Subgroups — #30606114

Order 15616.

478 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840649	C₂	1	2	yes	maximal
2	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×#6923	C₂	1	2	yes	maximal
4	#21461475	C₂	1	1	yes	maximal, characteristic
5	#346×D₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16840721	C₄	1	1	yes	characteristic
9	C₂×#14519831	C₄	1	2	yes	—
10	C₂×C₈×#6923	C₂²	1	2	yes	—
11	#16840698	C₄	1	1	yes	characteristic
12	#16840649	C₄	1	8	yes	complemented
13	#16840648	C₄	1	2	yes	complemented
14	C₄×#11758187	C₄	1	2	yes	complemented
15	C₁₆×#6923	C₄	1	8	yes	complemented
16	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
17	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
18	#16839538	C₄	1	1	yes	characteristic, complemented
19	#16839537	C₄	1	2	yes	complemented
20	C₆₁×#346	C₄	1	1	yes	characteristic, complemented
21	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
23	D₆₁⋊C₁₆	C₂×C₄	1	4	yes	—
24	D₆₁⋊C₁₆	C₂×C₄	1	4	yes	—
25	C₂×#11758187	C₂×C₄	1	2	yes	—
26	C₂×#11758187	C₂×C₄	1	2	yes	—
27	#14519832	C₈	1	4	yes	—
28	#14519831	C₈	1	8	yes	—
29	C₈×#6923	C₂×C₄	1	4	yes	—
30	C₈×#6923	C₂×C₄	1	4	yes	—
31	(C₂×#6923)⋊C₄	C₈	1	4	yes	—
32	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
34	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×#11758185	C₂×C₄	1	2	yes	—
36	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
37	C₁₆×D₆₁	C₂×C₄	1	4	yes	—
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×#6923	C₂×C₈	1	4	yes	—
44	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
45	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×#92420	C₂×C₈	1	4	yes	—
47	C₄×#6923	C₄²	1	4	yes	—
48	C₄×#6923	C₂×C₈	1	4	yes	—
49	D₆₁⋊C₈	C₄²	1	4	yes	—
50	D₆₁⋊C₈	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₆₁	C₄²	1	1	yes	characteristic
54	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
55	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
56	C₂×#92418	C₄²	1	1	yes	characteristic
57	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
58	C₈×D₆₁	C₂×C₈	1	4	yes	—
59	C₈×D₆₁	C₂²×C₄	1	2	yes	—
60	C₈×D₆₁	C₂²×C₄	1	2	yes	—
61	#11758187	C₄²	1	4	yes	—
62	#11758187	C₄²	1	4	yes	—
63	C₁₆×C₆₁	C₄²	1	4	yes	—
64	#11758185	C₄²	1	4	yes	—
65	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
66	C₂×#6923	—	2	8	no	—
67	C₂×#6923	C₄×C₈	1	8	yes	—
68	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
70	C₂×Dic₆₁	C₄×C₈	1	2	yes	—
71	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
72	C₄×D₆₁	—	2	4	no	—
73	C₄×D₆₁	C₂×C₄²	1	2	yes	—
74	C₄×D₆₁	C₂²×C₈	1	2	yes	—
75	#92420	C₄×C₈	1	8	yes	—
76	C₈×C₆₁	C₄×C₈	1	2	yes	—
77	C₈×C₆₁	C₂×C₄²	1	2	yes	—
78	#92418	C₄×C₈	1	2	yes	—
79	#92418	C₂×C₄²	1	1	yes	characteristic
80	#92418	C₂×C₄²	1	1	yes	characteristic
81	#12539	—	61	1	no	maximal, Sylow
82	C₂²×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂×D₆₁	C₂×C₄×C₈	1	2	yes	—
84	C₂×D₆₁	#504	1	4	yes	—
85	#6923	—	2	8	no	—
86	C₄×C₆₁	—	2	2	no	—
87	C₄×C₆₁	C₄³	1	1	yes	characteristic
88	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
89	Dic₆₁	—	2	2	no	—
90	Dic₆₁	C₂×C₄×C₈	1	2	yes	—
91	C₂×#346	—	61	2	no	—
92	C₂×#346	—	61	1	no	—
