Finite Group Databasebeta

Subgroups — #29600519

Order 15616.

194 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839595	C₂	1	1	yes	maximal, characteristic
2	#21447600	C₂	1	1	yes	maximal, characteristic
3	#366×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#21444020	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444019	C₂	1	1	yes	maximal, characteristic, complemented
6	#21444018	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
9	#16839799	—	2	1	no	—
10	#107×D₆₁	—	2	1	no	—
11	#16839595	C₄	1	4	yes	complemented
12	C₂×#14519683	C₂²	1	1	yes	characteristic
13	#16839589	—	2	1	no	—
14	#16839588	C₂²	1	1	yes	characteristic
15	#16839588	C₂²	1	1	yes	characteristic
16	#16839587	—	2	1	no	—
17	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#366	C₂²	1	1	yes	characteristic
19	C₆₁×#366	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
21	#14519735	—	4	1	no	—
22	C₄⋊C₄×D₆₁	—	4	1	no	—
23	C₂×#11758185	D₄	1	1	yes	characteristic
24	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
25	C₁₆×D₆₁	Q₈	1	2	yes	—
26	#14519711	—	2	1	no	—
27	#14519711	—	2	1	no	—
28	#14519710	—	4	1	no	—
29	#14519708	—	4	1	no	—
30	#14519683	C₂×C₄	1	2	yes	complemented
31	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
32	#107×C₆₁	—	2	1	no	—
33	#107×C₆₁	—	2	1	no	—
34	C₃₂×C₆₁	C₂×C₄	1	2	yes	complemented
35	C₂×C₄×D₆₁	—	8	1	no	—
36	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	—	4	1	no	—
39	C₄⋊C₄×C₆₁	—	4	1	no	—
40	#11758198	—	8	1	no	—
41	C₄⋊Dic₆₁	—	4	1	no	—
42	C₄⋊Dic₆₁	—	4	1	no	—
43	#11758196	—	8	1	no	—
44	C₄×Dic₆₁	—	8	1	no	—
45	C₂×#92418	D₈	1	1	yes	characteristic
46	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
47	C₈×D₆₁	Q₁₆	1	2	yes	—
48	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	#11758185	C₄⋊C₄	1	2	yes	—
51	C₂²×D₆₁	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₆₁	—	8	1	no	—
53	C₂×C₄×C₆₁	—	8	1	no	—
54	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₆₁	—	8	1	no	—
56	C₂×Dic₆₁	—	8	1	no	—
57	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
58	C₄×D₆₁	—	16	2	no	—
59	C₄×D₆₁	#107	1	2	yes	—
60	C₄×D₆₁	Q₃₂	1	2	yes	—
61	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
63	#92418	#107	1	2	yes	—
64	#13634	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
66	C₂×D₆₁	—	2	1	no	—
67	C₂×D₆₁	—	2	1	no	—
68	C₂×D₆₁	#366	1	2	yes	—
69	C₄×C₆₁	—	16	1	no	—
70	C₄×C₆₁	—	16	1	no	—
71	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
72	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
73	Dic₆₁	—	16	1	no	—
74	Dic₆₁	—	16	1	no	—
75	Dic₆₁	#366	1	2	yes	—
76	C₂×M₆₄	—	61	1	no	—
77	#2158	—	61	1	no	—
78	C₂×#366	—	61	1	no	—
79	#1424	—	61	1	no	—
80	#1424	D₆₁	1	1	yes	characteristic, complemented
81	#1423	—	61	1	no	—
82	#1423	—	61	1	no	—
83	C₂×C₆₁	C₂×#366	1	1	yes	characteristic
