Finite Group Databasebeta

← #29591208

Subgroups — #29591208

Order 15616.

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

normal subgroups only

This group has 274 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#21456198C211yesmaximal, characteristic
2#21447629C212yesmaximal
3#21443854C211yesmaximal, characteristic, complemented
4#21443853C212yesmaximal, complemented
5C61×#1356C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic
7C4⋊Q8×D61C411yescharacteristic
8#16840132C411yescharacteristic
9#16840131C2211yescharacteristic
10#16840130C411yescharacteristic
11#16840111—22no—
12#16839769—22no—
13#16839714C2211yescharacteristic
14#16839582—22no—
15#16839543—22no—
16#16839540C2212yes—
17C61×#491C2211yescharacteristic
18C61×#347C2212yes—
19C2×Q8×D61—21no—
20C2×#11758189D412yes—
21#14519777C2×C411yescharacteristic
22#14519776—21no—
23#14519776—41no—
24#14519775—21no—
25#14519769C2×C411yescharacteristic
26#14519755—41no—
27#14519742—41no—
28#14519735—21no—
29C4⋊C4×D61—41no—
30C42×D61C2×C411yescharacteristic
31#14519717—22no—
32#14519702—22no—
33#14519701—22no—
34#14519700—22no—
35#14519688C2×C411yescharacteristic
36C8×Dic61D412yes—
37C4⋊Q8×C61C2×C411yescharacteristic
38#125×C61C2×C411yescharacteristic
39M32×C61—22no—
40#103×C61—22no—
41#97×C61C2311yescharacteristic
42C2×Q8×C61—21no—
43Q8×D61—44no—
44C2×C4×D61—21no—
45C2×C4×D61—41no—
46C2×C4×D61C22⋊C411yescharacteristic
47C2×Dic122—21no—
48C2×Dic122—41no—
49C2×Dic122—41no—
50C2×C8×C61C2×D412yes—
51C4⋊C4×C61—21no—
52C4⋊C4×C61—41no—
53C4⋊C4×C61—41no—
54C42×C61C22×C411yescharacteristic
55#11758198—41no—
56C4⋊Dic61—21no—
57C4⋊Dic61—41no—
58#11758196—42no—
59#11758196—41no—
60#11758196—41no—
61C4×Dic61—21no—
62C4×Dic61—41no—
63C4×Dic61C22⋊C411yescharacteristic
64C2×#92418C22⋊C412yes—
65#11758189—42no—
66C16×C61—42no—
67#11758185—42no—
68C22×D61#10011yescharacteristic
69Q8×C61—42no—
70C2×C4×C61—21no—
71C2×C4×C61—41no—
72C2×C4×C61—41no—
73C2×C4×C61C2×C22⋊C411yescharacteristic
74C2×Dic61—21no—
75C2×Dic61—41no—
76C2×Dic61—41no—
77C2×Dic61#10011yescharacteristic
78C4×D61—22no—
79C4×D61—42no—
80C4×D61—82no—
81Dic122—42no—
82Dic122—82no—
83Dic122—82no—
84C8×C61—22no—
85#92418—22no—
86#11916—611nomaximal, Sylow
87C22×C61C2×#10011yescharacteristic
88C2×D61—21no—
89C2×D61—41no—
90C4×C61—41no—
91C4×C61—81no—
92C4×C61—81no—
93C4×C61#42112yes—
94Dic61—22no—
95Dic61—21no—
96Dic61—81no—
97Dic61—81no—
98#3139—611no—
99#2165—612no—
100#1356—611no—
101#1356—612no—
102#1356D6111yescharacteristic, complemented
103C2×C61—21no—
104C2×C61#153311yescharacteristic
105D61—41no—
106#533—611no—
107C2×C4⋊Q8—611no—
108#491—611no—
109#491—611no—
110#491—611no—
111#491C2×D6111yescharacteristic
112#487—1222no—
113#419—1222no—
114C4×M16—611no—
115#364—1222no—
116#349—1222no—
117#347—612no—
118#347C2×D6112yes—
119C61#1191611yescharacteristic, complemented, Sylow
120C22×Q8—1221no—
121C2×M16—612no—
122C4⋊Q8—611no—
123C4⋊Q8—1221no—
124C4⋊Q8—1221no—
125C4⋊Q8—2441no—
126C4⋊Q8C4×D6111yescharacteristic
127#125—611no—
128#125C4×D6111yescharacteristic
129C22⋊Q8—2441no—
130C4×Q8—2441no—
131#117—1221no—
132C2×C4⋊C4—2441no—
133C2×C42—611no—
134M32—22no—
135M32—1222no—
136#103—22no—
137#103—1222no—
138#103—1222no—
139#103—1222no—
140#97—611no—
141#97C22×D6111yescharacteristic
142C4×C8—612no—
143C2×Q8—21no—
144C2×Q8—1221no—
145C2×Q8—2441no—
146C2×Q8—2441no—
147C2×Q8—2444no—
148C22×C4—611no—
149C22×C4—1221no—
150C22×C4—2441no—
151M16—2442no—
152C2×C8—612no—
153C2×C8D4×D6112yes—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—1221no—
158C4⋊C4—2441no—
159C4⋊C4—2441no—
160C4⋊C4—2442no—
161C4⋊C4—2441no—
162C22⋊C4—2441no—
163C42—611no—
164C42—1221no—
165C42—2441no—
166C42C2×C4×D6111yescharacteristic
167C16—42no—
168C16—2442no—
169C23—611no—
170Q8—42no—
171Q8—4882no—
172Q8—4882no—
173Q8—2442no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—611no—
178C2×C4—1221no—
179C2×C4—1222no—
180C2×C4—4882no—
181C2×C4—2442no—
182C2×C4—2441no—
183C2×C4—2441no—
184C2×C4C22⋊C4×D6111yescharacteristic
185C8—22no—
186C8—1222no—
187C22—1221no—
188C22—2441no—
189C22#100×D6111yescharacteristic
190C4—41no—
191C4—81no—
192C4—81no—
193C4—1222no—
194C4—1221no—
195C4—4881no—
196C4—4881no—
197C4#1683977712yes—
198C2—21no—
199C2—2441no—
200C2#2144444111yescharacteristic
subgroupG/H
centerC2#21444441
commutatorC42×C61C22×C4
Frattini#97C22×D61
FittingC61×#1356C2
socleC2×C61#1533
prime radicalC22×D61#100
cosocle#97×C61C23
hypercenter#1356D61
hypocenterC61#11916
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
3. #21443854C2semidirect
4. #21443853C2semidirect
5. C61×#1356C2semidirect
102. #1356D61semidirect
119. C61#11916semidirect

Maximal subgroups

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

subgroupindex
#214561982
#214476292
#214438542
#214438532
C61×#13562
#1191661