Finite Group Databasebeta

← #29591205

Subgroups — #29591205

Order 15616.

326 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 61 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#21456192C211yesmaximal, characteristic
2#21447629C212yesmaximal, complemented
3#21443852C211yesmaximal, characteristic, complemented
4#21443851C212yesmaximal, complemented
5C61×#1355C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic
7#127×D61C411yescharacteristic
8#16840128C411yescharacteristic
9#16840127C2211yescharacteristic
10#16840126C411yescharacteristic
11#16840107—22no—
12#16839765—22no—
13#16839714C2211yescharacteristic
14#16839582—22no—
15#16839543—22no—
16#16839540C2212yescomplemented
17C61×#490C2211yescharacteristic
18C61×#347C2212yescomplemented
19C2×D4×D61—21no—
20C2×D4×D61—41no—
21C2×#11758189D412yes—
22#14519776—21no—
23#14519774C2×C411yescharacteristic
24#14519773—21no—
25#14519773—41no—
26#14519769C2×C411yescharacteristic
27#14519755—41no—
28#14519742—41no—
29#14519735—21no—
30C42×D61C2×C411yescharacteristic
31#14519717—22no—
32#14519699—22no—
33#14519698—22no—
34#14519697—22no—
35#14519688C2×C411yescharacteristic
36C8×Dic61D412yes—
37#127×C61C2×C411yescharacteristic
38#125×C61C2×C411yescharacteristic
39M32×C61—22no—
40#102×C61—22no—
41#97×C61C2311yescharacteristic
42C23×D61—41no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×C22⋊D61—42no—
46C2×C22⋊D61—41no—
47D4×D61—44no—
48D4×D61—84no—
49C2×D244—21no—
50C2×D244—41no—
51C2×C4×D61—21no—
52C2×C4×D61C22⋊C411yescharacteristic
53C2×Dic122—41no—
54C2×C8×C61C2×D412yes—
55C4⋊C4×C61—21no—
56C4⋊C4×C61—41no—
57C42×C61C22×C411yescharacteristic
58#11758198—41no—
59C4⋊Dic61—21no—
60#11758196—41no—
61#11758196—41no—
62C4×Dic61—21no—
63C4×Dic61—41no—
64C4×Dic61C22⋊C411yescharacteristic
65C2×#92418C22⋊C412yes—
66#11758189—42no—
67C16×C61—42no—
68#11758185—42no—
69C23×C61—41no—
70C22×D61—41no—
71C22×D61—82no—
72C22×D61—84no—
73C22×D61#10011yescharacteristic
74D4×C61—42no—
75D4×C61—82no—
76C2×C4×C61—21no—
77C2×C4×C61—41no—
78C2×C4×C61C2×C22⋊C411yescharacteristic
79C22⋊D61—82no—
80C22⋊D61—84no—
81C2×Dic61—21no—
82C2×Dic61—41no—
83C2×Dic61#10011yescharacteristic
84D244—42no—
85D244—82no—
86C4×D61—22no—
87C4×D61—42no—
88Dic122—82no—
89C8×C61—22no—
90#92418—22no—
91#11915—611nomaximal, Sylow
92C22×C61—82no—
93C22×C61—81no—
94C22×C61C2×#10011yescharacteristic
95C2×D61—21no—
96C2×D61—41no—
97C2×D61—82no—
98C2×D61—81no—
99C2×D61—164no—
100C4×C61—41no—
101C4×C61—81no—
102C4×C61#42112yes—
103Dic61—22no—
104Dic61—21no—
105Dic61—81no—
106#3138—611no—
107#2165—612no—
108#1355—611no—
109#1355—612no—
110#1355D6111yescharacteristic, complemented
111C2×C61—21no—
112C2×C61—161no—
113C2×C61#153311yescharacteristic
114D61—41no—
115D61—161no—
116#533—611no—
117C2×#127—611no—
118#490—611no—
119#490—611no—
120#490—611no—
121#490C2×D6111yescharacteristic
122#486—1222no—
123#418—1222no—
124C4×M16—611no—
125#364—1222no—
126#349—1222no—
127#347—612no—
128#347C2×D6112yescomplemented
129C61#1191511yescharacteristic, complemented, Sylow
130C22×D4—1221no—
131C22×D4—2441no—
132C2×M16—612no—
133C4⋊Q8—1221no—
134#127—611no—
135#127—1221no—
136#127—2441no—
137#127C4×D6111yescharacteristic
138#125—611no—
139#125C4×D6111yescharacteristic
140C22⋊Q8—2441no—
141C4×Q8—2441no—
142#117—1221no—
143C2×C42—611no—
144M32—22no—
145M32—1222no—
146#102—22no—
147#102—1222no—
148#102—1222no—
149#102—1222no—
150#97—611no—
151#97C22×D6111yescharacteristic
152C4×C8—612no—
153C24—2441no—
154C2×Q8—2441no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—1221no—
158C2×D4—4884no—
159C2×D4—2441no—
160C2×D4—2441no—
161C2×D4—2444no—
162C2×D4—2442no—
163C22×C4—611no—
164C22×C4—1221no—
165M16—2442no—
166C2×C8—612no—
167C2×C8D4×D6112yes—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—1221no—
171C4⋊C4—2441no—
172C4⋊C4—2441no—
173C22⋊C4—2441no—
174C42—611no—
175C42—1221no—
176C42—2441no—
177C42C2×C4×D6111yescharacteristic
178C16—42no—
179C16—2442no—
180C23—41no—
181C23—611no—
182C23—4882no—
183C23—4884no—
184C23—2441no—
185Q8—4882no—
186D4—42no—
187D4—82no—
188D4—4882no—
189D4—4884no—
190D4—4882no—
191D4—2442no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—611no—
195C2×C4—1221no—
196C2×C4—1222no—
197C2×C4—2442no—
198C2×C4—2441no—
199C2×C4C22⋊C4×D6111yescharacteristic
200C8—22no—
201C8—1222no—
202C22—82no—
203C22—81no—
204C22—1221no—
205C22—9764no—
206C22—4882no—
207C22—4881no—
208C22—2441no—
209C22#100×D6111yescharacteristic
210C4—41no—
211C4—81no—
212C4—1222no—
213C4—1221no—
214C4—4881no—
215C4#1683977712yes—
216C2—21no—
217C2—161no—
218C2—9761no—
219C2—2441no—
220C2#2144444111yescharacteristic
subgroupG/H
centerC2#21444441
commutatorC42×C61C22×C4
Frattini#97C22×D61
FittingC61×#1355C2
socleC2×C61#1533
prime radical#127×D61C4
cosocle#97×C61C23
hypercenter#1355D61
hypocenterC61#11915
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1355

Complements

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

NH ≅ G/Ntype
2. #21447629C2semidirect
3. #21443852C2semidirect
4. #21443851C2semidirect
5. C61×#1355C2semidirect
16. #16839540C22semidirect
18. C61×#347C22semidirect
110. #1355D61semidirect
128. #347C2×D61semidirect
129. C61#11915semidirect

Maximal subgroups

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

subgroupindex
#214561922
#214476292
#214438522
#214438512
C61×#13552
#1191561