Finite Group Databasebeta

← #30608390

Subgroups — #30608390

Order 15616.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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#21462092C212yesmaximal
2#21460960C214yesmaximal, complemented
3#21456165C211yesmaximal, characteristic
4#16840741—22no—
5#16840669—22no—
6#16840667C2214yes—
7#16840319C411yescharacteristic, complemented
8#16840312C2211yescharacteristic
9#16840131C412yescomplemented
10#16840127C2212yes—
11#16840122—22no—
12#16839759—22no—
13C2×#97×C61C411yescharacteristic, complemented
14#14519856—42no—
15#14519836—24no—
16#14519835—42no—
17C2×#11758222D412yes—
18#14519777—21no—
19#14519777C2×C411yescharacteristic
20#14519774C2311yescharacteristic
21#14519769C2×C412yes—
22#14519765D412yes—
23#14519756—41no—
24#14519751—41no—
25C4×Dic122—41no—
26C4×D244—41no—
27#14519736—21no—
28#14519702—24no—
29#14519699—24no—
30C22×C8×C61—21no—
31C2×C42×C61C2×C411yescharacteristic
32#97×C61—21no—
33#97×C61C2×C412yes—
34C22×C4×C61—21no—
35C22×C4×C61C22⋊C411yescharacteristic
36C2×C22⋊D61—22no—
37#11758222—42no—
38#11758222—42no—
39C2×D244—41no—
40C2×D244C2×D412yes—
41C2×C4×D61—22no—
42C2×Dic122—41no—
43C2×Dic122C22⋊C412yes—
44C22⋊#6923—44no—
45C4⋊#6923—82no—
46C4×#6923—82no—
47C2×C8×C61—22no—
48C2×C8×C61—41no—
49C2×C8×C61—41no—
50C42×C61—21no—
51C42×C61C22⋊C411yescharacteristic
52C42×C61C22×C411yescharacteristic
53C22⋊Dic61—41no—
54#11758198—42no—
55C4⋊Dic61—22no—
56C4⋊Dic61—41no—
57C4⋊Dic61—41no—
58#11758196—42no—
59C4×Dic61—41no—
60C23×C61#9911yescharacteristic
61C22×D61—22no—
62C2×#6923—84no—
63C2×C4×C61—21no—
64C2×C4×C61—21no—
65C2×C4×C61—21no—
66C2×C4×C61—21no—
67C2×C4×C61—81no—
68C2×C4×C61#9911yescharacteristic
69C2×C4×C61C2×C22⋊C411yescharacteristic
70C22⋊D61—82no—
71C2×Dic61—22no—
72C2×Dic61—42no—
73D244—42no—
74D244—82no—
75C4×D61—42no—
76C4×D61—42no—
77Dic122—42no—
78Dic122—82no—
79C8×C61—42no—
80#13594—611nomaximal, Sylow
81C22×C61—21no—
82C22×C61—41no—
83C22×C61C2×#9911yescharacteristic
84C2×D61—42no—
85C2×D61—82no—
86#6923—164no—
87C4×C61—21no—
88C4×C61—41no—
89C4×C61—41no—
90C4×C61#35411yescharacteristic
91C4×C61#35411yescharacteristic
92Dic61—42no—
93Dic61—82no—
94#3135—611no—
95#2126—612no—
96#1412—614no—
97C2×C61—21no—
98C2×C61—41no—
99C2×C61C2×#35411yescharacteristic
100D61—82no—
101#533—611no—
102#532—611no—
103#491—612no—
104#490—612no—
105#489—1222no—
106#416—1222no—
107#410—1222no—
108C2×#97#692311yescharacteristic, complemented
109#354—1222no—
110#353—614no—
111C61#1359411yescharacteristic, complemented, Sylow
112C2×#41—612no—
113C22×C8—21no—
114C4⋊Q8—611no—
115C4⋊Q8—1221no—
116#127—611no—
117#125—612no—
118#124—612no—
119C22⋊Q8—2441no—
120#121—2441no—
121C4×Q8—2441no—
122C4×D4—2441no—
123#117—1221no—
124#117—2442no—
125C2×C42C2×#692311yescharacteristic
126#103—1224no—
127#102—1224no—
128#99—1224no—
129#99—2442no—
130#97—21no—
131#97C2×#692312yes—
132#41—2442no—
133#41—2442no—
134C2×Q8—612no—
135C2×Q8—2441no—
136C2×D4—612no—
137C2×D4—1222no—
138C2×D4—2441no—
139C22×C4—21no—
140C22×C4—1222no—
141C22×C4C22⋊#692311yescharacteristic
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—41no—
145C4⋊C4—1222no—
146C4⋊C4—4882no—
147C4⋊C4—2441no—
148C4⋊C4—2442no—
149C4⋊C4—2441no—
150C22⋊C4—2442no—
151C22⋊C4—2444no—
152C22⋊C4—2441no—
153C42—21no—
154C42—4882no—
155C42—2441no—
156C42C22⋊#692311yescharacteristic
157C42C22×#692311yescharacteristic
158C23—1222no—
159C23#1451983611yescharacteristic
160Q8—4882no—
161Q8—2442no—
162D4—4882no—
163D4—4882no—
164D4—2442no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—81no—
170C2×C4—1222no—
171C2×C4—4884no—
172C2×C4—2442no—
173C2×C4—2442no—
174C2×C4—2442no—
175C2×C4#1451983611yescharacteristic
176C2×C4C2×C22⋊#692311yescharacteristic
177C8—42no—
178C22—21no—
179C22—41no—
180C22—4882no—
181C22—2442no—
182C22C2×#1451983611yescharacteristic
183C4—21no—
184C4—41no—
185C4—41no—
186C4—9764no—
187C4—4882no—
188C4—2442no—
189C4#1684066911yescharacteristic
190C4#1684066911yescharacteristic
191C2—21no—
192C2—41no—
193C2—4882no—
194C2C2×#1684066911yescharacteristic
subgroupG/H
centerC4#16840669
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC2×#97×C61C4
socleC2×C61C2×#354
prime radical#16840312C22
cosocle#14519774C23
hypercenterC2×#97#6923
hypocenterC61#13594
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
2. #21460960C2semidirect
7. #16840319C4semidirect
9. #16840131C4semidirect
13. C2×#97×C61C4semidirect
108. C2×#97#6923semidirect
111. C61#13594semidirect

Maximal subgroups

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

subgroupindex
#214620922
#214609602
#214561652
#1359461