Finite Group Databasebeta

← #29600377

Subgroups — #29600377

Order 15616.

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

normal subgroups only

This group has 318 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#21456221C211yesmaximal, characteristic
2#21447503C212yesmaximal
3#21443992C211yesmaximal, characteristic, complemented
4#21443991C212yesmaximal, complemented
5C61×#1413C211yesmaximal, characteristic, complemented
6#16840313C2211yescharacteristic
7#125×D61C411yescharacteristic
8#16840130C2211yescharacteristic
9#16840128C411yescharacteristic
10#16840127C411yescharacteristic
11#16840118—22no—
12#16839762—22no—
13#16839745—22no—
14#16839714C411yescharacteristic
15#16839561C2212yes—
16#16839558—22no—
17C61×#491C2211yescharacteristic
18C61×#356C2212yes—
19C2×#11758226D412yes—
20C2×#11758189—21no—
21M16×D61—42no—
22#14519774C2×C411yescharacteristic
23#14519769C2×C411yescharacteristic
24#14519768—21no—
25#14519767—41no—
26#14519766D412yes—
27#14519752—41no—
28#14519739—41no—
29C4⋊C4×D61—21no—
30C4⋊C4×D61—41no—
31C42×D61C2×C411yescharacteristic
32#14519701—22no—
33#14519699—22no—
34#14519697—22no—
35#14519696—22no—
36#14519693—42no—
37#14519688C2×C411yescharacteristic
38C8×Dic61—21no—
39C4⋊Q8×C61C2311yescharacteristic
40#125×C61C2×C411yescharacteristic
41#103×C61—22no—
42#101×C61—22no—
43#97×C61C2×C411yescharacteristic
44C2×Q8×C61C2×D412yes—
45#11758226—42no—
46#11758226—42no—
47C2×D244—22no—
48C2×D244—41no—
49C2×D244C22⋊C412yes—
50C2×C4×D61—22no—
51C2×C4×D61—21no—
52C2×C4×D61—41no—
53C2×C4×D61C22⋊C411yescharacteristic
54M16×C61—42no—
55C2×C8×C61—21no—
56C4⋊C4×C61—21no—
57C4⋊C4×C61—41no—
58C4⋊C4×C61—41no—
59C42×C61C22×C411yescharacteristic
60#11758198—42no—
61C4⋊Dic61—21no—
62C4⋊Dic61—41no—
63#11758196—41no—
64#11758196—41no—
65#11758196—41no—
66C4×Dic61—21no—
67C4×Dic61C22⋊C411yescharacteristic
68#11758194—42no—
69C2×#92418—21no—
70#11758189—42no—
71#11758189—82no—
72C8×D61—82no—
73C22×D61—22no—
74C22×D61#9911yescharacteristic
75Q8×C61—42no—
76C2×C4×C61—22no—
77C2×C4×C61—21no—
78C2×C4×C61—41no—
79C2×C4×C61C2×C22⋊C411yescharacteristic
80C2×Dic61—21no—
81C2×Dic61—41no—
82C2×Dic61#9911yescharacteristic
83D244—42no—
84D244—82no—
85D244—82no—
86C4×D61—22no—
87C4×D61—42no—
88C4×D61—42no—
89C4×D61—81no—
90C4×D61—82no—
91C8×C61—41no—
92C8×C61—82no—
93#92418—41no—
94#92418—82no—
95#13600—611nomaximal, Sylow
96C22×C61C2×#9911yescharacteristic
97C2×D61—21no—
98C2×D61—41no—
99C2×D61—42no—
100C2×D61—82no—
101C4×C61—21no—
102C4×C61—41no—
103C4×C61—42no—
104C4×C61—81no—
105Dic61—41no—
106Dic61—81no—
107Dic61#35412yes—
108#3142—611no—
109#2132—612no—
110#1413—611no—
111#1413—612no—
112#1413D6111yescharacteristic, complemented
113C2×C61—21no—
114C2×C61C2×#35411yescharacteristic
115D61—41no—
116D61—82no—
117#532—611no—
118C2×#125—611no—
119#491—611no—
120#491C2×D6111yescharacteristic
121#490—611no—
122#490—611no—
123#488—1222no—
124#417—1222no—
125#413—1222no—
126C4×M16—611no—
127#356—612no—
128#356C2×D6112yes—
129#355—1222no—
130C61#1360011yescharacteristic, complemented, Sylow
131C2×#41—612no—
132C2×M16—1221no—
133C2×M16—2442no—
134C4⋊Q8C22×D6111yescharacteristic
135#127—611no—
136#125—611no—
137#125—1221no—
138#125—2441no—
139#125C4×D6111yescharacteristic
140#124—612no—
141#121—2441no—
142C4×D4—2441no—
143C2×C4⋊C4—1221no—
144C2×C4⋊C4—2441no—
145C2×C42—611no—
146#103—22no—
147#103—1222no—
148#102—1222no—
149#102—1222no—
150#101—22no—
151#101—1222no—
152#100—2442no—
153#97—611no—
154#97C4×D6111yescharacteristic
155C4×C8—1221no—
156#41—2442no—
157#41—2442no—
158C2×Q8D4×D6112yes—
159C2×D4—612no—
160C2×D4—1222no—
161C2×D4—2441no—
162C22×C4—611no—
163C22×C4—1222no—
164C22×C4—1221no—
165C22×C4—2441no—
166M16—42no—
167M16—4882no—
168M16—2442no—
169M16—2442no—
170C2×C8—21no—
171C2×C8—1221no—
172C2×C8—4882no—
173C4⋊C4—21no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—1221no—
177C4⋊C4—2441no—
178C4⋊C4—2441no—
179C4⋊C4—2441no—
180C4⋊C4—2441no—
181C22⋊C4—2442no—
182C42—611no—
183C42—1221no—
184C42C2×C4×D6111yescharacteristic
185C23—611no—
186C23—1222no—
187Q8—42no—
188D4—4882no—
189D4—4882no—
190D4—2442no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—611no—
195C2×C4—1222no—
196C2×C4—1221no—
197C2×C4—4881no—
198C2×C4—4882no—
199C2×C4—2442no—
200C2×C4—2442no—
201C2×C4—2441no—
202C2×C4C22⋊C4×D6111yescharacteristic
203C8—41no—
204C8—82no—
205C8—4882no—
206C8—2441no—
207C22—1221no—
208C22—4882no—
209C22—2441no—
210C22—2442no—
211C22#99×D6111yescharacteristic
212C4—21no—
213C4—41no—
214C4—42no—
215C4—81no—
216C4—612no—
217C4—4881no—
218C4—2441no—
219C2—21no—
220C2—4882no—
221C2—2441no—
222C2#354×D6111yescharacteristic
subgroupG/H
centerC2#354×D61
commutatorC42×C61C22×C4
FrattiniC4⋊Q8C22×D61
FittingC61×#1413C2
socleC2×C61C2×#354
prime radical#16840313C22
cosocleC4⋊Q8×C61C23
hypercenter#1413D61
hypocenterC61#13600
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. #21443992C2semidirect
4. #21443991C2semidirect
5. C61×#1413C2semidirect
112. #1413D61semidirect
130. C61#13600semidirect

Maximal subgroups

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

subgroupindex
#214562212
#214475032
#214439922
#214439912
C61×#14132
#1360061