Finite Group Databasebeta

← D61×#1356 #29591185

Subgroups — D61×#1356 #29591185

Order 15616.

314 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 71 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#491×D61C211yesmaximal, characteristic
2#347×D61C212yesmaximal
3#21443854C211yesmaximal, characteristic, complemented
4#21443853C212yesmaximal, complemented
5C61×#1356C211yesmaximal, characteristic, complemented
6C4⋊Q8×D61C411yescharacteristic
7#125×D61C411yescharacteristic
8M32×D61—22no—
9#16840132C411yescharacteristic
10#16840131C2211yescharacteristic
11#16840130C411yescharacteristic
12#16840129—22no—
13#103×D61—22no—
14#97×D61C2211yescharacteristic
15#16839540C2212yes—
16#16839539—22no—
17C61×#491C2211yescharacteristic
18C61×#347C2212yes—
19C2×Q8×D61—21no—
20C2×C8×D61D412yes—
21#14519777C2×C411yescharacteristic
22#14519776—41no—
23#14519775—21no—
24#14519769C2×C411yescharacteristic
25#14519768—21no—
26#14519767—41no—
27C4⋊C4×D61—21no—
28C4⋊C4×D61—41no—
29C4⋊C4×D61—41no—
30C42×D61C2×C411yescharacteristic
31#14519717—22no—
32#14519716—24no—
33C16×D61—24no—
34#14519702—22no—
35#14519701—22no—
36#14519700—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—
45Q8×D61—42no—
46Q8×D61—42no—
47C2×C4×D61—21no—
48C2×C4×D61—41no—
49C2×C4×D61—41no—
50C2×C4×D61C22⋊C411yescharacteristic
51C2×Dic122—21no—
52C2×Dic122—41no—
53C2×C8×C61C2×D412yes—
54C4⋊C4×C61—21no—
55C4⋊C4×C61—41no—
56C4⋊C4×C61—41no—
57C42×C61C22×C411yescharacteristic
58C4⋊Dic61—21no—
59C4⋊Dic61—41no—
60C4⋊Dic61—41no—
61#11758196—41no—
62#11758196—42no—
63#11758196—41no—
64#11758196—41no—
65C4×Dic61—21no—
66C4×Dic61C22⋊C411yescharacteristic
67C2×#92418C22⋊C412yes—
68C8×D61—22no—
69C8×D61—22no—
70C16×C61—24no—
71#11758185—24no—
72C22×D61#10011yescharacteristic
73Q8×C61—42no—
74C2×C4×C61—21no—
75C2×C4×C61—41no—
76C2×C4×C61—41no—
77C2×C4×C61C2×C22⋊C411yescharacteristic
78C2×Dic61—21no—
79C2×Dic61—41no—
80C2×Dic61—41no—
81C2×Dic61#10011yescharacteristic
82C4×D61—42no—
83C4×D61—82no—
84C4×D61—81no—
85C4×D61—81no—
86C4×D61#10412yes—
87C4×D61#10412yes—
88Dic122—42no—
89Dic122—82no—
90C8×C61—22no—
91#92418—22no—
92C2×#1356—611nomaximal, Sylow
93C22×C61C2×#10011yescharacteristic
94C2×D61—22no—
95C2×D61#32911yescharacteristic
96C2×D61#32911yescharacteristic
97C4×C61—41no—
98C4×C61—81no—
99C4×C61—81no—
100C4×C61C2×#10412yes—
101Dic61—41no—
102Dic61—81no—
103Dic61—81no—
104Dic61#32912yes—
105C2×#491—611no—
106C2×#347—612no—
107#1356—611no—
108#1356—612no—
109#1356D6111yescharacteristic, direct product factor
110C2×C61—21no—
111C2×C61C2×#32911yescharacteristic
112D61—21no—
113D61#135612yesdirect product factor
114C2×C4⋊Q8—611no—
115C2×#125—611no—
116C2×M32—1222no—
117#491—611no—
118#491—611no—
119#491—611no—
120#491—1222no—
121#491C2×D6111yescharacteristic
122C2×#103—1222no—
123C2×#97—611no—
124#347—612no—
125#347—1222no—
126#347C2×D6112yes—
127C61C2×#135611yescharacteristic, complemented, Sylow
128C22×Q8—1221no—
129C22×C8—612no—
130C4⋊Q8—611no—
131C4⋊Q8—1221no—
132C4⋊Q8—2441no—
133C4⋊Q8C4×D6111yescharacteristic
134#125—611no—
135#125—1221no—
136#125—2441no—
137#125C4×D6111yescharacteristic
138C2×C4⋊C4—1221no—
139C2×C4⋊C4—2441no—
140C2×C4⋊C4—2441no—
141C2×C42—611no—
142M32—22no—
143M32—1224no—
144M32—1222no—
145C2×C16—1224no—
146#103—22no—
147#103—1222no—
148#103—1222no—
149#103—1222no—
150#97—612no—
151#97—611no—
152#97C22×D6111yescharacteristic
153C2×Q8—21no—
154C2×Q8—1221no—
155C2×Q8—2441no—
156C2×Q8—2442no—
157C2×Q8—2442no—
158C22×C4—611no—
159C22×C4—1221no—
160C22×C4—2441no—
161C22×C4—2441no—
162C2×C8—612no—
163C2×C8—1222no—
164C2×C8—1222no—
165C2×C8D4×D6112yes—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—1221no—
170C4⋊C4—2441no—
171C4⋊C4—2441no—
172C4⋊C4—2441no—
173C4⋊C4—2441no—
174C4⋊C4—2442no—
175C4⋊C4—2441no—
176C42—611no—
177C42—1221no—
178C42C2×C4×D6111yescharacteristic
179C16—24no—
180C16—1224no—
181C23—611no—
182Q8—42no—
183Q8—4882no—
184Q8—2442no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—611no—
189C2×C4—612no—
190C2×C4—612no—
191C2×C4—1221no—
192C2×C4—4881no—
193C2×C4—4881no—
194C2×C4—4882no—
195C2×C4—2442no—
196C2×C4—2441no—
197C2×C4—2441no—
198C2×C4C22⋊C4×D6111yescharacteristic
199C8—22no—
200C8—1222no—
201C22—611no—
202C22—611no—
203C22—1222no—
204C22#100×D6111yescharacteristic
205C4—41no—
206C4—81no—
207C4—81no—
208C4—612no—
209C4—4881no—
210C4—4881no—
211C4—2441no—
212C4#104×D6112yes—
213C2—21no—
214C2—612no—
215C2—1221no—
216C2#329×D6111yescharacteristic
subgroupG/H
centerC2#329×D61
commutatorC42×C61C22×C4
Frattini#97C22×D61
FittingC61×#1356C2
socleC2×C61C2×#329
prime radicalC22×D61#100
cosocle#97×C61C23
hypercenter#1356D61
hypocenterC61C2×#1356
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
109. #1356D61direct (N × H)
113. D61#1356direct (N × H)
127. C61C2×#1356semidirect

Maximal subgroups

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

subgroupindex
#491×D612
#347×D612
#214438542
#214438532
C61×#13562
C2×#135661