Subgroups — #29569371

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445564	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839483	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#92418	C₂	1	2	yes	maximal
5	C₂×C₆₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	C₂×C₈×Dic₆₁	C₄	1	4	yes	—
9	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
10	#16839516	C₄	1	8	yes	—
11	#16839483	C₄	1	4	yes	complemented
12	C₈×#92418	C₄	1	8	yes	complemented
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#322	C₄	1	4	yes	complemented
16	C₂²×#92418	C₂×C₄	1	4	yes	—
17	C₂²×#92418	C₂×C₄	1	2	yes	—
18	C₂×C₄×Dic₆₁	C₂×C₄	1	4	yes	—
19	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₄×#92418	C₂×C₄	1	4	yes	—
21	C₄×#92418	C₂×C₄	1	4	yes	—
22	C₄×#92418	C₂×C₄	1	4	yes	—
23	C₈×Dic₆₁	C₈	1	16	yes	complemented
24	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
33	C₂²×Dic₆₁	C₄²	1	4	yes	—
34	C₂×C₈×C₆₁	—	2	8	no	—
35	C₂×C₈×C₆₁	C₄²	1	8	yes	—
36	C₂×C₈×C₆₁	C₄²	1	4	yes	—
37	C₂×C₈×C₆₁	C₂×C₈	1	8	yes	—
38	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
39	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	C₄×Dic₆₁	—	2	8	no	—
42	C₄×Dic₆₁	C₂×C₈	1	8	yes	—
43	C₂×#92418	—	2	8	no	—
44	C₂×#92418	C₄²	1	8	yes	—
45	C₂×#92418	C₄²	1	4	yes	—
46	C₂×#92418	C₂×C₈	1	8	yes	—
47	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
51	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
52	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
53	C₂×C₄×C₆₁	#131	1	8	yes	—
54	C₂×Dic₆₁	—	2	8	no	—
55	C₂×Dic₆₁	C₄×C₈	1	8	yes	—
56	C₈×C₆₁	—	2	8	no	—
57	C₈×C₆₁	C₄×C₈	1	8	yes	—
58	#92418	—	2	8	no	—
59	#92418	C₄×C₈	1	8	yes	—
60	#7841	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	C₄³	1	1	yes	characteristic
62	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
63	C₂²×C₆₁	#405	1	4	yes	—
64	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
65	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
66	C₄×C₆₁	#405	1	4	yes	—
67	C₄×C₆₁	#405	1	4	yes	—
68	Dic₆₁	—	2	8	no	—
69	#1744	—	61	2	no	—
70	C₄²×C₈	—	61	1	no	—
71	C₂×#322	—	61	1	no	—
72	C₂×#322	D₆₁	1	1	yes	characteristic
73	C₂×C₈²	—	61	2	no	—
74	C₂×C₆₁	#1450	1	4	yes	—
75	C₂×C₆₁	C₄²×C₈	1	1	yes	characteristic
76	C₂×C₆₁	#1726	1	2	yes	—
77	C₂×C₄×C₈	—	61	4	no	—
78	C₂×C₄×C₈	—	61	1	no	—
79	C₂×C₄×C₈	—	61	2	no	—
80	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
82	C₄³	—	61	1	no	—
83	#336	—	61	8	no	—
84	#322	—	61	4	no	—
85	#322	Dic₆₁	1	4	yes	complemented
86	C₈²	—	61	8	no	—
87	C₆₁	#7841	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	61	2	no	—
89	C₂²×C₈	—	61	4	no	—
90	C₂²×C₈	C₄×D₆₁	1	4	yes	—
91	C₂²×C₈	C₄×D₆₁	1	2	yes	—
92	C₂×C₄²	—	61	2	no	—
93	C₂×C₄²	—	61	4	no	—
94	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
95	C₄×C₈	—	61	16	no	—
96	C₄×C₈	—	61	4	no	—
97	C₄×C₈	—	61	4	no	—
98	C₄×C₈	—	61	4	no	—
99	C₄×C₈	C₄×D₆₁	1	4	yes	—
100	C₄×C₈	C₄×D₆₁	1	2	yes	—
101	C₄×C₈	C₂×Dic₆₁	1	4	yes	—
102	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
103	C₂²×C₄	—	61	4	no	—
104	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	61	4	no	—
108	C₂×C₈	—	61	8	no	—
109	C₂×C₈	—	61	8	no	—
110	C₂×C₈	—	122	8	no	—
111	C₂×C₈	C₈×D₆₁	1	8	yes	—
112	C₂×C₈	C₄×Dic₆₁	1	8	yes	—
113	C₂×C₈	C₄×Dic₆₁	1	4	yes	—
114	C₄²	—	61	8	no	—
115	C₄²	—	122	8	no	—
116	C₄²	C₂×C₄×D₆₁	1	2	yes	—
117	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
118	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
119	C₂³	C₄²×D₆₁	1	1	yes	characteristic
120	C₂×C₄	—	61	8	no	—
121	C₂×C₄	—	122	8	no	—
122	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
123	C₂×C₄	C₄²×D₆₁	1	2	yes	—
124	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×Dic₆₁	1	2	yes	—
126	C₂×C₄	C₂×C₈×D₆₁	1	4	yes	—
127	C₂×C₄	#14519785	1	8	yes	—
128	C₈	—	2	8	no	—
129	C₈	—	61	8	no	—
130	C₈	—	122	8	no	—
131	C₈	C₈×Dic₆₁	1	8	yes	—
132	C₂²	C₄²×Dic₆₁	1	1	yes	characteristic
133	C₂²	C₄×C₈×D₆₁	1	2	yes	—
134	C₂²	#16839719	1	4	yes	—
135	C₄	—	122	8	no	—
136	C₄	C₄×C₈×D₆₁	1	2	yes	—
137	C₄	C₂×C₈×Dic₆₁	1	2	yes	—
138	C₄	#16839719	1	4	yes	—
139	C₄	#16839720	1	4	yes	—
140	C₂	#21444104	1	4	yes	—
141	C₂	C₄×C₈×Dic₆₁	1	1	yes	characteristic
142	C₂	#21445507	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator	C₂×C₆₁	C₄²×C₈
Fitting	C₂×C₆₁×#322	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#322	D₆₁
hypocenter	C₆₁	#7841
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
11. #16839483	C₄	semidirect
12. C₈×#92418	C₄	semidirect
15. C₆₁×#322	C₄	semidirect
23. C₈×Dic₆₁	C₈	semidirect
85. #322	Dic₆₁	semidirect
87. C₆₁	#7841	semidirect

Maximal subgroups

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

subgroup	index
#21445564	2
C₄×C₈×Dic₆₁	2
C₂×#16839483	2
C₂×C₈×#92418	2
C₂×C₆₁×#322	2
#7841	61