Subgroups — #29570438

Order 15616.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445565	C₂	1	1	yes	maximal, characteristic
2	C₄²×#92418	C₂	1	1	yes	maximal, characteristic
3	#21443646	C₂	1	4	yes	maximal
4	C₄×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#92418	C₄	1	4	yes	—
6	C₂×C₄×#92418	C₂²	1	2	yes	—
7	C₂×C₄×#92418	C₂²	1	2	yes	—
8	#16839517	C₄	1	4	yes	—
9	C₂×#97×C₆₁	C₄	1	2	yes	—
10	C₂×#97×C₆₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
12	C₄³×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×#92418	C₂×C₄	1	4	yes	—
14	C₂²×#92418	C₂×C₄	1	4	yes	—
15	C₄×#92418	—	2	4	no	—
16	C₄×#92418	—	2	4	no	—
17	C₄×#92418	—	2	8	no	—
18	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
24	#97×C₆₁	—	2	4	no	—
25	#97×C₆₁	C₈	1	8	yes	complemented
26	C₄×C₈×C₆₁	—	2	4	no	—
27	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
28	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
29	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	—	2	4	no	—
33	C₂×C₈×C₆₁	—	2	4	no	—
34	C₂×C₈×C₆₁	C₂×C₈	1	8	yes	—
35	C₄²×C₆₁	—	2	4	no	—
36	C₄²×C₆₁	—	2	4	no	—
37	C₄²×C₆₁	C₂×C₈	1	4	yes	—
38	C₄²×C₆₁	M₁₆	1	4	yes	—
39	C₄²×C₆₁	#41	1	4	yes	—
40	C₂×#92418	—	2	4	no	—
41	C₂×#92418	—	2	4	no	—
42	C₂×#92418	—	2	4	no	—
43	C₂×#92418	—	2	8	no	—
44	C₂×#92418	M₁₆	1	8	yes	—
45	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₆₁	#97	1	4	yes	—
52	C₂×C₄×C₆₁	#117	1	2	yes	—
53	C₂×C₄×C₆₁	#117	1	2	yes	—
54	C₂×C₄×C₆₁	#117	1	2	yes	—
55	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₆₁	#131	1	4	yes	—
59	C₈×C₆₁	—	2	8	no	—
60	#92418	—	2	8	no	—
61	#92418	—	4	4	no	—
62	#8132	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	#376	1	1	yes	characteristic
64	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
66	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
67	C₂²×C₆₁	#405	1	2	yes	—
68	C₄×C₆₁	—	2	4	no	—
69	C₄×C₆₁	—	2	4	no	—
70	C₄×C₆₁	#431	1	4	yes	—
71	C₄×C₆₁	#432	1	4	yes	—
72	C₄×C₆₁	#433	1	4	yes	—
73	#1744	—	61	1	no	—
74	C₄×#97	D₆₁	1	1	yes	characteristic
75	C₄²×C₈	—	61	1	no	—
76	#1273	—	61	4	no	—
77	C₂×C₆₁	C₈×M₁₆	1	2	yes	—
78	C₂×C₆₁	#1451	1	2	yes	—
79	C₂×C₆₁	#1744	1	1	yes	characteristic
80	C₂×C₆₁	#1745	1	1	yes	characteristic
81	C₂×C₆₁	#1746	1	1	yes	characteristic
82	C₂×#97	Dic₆₁	1	2	yes	—
83	C₂×#97	Dic₆₁	1	2	yes	—
84	C₂×C₄×C₈	—	61	4	no	—
85	C₂×C₄×C₈	—	61	2	no	—
86	C₂×C₄×C₈	—	61	2	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
88	C₄³	C₂×D₆₁	1	1	yes	characteristic
89	#336	—	61	4	no	—
90	C₆₁	#8132	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	61	4	no	—
92	C₂²×C₈	—	61	4	no	—
93	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
94	C₂×C₄²	C₄×D₆₁	1	2	yes	—
95	C₂×C₄²	C₄×D₆₁	1	2	yes	—
96	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
98	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
99	#97	—	2	4	no	—
100	#97	#92418	1	8	yes	complemented
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	122	8	no	—
103	C₄×C₈	—	122	4	no	—
104	C₄×C₈	—	122	4	no	—
105	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
106	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	61	8	no	—
113	C₂×C₈	—	122	4	no	—
114	C₂×C₈	—	122	4	no	—
115	C₂×C₈	—	122	4	no	—
116	C₂×C₈	—	122	8	no	—
117	C₂×C₈	C₂×#92418	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₂×#92418	1	4	yes	—
121	C₄²	#11758194	1	4	yes	—
122	C₄²	#11758222	1	4	yes	—
123	C₂³	C₂×C₄×Dic₆₁	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₄×#92418	1	4	yes	—
129	C₂×C₄	#14519688	1	4	yes	—
130	C₂×C₄	#14519734	1	2	yes	—
131	C₂×C₄	#14519734	1	2	yes	—
132	C₂×C₄	#14519736	1	2	yes	—
133	C₂×C₄	C₂²×#92418	1	2	yes	—
134	C₂×C₄	M₁₆×D₆₁	1	4	yes	—
135	C₂×C₄	C₂×#11758194	1	2	yes	—
136	C₂×C₄	#14519785	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	122	8	no	—
139	C₈	—	244	4	no	—
140	C₂²	#16839608	1	1	yes	characteristic
141	C₂²	C₂×C₄×#92418	1	1	yes	characteristic
142	C₂²	C₂×#14519688	1	1	yes	characteristic
143	C₂²	M₁₆×Dic₆₁	1	2	yes	—
144	C₂²	#16839720	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#16839815	1	4	yes	—
148	C₄	#16839817	1	4	yes	—
149	C₄	#16839824	1	4	yes	—
150	C₂	M₁₆×#92418	1	2	yes	—
151	C₂	#21444106	1	2	yes	—
152	C₂	#21445565	1	1	yes	characteristic
153	C₂	#21445567	1	1	yes	characteristic
154	C₂	#21445572	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂×C₄×C₈
Fitting	C₄×#97×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₄²
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#97	D₆₁
hypocenter	C₆₁	#8132
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
25. #97×C₆₁	C₈	semidirect
90. C₆₁	#8132	semidirect
100. #97	#92418	semidirect

Maximal subgroups

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

subgroup	index
#21445565	2
C₄²×#92418	2
#21443646	2
C₄×#97×C₆₁	2
#8132	61