Finite Group Databasebeta

← #29584157

Subgroups — #29584157

Order 15616.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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#21457200C211yesmaximal, characteristic
2#21445959C211yesmaximal, characteristic, complemented
3#21444542C212yesmaximal, complemented
4#21444431C212yesmaximal, complemented
5C61×#1307C211yesmaximal, characteristic, complemented
6#16840312C2212yes—
7#16840053—22no—
8#16840015—22no—
9#16839958—22no—
10#16839786C2211yescharacteristic
11C2×#14519688C2211yescharacteristic
12#16839526—22no—
13#16839509—22no—
14#16839505—22no—
15#16839501—24no—
16C61×#432C2212yes—
17C61×#423C2211yescharacteristic
18C2×#11758222—22no—
19C2×D488—22no—
20C2×#11758194—21no—
21C22×#92418—21no—
22#14519777D412yes—
23#14519774D412yes—
24#14519765—22no—
25#14519751—22no—
26#14519751—22no—
27C4×D244—22no—
28C4×D244—22no—
29#14519709—22no—
30C4⋊#92418—22no—
31#14519699—24no—
32#14519688—21no—
33#14519688D412yes—
34C2×M16×C61D412yes—
35C2×C42×C61C2311yescharacteristic
36C4⋊C8×C61—22no—
37C4⋊C8×C61—22no—
38C22⋊C8×C61—22no—
39#97×C61—22no—
40C22×C4×C61C2×D411yescharacteristic
41C22×C4×C61#4112yes—
42C2×C22⋊D61—42no—
43#11758222—84no—
44C2×D244—22no—
45C2×D244—22no—
46C2×D244—22no—
47C2×C4×D61—42no—
48C2×Dic122—22no—
49M16×C61—42no—
50C2×C8×C61—22no—
51C2×C8×C61—42no—
52C42×C61C2×D412yes—
53C42×C61C2×D411yescharacteristic
54C42×C61#4111yescharacteristic
55#11758198—42no—
56#11758198—42no—
57C4⋊Dic61—22no—
58C4⋊Dic61—22no—
59#11758194—42no—
60C2×#92418—22no—
61C2×#92418—41no—
62C2×#92418—41no—
63C2×#92418—41no—
64D488—82no—
65C23×C61#12411yescharacteristic
66C22×D61—42no—
67C2×C4×C61—21no—
68C2×C4×C61—21no—
69C2×C4×C61—42no—
70C2×C4×C61#12011yescharacteristic
71C2×C4×C61#12112yes—
72C2×C4×C61#12111yescharacteristic
73C2×C4×C61#12312yes—
74C22⋊D61—84no—
75C2×Dic61—42no—
76D244—42no—
77D244—42no—
78D244—82no—
79D244—82no—
80C4×D61—84no—
81Dic122—82no—
82C8×C61—42no—
83C8×C61—82no—
84#92418—42no—
85#92418—81no—
86#10525—611nomaximal, Sylow
87C22×C61—22no—
88C22×C61—21no—
89C22×C61#39411yescharacteristic
90C2×D61—82no—
91C2×D61—82no—
92C2×D61—82no—
93C4×C61—22no—
94C4×C61—22no—
95C4×C61—21no—
96C4×C61#45411yescharacteristic
97C4×C61#45511yescharacteristic
98Dic61—82no—
99#3253—611no—
100#1838—611no—
101#1557—612no—
102#1530—612no—
103#1307D6111yescharacteristic, complemented
104C2×C61—41no—
105C2×C61#204011yescharacteristic
106C2×C61#204812yes—
107D61—162no—
108#532—612no—
109#476—1222no—
110#469—1222no—
111#459—1222no—
112#432C2×D6112yes—
113#423—611no—
114#423C2×D6111yescharacteristic
115C2×#97—611no—
116#341—1222no—
117#332—1222no—
118#331—1222no—
119#329—1224no—
120C61#1052511yescharacteristic, complemented, Sylow
121C2×#41—1222no—
122C2×D8—1222no—
123C2×M16—1221no—
124C2×M16D24412yes—
125C22×C8—1221no—
126C4⋊Q8—612no—
127#127—612no—
128#124—1222no—
129#121—1222no—
130#121—1222no—
131C4×D4—1222no—
132C4×D4—1222no—
133C2×C42C22×D6111yescharacteristic
134#106—1222no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—1222no—
138#102—1224no—
139C22⋊C8—22no—
140#97—22no—
141#97—612no—
142#97—1221no—
143#41—4884no—
144C2×Q8—1222no—
145C2×D4—1222no—
146C2×D4—1222no—
147C2×D4—1222no—
148C2×D4—2442no—
149C22×C4—2442no—
150C22×C4C2×D24411yescharacteristic
151C22×C4#1175822212yes—
152D8—4882no—
153M16—42no—
154M16—2442no—
155C2×C8—22no—
156C2×C8—42no—
157C2×C8—1222no—
158C2×C8—2441no—
159C2×C8—2441no—
160C2×C8—2441no—
161C4⋊C4—1222no—
162C4⋊C4—1222no—
163C22⋊C4—2442no—
164C22⋊C4—2442no—
165C42D4×D6112yes—
166C42D4×D6111yescharacteristic
167C42#1175822611yescharacteristic
168C23—2442no—
169C23#1451976511yescharacteristic
170Q8—4882no—
171D4—4882no—
172D4—4882no—
173D4—4884no—
174D4—2442no—
175D4—2442no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—4884no—
180C2×C4—2442no—
181C2×C4#1451974711yescharacteristic
182C2×C4#1451975012yes—
183C2×C4#1451975211yescharacteristic
184C2×C4#1451976212yes—
185C8—42no—
186C8—82no—
187C8—4881no—
188C8—2442no—
189C22—21no—
190C22—22no—
191C22—4882no—
192C22—4882no—
193C22—4882no—
194C22#1683968011yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—4882no—
199C4#1683993711yescharacteristic
200C4#1683994211yescharacteristic
201C2—41no—
202C2—9762no—
203C2#2144704911yescharacteristic
204C2#2144709512yes—
subgroupG/H
centerC22#16839680
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1307C2
socleC22×C61#394
prime radical#21457200C2
hypercenter#1307D61
hypocenterC61#10525
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #21445959C2semidirect
3. #21444542C2semidirect
4. #21444431C2semidirect
5. C61×#1307C2semidirect
103. #1307D61semidirect
120. C61#10525semidirect

Maximal subgroups

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

subgroupindex
#214572002
#214459592
#214445422
#214444312
C61×#13072
#1052561