Finite Group Databasebeta

← #30594061

Subgroups — #30594061

Order 15616.

286 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 42 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#21460740C211yesmaximal, characteristic
2#21460682C211yesmaximal, characteristic
3#21446826C211yesmaximal, characteristic
4#16840704—21no—
5C22⋊#11758187—21no—
6C2×#14519753—21no—
7C2×#14519701—21no—
8C2×#14519689C2211yescharacteristic
9#16839525C411yescharacteristic, complemented
10C22×Q16×C61C411yescharacteristic, complemented
11C2×C4⋊#6923—41no—
12C2×C22⋊#6923—41no—
13C2×#11758187—41no—
14(C2×#6923)⋊C4—22no—
15C22×C4×D61D411yescharacteristic
16C22×#92418D411yescharacteristic
17#14519753—42no—
18#14519753—42no—
19C2×C4⋊Dic61—21no—
20C2×#11758196—41no—
21C2×#11758198—41no—
22#14519701—42no—
23#14519689—41no—
24C22×Q8×C61—21no—
25C2×Q16×C61—41no—
26C2×Q16×C61—42no—
27C22×C8×C61C2×C411yescharacteristic
28C23×D61D811yescharacteristic
29C22×#6923—42no—
30C2×Q8×C61—22no—
31C2×Q8×C61—41no—
32C2×Q8×C61—41no—
33C2×Q8×C61—41no—
34C2×Q8×C61—41no—
35C2×Q8×C61—41no—
36C22×C4×C61—41no—
37C22×C4×C61C22⋊C411yescharacteristic
38C22×Dic61—41no—
39C22×Dic61SD1611yescharacteristic
40C2×C4×D61—21no—
41C2×C4×D61—21no—
42C2×C4×D61—41no—
43C2×C4×D61—41no—
44C22⋊#6923—81no—
45C22⋊#6923—81no—
46C4⋊#6923—81no—
47C4⋊#6923—81no—
48Q16×C61—42no—
49Q16×C61—42no—
50C2×C8×C61—21no—
51C2×C8×C61—41no—
52#11758198—44no—
53C4⋊Dic61—22no—
54C4⋊Dic61—41no—
55#11758196—44no—
56C2×#92418—21no—
57C2×#92418—41no—
58#11758187—81no—
59C23×C61#10211yescharacteristic
60C22×D61—21no—
61C22×D61—21no—
62C22×D61—41no—
63C22×D61—41no—
64C22×D61SD3211yescharacteristic
65C22×D61Q3211yescharacteristic
66C2×#6923—82no—
67C2×#6923—82no—
68C2×#6923—82no—
69Q8×C61—42no—
70Q8×C61—42no—
71Q8×C61—81no—
72Q8×C61—81no—
73C2×C4×C61—21no—
74C2×C4×C61—21no—
75C2×C4×C61—42no—
76C2×C4×C61—81no—
77C2×C4×C61—81no—
78C2×C4×C61#9911yescharacteristic
79C2×C4×C61#10411yescharacteristic
80C2×Dic61—21no—
81C2×Dic61—41no—
82C2×Dic61—42no—
83C2×Dic61—81no—
84C2×Dic61—81no—
85C4×D61—41no—
86C4×D61—41no—
87C4×D61—81no—
88C8×C61—41no—
89#92418—41no—
90#7078—611nomaximal, Sylow
91C22×C61—21no—
92C22×C61—21no—
93C22×C61#32711yescharacteristic
94C22×C61#35811yescharacteristic
95C22×C61#36011yescharacteristic
96C2×D61—21no—
97C2×D61—21no—
98C2×D61—41no—
99C2×D61—41no—
100C2×D61—41no—
101C2×D61—41no—
102C2×D61—81no—
103#6923—162no—
104C4×C61—21no—
105C4×C61—82no—
106C4×C61AΣL(1,𝔽16)11yescharacteristic
107C4×C61#35311yescharacteristic
108Dic61—41no—
109Dic61—82no—
110#2001—611no—
111#1314—611no—
112#1289—611no—
113C2×C61—21no—
114C2×C61—21no—
115C2×C61#134011yescharacteristic
116C2×C61#134311yescharacteristic
