Finite Group Databasebeta

← #29589449

Subgroups — #29589449

Order 15616.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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#21447631C212yesmaximal, complemented
2#21443770C212yesmaximal, complemented
3#21443755C212yesmaximal
4C61×#2921C211yesmaximal, characteristic
5#16839583—22no—
6#16839544C2214yes—
7#16839540—22no—
8C2×#117×C61C411yescharacteristic
9C61×#433—22no—
10C61×#433—22no—
11C61×#423C411yescharacteristic
12C61×#423C2211yescharacteristic
13C61×#407C411yescharacteristic
14C61×#407C2211yescharacteristic
15C4×M16×C61C411yescharacteristic
16C4×M16×C61C2211yescharacteristic
17#14519717—24no—
18C23×C4×C61C2×C411yescharacteristic
19C2×M16×C61C2×C411yescharacteristic
20C2×M16×C61C2×C411yescharacteristic
21C2×M16×C61C2×C411yescharacteristic
22C2×M16×C61C2311yescharacteristic
23#117×C61—21no—
24#117×C61—21no—
25#117×C61—22no—
26C2×C4⋊C4×C61—21no—
27C2×C22⋊C4×C61—21no—
28C2×C42×C61C2×C411yescharacteristic
29C2×C42×C61C2×C411yescharacteristic
30C4⋊C8×C61—22no—
31C4⋊C8×C61—22no—
32C22⋊C8×C61—22no—
33C22⋊C8×C61—22no—
34#97×C61—21no—
35#97×C61D412yes—
36C4×C8×C61—21no—
37C4×C8×C61D412yes—
38C24×C61M1611yescharacteristic
39C22×C4×C61—21no—
40C22×C4×C61—21no—
41C22×C4×C61—21no—
42C22×C4×C61—21no—
43C22×C4×C61—41no—
44C22×C4×C61—41no—
45C22×C4×C61M1611yescharacteristic
46C22×C4×C61C22×C411yescharacteristic
47M16×C61—41no—
48M16×C61—41no—
49M16×C61—41no—
50M16×C61—41no—
51C2×C8×C61—21no—
52C2×C8×C61—21no—
53C2×C8×C61C22⋊C412yes—
54C2×C8×C61C2×D412yes—
55C4⋊C4×C61—22no—
56C4⋊C4×C61—22no—
57C22⋊C4×C61—22no—
58C22⋊C4×C61—22no—
59C42×C61—21no—
60C42×C61—21no—
61C42×C61C22⋊C411yescharacteristic
62C42×C61C22⋊C411yescharacteristic
63C42×C61M1612yes—
64#11758185—44no—
65C23×C61—21no—
66C23×C61—41no—
67C23×C61—81no—
68C23×C61C2×M1611yescharacteristic
69C2×C4×C61—21no—
70C2×C4×C61—21no—
71C2×C4×C61—21no—
72C2×C4×C61—21no—
73C2×C4×C61—22no—
74C2×C4×C61—21no—
75C2×C4×C61—42no—
76C2×C4×C61—41no—
77C2×C4×C61—41no—
78C2×C4×C61—41no—
79C2×C4×C61—81no—
80C2×C4×C61—81no—
81C2×C4×C61C2×C22⋊C411yescharacteristic
82C2×C4×C61C2×M1611yescharacteristic
83C8×C61—22no—
84C8×C61—22no—
85C8×C61—41no—
86C8×C61—41no—
87#11522—611nomaximal, Sylow
88C22×C61—21no—
89C22×C61—41no—
90C22×C61—42no—
91C22×C61—41no—
92C22×C61—82no—
93C22×C61#40711yescharacteristic
94C4×C61—21no—
95C4×C61—41no—
96C4×C61—41no—
97C4×C61—42no—
98C4×C61—41no—
99C4×C61#41011yescharacteristic
100C4×C61#41311yescharacteristic
101#2921D6111yescharacteristic
102#2165—612no—
103#1327—612no—
104#1320—612no—
105C2×C61—21no—
106C2×C61—41no—
107C2×C61—81no—
108C2×C61#146411yescharacteristic
109C2×#117Dic6111yescharacteristic
110#433—22no—
111#433—22no—
112#423Dic6111yescharacteristic
113#423C2×D6111yescharacteristic
114#407Dic6111yescharacteristic
115#407C2×D6111yescharacteristic
116C4×M16Dic6111yescharacteristic
117C4×M16C2×D6111yescharacteristic
118#364—1222no—
119#349—614no—
120#347—1222no—
121C61#1152211yescharacteristic, complemented, Sylow
122C23×C4C2×Dic6111yescharacteristic
123C2×M16C2×Dic6111yescharacteristic
124C2×M16C2×Dic6111yescharacteristic
125C2×M16C2×Dic6111yescharacteristic
126C2×M16C22×D6111yescharacteristic
127#117—22no—
128#117—21no—
129#117—21no—
130C2×C4⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C42C2×Dic6111yescharacteristic
133C2×C42C2×Dic6111yescharacteristic
134M32—1224no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139#97—21no—
140#97C22⋊D6112yes—
141C4×C8—21no—
142C4×C8C22⋊D6112yes—
143C24#1175819411yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#1175819411yescharacteristic
151C22×C4C22×Dic6111yescharacteristic
152M16—41no—
153M16—41no—
154M16—41no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C22⋊Dic6112yes—
159C2×C8C2×C22⋊D6112yes—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—21no—
165C42—21no—
166C42#1175819412yes—
167C42C22⋊Dic6111yescharacteristic
168C42C22⋊Dic6111yescharacteristic
169C16—2444no—
170C23—21no—
171C23—41no—
172C23—81no—
173C23C2×#1175819411yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4C2×C22⋊Dic6111yescharacteristic
187C2×C4C2×#1175819411yescharacteristic
188C8—22no—
189C8—22no—
190C8—41no—
191C8—41no—
192C22—21no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—82no—
197C22#1683972611yescharacteristic
198C4—21no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—41no—
203C4#1683973611yescharacteristic
204C4#1683974611yescharacteristic
205C2—21no—
206C2—41no—
207C2—81no—
208C2#2144415311yescharacteristic
subgroupG/H
centerC4#16839736
commutatorC22×C4×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#2921C2
socleC2×C61#1464
prime radicalC24×C61M16
cosocleC2×M16×C61C23
hypercenter#2921D61
hypocenterC61#11522
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C42 ◁ C4×M16 ◁ #2921

Complements

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

NH ≅ G/Ntype
1. #21447631C2semidirect
2. #21443770C2semidirect
121. C61#11522semidirect

Maximal subgroups

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

subgroupindex
#214476312
#214437702
#214437552
C61×#29212
#1152261