Subgroups — C₂²×#16840603 #30609657

Order 15616.

1182 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#92420	C₂	1	2	yes	maximal
2	C₂×#16840603	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×#92420	C₄	1	4	yes	—
5	C₂×C₄×#92420	C₄	1	12	yes	—
6	C₂×C₄×#92420	C₂²	1	12	yes	—
7	#16840603	C₂²	1	16	yes	direct product factor
8	C₂³×#92418	C₄	1	1	yes	characteristic
9	C₂²×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	6	yes	—
11	C₂×C₈×Dic₆₁	C₂²	1	6	yes	—
12	C₂³×C₈×C₆₁	C₄	1	1	yes	characteristic
13	C₂²×#92420	—	2	16	no	—
14	C₂²×#92420	C₂×C₄	1	24	yes	—
15	C₄×#92420	C₂×C₄	1	24	yes	—
16	C₄×#92420	C₂³	1	8	yes	—
17	C₂³×Dic₆₁	C₂×C₄	1	2	yes	—
18	C₂²×#92418	C₈	1	8	yes	complemented
19	C₂²×#92418	C₂×C₄	1	6	yes	—
20	C₂×C₄×Dic₆₁	C₂×C₄	1	3	yes	—
21	C₂×C₄×Dic₆₁	C₂×C₄	1	6	yes	—
22	C₂×C₄×Dic₆₁	C₂³	1	3	yes	—
23	C₈×Dic₆₁	C₂×C₄	1	12	yes	—
24	C₈×Dic₆₁	C₂³	1	4	yes	—
25	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₈	1	8	yes	complemented
27	C₂²×C₈×C₆₁	C₂×C₄	1	6	yes	—
28	C₂⁴×C₆₁	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₄²	1	3	yes	—
30	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
31	C₂²×C₄×C₆₁	C₂×C₈	1	4	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	3	yes	—
33	C₂²×Dic₆₁	C₄²	1	6	yes	—
34	C₂²×Dic₆₁	M₁₆	1	16	yes	—
35	C₂²×Dic₆₁	C₂²×C₄	1	6	yes	—
36	C₂×#92420	—	2	48	no	—
37	C₂×#92420	C₂²×C₄	1	16	yes	—
38	C₂×C₈×C₆₁	C₂×C₈	1	24	yes	complemented
39	C₂×C₈×C₆₁	C₂²×C₄	1	4	yes	—
40	C₄×Dic₆₁	C₄²	1	6	yes	—
41	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
42	C₄×Dic₆₁	C₂²×C₄	1	6	yes	—
43	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
44	C₂×#92418	C₂×C₈	1	24	yes	complemented
45	C₂×#92418	C₂²×C₄	1	4	yes	—
46	C₂³×C₆₁	C₄×C₈	1	4	yes	—
47	C₂³×C₆₁	#97	1	8	yes	—
48	C₂³×C₆₁	C₂×C₄²	1	3	yes	—
49	C₂×C₄×C₆₁	C₄×C₈	1	12	yes	—
50	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
51	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
52	C₂×C₄×C₆₁	C₂²×C₈	1	6	yes	—
53	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
54	C₂×Dic₆₁	#97	1	24	yes	—
55	C₂×Dic₆₁	C₂×C₄²	1	6	yes	—
56	C₂×Dic₆₁	C₂×M₁₆	1	24	yes	—
57	C₂×Dic₆₁	C₂³×C₄	1	2	yes	—
58	#92420	—	2	16	no	—
59	C₈×C₆₁	C₂²×C₈	1	8	yes	complemented
60	#92418	C₂²×C₈	1	8	yes	complemented
61	C₂²×#322	—	61	1	no	maximal, Sylow
62	C₂²×C₆₁	#322	1	16	yes	complemented
63	C₂²×C₆₁	C₂×C₄×C₈	1	6	yes	—
64	C₂²×C₆₁	C₂×#97	1	12	yes	—
65	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
66	C₄×C₆₁	C₂×C₄×C₈	1	6	yes	—
67	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
68	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
69	Dic₆₁	C₂×#97	1	12	yes	—
70	Dic₆₁	C₂²×M₁₆	1	4	yes	—
71	C₂²×C₄×C₈	—	61	2	no	—
72	C₂²×C₄×C₈	—	61	1	no	—
73	C₂×#322	—	61	12	no	—
74	C₂×C₆₁	C₂×#322	1	12	yes	complemented
75	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
76	C₂×C₆₁	C₂²×#97	1	2	yes	—
77	C₂³×C₈	—	61	1	no	—
78	C₂³×C₈	—	61	4	no	—
79	C₂³×C₈	#6923	1	1	yes	characteristic
