Finite Group Databasebeta

← #30595039

Subgroups — #30595039

Order 15616.

198 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 36 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#21460762C211yesmaximal, characteristic
2#21460697C211yesmaximal, characteristic
3#21447161C211yesmaximal, characteristic
4C2×#14519844—21no—
5#16840607—21no—
6#16839805—21no—
7#16839631C2211yescharacteristic
8#16839531C411yescharacteristic
9#16839521—21no—
10C61×#544C411yescharacteristic
11C22×#92420—41no—
12#14519844—22no—
13#14519844—41no—
14C22⋊#92420—41no—
15C2×#11758194—21no—
16C2×C4⋊Dic61D411yescharacteristic
17C2×#11758196—41no—
18C2×C4×Dic61D411yescharacteristic
19#14519710—41no—
20#14519708—41no—
21C4⋊Q8×C61—41no—
22#125×C61—41no—
23C22⋊Q8×C61—41no—
24#117×C61—21no—
25C2×C4⋊C4×C61C2×C411yescharacteristic
26C2×Q8×C61—41no—
27C22×C4×C61—21no—
28C22×C4×C61C22⋊C411yescharacteristic
29C22×Dic61—21no—
30C22×Dic61—21no—
31C2×#92420—42no—
32C2×#92420—81no—
33C2×#92420—81no—
34C2×#92420—81no—
35C4⋊C4×C61—21no—
36C4⋊C4×C61—21no—
37C4⋊C4×C61—41no—
38C4⋊C4×C61—42no—
39C4⋊C4×C61—41no—
40C22⋊C4×C61—41no—
41C42×C61—41no—
42C4⋊Dic61—41no—
43#11758196—81no—
44#11758196—81no—
45C4×Dic61—21no—
46C4×Dic61SD1611yescharacteristic
47C4×Dic61Q1611yescharacteristic
48#11758194—22no—
49#11758194—41no—
50C2×#92418—41no—
51C23×C61#9911yescharacteristic
52Q8×C61—161no—
53C2×C4×C61—21no—
54C2×C4×C61—21no—
55C2×C4×C61—41no—
56C2×C4×C61—41no—
57C2×C4×C61—41no—
58C2×C4×C61—81no—
59C2×C4×C61#10311yescharacteristic
60C2×C4×C61#10411yescharacteristic
61C2×Dic61—21no—
62C2×Dic61—21no—
63C2×Dic61—21no—
64C2×Dic61—21no—
65C2×Dic61—21no—
66C2×Dic61—21no—
67C2×Dic61—41no—
68C2×Dic61—81no—
69#92420—82no—
70#92420—161no—
71#92418—42no—
72#7443—611nomaximal, Sylow
73C22×C61—21no—
74C22×C61—21no—
75C22×C61#32811yescharacteristic
76C22×C61AΣL(1,𝔽16)11yescharacteristic
77C22×C61#35611yescharacteristic
78C4×C61—21no—
79C4×C61—21no—
80C4×C61—81no—
81C4×C61—81no—
82C4×C61—81no—
83Dic61—21no—
84Dic61—21no—
85Dic61—21no—
86Dic61—21no—
87Dic61—81no—
88#2060—611no—
89#1325—611no—
90#1296—611no—
91C2×C61—21no—
92C2×C61—21no—
93C2×C61#135411yescharacteristic
94C2×C61#140211yescharacteristic
95C2×C61#140511yescharacteristic
96#544#692311yescharacteristic
97#428—1221no—
98C2×C4⋊C8—1221no—
99#384—611no—
100#343—611no—
101#339—1221no—
102#324—1221no—
103C61#744311yescharacteristic, complemented, Sylow
104C2×M16—1221no—
105C22×C8—2441no—
106C4⋊Q8—41no—
107#125—41no—
108C22⋊Q8—41no—
109#117—21no—
110C2×C4⋊C4—611no—
111C2×C4⋊C4—2441no—
112C2×C4⋊C4C2×#692311yescharacteristic
113C2×C42—611no—
114#107—2441no—
115#106—2441no—
116C4⋊C8—1222no—
117C4⋊C8—2441no—
118C22⋊C8—2441no—
119C2×Q8—41no—
120C22×C4—21no—
121C22×C4—1221no—
122C22×C4—1221no—
123C22×C4C22⋊#692311yescharacteristic
124M16—1222no—
125M16—2441no—
126C2×C8—4881no—
127C2×C8—4881no—
128C2×C8—4881no—
129C2×C8—2442no—
130C2×C8—2441no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—42no—
136C4⋊C4—4881no—
137C4⋊C4—4881no—
138C4⋊C4—2441no—
139C22⋊C4—41no—
140C42—41no—
141C42—611no—
142C42—611no—
143C42—1221no—
144C23#1451983511yescharacteristic
145Q8—161no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—81no—
152C2×C4—1221no—
153C2×C4—1221no—
154C2×C4—1221no—
155C2×C4—1221no—
156C2×C4—1221no—
157C2×C4—1221no—
158C2×C4—4881no—
159C2×C4—2441no—
160C2×C4#1451984011yescharacteristic
161C2×C4#1451984211yescharacteristic
162C8—9761no—
163C8—4882no—
164C8—2442no—
165C22—21no—
166C22—21no—
167C22#1684061611yescharacteristic
168C22#1684066211yescharacteristic
169C22#1684067211yescharacteristic
170C4—21no—
171C4—21no—
172C4—81no—
173C4—81no—
174C4—81no—
175C4—1221no—
176C4—1221no—
177C4—1221no—
178C4—1221no—
179C4—4881no—
180C2—21no—
181C2—21no—
182C2#2146082011yescharacteristic
183C2#2146093811yescharacteristic
184C2#2146094411yescharacteristic
subgroupG/H
centerC22#16840662
commutatorC2×C4⋊C4×C61C2×C4
FrattiniC2×C4⋊C4C2×#6923
FittingC61×#544C4
socleC22×C61AΣL(1,𝔽16)
prime radicalC23×C61#99
cosocle#16839631C22
hypercenter#544#6923
hypocenterC61#7443
LCSG ▷ C2×C4⋊C4×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4 ◁ #544

Complements

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

NH ≅ G/Ntype
103. C61#7443semidirect

Maximal subgroups

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

subgroupindex
#214607622
#214606972
#214471612
#744361