Subgroups — #30624533

Order 15616.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462243	C₂	1	8	yes	maximal, complemented
2	D₆₁⋊M₆₄	C₂	1	4	yes	maximal, complemented
3	C₂×#16840694	C₂	1	2	yes	maximal, complemented
4	C₂×#16840180	C₂	1	1	yes	maximal, characteristic
5	#16840695	C₂²	1	8	yes	complemented
6	#16840694	C₂²	1	4	yes	—
7	#16840694	C₂²	1	4	yes	complemented
8	D₆₁⋊C₃₂	C₂²	1	8	yes	complemented
9	C₂×#14519827	C₂²	1	4	yes	complemented
10	C₂×#14519783	C₄	1	1	yes	characteristic
11	#16840180	C₄	1	4	yes	—
12	#16840180	C₂²	1	4	yes	—
13	C₂×#14519716	C₄	1	2	yes	—
14	C₂²×#11758185	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
16	C₂×M₃₂×C₆₁	C₄	1	1	yes	characteristic
17	#14519827	C₂³	1	8	yes	—
18	C₂×#11758222	C₈	1	1	yes	characteristic
19	#14519783	C₈	1	4	yes	—
20	#14519783	C₂×C₄	1	4	yes	—
21	C₂×#11758194	C₈	1	1	yes	characteristic
22	C₂×#11758189	C₈	1	2	yes	—
23	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
24	#14519716	C₂×C₄	1	8	yes	—
25	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
26	C₂×#11758185	C₂×C₄	1	2	yes	—
27	C₂×#11758185	C₂³	1	1	yes	characteristic
28	C₂×#11758185	C₂³	1	2	yes	—
29	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
30	C₁₆×D₆₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₆₁	C₁₆	1	2	yes	—
36	#11758222	C₁₆	1	4	yes	—
37	#11758222	C₂×C₈	1	4	yes	—
38	C₂×D₂₄₄	C₁₆	1	1	yes	characteristic
39	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
40	C₂×Dic₁₂₂	C₁₆	1	1	yes	characteristic
41	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
45	#11758194	C₂×C₈	1	4	yes	—
46	C₂×#92418	C₂×C₈	1	2	yes	—
47	#11758189	C₂×C₈	1	8	yes	—
48	C₈×D₆₁	C₂×C₈	1	4	yes	—
49	C₈×D₆₁	C₂²×C₄	1	4	yes	—
50	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
51	#11758185	C₂²×C₄	1	1	yes	characteristic
52	#11758185	C₂²×C₄	1	2	yes	—
53	#11758185	C₂⁴	1	1	yes	characteristic
54	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₆₁	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
60	C₂²⋊D₆₁	C₂×C₁₆	1	8	yes	—
61	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
62	D₂₄₄	C₂×C₁₆	1	4	yes	—
63	C₄×D₆₁	C₂×C₁₆	1	4	yes	—
64	C₄×D₆₁	C₂²×C₈	1	4	yes	—
65	Dic₁₂₂	C₂×C₁₆	1	4	yes	—
66	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₆₁	C₂²×C₈	1	2	yes	—
68	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
69	#92418	C₂²×C₈	1	4	yes	—
70	#33974	—	61	1	no	maximal, Sylow
71	C₂²×C₆₁	—	4	1	no	—
72	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
73	C₂²×C₆₁	C₂²×C₁₆	1	2	yes	—
74	C₂×D₆₁	—	4	2	no	—
75	C₂×D₆₁	C₂²×C₁₆	1	4	yes	—
76	C₄×C₆₁	C₂²×C₁₆	1	2	yes	—
77	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
79	Dic₆₁	C₂²×C₁₆	1	4	yes	—
80	C₂×#504	—	61	1	no	—
81	#2258	—	61	8	no	—
82	C₂×M₆₄	—	61	2	no	—
83	C₂×M₆₄	—	61	4	no	—
84	C₂×C₆₁	—	2	1	no	—
85	C₂×C₆₁	—	2	2	no	—
86	C₂×C₆₁	C₂³×C₁₆	1	1	yes	characteristic
87	D₆₁	—	2	4	no	—
88	C₂×#131	—	61	1	no	—
