Finite Group Databasebeta

← #29591190

Subgroups — #29591190

Order 15616.

318 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 69 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#21456166C211yesmaximal, characteristic
2#347×D61C212yesmaximal, complemented
3#21443854C211yesmaximal, characteristic, complemented
4#21443851C212yesmaximal, complemented
5C61×#1356C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic
7#16840313C411yescharacteristic
8M32×D61—22no—
9#16840130C411yescharacteristic
10#16840128C411yescharacteristic
11#16840127C2211yescharacteristic
12#16840123—22no—
13#16839757—22no—
14#97×D61C2211yescharacteristic
15#16839540C2212yes—
16#16839539—22no—
17C61×#491C2211yescharacteristic
18C61×#347C2212yes—
19C2×#11758226—21no—
20C2×C8×D61D412yes—
21#14519776—21no—
22#14519774C2×C411yescharacteristic
23#14519769C2×C411yescharacteristic
24#14519766—21no—
25#14519755—41no—
26#14519752—41no—
27#14519742—41no—
28#14519739—41no—
29#14519735—21no—
30C42×D61C2×C411yescharacteristic
31#14519717—22no—
32#14519716—24no—
33C16×D61—24no—
34#14519701—22no—
35#14519699—22no—
36#14519697—22no—
37#14519688C2×C411yescharacteristic
38#14519687D412yes—
39C4⋊Q8×C61C2×C411yescharacteristic
40#125×C61C2×C411yescharacteristic
41M32×C61—22no—
42#103×C61—22no—
43#97×C61C2311yescharacteristic
44C2×Q8×C61—21no—
45#11758226—42no—
46#11758226—42no—
47C2×D244—21no—
48C2×D244—41no—
49C2×D244—41no—
50C2×C4×D61—21no—
51C2×C4×D61—41no—
52C2×C4×D61C22⋊C411yescharacteristic
53C2×Dic122—41no—
54C2×C8×C61C2×D412yes—
55C4⋊C4×C61—21no—
56C4⋊C4×C61—41no—
57C4⋊C4×C61—41no—
58C42×C61C22×C411yescharacteristic
59#11758198—42no—
60#11758198—41no—
61C4⋊Dic61—21no—
62#11758196—41no—
63#11758196—41no—
64C4×Dic61—21no—
65C4×Dic61—41no—
66C4×Dic61C22⋊C411yescharacteristic
67C2×#92418C22⋊C412yes—
68C8×D61—22no—
69C8×D61—22no—
70C16×C61—24no—
71#11758185—24no—
72C22×D61—41no—
73C22×D61#10011yescharacteristic
74Q8×C61—42no—
75C2×C4×C61—21no—
76C2×C4×C61—41no—
77C2×C4×C61—41no—
78C2×C4×C61C2×C22⋊C411yescharacteristic
79C2×Dic61—21no—
80C2×Dic61—41no—
81C2×Dic61#10011yescharacteristic
82D244—42no—
83D244—82no—
84D244—82no—
85C4×D61—81no—
86C4×D61—81no—
87C4×D61—81no—
88C4×D61#10412yes—
89C4×D61#10412yes—
90Dic122—82no—
91C8×C61—22no—
92#92418—22no—
93#11911—611nomaximal, Sylow
94C22×C61C2×#10011yescharacteristic
95C2×D61—41no—
96C2×D61—82no—
97C2×D61—81no—
98C2×D61#32912yes—
99C4×C61—41no—
100C4×C61—81no—
101C4×C61—81no—
102C4×C61C2×#10412yes—
103Dic61—41no—
104Dic61—81no—
105Dic61#32911yescharacteristic
106Dic61#32911yescharacteristic
107#3135—611no—
108C2×#347—612no—
109#1356—611no—
110#1356D6111yescharacteristic, complemented
111#1355—612no—
112C2×C61—21no—
113C2×C61C2×#32911yescharacteristic
114D61—22no—
115D61—161no—
116#533—611no—
117#532—611no—
118C2×M32—1222no—
119#491—611no—
120#491C2×D6111yescharacteristic
121#490—611no—
122#490—611no—
123#489—1222no—
124#416—1222no—
125C2×#97—611no—
126#347—612no—
127#347—1222no—
128#347C2×D6112yes—
129C61#1191111yescharacteristic, complemented, Sylow
130C2×#41—1221no—
131C22×C8—612no—
132C4⋊Q8—1221no—
133C4⋊Q8C4×D6111yescharacteristic
134#127—611no—
135#125—611no—
136#125C4×D6111yescharacteristic
137#124—1221no—
138C22⋊Q8—2441no—
139#121—2441no—
140C4×Q8—2441no—
141C4×D4—2441no—
142#117—1221no—
143C2×C42—611no—
144M32—22no—
145M32—1224no—
146M32—1222no—
147C2×C16—1224no—
148#103—22no—
149#103—1222no—
150#102—1222no—
151#102—1222no—
152#97—612no—
153#97—611no—
154#97C22×D6111yescharacteristic
155#41—2442no—
156#41—2442no—
157C2×Q8—21no—
158C2×Q8—2441no—
159C2×D4—1221no—
160C2×D4—2441no—
161C2×D4—2441no—
162C22×C4—611no—
163C22×C4—1221no—
164C22×C4—2441no—
165C2×C8—612no—
166C2×C8—1222no—
167C2×C8—1222no—
168C2×C8D4×D6112yes—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—1221no—
173C4⋊C4—2441no—
174C4⋊C4—2441no—
175C22⋊C4—2442no—
176C22⋊C4—2441no—
177C42—611no—
178C42—1221no—
179C42—2441no—
180C42C2×C4×D6111yescharacteristic
181C16—24no—
182C16—1224no—
183C23—611no—
184C23—2441no—
185Q8—42no—
186Q8—4882no—
187D4—4882no—
188D4—4882no—
189D4—2442no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—611no—
194C2×C4—612no—
195C2×C4—612no—
196C2×C4—1221no—
197C2×C4—4881no—
198C2×C4—4881no—
199C2×C4—4881no—
200C2×C4—2441no—
201C2×C4C22⋊C4×D6111yescharacteristic
202C8—22no—
203C8—1222no—
204C22—612no—
205C22—4882no—
206C22—4881no—
207C22—2441no—
208C22#100×D6111yescharacteristic
209C4—41no—
210C4—81no—
211C4—81no—
212C4—611no—
213C4—611no—
214C4—4881no—
215C4—2441no—
216C4#104×D6112yes—
217C2—21no—
218C2—1222no—
219C2—9761no—
220C2#329×D6111yescharacteristic
subgroupG/H
centerC2#329×D61
commutatorC42×C61C22×C4
Frattini#97C22×D61
FittingC61×#1356C2
socleC2×C61C2×#329
prime radical#16840313C4
cosocle#97×C61C23
hypercenter#1356D61
hypocenterC61#11911
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1356

Complements

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

NH ≅ G/Ntype
2. #347×D61C2semidirect
3. #21443854C2semidirect
4. #21443851C2semidirect
5. C61×#1356C2semidirect
110. #1356D61semidirect
129. C61#11911semidirect

Maximal subgroups

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

subgroupindex
#214561662
#347×D612
#214438542
#214438512
C61×#13562
#1191161