80	C₂²×C₄²	—	61	1	no	—
81	C₂×C₄×C₈	—	61	12	no	—
82	C₂×C₄×C₈	—	61	12	no	—
83	C₂×C₄×C₈	—	61	6	no	—
84	C₂×C₄×C₈	—	61	6	no	—
85	#322	—	61	16	no	—
86	C₆₁	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₄	—	61	2	no	—
88	C₂³×C₄	C₂×#6923	1	1	yes	characteristic
89	C₂²×C₈	—	61	8	no	—
90	C₂²×C₈	—	61	6	no	—
91	C₂²×C₈	—	61	24	no	—
92	C₂²×C₈	—	122	16	no	—
93	C₂²×C₈	#92420	1	8	yes	complemented
94	C₂²×C₈	C₂×#6923	1	6	yes	—
95	C₂×C₄²	—	61	6	no	—
96	C₂×C₄²	—	61	3	no	—
97	C₂×C₄²	—	61	3	no	—
98	C₄×C₈	—	61	24	no	—
99	C₄×C₈	—	61	8	no	—
100	C₄×C₈	—	61	4	no	—
101	C₄×C₈	—	61	12	no	—
102	C₂⁴	C₄×#6923	1	1	yes	characteristic
103	C₂²×C₄	—	61	16	no	—
104	C₂²×C₄	—	61	6	no	—
105	C₂²×C₄	—	61	6	no	—
106	C₂²×C₄	D₆₁⋊C₈	1	4	yes	—
107	C₂²×C₄	C₄×#6923	1	3	yes	—
108	C₂²×C₄	C₂×#92420	1	4	yes	—
109	C₂²×C₄	C₂²×#6923	1	3	yes	—
110	C₂×C₈	—	61	24	no	—
111	C₂×C₈	—	61	4	no	—
112	C₂×C₈	—	61	16	no	—
113	C₂×C₈	—	122	48	no	—
114	C₂×C₈	C₂×#92420	1	24	yes	complemented
115	C₂×C₈	C₂²×#6923	1	4	yes	—
116	C₄²	—	61	6	no	—
117	C₄²	—	61	6	no	—
118	C₄²	—	61	1	no	—
119	C₄²	—	61	3	no	—
120	C₂³	C₄×#92420	1	4	yes	—
121	C₂³	#14519831	1	8	yes	—
122	C₂³	C₂×C₄×#6923	1	3	yes	—
123	C₂×C₄	—	61	24	no	—
124	C₂×C₄	—	61	24	no	—
125	C₂×C₄	—	61	6	no	—
126	C₂×C₄	—	61	2	no	—
127	C₂×C₄	C₄×#92420	1	12	yes	—
128	C₂×C₄	C₂×C₄×#6923	1	3	yes	—
129	C₂×C₄	C₂²×#92420	1	6	yes	—
130	C₂×C₄	C₂×D₆₁⋊C₈	1	6	yes	—
131	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
132	C₈	—	61	8	no	—
133	C₈	—	122	16	no	—
134	C₈	C₂²×#92420	1	8	yes	complemented
135	C₂²	#16840603	1	16	yes	direct product factor
136	C₂²	C₂×C₄×#92420	1	6	yes	—
137	C₂²	C₂×#14519831	1	12	yes	—
138	C₂²	C₂²×C₄×#6923	1	1	yes	characteristic
139	C₄	—	61	4	no	—
140	C₄	—	61	12	no	—
141	C₄	C₂×C₄×#92420	1	6	yes	—
142	C₄	C₂³×#92420	1	1	yes	characteristic
143	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
144	C₂	C₂×#16840603	1	12	yes	direct product factor
145	C₂	C₂²×C₄×#92420	1	1	yes	characteristic
146	C₂	C₂²×#14519831	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂³×C₈×C₆₁	C₄
socle, prime radical	C₂⁴×C₆₁	C₄²
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂³×C₈	#6923
hypocenter	C₆₁	C₂²×#322
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
2. C₂×#16840603	C₂	direct (N × H)
7. #16840603	C₂²	direct (N × H)
18. C₂²×#92418	C₈	semidirect
26. C₂²×C₈×C₆₁	C₈	semidirect
38. C₂×C₈×C₆₁	C₂×C₈	semidirect
44. C₂×#92418	C₂×C₈	semidirect
59. C₈×C₆₁	C₂²×C₈	semidirect
60. #92418	C₂²×C₈	semidirect
62. C₂²×C₆₁	#322	semidirect
74. C₂×C₆₁	C₂×#322	semidirect
86. C₆₁	C₂²×#322	semidirect
93. C₂²×C₈	#92420	semidirect
114. C₂×C₈	C₂×#92420	semidirect
134. C₈	C₂²×#92420	semidirect
135. C₂²	#16840603	direct (N × H)
144. C₂	C₂×#16840603	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×#92420	2
C₂×#16840603	2
C₂²×C₈×Dic₆₁	2
C₂²×#322	61

Subgroups — C22×#16840603 #30609657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840603 #30609657