Subgroups — C₂³×#14519831 #30625090

Order 15616.

4126 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×#6923	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14519831	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×#6923	C₄	1	2	yes	—
6	C₂²×D₆₁⋊C₈	C₄	1	14	yes	complemented
7	C₂²×D₆₁⋊C₈	C₂²	1	14	yes	—
8	C₂³×#92420	C₄	1	2	yes	complemented
9	C₂²×C₄×#6923	C₄	1	14	yes	—
10	C₂²×C₄×#6923	C₂²	1	14	yes	—
11	C₂×#14519831	C₂²	1	112	yes	direct product factor
12	C₂³×C₄×D₆₁	C₂²	1	1	yes	characteristic
13	C₂³×#92418	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₈×D₆₁	C₄	1	14	yes	complemented
15	C₂²×C₈×D₆₁	C₂²	1	14	yes	—
16	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic, complemented
17	C₂³×#6923	—	2	16	no	—
18	C₂³×#6923	C₂×C₄	1	28	yes	—
19	C₂×D₆₁⋊C₈	C₂×C₄	1	84	yes	complemented
20	C₂×D₆₁⋊C₈	C₂³	1	28	yes	—
21	C₂²×#92420	C₂×C₄	1	28	yes	complemented
22	C₂×C₄×#6923	C₂×C₄	1	84	yes	—
23	C₂×C₄×#6923	C₂³	1	28	yes	—
24	#14519831	C₂³	1	64	yes	direct product factor
25	C₂⁴×D₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₆₁	C₂×C₄	1	7	yes	—
28	C₂²×C₄×D₆₁	C₂×C₄	1	7	yes	—
29	C₂²×C₄×D₆₁	C₂×C₄	1	7	yes	—
30	C₂²×C₄×D₆₁	C₂³	1	7	yes	—
31	C₂²×#92418	C₂×C₄	1	14	yes	complemented
32	C₂×C₈×D₆₁	C₂×C₄	1	84	yes	complemented
33	C₂×C₈×D₆₁	C₂³	1	28	yes	—
34	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂×C₄	1	14	yes	complemented
36	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
37	C₂³×D₆₁	C₄²	1	7	yes	—
38	C₂³×D₆₁	M₁₆	1	16	yes	—
39	C₂³×D₆₁	C₂²×C₄	1	7	yes	—
40	C₂²×#6923	—	2	112	no	—
41	C₂²×#6923	C₂²×C₄	1	56	yes	—
42	C₂²×C₄×C₆₁	C₄²	1	7	yes	—
43	C₂²×C₄×C₆₁	C₂²×C₄	1	7	yes	—
44	C₂²×Dic₆₁	C₄²	1	7	yes	—
45	C₂²×Dic₆₁	C₂²×C₄	1	7	yes	—
46	C₂×C₄×D₆₁	C₄²	1	42	yes	—
47	C₂×C₄×D₆₁	C₂²×C₄	1	21	yes	—
48	C₂×C₄×D₆₁	C₂²×C₄	1	21	yes	—
49	C₂×C₄×D₆₁	C₂²×C₄	1	21	yes	—
50	C₂×C₄×D₆₁	C₂⁴	1	7	yes	—
51	C₂×#92420	C₂²×C₄	1	56	yes	complemented
52	C₄×#6923	C₂²×C₄	1	56	yes	—
53	C₄×#6923	C₂⁴	1	8	yes	—
54	D₆₁⋊C₈	C₂²×C₄	1	56	yes	complemented
55	D₆₁⋊C₈	C₂⁴	1	8	yes	—
56	C₂×C₈×C₆₁	C₂²×C₄	1	28	yes	complemented
57	C₂×#92418	C₂²×C₄	1	28	yes	complemented
58	C₈×D₆₁	C₂²×C₄	1	56	yes	complemented
59	C₈×D₆₁	C₂⁴	1	8	yes	—
60	C₂³×C₆₁	#97	1	8	yes	complemented
61	C₂³×C₆₁	C₂×C₄²	1	7	yes	—
62	C₂²×D₆₁	#97	1	56	yes	complemented
63	C₂²×D₆₁	C₂×C₄²	1	21	yes	—
64	C₂²×D₆₁	C₂×M₁₆	1	56	yes	—
65	C₂²×D₆₁	C₂³×C₄	1	7	yes	—
66	C₂×#6923	—	2	112	no	—
67	C₂×#6923	C₂³×C₄	1	16	yes	—
68	C₂×C₄×C₆₁	C₂×C₄²	1	21	yes	—
69	C₂×C₄×C₆₁	C₂³×C₄	1	7	yes	—
70	C₂×Dic₆₁	C₂×C₄²	1	21	yes	—
71	C₂×Dic₆₁	C₂³×C₄	1	7	yes	—