89	#504	—	61	4	no	—
90	#504	—	61	4	no	—
91	C₂×M₃₂	—	61	2	no	—
92	C₂×M₃₂	#6923	1	1	yes	characteristic
93	C₂²×C₁₆	—	61	1	no	—
94	C₂²×C₁₆	—	61	2	no	—
95	M₆₄	—	61	4	no	—
96	M₆₄	—	61	4	no	—
97	M₆₄	—	61	8	no	—
98	C₂×C₃₂	—	61	4	no	—
99	C₂×C₃₂	—	61	8	no	—
100	C₆₁	#33974	1	1	yes	characteristic, complemented, Sylow
101	C₂×#41	—	61	1	no	—
102	#131	—	61	4	no	—
103	#131	—	61	4	no	—
104	C₂×M₁₆	—	61	1	no	—
105	C₂×M₁₆	—	61	2	no	—
106	C₂²×C₈	—	61	2	no	—
107	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
108	M₃₂	—	61	8	no	—
109	M₃₂	C₂×#6923	1	4	yes	—
110	C₂×C₁₆	—	61	2	no	—
111	C₂×C₁₆	—	61	1	no	—
112	C₂×C₁₆	—	61	1	no	—
113	C₂×C₁₆	—	61	2	no	—
114	C₂×C₁₆	—	61	4	no	—
115	C₂×C₁₆	—	61	4	no	—
116	C₂×C₁₆	C₂×#6923	1	2	yes	—
117	C₃₂	—	61	8	no	—
118	#41	—	61	4	no	—
119	#41	—	61	4	no	—
120	C₂×Q₈	—	61	1	no	—
121	C₂×D₄	—	61	2	no	—
122	C₂×D₄	—	61	1	no	—
123	C₂²×C₄	—	61	2	no	—
124	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
125	M₁₆	—	61	4	no	—
126	M₁₆	—	61	8	no	—
127	C₂×C₈	—	61	2	no	—
128	C₂×C₈	—	61	4	no	—
129	C₂×C₈	—	61	4	no	—
130	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
131	C₂×C₈	D₆₁⋊C₈	1	2	yes	—
132	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
133	C₂×C₈	C₂²×#6923	1	2	yes	—
134	C₁₆	—	61	1	no	—
135	C₁₆	—	61	1	no	—
136	C₁₆	—	61	2	no	—
137	C₁₆	C₂²×#6923	1	4	yes	—
138	C₂³	—	61	2	no	—
139	C₂³	D₆₁⋊C₁₆	1	1	yes	characteristic
140	Q₈	—	61	4	no	—
141	D₄	—	61	8	no	—
142	D₄	—	61	4	no	—
143	C₂×C₄	—	61	2	no	—
144	C₂×C₄	—	61	4	no	—
145	C₂×C₄	—	61	4	no	—
146	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
147	C₂×C₄	D₆₁⋊C₁₆	1	2	yes	—
148	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
149	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
150	C₈	—	61	4	no	—
151	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
152	C₈	C₂×D₆₁⋊C₈	1	2	yes	—
153	C₈	C₂³×#6923	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	—	61	4	no	—
156	C₂²	—	244	2	no	—
157	C₂²	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
158	C₂²	C₂×D₆₁⋊C₁₆	1	2	yes	—
159	C₄	—	61	4	no	—
160	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
161	C₄	C₂×D₆₁⋊C₁₆	1	2	yes	—
162	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	2	no	—
165	C₂	—	122	4	no	—
166	C₂	C₂²×D₆₁⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#6923
commutator, socle	C₂×C₆₁	C₂³×C₁₆
Fitting	C₂×M₃₂×C₆₁	C₄
prime radical	C₂×#11758222	C₈
cosocle	#11758185	C₂⁴
hypercenter	C₂×M₃₂	#6923
hypocenter	C₆₁	#33974
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. #21462243	C₂	semidirect
2. D₆₁⋊M₆₄	C₂	semidirect
3. C₂×#16840694	C₂	semidirect
5. #16840695	C₂²	semidirect
7. #16840694	C₂²	semidirect
8. D₆₁⋊C₃₂	C₂²	semidirect
9. C₂×#14519827	C₂²	semidirect
100. C₆₁	#33974	semidirect

Maximal subgroups

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

subgroup	index
#21462243	2
D₆₁⋊M₆₄	2
C₂×#16840694	2
C₂×#16840180	2
#33974	61