Subgroups — #29589014

Order 15616.

298 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
3	#21443740	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₄	1	2	yes	—
6	C₂×#14519717	C₄	1	2	yes	—
7	#16839538	C₂²	1	2	yes	—
8	#16839538	C₂²	1	2	yes	—
9	C₄×#11758185	C₄	1	2	yes	—
10	C₄×#11758185	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₆₁×#322	C₄	1	2	yes	—
14	C₆₁×#322	C₂²	1	2	yes	—
15	#14519717	—	2	4	no	—
16	#14519717	C₈	1	8	yes	complemented
17	C₂×#11758185	C₂×C₄	1	4	yes	—
18	C₂×#11758185	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₆₁	—	2	2	no	—
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₆₁	—	2	2	no	—
24	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
27	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
31	C₂×C₈×C₆₁	—	2	4	no	—
32	C₂×C₈×C₆₁	—	4	4	no	—
33	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₆₁	M₁₆	1	4	yes	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₆₁	C₄²	1	1	yes	characteristic
40	C₄²×C₆₁	M₁₆	1	2	yes	—
41	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	#11758185	—	2	4	no	—
43	#11758185	C₂×C₈	1	8	yes	complemented
44	C₂³×C₆₁	#97	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#97	1	2	yes	—
49	C₂×C₄×C₆₁	#97	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
52	C₈×C₆₁	—	4	4	no	—
53	C₈×C₆₁	C₄×C₈	1	2	yes	—
54	C₈×C₆₁	#97	1	2	yes	—
55	C₈×C₆₁	C₂²×C₈	1	2	yes	—
56	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
57	#11404	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	—	2	2	no	—
59	C₂²×C₆₁	#322	1	2	yes	—
60	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	#322	1	2	yes	—
63	C₄×C₆₁	#347	1	4	yes	—
64	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
65	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
66	C₄×M₃₂	—	61	1	no	—
67	C₄×M₃₂	—	61	1	no	—
68	C₂×#322	D₆₁	1	1	yes	characteristic
69	#1313	—	61	4	no	—
70	C₂×C₆₁	—	2	2	no	—
71	C₂×C₆₁	C₂×#322	1	1	yes	characteristic
72	C₂×C₆₁	C₂×#347	1	2	yes	—
73	C₂×M₃₂	—	61	2	no	—
74	C₂×M₃₂	—	61	2	no	—
75	C₂×C₄×C₈	Dic₆₁	1	2	yes	—
76	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
77	#346	—	61	2	no	—
78	#346	—	61	2	no	—
79	C₄×C₁₆	—	61	2	no	—
80	C₄×C₁₆	—	61	2	no	—
81	#322	Dic₆₁	1	2	yes	—
82	#322	C₂×D₆₁	1	2	yes	—
83	C₆₁	#11404	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	2	2	no	—
85	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
86	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
87	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
88	M₃₂	—	61	8	no	—
89	M₃₂	—	122	4	no	—
90	C₂×C₁₆	—	61	4	no	—
91	C₂×C₁₆	—	61	4	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
94	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
95	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
96	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
98	C₂²×C₄	#11758194	1	2	yes	—
99	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₂×C₈	C₈×D₆₁	1	4	yes	—
103	C₂×C₈	#11758189	1	4	yes	—
104	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
105	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
106	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
107	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
108	C₄²	#11758194	1	2	yes	—
109	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
110	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
111	C₁₆	—	61	8	no	—
112	C₁₆	—	122	4	no	—
113	C₂³	#14519688	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₈×Dic₆₁	1	2	yes	—
116	C₂×C₄	#14519687	1	2	yes	—
117	C₂×C₄	#14519688	1	1	yes	characteristic
118	C₂×C₄	#14519688	1	2	yes	—
119	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
120	C₂×C₄	C₂×#11758194	1	2	yes	—
121	C₈	—	4	4	no	—
122	C₈	C₈×Dic₆₁	1	2	yes	—
123	C₈	#14519687	1	2	yes	—
124	C₈	C₂×C₈×D₆₁	1	2	yes	—
125	C₈	C₂×#11758189	1	2	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	#16839484	1	2	yes	—
128	C₂²	C₂×#14519688	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	#16839484	1	2	yes	—
131	C₄	#16839540	1	4	yes	—
132	C₄	C₂×C₈×Dic₆₁	1	1	yes	characteristic
133	C₄	C₂×#14519687	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×#16839484	1	1	yes	characteristic
136	C₂	C₂×#16839540	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₂×C₆₁×#322	C₂
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	#97
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#11404
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
3. #21443740	C₂	semidirect
16. #14519717	C₈	semidirect
43. #11758185	C₂×C₈	semidirect
83. C₆₁	#11404	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14519717	2
#21443740	2
C₂×C₆₁×#322	2
#11404	61