Dic61—41no—
67D244—41no—
68D244—41no—
69D244—82no—
70C4×D61—81no—
71C4×D61—81no—
72Dic122—41no—
73Dic122—41no—
74C8×C61—41no—
75C8×C61—41no—
76#7357—611nomaximal, Sylow
77C22×C61—21no—
78C22×C61—41no—
79C22×C61#32311yescharacteristic
80C2×D61—82no—
81C2×D61—81no—
82C4×C61—22no—
83C4×C61—21no—
84C4×C61—41no—
85C4×C61#35311yescharacteristic
86C4×C61#35511yescharacteristic
87Dic61—44no—
88Dic61—81no—
89#1962—611no—
90#1327—611no—
91#1327D6111yescharacteristic, complemented
92C2×C61—21no—
93C2×C61—41no—
94C2×C61#131811yescharacteristic
95D61—161no—
96#532—611no—
97#423C2×D6111yescharacteristic
98C2×#104—1221no—
99C2×#104—1221no—
100C43—611no—
101#349—21no—
102#349—1221no—
103C61#735711yescharacteristic, complemented, Sylow
104C2×#41—1221no—
105C2×M16D24411yescharacteristic
106C2×M16C22⋊D6111yescharacteristic
107C4⋊Q8—611no—
108#127—611no—
109#124—1221no—
110#121—2441no—
111C4×D4—2441no—
112C2×C42—1221no—
113C2×C42—2441no—
114C2×C42C4×D6111yescharacteristic
115M32—41no—
116M32—2441no—
117C4⋊C8—41no—
118#104—2442no—
119#104—2442no—
120#41—2441no—
121#41—2441no—
122#41—2441no—
123#41—2441no—
124C2×Q8—1221no—
125C2×D4—1221no—
126C2×D4—2441no—
127C2×D4—2441no—
128C22×C4—21no—
129C22×C4—2441no—
130C22×C4—2441no—
131C22×C4#1175819811yescharacteristic
132M16—41no—
133M16—41no—
134C2×C8—21no—
135C2×C8—21no—
136C4⋊C4—2441no—
137C22⋊C4—2442no—
138C42—21no—
139C42—1222no—
140C42—2441no—
141C42—2444no—
142C42C8×D6111yescharacteristic
143C42#1175818911yescharacteristic
144C16—81no—
145C16—4881no—
146C23—2441no—
147C23#1451969311yescharacteristic
148Q8—2441no—
149Q8—2441no—
150D4—4882no—
151D4—4881no—
152D4—4881no—
153D4—2441no—
154D4—2441no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—4881no—
161C2×C4—4881no—
162C2×C4—2441no—
163C2×C4—2442no—
164C2×C4—2444no—
165C2×C4#1451968911yescharacteristic
166C2×C4#1451969111yescharacteristic
167C8—41no—
168C8—41no—
169C22—21no—
170C22—41no—
171C22—4882no—
172C22—4881no—
173C22#1683948511yescharacteristic
174C4—22no—
175C4—21no—
176C4—41no—
177C4—4881no—
178C4—2444no—
179C4#1683955411yescharacteristic
180C4#1683955811yescharacteristic
181C2—21no—
182C2—41no—
183C2—9761no—
184C2#2144375011yescharacteristic
subgroupG/H
centerC4#16839554
commutatorC42×C61C2×C8
Frattini#423C2×D61
FittingC61×#1327C2
socleC2×C61#1318
prime radical#16840312C4
cosocleC61×#423C22
hypercenter#1327D61
hypocenterC61#7357
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #21443770C2semidirect
3. C61×#1327C2semidirect
91. #1327D61semidirect
103. C61#7357semidirect

Maximal subgroups

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

subgroupindex
#214466462
#214437702
C61×#13272
#735761