Finite Group Databasebeta

← #29599904

Subgroups — #29599904

Order 15616.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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#21447429C211yesmaximal, characteristic
2#21445854C212yesmaximal
3#21443930C212yesmaximal, complemented
4#21443923C211yesmaximal, characteristic, complemented
5C61×#2161C211yesmaximal, characteristic, complemented
6C22×C8×D61C2211yescharacteristic
7C2×#14519717C2211yescharacteristic
8C2×#14519713C2212yes—
9#16839725—22no—
10#16839541—22no—
11#16839526—22no—
12C2×M32×C61C2211yescharacteristic
13C2×#108×C61C2212yes—
14C61×#368—22no—
15C22×C4×D61D411yescharacteristic
16C2×#11758194—22no—
17C22×#92418D411yescharacteristic
18C2×C8×D61—21no—
19C2×C8×D61—22no—
20C2×C8×D61—22no—
21C2×C8×D61—21no—
22#14519717—22no—
23C2×#11758185—21no—
24#14519713—22no—
25#14519713—22no—
26#14519689—44no—
27C2×M16×C61—22no—
28C22×C8×C61C2311yescharacteristic
29M32×C61—22no—
30C2×C16×C61D412yes—
31#108×C61—22no—
32#108×C61—22no—
33C23×D61Q1611yescharacteristic
34C22×C4×C61C2×D411yescharacteristic
35C22×Dic61Q1611yescharacteristic
36C2×C4×D61—21no—
37C2×C4×D61—22no—
38C2×C4×D61—21no—
39C2×C4×D61—22no—
40M16×C61—42no—
41M16×C61—42no—
42C2×C8×C61—42no—
43C2×C8×C61C2×D411yescharacteristic
44C2×C8×C61#4112yes—
45C2×C8×C61#4112yes—
46C2×C8×C61#4111yescharacteristic
47#11758194—42no—
48#11758194—42no—
49C2×#92418—21no—
50C2×#92418—22no—
51C2×#92418—42no—
52C8×D61—41no—
53C8×D61—41no—
54C8×D61—42no—
55C16×C61—22no—
56#11758185—41no—
57C23×C61C2×Q1611yescharacteristic
58C22×D61—21no—
59C22×D61—22no—
60C22×D61—42no—
61C2×C4×C61#12111yescharacteristic
62C2×C4×C61#12312yes—
63C2×C4×C61#13511yescharacteristic
64C2×C4×C61AGL(1,ℤ/8)12yes—
65C2×Dic61—21no—
66C2×Dic61—22no—
67C4×D61—41no—
68C4×D61—41no—
69C4×D61—42no—
70C8×C61—82no—
71C8×C61#12311yescharacteristic
72C8×C61#12411yescharacteristic
73C8×C61#12612yes—
74#92418—41no—
75#92418—82no—
76#13498—611nomaximal, Sylow
77C22×C61—41no—
78C22×C61#46711yescharacteristic
79C22×C61#48412yes—
80C2×D61—22no—
81C2×D61—24no—
82C2×D61—41no—
83C2×D61—42no—
84C2×D61—41no—
85C4×C61#39711yescharacteristic
86C4×C61#46911yescharacteristic
87C4×C61#48212yes—
88Dic61—41no—
89#2161D6111yescharacteristic, complemented
90#2112—611no—
91#1810—612no—
92#1388—612no—
93#1385—611no—
94C2×C61—21no—
95C2×C61—22no—
96C2×C61#207611yescharacteristic
97D61—42no—
98C23×C8—611no—
99C2×M32—611no—
100C2×M32C2×D6111yescharacteristic
101C2×#108—612no—
102C2×#108C2×D6112yes—
103#407—1222no—
104#368—22no—
105C22⋊C16—1222no—
106#341—1222no—
107C61#1349811yescharacteristic, complemented, Sylow
108C23×C4—611no—
109C2×M16—22no—
110C2×M16—1222no—
111C22×C8—611no—
112C22×C8—1222no—
113C22×C8—1221no—
114C22×C8—1222no—
115C22×C8—1221no—
116C22×C8C22×D6111yescharacteristic
117M32—22no—
118M32—1222no—
119C2×C16—1221no—
120C2×C16C22⋊D6112yes—
121#108—22no—
122#108—22no—
123#108—1222no—
124#108—1222no—
125C22⋊C8—2444no—
126C24—611no—
127C22×C4—611no—
128C22×C4—1221no—
129C22×C4—1222no—
130C22×C4—1221no—
131C22×C4—1222no—
132C22×C4D4×D6111yescharacteristic
133M16—42no—
134M16—42no—
135M16—2442no—
136M16—2442no—
137C2×C8—42no—
138C2×C8—1221no—
139C2×C8—1222no—
140C2×C8—2442no—
141C2×C8—2441no—
142C2×C8—2441no—
143C2×C8—2442no—
144C2×C8#1175822212yes—
145C2×C8#1175822411yescharacteristic
146C2×C8#1175822612yes—
147C2×C8C2×C22⋊D6111yescharacteristic
148C16—22no—
149C16—2441no—
150C23—1221no—
151C23—1222no—
152C23—2442no—
153C23Q16×D6111yescharacteristic
154C2×C4—1221no—
155C2×C4—1222no—
156C2×C4—2441no—
157C2×C4—2441no—
158C2×C4—2442no—
159C2×C4#1451974811yescharacteristic
160C2×C4#1451976212yes—
161C2×C4#1451980011yescharacteristic
162C2×C4#1451980112yes—
163C8—82no—
164C8—4882no—
165C8—2441no—
166C8#1451976011yescharacteristic
167C8#1451976611yescharacteristic
168C8#1451977212yes—
169C22—41no—
170C22—1222no—
171C22—1224no—
172C22—2441no—
173C22—2442no—
174C22—2441no—
175C22#1684000011yescharacteristic
176C22#1684009912yes—
177C4—2441no—
178C4#1683968811yescharacteristic
179C4#1684001211yescharacteristic
180C4#1684008912yes—
181C2—21no—
182C2—22no—
183C2—2442no—
184C2#2144724811yescharacteristic
subgroupG/H
centerC4#16840012
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#2161C2
socleC2×C61#2076
prime radicalC23×D61Q16
hypercenter#2161D61
hypocenterC61#13498
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2161

Complements

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

NH ≅ G/Ntype
3. #21443930C2semidirect
4. #21443923C2semidirect
5. C61×#2161C2semidirect
89. #2161D61semidirect
107. C61#13498semidirect

Maximal subgroups

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

subgroupindex
#214474292
#214458542
#214439302
#214439232
C61×#21612
#1349861