84	C₂×C₆₁	#2263	1	1	yes	characteristic
85	C₂×C₆₁	#2264	1	1	yes	characteristic
86	D₆₁	—	2	2	no	—
87	C₂²×C₁₆	—	61	1	no	—
88	#427	—	122	1	no	—
89	C₂×#107	—	122	1	no	—
90	M₆₄	—	61	4	no	—
91	C₂×C₃₂	—	61	1	no	—
92	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
93	#367	—	122	1	no	—
94	#366	—	61	1	no	—
95	#366	—	61	1	no	—
96	#366	—	122	1	no	—
97	#366	C₂×D₆₁	1	1	yes	characteristic
98	#366	C₂×D₆₁	1	1	yes	characteristic
99	C₆₁	#13634	1	1	yes	characteristic, complemented, Sylow
100	C₂²×C₈	—	61	1	no	—
101	#117	—	244	1	no	—
102	C₂×C₄⋊C₄	—	244	1	no	—
103	C₂×C₁₆	—	61	1	no	—
104	C₂×C₁₆	—	61	2	no	—
105	C₂×C₁₆	—	61	2	no	—
106	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
107	#107	—	2	1	no	—
108	#107	—	2	1	no	—
109	#107	—	122	1	no	—
110	#107	—	122	1	no	—
111	#107	—	244	1	no	—
112	#106	—	244	1	no	—
113	C₃₂	—	61	2	no	—
114	C₃₂	C₄×D₆₁	1	2	yes	complemented
115	C₂²×C₄	—	61	1	no	—
116	C₂²×C₄	—	488	1	no	—
117	C₂×C₈	—	61	1	no	—
118	C₂×C₈	—	61	2	no	—
119	C₂×C₈	—	61	2	no	—
120	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	1	no	—
123	C₄⋊C₄	—	488	1	no	—
124	C₄⋊C₄	—	244	1	no	—
125	C₄⋊C₄	—	244	1	no	—
126	C₂²⋊C₄	—	488	1	no	—
127	C₄²	—	488	1	no	—
128	C₁₆	—	61	2	no	—
129	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₁₆	Q₈×D₆₁	1	1	yes	characteristic
131	C₂³	—	61	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	—	61	1	no	—
135	C₂×C₄	—	61	2	no	—
136	C₂×C₄	—	61	2	no	—
137	C₂×C₄	—	976	2	no	—
138	C₂×C₄	—	488	1	no	—
139	C₂×C₄	—	488	1	no	—
140	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
141	C₈	—	61	2	no	—
142	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
143	C₈	Q₁₆×D₆₁	1	1	yes	characteristic
144	C₂²	—	61	2	no	—
145	C₂²	—	122	1	no	—
146	C₂²	—	122	1	no	—
147	C₂²	D₁₆×D₆₁	1	1	yes	characteristic
148	C₄	—	16	1	no	—
149	C₄	—	16	1	no	—
150	C₄	—	61	2	no	—
151	C₄	—	976	1	no	—
152	C₄	—	976	1	no	—
153	C₄	#107×D₆₁	1	1	yes	characteristic
154	C₄	Q₃₂×D₆₁	1	1	yes	characteristic
155	C₂	—	122	2	no	—
156	C₂	#366×D₆₁	1	1	yes	characteristic
157	C₂	#21448133	1	1	yes	characteristic
158	C₂	#21448135	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₆₁
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#1424	C₂
socle	C₂²×C₆₁	C₂×D₁₆
prime radical	C₂²×D₆₁	D₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	#1424	D₆₁
hypocenter	C₆₁	#13634
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
4. #21444020	C₂	semidirect
5. #21444019	C₂	semidirect
6. #21444018	C₂	semidirect
7. C₆₁×#1424	C₂	semidirect
11. #16839595	C₄	semidirect
30. #14519683	C₂×C₄	semidirect
34. C₃₂×C₆₁	C₂×C₄	semidirect
80. #1424	D₆₁	semidirect
99. C₆₁	#13634	semidirect
114. C₃₂	C₄×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839595	2
#21447600	2
#366×D₆₁	2
#21444020	2
#21444019	2
#21444018	2
C₆₁×#1424	2
#13634	61