Finite Group Databasebeta

← #29594251

Subgroups — #29594251

Order 15616.

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

normal subgroups only

This group has 234 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#21448068C211yesmaximal, characteristic, complemented
2#21447895C211yesmaximal, characteristic, complemented
3#21447656C211yesmaximal, characteristic, complemented
4#21443894C211yesmaximal, characteristic, complemented
5#21443856C211yesmaximal, characteristic, complemented
6#21443855C211yesmaximal, characteristic, complemented
7C61×#2213C211yesmaximal, characteristic, complemented
8#16840197—21no—
9C2×#14519718—21no—
10C2×#14519718—21no—
11#16840179—21no—
12#16840139C2211yescharacteristic
13#16839583C2211yescharacteristic
14#16839583C2211yescharacteristic
15#16839574—21no—
16#16839546—21no—
17C4×#11758185C2211yescharacteristic
18C61×#509—21no—
19C61×#493C2211yescharacteristic
20C61×#443C2211yescharacteristic
21C61×#364C2211yescharacteristic
22C61×#361—21no—
23C2×D488—21no—
24C2×D488—21no—
25#14519774—21no—
26#14519720—41no—
27#14519718—42no—
28#14519718—41no—
29#14519718—41no—
30#14519718—41no—
31#14519717—21no—
32#14519717—21no—
33#14519717—41no—
34C2×#11758185—41no—
35C2×#11758185D411yescharacteristic
36C2×#11758185D411yescharacteristic
37#135×C61D411yescharacteristic
38C2×D8×C61—21no—
39C2×D8×C61—21no—
40#131×C61—21no—
41#127×C61—21no—
42SD32×C61—41no—
43D16×C61—41no—
44M32×C61—21no—
45#108×C61D411yescharacteristic
46#104×C61—21no—
47C4×C8×C61C2311yescharacteristic
48#41×C61—21no—
49C2×D4×C61—41no—
50C2×D4×C61—81no—
51C2×D244—41no—
52C2×D244—81no—
53Q16×C61D811yescharacteristic
54SD16×C61—21no—
55D8×C61—41no—
56D8×C61—41no—
57D8×C61—41no—
58D8×C61—41no—
59D8×C61D811yescharacteristic
60M16×C61—21no—
61M16×C61—41no—
62C2×C8×C61—41no—
63C2×C8×C61C2×D411yescharacteristic
64C2×C8×C61C2×D411yescharacteristic
65C42×C61#4111yescharacteristic
66D488—41no—
67D488—41no—
68D488—41no—
69D488—41no—
70C16×C61—41no—
71#11758185—21no—
72#11758185—41no—
73#11758185—41no—
74#11758185D811yescharacteristic
75#11758185D811yescharacteristic
76C23×C61—81no—
77C22×D61—81no—
78Q8×C61—21no—
79D4×C61—21no—
80D4×C61—41no—
81D4×C61—81no—
82D4×C61—81no—
83D4×C61—162no—
84C2×C4×C61—21no—
85C2×C4×C61—41no—
86C2×C4×C61#12111yescharacteristic
87D244—81no—
88D244—81no—
89D244—162no—
90C8×C61—41no—
91C8×C61C2×D811yescharacteristic
92C8×C61C2×D811yescharacteristic
93C8×C61#13511yescharacteristic
94C8×C61AGL(1,ℤ/8)11yescharacteristic
95#12436—611nomaximal, Sylow
96C22×C61—41no—
97C22×C61—162no—
98C22×C61—161no—
99C22×C61#47211yescharacteristic
100C2×D61—162no—
101C2×D61—161no—
102C4×C61—22no—
103C4×C61—41no—
104C4×C61#45911yescharacteristic
105C4×C61#46611yescharacteristic
106#2246—611no—
107#2213—611no—
108#2213D6111yescharacteristic, complemented
109#2170—611no—
110#1373—611no—
111#1357—611no—
112#1357—611no—
113C2×C61—21no—
114C2×C61—81no—
115C2×C61—321no—
116C2×C61#166711yescharacteristic
117D61—321no—
118#509—21no—
119#509—1221no—
120C2×D16—1221no—
121C2×D16—1221no—
122#504—1221no—
123#493—611no—
124#493C2×D6111yescharacteristic
125#443C2×D6111yescharacteristic
126#364—611no—
127#364—611no—
128#364C2×D6111yescharacteristic
129#361—21no—
130#361—1221no—
131#350—1221no—
132C4×C16—611no—
133C61#1243611yescharacteristic, complemented, Sylow
134#135C22⋊D6111yescharacteristic
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—1221no—
138C2×D8—1221no—
139#131—21no—
140#127—21no—
141#127—1221no—
142SD32—41no—
143SD32—2441no—
144D16—41no—
145D16—2441no—
146D16—2441no—
147D16—2442no—
148D16—2441no—
149M32—21no—
150M32—1221no—
151M32—1221no—
152M32—2441no—
153C2×C16—611no—
154C2×C16—611no—
155C2×C16—2441no—
156#108C22⋊D6111yescharacteristic
157#104—21no—
158C4×C8C22×D6111yescharacteristic
159#41—21no—
160C2×D4—41no—
161C2×D4—81no—
162C2×D4—4881no—
163C2×D4—2441no—
164Q16#1175819911yescharacteristic
165SD16—21no—
166D8—41no—
167D8—41no—
168D8—41no—
169D8—41no—
170D8—2441no—
171D8—2441no—
172D8—2441no—
173D8—2441no—
174D8#1175819911yescharacteristic
175M16—21no—
176M16—41no—
177C2×C8—41no—
178C2×C8D4×D6111yescharacteristic
179C2×C8C2×C22⋊D6111yescharacteristic
180C42#1175822411yescharacteristic
181C16—41no—
182C16—611no—
183C16—611no—
184C16—1221no—
185C16—2441no—
186C16—2441no—
187C23—81no—
188C23—4881no—
189Q8—21no—
190D4—21no—
191D4—41no—
192D4—81no—
193D4—81no—
194D4—162no—
195D4—9762no—
196D4—4881no—
197D4—4881no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4#1451974911yescharacteristic
201C8—41no—
202C8D8×D6111yescharacteristic
203C8C2×#1175819911yescharacteristic
204C8#1451979711yescharacteristic
205C8#1451980111yescharacteristic
206C22—41no—
207C22—162no—
208C22—161no—
209C22—9762no—
210C22—9761no—
211C22#1684003011yescharacteristic
212C4—22no—
213C4—41no—
214C4#1683995511yescharacteristic
215C4#1683999611yescharacteristic
216C2—21no—
217C2—81no—
218C2—321no—
219C2—19521no—
220C2#2144517211yescharacteristic
subgroupG/H
centerC2#21445172
commutator, cosocleC4×C8×C61C23
FrattiniC4×C8C22×D61
FittingC61×#2213C2
socleC2×C61#1667
hypercenter#2213D61
hypocenterC61#12436
LCSG ▷ C4×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2213

Complements

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

NH ≅ G/Ntype
1. #21448068C2semidirect
2. #21447895C2semidirect
3. #21447656C2semidirect
4. #21443894C2semidirect
5. #21443856C2semidirect
6. #21443855C2semidirect
7. C61×#2213C2semidirect
108. #2213D61semidirect
133. C61#12436semidirect

Maximal subgroups

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

subgroupindex
#214480682
#214478952
#214476562
#214438942
#214438562
#214438552
C61×#22132
#1243661