Finite Group Databasebeta

← #29600332

Subgroups — #29600332

Order 15616.

458 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 79 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#21447484C212yesmaximal, complemented
2#21443982C214yesmaximal, complemented
3C61×#3151C211yesmaximal, characteristic
4#16839736—22no—
5#16839557—22no—
6#16839555C2214yescomplemented
7C2×AGL(1,ℤ/8)×C61—22no—
8C61×#532C2211yescharacteristic
9C2×#127×C61C411yescharacteristic, complemented
10C61×#496—22no—
11C61×#493C2212yes—
12C61×#492C412yescomplemented
13C4×M16×C61C411yescharacteristic, complemented
14#14519736—42no—
15#14519692—24no—
16#14519692—42no—
17C2×#41×C61D412yes—
18C22×D4×C61—21no—
19C22×D4×C61—41no—
20AGL(1,ℤ/8)×C61—44no—
21AGL(1,ℤ/8)×C61—44no—
22C2×SD16×C61—24no—
23C2×D8×C61—24no—
24C2×M16×C61—21no—
25C4⋊Q8×C61C2×C411yescharacteristic
26#127×C61—21no—
27#127×C61—41no—
28#127×C61C2×C412yes—
29#127×C61C2311yescharacteristic
30#124×C61D412yes—
31#121×C61—41no—
32C4×D4×C61—41no—
33C2×C42×C61C2×C411yescharacteristic
34#97×C61—21no—
35C4×C8×C61C2×C412yes—
36C24×C61—41no—
37#41×C61—42no—
38#41×C61—42no—
39C2×Q8×C61C22⋊C412yes—
40C2×D4×C61—22no—
41C2×D4×C61—22no—
42C2×D4×C61—42no—
43C2×D4×C61—42no—
44C2×D4×C61—41no—
45C2×D4×C61—42no—
46C2×D4×C61—42no—
47C2×D4×C61—41no—
48C2×D4×C61—84no—
49C2×D4×C61C2×D412yes—
50C22×C4×C61—21no—
51C22×C4×C61—22no—
52C22×C4×C61C22⋊C411yescharacteristic
53SD16×C61—48no—
54D8×C61—48no—
55M16×C61—24no—
56C2×C8×C61—22no—
57C4⋊C4×C61—41no—
58C22⋊C4×C61—42no—
59C42×C61—21no—
60C42×C61C22⋊C411yescharacteristic
61C42×C61C22×C411yescharacteristic
62C22⋊Dic61—44no—
63C4⋊Dic61—82no—
64C4×Dic61—82no—
65C23×C61—22no—
66C23×C61—42no—
67C23×C61—81no—
68C23×C61—81no—
69C23×C61—84no—
70C23×C61#9911yescharacteristic
71Q8×C61—42no—
72D4×C61—42no—
73D4×C61—44no—
74D4×C61—82no—
75D4×C61—84no—
76D4×C61—82no—
77D4×C61—84no—
78D4×C61—82no—
79C2×C4×C61—21no—
80C2×C4×C61—22no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61#9911yescharacteristic
87C2×C4×C61C2×C22⋊C411yescharacteristic
88C2×Dic61—84no—
89C8×C61—24no—
90#13585—611nomaximal, Sylow
91C22×C61—21no—
92C22×C61—41no—
93C22×C61—42no—
94C22×C61—82no—
95C22×C61—84no—
96C22×C61—82no—
97C22×C61—162no—
98C22×C61—162no—
99C22×C61C2×#9911yescharacteristic
100C4×C61—22no—
101C4×C61—21no—
102C4×C61—41no—
103C4×C61—42no—
104C4×C61#35312yes—
105Dic61—164no—
106#3151D6111yescharacteristic
107#2126—612no—
108#1408—614no—
109C2×C61—21no—
110C2×C61—41no—
111C2×C61—82no—
112C2×C61—162no—
113C2×C61C2×#35311yescharacteristic
114C2×AGL(1,ℤ/8)—22no—
115#532C2×D6111yescharacteristic
116C2×#127Dic6111yescharacteristic, complemented
117#496—22no—
118#493C2×D6112yes—
119#492Dic6112yescomplemented
120#410—1222no—
121C4×M16Dic6111yescharacteristic, complemented
122#354—1222no—
123#353—614no—
124C61#1358511yescharacteristic, complemented, Sylow
125C2×#41C22⋊D6112yes—
126C22×D4—21no—
127C22×D4—41no—
128AGL(1,ℤ/8)—44no—
129AGL(1,ℤ/8)—44no—
130C2×SD16—24no—
131C2×D8—24no—
132C2×M16—21no—
133C4⋊Q8C2×Dic6111yescharacteristic
134#127—21no—
135#127—41no—
136#127C2×Dic6112yes—
137#127C22×D6111yescharacteristic
138#124C22⋊D6112yes—
139#121—41no—
140C4×D4—41no—
141#117—2442no—
142C2×C42C2×Dic6111yescharacteristic
143#99—1224no—
144#99—2442no—
145#97—21no—
146C4×C8C2×Dic6112yes—
147C24—41no—
148#41—42no—
149#41—42no—
150C2×Q8C22⋊Dic6112yes—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—41no—
159C2×D4—84no—
160C2×D4C2×C22⋊D6112yes—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4C22⋊Dic6111yescharacteristic
164SD16—48no—
165D8—48no—
166M16—24no—
167C2×C8—22no—
168C4⋊C4—41no—
169C4⋊C4—4882no—
170C22⋊C4—42no—
171C22⋊C4—2444no—
172C42—21no—
173C42—4882no—
174C42C22⋊Dic6111yescharacteristic
175C42C22×Dic6111yescharacteristic
176C23—22no—
177C23—42no—
178C23—81no—
179C23—81no—
180C23—84no—
181C23#1451969211yescharacteristic
182Q8—42no—
183D4—42no—
184D4—44no—
185D4—82no—
186D4—84no—
187D4—82no—
188D4—84no—
189D4—82no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—4884no—
198C2×C4#1451969211yescharacteristic
199C2×C4C2×C22⋊Dic6111yescharacteristic
200C8—24no—
201C22—21no—
202C22—41no—
203C22—42no—
204C22—82no—
205C22—84no—
206C22—82no—
207C22—162no—
208C22—162no—
209C22C2×#1451969211yescharacteristic
210C4—22no—
211C4—21no—
212C4—41no—
213C4—42no—
214C4—9764no—
215C4#1683955512yes—
216C2—21no—
217C2—41no—
218C2—82no—
219C2—162no—
220C2C2×#1683955511yescharacteristic
subgroupG/H
centerC2C2×#16839555
commutatorC42×C61C22×C4
Frattini#127C22×D61
Fitting, prime radicalC61×#3151C2
socleC2×C61C2×#353
cosocle#127×C61C23
hypercenter#3151D61
hypocenterC61#13585
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. #21447484C2semidirect
2. #21443982C2semidirect
6. #16839555C22semidirect
9. C2×#127×C61C4semidirect
12. C61×#492C4semidirect
13. C4×M16×C61C4semidirect
116. C2×#127Dic61semidirect
119. #492Dic61semidirect
121. C4×M16Dic61semidirect
124. C61#13585semidirect

Maximal subgroups

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

subgroupindex
#214474842
#214439822
C61×#31512
#1358561