Finite Group Databasebeta

← #29567748

Subgroups — #29567748

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#21446203C211yesmaximal, characteristic
2#21444454C211yesmaximal, characteristic
3C61×#2132C211yesmaximal, characteristic
4#16840529C411yescharacteristic
5#16839819—21no—
6#16839779—21no—
7#16839736C411yescharacteristic
8#16839522—21no—
9#16839502—21no—
10#16839502C412yescomplemented
11C61×#532C2211yescharacteristic
12C61×#413—21no—
13C61×#356C412yescomplemented
14C61×#355—21no—
15#14519784—42no—
16C2×#11758194—21no—
17C2×#11758194—41no—
18#14519736—21no—
19C4⋊#92418—41no—
20#14519704—42no—
21#14519692—41no—
22C22⋊#92418—41no—
23#14519688—22no—
24C2×#41×C61C2×C411yescharacteristic
25C2×#41×C61D411yescharacteristic
26C2×M16×C61—41no—
27C4⋊Q8×C61C2×C411yescharacteristic
28#127×C61Q811yescharacteristic
29#124×C61C2×C411yescharacteristic
30#124×C61D411yescharacteristic
31#121×C61—41no—
32C4×D4×C61—41no—
33C2×C42×C61D411yescharacteristic
34#101×C61—22no—
35#100×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—42no—
52M16×C61—81no—
53C2×C8×C61—81no—
54C4⋊C4×C61—41no—
55C22⋊C4×C61—41no—
56C22⋊C4×C61—41no—
57C42×C61—21no—
58C42×C61C22⋊C411yescharacteristic
59C42×C61C4⋊C411yescharacteristic
60C22⋊Dic61—41no—
61C4⋊Dic61—41no—
62C4×Dic61—41no—
63#11758194—22no—
64#11758194—42no—
65#11758194—41no—
66C2×#92418—41no—
67C2×#92418—42no—
68C2×#92418—42no—
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#9511yescharacteristic
91C2×C4×C61#9911yescharacteristic
92C2×Dic61—81no—
93C8×C61—82no—
94#92418—42no—
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×C61AΣL(1,𝔽16)11yescharacteristic
110C4×C61#35211yescharacteristic
111Dic61—161no—
112#2132D6111yescharacteristic
113#1884—611no—
114#1536—611no—
115C2×C61—21no—
116C2×C61—41no—
117C2×C61—81no—
118C2×C61—81no—
119C2×C61#139011yescharacteristic
120#568—611no—
121#532C2×D6111yescharacteristic
122#432—1221no—
123#421—1221no—
124#413—21no—
125#410—611no—
126#356Dic6112yescomplemented
127#355—21no—
128#339—1221no—
129#329—612no—
130#329—1221no—
131C61#748411yescharacteristic, complemented, Sylow
132C2×#41C4×D6111yescharacteristic
133C2×#41C22⋊D6111yescharacteristic
134#131—2442no—
135C2×M16—41no—
136C2×M16—1221no—
137C2×M16—2441no—
138C4⋊Q8C2×Dic6111yescharacteristic
139#127Dic12211yescharacteristic
140#124C4×D6111yescharacteristic
141#124C22⋊D6111yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—1221no—
145C2×C42D24411yescharacteristic
146C4⋊C8—2441no—
147#104—2442no—
148#101—22no—
149#100—41no—
150#99—2441no—
151C22⋊C8—2441no—
152#97—1222no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8C4×Dic6111yescharacteristic
158C2×Q8C22⋊Dic6111yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4#1175819611yescharacteristic
163C2×D4#1175819611yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4#1175819811yescharacteristic
168M16—42no—
169M16—81no—
170M16—1222no—
171M16—2442no—
172M16—2441no—
173C2×C8—81no—
174C2×C8—2442no—
175C2×C8—2441no—
176C2×C8—2442no—
177C4⋊C4—41no—
178C4⋊C4—2441no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—2441no—
182C42—21no—
183C42—2441no—
184C42C4⋊Dic6111yescharacteristic
185C42#1175819811yescharacteristic
186C23—21no—
187C23—21no—
188C23#1451969111yescharacteristic
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—4881no—
208C2×C4#1451968411yescharacteristic
209C2×C4#1451969111yescharacteristic
210C8—82no—
211C8—4882no—
212C8—2442no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22#1683952811yescharacteristic
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—9761no—
226C4#1683954811yescharacteristic
227C4#1683955111yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2#2144393611yescharacteristic
subgroupG/H
centerC2#21443936
commutatorC2×Q8×C61C42
Frattini#532C2×D61
FittingC61×#2132C2
socleC2×C61#1390
prime radical, cosocleC61×#532C22
hypercenter#2132D61
hypocenterC61#7484
LCSG ▷ C2×Q8×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
10. #16839502C4semidirect
13. C61×#356C4semidirect
126. #356Dic61semidirect
131. C61#7484semidirect

Maximal subgroups

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

subgroupindex
#214462032
#214444542
C61×#21322
#748461