Finite Group Databasebeta

← #29584191

Subgroups — #29584191

Order 15616.

386 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 81 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#21445956C211yesmaximal, characteristic, complemented
3#21444552C212yesmaximal, complemented
4#21444269C212yesmaximal, complemented
5C61×#1309C211yesmaximal, characteristic, complemented
6#16840312C2212yes—
7#16840053—22no—
8#16840015—22no—
9#16839958—22no—
10#16839786C2211yescharacteristic
11C2×C4×#92418C2211yescharacteristic
12#16839526—22no—
13#16839510—22no—
14#16839505—22no—
15#16839489—24no—
16C61×#423C2212yes—
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—
32C4×#92418—21no—
33C4×#92418D412yes—
34C2×M16×C61—22no—
35C2×M16×C61D412yes—
36C2×C42×C61C2311yescharacteristic
37C4⋊C8×C61—22no—
38C4⋊C8×C61—22no—
39C4⋊C8×C61—22no—
40C22×C4×C61C2×D411yescharacteristic
41C22×C4×C61#4112yes—
42C2×C22⋊D61—42no—
43#11758222—82no—
44#11758222—82no—
45C2×D244—22no—
46C2×D244—22no—
47C2×D244—22no—
48C2×C4×D61—42no—
49C2×Dic122—22no—
50M16×C61—42no—
51M16×C61—82no—
52C2×C8×C61—22no—
53C2×C8×C61—42no—
54C42×C61C2×D412yes—
55C42×C61C2×D411yescharacteristic
56C42×C61#4111yescharacteristic
57#11758198—42no—
58#11758198—42no—
59C4⋊Dic61—22no—
60C4⋊Dic61—22no—
61#11758194—42no—
62C2×#92418—22no—
63C2×#92418—41no—
64C2×#92418—41no—
65C2×#92418—41no—
66D488—82no—
67C23×C61#12411yescharacteristic
68C22×D61—42no—
69C2×C4×C61—21no—
70C2×C4×C61—21no—
71C2×C4×C61—22no—
72C2×C4×C61—22no—
73C2×C4×C61#12011yescharacteristic
74C2×C4×C61#12111yescharacteristic
75C2×C4×C61#12112yes—
76C2×C4×C61#12312yes—
77C22⋊D61—84no—
78C2×Dic61—42no—
79D244—42no—
80D244—42no—
81D244—42no—
82D244—42no—
83D244—82no—
84C4×D61—82no—
85C4×D61—82no—
86Dic122—82no—
87C8×C61—42no—
88C8×C61—82no—
89#92418—42no—
90#92418—81no—
91#10531—611nomaximal, Sylow
92C22×C61—22no—
93C22×C61—21no—
94C22×C61#39411yescharacteristic
95C2×D61—82no—
96C2×D61—82no—
97C2×D61—82no—
98C4×C61—22no—
99C4×C61—21no—
100C4×C61#45411yescharacteristic
101C4×C61#45511yescharacteristic
102C4×C61#47112yes—
103C4×C61#47212yes—
104Dic61—82no—
105#3253—611no—
106#1837—611no—
107#1560—612no—
108#1492—612no—
109#1309D6111yescharacteristic, complemented
110C2×C61—41no—
111C2×C61#204011yescharacteristic
112C2×C61#205112yes—
113D61—162no—
114#532—612no—
115#476—1222no—
116#469—1222no—
117#459—1222no—
118#423—611no—
119#423C2×D6112yes—
120#423C2×D6111yescharacteristic
121C2×C4×C8—611no—
122#341—1222no—
123#333—1222no—
124#331—1222no—
125D4⋊C8—1224no—
126C61#1053111yescharacteristic, complemented, Sylow
127C2×#41—1222no—
128C2×D8—1222no—
129C2×M16—22no—
130C2×M16—1221no—
131C2×M16D24412yes—
132C22×C8—1221no—
133C4⋊Q8—612no—
134#127—612no—
135#124—1222no—
136#121—1222no—
137#121—1222no—
138C4×D4—1222no—
139C4×D4—1222no—
140C2×C42C22×D6111yescharacteristic
141#106—1222no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—1222no—
146#102—1224no—
147C4×C8—612no—
148C4×C8—1221no—
149#41—4882no—
150#41—4882no—
151C2×Q8—1222no—
152C2×D4—1222no—
153C2×D4—1222no—
154C2×D4—1222no—
155C2×D4—2442no—
156C22×C4—2442no—
157C22×C4C2×D24411yescharacteristic
158C22×C4#1175822212yes—
159D8—4882no—
160M16—42no—
161M16—82no—
162M16—2442no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—1222no—
166C2×C8—2441no—
167C2×C8—2441no—
168C2×C8—2441no—
169C4⋊C4—1222no—
170C4⋊C4—1222no—
171C22⋊C4—2442no—
172C22⋊C4—2442no—
173C42D4×D6112yes—
174C42D4×D6111yescharacteristic
175C42#1175822611yescharacteristic
176C23—2442no—
177C23#1451976511yescharacteristic
178Q8—4882no—
179D4—4882no—
180D4—4884no—
181D4—2442no—
182D4—2442no—
183D4—2442no—
184D4—2442no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—4882no—
190C2×C4—4882no—
191C2×C4—2442no—
192C2×C4#1451974711yescharacteristic
193C2×C4#1451975012yes—
194C2×C4#1451975211yescharacteristic
195C2×C4#1451976212yes—
196C8—42no—
197C8—82no—
198C8—4881no—
199C8—2442no—
200C22—21no—
201C22—22no—
202C22—4882no—
203C22—4882no—
204C22—4882no—
205C22#1683968011yescharacteristic
206C4—21no—
207C4—22no—
208C4—4882no—
209C4#1683993711yescharacteristic
210C4#1683994211yescharacteristic
211C4#1684002812yes—
212C4#1684003112yes—
213C2—41no—
214C2—9762no—
215C2#2144704911yescharacteristic
216C2#2144711412yes—
subgroupG/H
centerC22#16839680
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1309C2
socleC22×C61#394
prime radical#21457200C2
hypercenter#1309D61
hypocenterC61#10531
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
2. #21445956C2semidirect
3. #21444552C2semidirect
4. #21444269C2semidirect
5. C61×#1309C2semidirect
109. #1309D61semidirect
126. C61#10531semidirect

Maximal subgroups

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

subgroupindex
#214572002
#214459562
#214445522
#214442692
C61×#13092
#1053161