72	C₄×D₆₁	C₂×C₄²	1	42	yes	—
73	C₄×D₆₁	C₂³×C₄	1	7	yes	—
74	C₄×D₆₁	C₂³×C₄	1	7	yes	—
75	C₄×D₆₁	C₂³×C₄	1	7	yes	—
76	C₄×D₆₁	C₂⁵	1	1	yes	characteristic
77	#92420	C₂³×C₄	1	16	yes	complemented
78	C₈×C₆₁	C₂³×C₄	1	8	yes	complemented
79	#92418	C₂³×C₄	1	8	yes	complemented
80	C₂³×#97	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₂×#97	1	28	yes	complemented
82	C₂²×C₆₁	C₂²×C₄²	1	7	yes	—
83	C₂×D₆₁	C₂×#97	1	84	yes	complemented
84	C₂×D₆₁	C₂²×C₄²	1	7	yes	—
85	C₂×D₆₁	C₂²×M₁₆	1	28	yes	—
86	C₂×D₆₁	C₂⁴×C₄	1	1	yes	characteristic
87	#6923	—	2	16	no	—
88	C₄×C₆₁	C₂²×C₄²	1	7	yes	—
89	C₄×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
90	Dic₆₁	C₂²×C₄²	1	7	yes	—
91	Dic₆₁	C₂⁴×C₄	1	1	yes	characteristic
92	C₂⁴×C₈	—	61	1	no	—
93	C₂⁴×C₈	—	61	1	no	—
94	C₂³×C₄²	—	61	1	no	—
95	C₂²×#97	—	61	28	no	—
96	C₂×C₆₁	C₂²×#97	1	14	yes	complemented
97	C₂×C₆₁	C₂³×C₄²	1	1	yes	characteristic
98	D₆₁	C₂²×#97	1	14	yes	complemented
99	D₆₁	C₂³×M₁₆	1	2	yes	—
100	C₂⁴×C₄	—	61	1	no	—
101	C₂⁴×C₄	—	61	2	no	—
102	C₂³×C₈	—	61	14	no	—
103	C₂³×C₈	—	61	2	no	—
104	C₂³×C₈	—	61	14	no	—
105	C₂³×C₈	—	61	1	no	—
106	C₂³×C₈	—	61	14	no	—
107	C₂³×C₈	—	61	14	no	—
108	C₂³×C₈	#6923	1	1	yes	characteristic, complemented
109	C₂²×C₄²	—	61	14	no	—
110	C₂²×C₄²	—	61	14	no	—
111	C₂×#97	—	61	112	no	—
112	C₆₁	C₂³×#97	1	1	yes	characteristic, complemented, Sylow
113	C₂⁵	—	61	1	no	—
114	C₂³×C₄	—	61	1	no	—
115	C₂³×C₄	—	61	7	no	—
116	C₂³×C₄	—	61	7	no	—
117	C₂³×C₄	—	61	7	no	—
118	C₂³×C₄	—	61	7	no	—
119	C₂³×C₄	—	61	28	no	—
120	C₂³×C₄	—	122	16	no	—
121	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
122	C₂²×C₈	—	61	28	no	—
123	C₂²×C₈	—	61	84	no	—
124	C₂²×C₈	—	61	28	no	—
125	C₂²×C₈	—	61	14	no	—
126	C₂²×C₈	—	61	28	no	—
127	C₂²×C₈	—	61	84	no	—
128	C₂²×C₈	C₂×#6923	1	14	yes	complemented
129	C₂×C₄²	—	61	84	no	—
130	C₂×C₄²	—	61	28	no	—
131	#97	—	61	64	no	—
132	C₂⁴	—	61	16	no	—
133	C₂⁴	—	61	7	no	—
134	C₂⁴	—	61	7	no	—
135	C₂⁴	C₄×#6923	1	1	yes	characteristic
136	C₂²×C₄	—	61	7	no	—
137	C₂²×C₄	—	61	7	no	—
138	C₂²×C₄	—	61	21	no	—
139	C₂²×C₄	—	61	21	no	—
140	C₂²×C₄	—	61	42	no	—
141	C₂²×C₄	—	61	7	no	—
142	C₂²×C₄	—	61	21	no	—
143	C₂²×C₄	—	61	56	no	—
144	C₂²×C₄	—	122	112	no	—
145	C₂²×C₄	C₄×#6923	1	7	yes	—
146	C₂²×C₄	C₂²×#6923	1	7	yes	—
147	C₂×C₈	—	61	8	no	—
148	C₂×C₈	—	61	56	no	—
149	C₂×C₈	—	61	56	no	—
150	C₂×C₈	—	61	28	no	—
151	C₂×C₈	—	61	8	no	—
152	C₂×C₈	—	61	56	no	—
153	C₂×C₈	C₂²×#6923	1	28	yes	complemented
154	C₄²	—	61	8	no	—
155	C₄²	—	61	56	no	—
156	C₂³	—	61	56	no	—
157	C₂³	—	61	56	no	—
