Finite Group Databasebeta

← #29593345

Subgroups — #29593345

Order 15616.

324 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 55 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#21456643C211yesmaximal, characteristic
2#21448039C212yesmaximal, complemented
3#21447335C211yesmaximal, characteristic
4#21443814C212yesmaximal, complemented
5C61×#1320C211yesmaximal, characteristic
6C2×#14519756—21no—
7#16840093—22no—
8#16840020—21no—
9#16840020C2212yes—
10#16839920—22no—
11#16839811C2211yescharacteristic
12#16839810—21no—
13#16839726—21no—
14#16839585—22no—
15#16839579—22no—
16#16839520C2211yescharacteristic
17C61×#407C2211yescharacteristic
18C61×#349C2212yes—
19C22×Dic122—21no—
20C2×Dic244—21no—
21C2×Dic244D412yes—
22C2×#11758194—21no—
23C2×#11758194—21no—
24#14519756—22no—
25#14519756—22no—
26#14519756—41no—
27#14519756—41no—
28C2×C4⋊Dic61—21no—
29C2×#11758196—41no—
30C2×C22⋊Dic61—41no—
31#14519713—22no—
32#14519712—42no—
33#14519709—21no—
34#14519709D412yes—
35#14519702—22no—
36#14519702—22no—
37C22⋊#92418—42no—
38C23×C4×C61D411yescharacteristic
39C2×M16×C61D411yescharacteristic
40C2×M16×C61C2311yescharacteristic
41M32×C61—22no—
42C22⋊C8×C61—22no—
43C24×C61SD1611yescharacteristic
44C22×C4×C61—21no—
45C22×C4×C61—21no—
46C22×C4×C61—41no—
47C22×C4×C61—41no—
48C22×C4×C61SD1611yescharacteristic
49C22×C4×C61C2×D411yescharacteristic
50C22×Dic61—41no—
51C22×Dic61—41no—
52C2×Dic122—22no—
53C2×Dic122—41no—
54C2×Dic122—81no—
55C2×Dic122—81no—
56M16×C61—41no—
57M16×C61—41no—
58C2×C8×C61—21no—
59C2×C8×C61C2×D412yes—
60C22⋊Dic61—42no—
61C22⋊Dic61—42no—
62C4⋊Dic61—22no—
63C4⋊Dic61—41no—
64#11758196—42no—
65#11758196—42no—
66#11758194—41no—
67#11758194—41no—
68#11758194—41no—
69#11758194—41no—
70C2×#92418—22no—
71C2×#92418—41no—
72Dic244—42no—
73Dic244—81no—
74C16×C61—42no—
75C23×C61—21no—
76C23×C61—41no—
77C23×C61—42no—
78C23×C61C2×SD1611yescharacteristic
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—41no—
83C2×C4×C61—41no—
84C2×C4×C61—41no—
85C2×C4×C61—81no—
86C2×C4×C61#12011yescharacteristic
87C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
88C2×Dic61—42no—
89C2×Dic61—42no—
90C2×Dic61—82no—
91C2×Dic61—82no—
92Dic122—42no—
93Dic122—42no—
94Dic122—82no—
95C8×C61—22no—
96C8×C61—41no—
97#92418—42no—
98#92418—81no—
99#12299—611nomaximal, Sylow
100C22×C61—22no—
101C22×C61—21no—
102C22×C61—41no—
103C22×C61—41no—
104C22×C61—42no—
105C22×C61—41no—
106C22×C61—81no—
107C22×C61#45011yescharacteristic
108C4×C61—21no—
109C4×C61—41no—
110C4×C61#45311yescharacteristic
111C4×C61#45811yescharacteristic
112Dic61—82no—
113Dic61—82no—
114#3195—611no—
115#2239—612no—
116#2091—611no—
117#1342—612no—
118#1320D6111yescharacteristic
119C2×C61—21no—
120C2×C61—42no—
121C2×C61—41no—
122C2×C61#161311yescharacteristic
123C2×C22⋊Q8—1221no—
124#483—1222no—
125#470—612no—
126#470—1221no—
127#451—1222no—
128#430—611no—
129#430—1221no—
130#407—1221no—
131#407C2×D6111yescharacteristic
132#365—1222no—
133#362—1222no—
134#349C2×D6112yes—
135#338—611no—
136C61#1229911yescharacteristic, complemented, Sylow
137C22×Q8—1221no—
138C23×C4D24411yescharacteristic
139C2×Q16—612no—
140C2×Q16—1221no—
141C2×M16—1221no—
142C2×M16—1221no—
143C2×M16D24411yescharacteristic
144C2×M16C22×D6111yescharacteristic
145C22⋊Q8—1222no—
146C22⋊Q8—1222no—
147C22⋊Q8—2441no—
148C22⋊Q8—2441no—
149C2×C4⋊C4—1221no—
150C2×C4⋊C4—2441no—
151C2×C22⋊C4—2441no—
152M32—22no—
153#108—1222no—
154#108—2442no—
155#106—612no—
156#106—1221no—
157#103—1222no—
158#103—1222no—
159C22⋊C8—22no—
160C22⋊C8—2442no—
161C24#1175819011yescharacteristic
162C2×Q8—1222no—
163C2×Q8—4881no—
164C2×Q8—4881no—
165C2×Q8—2441no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—2441no—
171C22×C4—2441no—
172C22×C4#1175819011yescharacteristic
173C22×C4C2×D24411yescharacteristic
174Q16—4881no—
175Q16—2442no—
176M16—41no—
177M16—41no—
178M16—2441no—
179M16—2441no—
180M16—2441no—
181M16—2441no—
182C2×C8—21no—
183C2×C8—1222no—
184C2×C8—2441no—
185C2×C8D4×D6112yes—
186C4⋊C4—1222no—
187C4⋊C4—2441no—
188C4⋊C4—2442no—
189C4⋊C4—2442no—
190C22⋊C4—2442no—
191C22⋊C4—2442no—
192C16—42no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1175819011yescharacteristic
197Q8—4882no—
198Q8—2442no—
199Q8—2442no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—4882no—
208C2×C4—4882no—
209C2×C4—2442no—
210C2×C4—2442no—
211C2×C4#1451974711yescharacteristic
212C2×C4#1451980511yescharacteristic
213C8—22no—
214C8—41no—
215C8—4881no—
216C8—2442no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1683991511yescharacteristic
225C4—21no—
226C4—41no—
227C4—4882no—
228C4—4882no—
229C4#1683993211yescharacteristic
230C4#1683995311yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2144484111yescharacteristic
subgroupG/H
centerC2#21444841
commutator, cosocleC2×M16×C61C23
FrattiniC2×M16C22×D61
FittingC61×#1320C2
socleC2×C61#1613
prime radicalC24×C61SD16
hypercenter#1320D61
hypocenterC61#12299
LCSG ▷ C2×M16×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1320

Complements

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

NH ≅ G/Ntype
2. #21448039C2semidirect
4. #21443814C2semidirect
136. C61#12299semidirect

Maximal subgroups

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

subgroupindex
#214566432
#214480392
#214473352
#214438142
C61×#13202
#1229961