Finite Group Databasebeta

← #29567749

Subgroups — #29567749

Order 15616.

266 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 59 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#21447504C211yesmaximal, characteristic
2#21444454C211yesmaximal, characteristic
3C61×#1884C211yesmaximal, characteristic
4#16839819—21no—
5#16839746—21no—
6#16839561C412yescomplemented
7#16839559—21no—
8#16839502—21no—
9#16839502C412yescomplemented
10C61×#568C411yescharacteristic
11C61×#532C2211yescharacteristic
12C61×#421—21no—
13C61×#410C411yescharacteristic
14C61×#339—21no—
15C2×#11758194—41no—
16C4⋊#92418—41no—
17#14519696—22no—
18#14519694—41no—
19C22⋊#92418—41no—
20#14519688—22no—
21C2×#41×C61C2×C411yescharacteristic
22C2×#41×C61D411yescharacteristic
23#131×C61—42no—
24C2×M16×C61—21no—
25C2×M16×C61—41no—
26C4⋊Q8×C61C2×C411yescharacteristic
27#127×C61Q811yescharacteristic
28#124×C61C2×C411yescharacteristic
29#124×C61D411yescharacteristic
30#121×C61—41no—
31C4×D4×C61—41no—
32#117×C61—21no—
33C2×C42×C61D411yescharacteristic
34#104×C61—42no—
35#99×C61—41no—
36#41×C61—41no—
37#41×C61—41no—
38#41×C61—41no—
39#41×C61—41no—
40C2×Q8×C61C4211yescharacteristic
41C2×Q8×C61C22⋊C411yescharacteristic
42C2×D4×C61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×D4×C61C4⋊C411yescharacteristic
46C2×D4×C61C4⋊C411yescharacteristic
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C411yescharacteristic
51M16×C61—22no—
52M16×C61—42no—
53M16×C61—41no—
54C2×C8×C61—41no—
55C2×C8×C61—42no—
56C4⋊C4×C61—41no—
57C4⋊C4×C61—41no—
58C22⋊C4×C61—41no—
59C22⋊C4×C61—41no—
60C22⋊C4×C61—41no—
61C42×C61—21no—
62C42×C61—41no—
63C42×C61C22⋊C411yescharacteristic
64C42×C61C4⋊C411yescharacteristic
65#11758194—42no—
66#11758194—81no—
67C2×#92418—42no—
68C2×#92418—81no—
69C23×C61—21no—
70C23×C61—21no—
71C23×C61#9911yescharacteristic
72Q8×C61—41no—
73Q8×C61—41no—
74D4×C61—41no—
75D4×C61—41no—
76D4×C61—42no—
77D4×C61—81no—
78D4×C61—82no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—41no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—81no—
90C2×C4×C61—81no—
91C2×C4×C61#9511yescharacteristic
92C2×C4×C61#9911yescharacteristic
93C8×C61—42no—
94#92418—82no—
95#92418—82no—
96#7484—611nomaximal, Sylow
97C22×C61—21no—
98C22×C61—41no—
99C22×C61—41no—
100C22×C61—41no—
101C22×C61—81no—
102C22×C61—81no—
103C22×C61#34211yescharacteristic
104C4×C61—21no—
105C4×C61—41no—
106C4×C61—41no—
107C4×C61—41no—
108C4×C61—41no—
109C4×C61—161no—
110C4×C61AΣL(1,𝔽16)11yescharacteristic
111C4×C61#35211yescharacteristic
112#2132—611no—
113#1884D6111yescharacteristic
114#1536—611no—
115C2×C61—21no—
116C2×C61—41no—
117C2×C61—81no—
118C2×C61—81no—
119C2×C61#139011yescharacteristic
120#568Dic6111yescharacteristic
121#532C2×D6111yescharacteristic
122#432—1221no—
123#421—21no—
124#413—1221no—
125#410Dic6111yescharacteristic
126#356—612no—
127#355—1221no—
128#339—21no—
129#329—612no—
130#329—1221no—
131C61#748411yescharacteristic, complemented, Sylow
132C2×#41C2×Dic6111yescharacteristic
133C2×#41C22⋊D6111yescharacteristic
134#131—42no—
135C2×M16—21no—
136C2×M16—41no—
137C2×M16—2441no—
138C4⋊Q8C4×D6111yescharacteristic
139#127Dic12211yescharacteristic
140#124C4×D6111yescharacteristic
141#124D24411yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—21no—
145C2×C42C22⋊D6111yescharacteristic
146C4⋊C8—2441no—
147#104—42no—
148#101—1222no—
149#100—2441no—
150#99—41no—
151C22⋊C8—2441no—
152#97—1222no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8C4×Dic6111yescharacteristic
158C2×Q8#1175819811yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4#1175819611yescharacteristic
163C2×D4C4⋊Dic6111yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C22⋊Dic6111yescharacteristic
168M16—22no—
169M16—42no—
170M16—41no—
171M16—4881no—
172M16—2442no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—4881no—
176C2×C8—2442no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—21no—
183C42—41no—
184C42#1175819611yescharacteristic
185C42#1175819811yescharacteristic
186C23—21no—
187C23—21no—
188C23#1451969211yescharacteristic
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—42no—
194D4—81no—
195D4—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4#1451968411yescharacteristic
209C2×C4#1451969211yescharacteristic
210C8—42no—
211C8—4882no—
212C8—4882no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22#1683952911yescharacteristic
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—161no—
226C4#1683954911yescharacteristic
227C4#1683955211yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2#2144393711yescharacteristic
subgroupG/H
centerC2#21443937
commutatorC2×Q8×C61C42
Frattini#532C2×D61
FittingC61×#1884C2
socleC2×C61#1390
prime radical, cosocleC61×#532C22
hypercenter#1884D61
hypocenterC61#7484
LCSG ▷ C2×Q8×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1884

Complements

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

NH ≅ G/Ntype
6. #16839561C4semidirect
9. #16839502C4semidirect
131. C61#7484semidirect

Maximal subgroups

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

subgroupindex
#214475042
#214444542
C61×#18842
#748461