93	C₂×C₄×C₁₆	—	61	1	no	—
94	C₂×C₄×C₁₆	—	61	2	no	—
95	#1726	—	61	1	no	—
96	C₂×C₆₁	C₄²×C₈	1	1	yes	characteristic
97	C₂×C₆₁	#2108	1	2	yes	—
98	D₆₁	#2108	1	4	yes	—
99	C₂²×C₁₆	—	61	1	no	—
100	C₂²×C₁₆	—	61	1	no	—
101	C₂²×C₁₆	—	61	2	no	—
102	C₂×#97	—	61	2	no	—
103	C₂×#97	—	61	1	no	—
104	C₂×C₄×C₈	—	61	2	no	—
105	C₂×C₄×C₈	—	61	1	no	—
106	#376	—	61	1	no	—
107	#346	—	61	8	no	—
108	#346	—	61	2	no	—
109	#346	—	61	1	no	—
110	#346	#6923	1	1	yes	characteristic, complemented
111	C₄×C₁₆	—	61	8	no	—
112	C₄×C₁₆	—	61	2	no	—
113	C₄×C₁₆	—	61	2	no	—
114	C₆₁	#12539	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	—	61	2	no	—
116	C₂²×C₈	—	61	1	no	—
117	C₂²×C₈	—	61	1	no	—
118	C₂×C₄²	—	61	2	no	—
119	C₂×C₄²	—	61	1	no	—
120	C₂×C₁₆	—	61	2	no	—
121	C₂×C₁₆	—	61	2	no	—
122	C₂×C₁₆	—	61	4	no	—
123	C₂×C₁₆	—	61	4	no	—
124	C₂×C₁₆	—	61	2	no	—
125	C₂×C₁₆	—	61	4	no	—
126	C₂×C₁₆	—	61	4	no	—
127	C₂×C₁₆	C₂×#6923	1	2	yes	—
128	#97	—	61	8	no	—
129	#97	—	61	4	no	—
130	C₄×C₈	—	61	4	no	—
131	C₄×C₈	—	61	4	no	—
132	C₄×C₈	—	61	1	no	—
133	C₄×C₈	—	61	1	no	—
134	C₄×C₈	—	61	1	no	—
135	C₄×C₈	C₂×#6923	1	1	yes	characteristic
136	#95	—	61	4	no	—
137	C₂²×C₄	—	61	1	no	—
138	C₂²×C₄	—	61	4	no	—
139	C₂²×C₄	—	61	2	no	—
140	C₂×C₈	—	61	4	no	—
141	C₂×C₈	—	61	4	no	—
142	C₂×C₈	—	61	4	no	—
143	C₂×C₈	—	61	1	no	—
144	C₂×C₈	—	61	1	no	—
145	C₂×C₈	—	61	4	no	—
146	C₂×C₈	—	61	2	no	—
147	C₂×C₈	—	61	2	no	—
148	C₂×C₈	C₄×#6923	1	1	yes	characteristic
149	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
150	C₄²	—	61	4	no	—
151	C₄²	—	61	4	no	—
152	C₄²	—	61	2	no	—
153	C₄²	—	61	1	no	—
154	C₄²	C₄×#6923	1	1	yes	characteristic
155	C₁₆	—	61	4	no	—
156	C₁₆	—	61	4	no	—
157	C₁₆	—	61	4	no	—
158	C₁₆	C₄×#6923	1	4	yes	—
159	C₂³	—	61	1	no	—
160	C₂×C₄	—	61	1	no	—
161	C₂×C₄	—	61	2	no	—
162	C₂×C₄	—	61	2	no	—
163	C₂×C₄	—	61	8	no	—
164	C₂×C₄	—	61	2	no	—
165	C₂×C₄	—	122	4	no	—
166	C₂×C₄	—	122	8	no	—
167	C₂×C₄	C₈×#6923	1	2	yes	—
168	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
169	C₈	—	61	8	no	—
170	C₈	—	61	2	no	—
171	C₈	—	61	1	no	—
172	C₈	—	61	1	no	—
173	C₈	C₈×#6923	1	2	yes	—
174	C₈	C₂×C₄×#6923	1	2	yes	—
175	C₂²	—	61	4	no	—
176	C₂²	—	61	2	no	—
177	C₂²	C₂×C₈×#6923	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	61	2	no	—
180	C₄	—	122	8	no	—
181	C₄	—	122	2	no	—
182	C₄	C₄²×#6923	1	1	yes	characteristic
183	C₄	C₂×C₈×#6923	1	1	yes	characteristic
184	C₂	—	61	4	no	—
185	C₂	C₄×C₈×#6923	1	1	yes	characteristic
186	C₂	#21462031	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6923
commutator	C₂×C₆₁	C₄²×C₈
Fitting	C₆₁×#346	C₄
socle	C₂²×C₆₁	C₂×C₄×C₈
prime radical	C₂²×D₆₁	C₂²×C₈
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	#346	#6923
hypocenter	C₆₁	#12539
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. #16840649	C₄	semidirect
13. #16840648	C₄	semidirect
14. C₄×#11758187	C₄	semidirect
15. C₁₆×#6923	C₄	semidirect
18. #16839538	C₄	semidirect
19. #16839537	C₄	semidirect
20. C₆₁×#346	C₄	semidirect
110. #346	#6923	semidirect
114. C₆₁	#12539	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840649	2
C₄×D₆₁⋊C₁₆	2
C₂×C₁₆×#6923	2
#21461475	2
#346×D₆₁	2
#12539	61