117C2×C61#134811yescharacteristic
118D61—41no—
119D61—41no—
120C22×Q16#692311yescharacteristic, complemented
121C2×C22⋊Q8—1221no—
122C2×#103—1221no—
123C2×C22⋊C8—611no—
124#380—1221no—
125C22⋊C16—1221no—
126#340—611no—
127C61#707811yescharacteristic, complemented, Sylow
128C22×Q8—21no—
129C23×C4—611no—
130C2×Q16—41no—
131C2×Q16—42no—
132C22×C8—611no—
133C22×C8C2×#692311yescharacteristic
134C22⋊Q8—2442no—
135C22⋊Q8—2442no—
136C2×C4⋊C4—1221no—
137C2×C4⋊C4—2441no—
138C2×C4⋊C4—2441no—
139C2×C22⋊C4—2441no—
140C2×C22⋊C4—2441no—
141C2×C16—2441no—
142#103—2442no—
143C22⋊C8—2441no—
144#95—1222no—
145C24—611no—
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C22×C4—41no—
153C22×C4—611no—
154C22×C4—1221no—
155C22×C4—1221no—
156C22×C4—2441no—
157C22×C4—2441no—
158C22×C4—2442no—
159C22×C4—2441no—
160C22×C4C22⋊#692311yescharacteristic
161Q16—42no—
162Q16—42no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—1221no—
166C2×C8—2441no—
167C4⋊C4—1222no—
168C4⋊C4—4881no—
169C4⋊C4—4881no—
170C4⋊C4—2441no—
171C4⋊C4—2444no—
172C22⋊C4—4881no—
173C22⋊C4—4881no—
174C22⋊C4—2444no—
175C16—4881no—
176C23—611no—
177C23—611no—
178C23—1221no—
179C23—1221no—
180C23—2441no—
181C23—2441no—
182C23#1451983911yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—81no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—1221no—
193C2×C4—4881no—
194C2×C4—4882no—
195C2×C4—4882no—
196C2×C4—4882no—
197C2×C4—4881no—
198C2×C4—4881no—
199C2×C4—2442no—
200C2×C4—2441no—
201C2×C4—2441no—
202C2×C4—2441no—
203C2×C4#1451983511yescharacteristic
204C2×C4#1451984111yescharacteristic
205C8—41no—
206C8—2441no—
207C22—21no—
208C22—21no—
209C22—1221no—
210C22—1221no—
211C22—4881no—
212C22—2441no—
213C22—2441no—
214C22—2441no—
215C22—2441no—
216C22#1684061311yescharacteristic
217C22#1684067511yescharacteristic
218C22#1684067911yescharacteristic
219C4—21no—
220C4—82no—
221C4—9762no—
222C4—4882no—
223C4—2441no—
224C4#1684066211yescharacteristic
225C4#1684066611yescharacteristic
226C2—21no—
227C2—21no—
228C2—2441no—
229C2—2441no—
230C2#2146079111yescharacteristic
231C2#2146079711yescharacteristic
232C2#2146080711yescharacteristic
subgroupG/H
centerC22#16840613
commutatorC22×C8×C61C2×C4
FrattiniC22×C8C2×#6923
FittingC22×Q16×C61C4
socleC22×C61#327
prime radicalC23×D61D8
cosocleC2×#14519689C22
hypercenterC22×Q16#6923
hypocenterC61#7078
LCSG ▷ C22×C8×C61 ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
9. #16839525C4semidirect
10. C22×Q16×C61C4semidirect
120. C22×Q16#6923semidirect
127. C61#7078semidirect

Maximal subgroups

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

subgroupindex
#214607402
#214606822
#214468262
#707861