158	C₂³	—	61	21	no	—
159	C₂³	—	61	7	no	—
160	C₂³	#14519831	1	8	yes	direct product factor
161	C₂³	C₂×C₄×#6923	1	7	yes	—
162	C₂×C₄	—	61	21	no	—
163	C₂×C₄	—	61	7	no	—
164	C₂×C₄	—	61	42	no	—
165	C₂×C₄	—	61	7	no	—
166	C₂×C₄	—	61	7	no	—
167	C₂×C₄	—	61	7	no	—
168	C₂×C₄	—	61	1	no	—
169	C₂×C₄	—	61	16	no	—
170	C₂×C₄	—	122	112	no	—
171	C₂×C₄	C₂×C₄×#6923	1	21	yes	—
172	C₂×C₄	C₂³×#6923	1	7	yes	—
173	C₈	—	61	16	no	—
174	C₈	—	61	8	no	—
175	C₈	C₂³×#6923	1	8	yes	complemented
176	C₂²	—	61	1	no	—
177	C₂²	—	61	7	no	—
178	C₂²	—	61	28	no	—
179	C₂²	—	61	84	no	—
180	C₂²	C₂×#14519831	1	28	yes	direct product factor
181	C₂²	C₂²×C₄×#6923	1	7	yes	—
182	C₄	—	61	1	no	—
183	C₄	—	61	7	no	—
184	C₄	—	122	16	no	—
185	C₄	C₂²×C₄×#6923	1	7	yes	—
186	C₄	C₂⁴×#6923	1	1	yes	characteristic
187	C₂	—	61	2	no	—
188	C₂	—	61	14	no	—
189	C₂	C₂²×#14519831	1	14	yes	direct product factor
190	C₂	C₂³×C₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×#6923
commutator	C₂×C₆₁	C₂³×C₄²
Frattini	C₄	C₂⁴×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle	C₂⁴×C₆₁	C₄²
prime radical	C₂⁴×D₆₁	C₂×C₄
cosocle	C₄×D₆₁	C₂⁵
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂³×#97
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#14519831	C₂	direct (N × H)
6. C₂²×D₆₁⋊C₈	C₄	semidirect
8. C₂³×#92420	C₄	semidirect
11. C₂×#14519831	C₂²	direct (N × H)
13. C₂³×#92418	C₄	semidirect
14. C₂²×C₈×D₆₁	C₄	semidirect
16. C₂³×C₈×C₆₁	C₄	semidirect
19. C₂×D₆₁⋊C₈	C₂×C₄	semidirect
21. C₂²×#92420	C₂×C₄	semidirect
24. #14519831	C₂³	direct (N × H)
31. C₂²×#92418	C₂×C₄	semidirect
32. C₂×C₈×D₆₁	C₂×C₄	semidirect
35. C₂²×C₈×C₆₁	C₂×C₄	semidirect
51. C₂×#92420	C₂²×C₄	semidirect
54. D₆₁⋊C₈	C₂²×C₄	semidirect
56. C₂×C₈×C₆₁	C₂²×C₄	semidirect
57. C₂×#92418	C₂²×C₄	semidirect
58. C₈×D₆₁	C₂²×C₄	semidirect
60. C₂³×C₆₁	#97	semidirect
62. C₂²×D₆₁	#97	semidirect
77. #92420	C₂³×C₄	semidirect
78. C₈×C₆₁	C₂³×C₄	semidirect
79. #92418	C₂³×C₄	semidirect
81. C₂²×C₆₁	C₂×#97	semidirect
83. C₂×D₆₁	C₂×#97	semidirect
96. C₂×C₆₁	C₂²×#97	semidirect
98. D₆₁	C₂²×#97	semidirect
108. C₂³×C₈	#6923	semidirect
112. C₆₁	C₂³×#97	semidirect
128. C₂²×C₈	C₂×#6923	semidirect
153. C₂×C₈	C₂²×#6923	semidirect
160. C₂³	#14519831	direct (N × H)
175. C₈	C₂³×#6923	semidirect
180. C₂²	C₂×#14519831	direct (N × H)
189. C₂	C₂²×#14519831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₆₁⋊C₈	2
C₂³×C₄×#6923	2
C₂²×#14519831	2
C₂³×C₈×D₆₁	2
C₂³×#97	61

Subgroups — C23×#14519831 #30625090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14519831 #30625090