Subgroups — C₄²×C₄⋊Dic₆₁ #29601092

Order 15616.

2478 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	C₂×C₄×C₄⋊Dic₆₁	C₂	1	6	yes	maximal
2	C₂×C₄²×Dic₆₁	C₂	1	2	yes	maximal
3	C₄×#14519684	C₂	1	6	yes	maximal
4	C₂×C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₆₁	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₆₁	C₂²	1	12	yes	—
7	C₄×C₄⋊Dic₆₁	C₄	1	48	yes	direct product factor
8	C₂×#14519684	C₂²	1	12	yes	—
9	C₄²×Dic₆₁	—	2	8	no	—
10	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
11	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
12	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₄³×C₆₁	C₄	1	8	yes	complemented
14	C₂³×Dic₆₁	C₂³	1	8	yes	—
15	C₂×C₄⋊Dic₆₁	C₂×C₄	1	48	yes	—
16	C₂×C₄×Dic₆₁	—	2	24	no	—
17	C₂×C₄×Dic₆₁	—	2	24	no	—
18	C₂×C₄×Dic₆₁	C₂×C₄	1	48	yes	—
19	#14519684	C₂×C₄	1	48	yes	—
20	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
21	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
23	C₂×C₄²×C₆₁	—	2	12	no	—
24	C₂×C₄²×C₆₁	C₂×C₄	1	24	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
27	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₁	D₄	1	4	yes	—
29	C₂×C₄²×C₆₁	Q₈	1	4	yes	—
30	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	—	2	12	no	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
35	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
36	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
37	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₆₁	C₂×D₄	1	6	yes	—
39	C₂²×C₄×C₆₁	C₂×Q₈	1	6	yes	—
40	C₂²×C₄×C₆₁	#41	1	12	yes	—
41	C₂²×Dic₆₁	—	2	8	no	—
42	C₂²×Dic₆₁	—	2	24	no	—
43	C₂²×Dic₆₁	C₂²×C₄	1	48	yes	—
44	C₄²×C₆₁	—	2	24	no	—
45	C₄²×C₆₁	C₄²	1	48	yes	complemented
46	C₄²×C₆₁	C₄⋊C₄	1	16	yes	complemented
47	C₄⋊Dic₆₁	C₄²	1	64	yes	direct product factor
48	C₄×Dic₆₁	—	2	96	no	—
49	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
50	C₂³×C₆₁	C₂³×C₄	1	6	yes	—
51	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
52	C₂³×C₆₁	C₂²×Q₈	1	1	yes	characteristic
53	C₂³×C₆₁	C₂×#41	1	6	yes	—
54	C₂×C₄×C₆₁	—	2	12	no	—
55	C₂×C₄×C₆₁	—	2	24	no	—
56	C₂×C₄×C₆₁	C₂×C₄²	1	12	yes	—
57	C₂×C₄×C₆₁	C₂×C₄²	1	16	yes	—
58	C₂×C₄×C₆₁	C₂×C₄²	1	12	yes	—
59	C₂×C₄×C₆₁	C₂×C₄²	1	12	yes	—
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	12	yes	—
61	C₂×C₄×C₆₁	#117	1	12	yes	—
62	C₂×C₄×C₆₁	C₄×D₄	1	24	yes	—
63	C₂×C₄×C₆₁	C₄×Q₈	1	24	yes	—
64	C₂×Dic₆₁	—	2	48	no	—
65	C₂×Dic₆₁	—	2	48	no	—
66	C₂×Dic₆₁	C₂×C₄²	1	32	yes	—
67	C₄²×C₄⋊C₄	—	61	1	no	maximal, Sylow
68	C₂²×C₆₁	C₂²×C₄²	1	4	yes	—
69	C₂²×C₆₁	C₂²×C₄²	1	3	yes	—
70	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₆₁	C₂×#117	1	3	yes	—
72	C₂²×C₆₁	C₂×C₄×D₄	1	6	yes	—
73	C₂²×C₆₁	C₂×C₄×Q₈	1	6	yes	—
74	C₂²×C₆₁	C₄×#41	1	12	yes	—
75	C₄×C₆₁	—	2	12	no	—
76	C₄×C₆₁	C₄³	1	8	yes	complemented
77	C₄×C₆₁	C₄×C₄⋊C₄	1	24	yes	complemented
78	Dic₆₁	—	2	32	no	—
79	C₂×C₄×C₄⋊C₄	—	61	6	no	—
80	C₂×C₄³	—	61	2	no	—
81	C₂×C₄³	D₆₁	1	1	yes	characteristic
82	C₄×#95	—	61	6	no	—
83	C₂×C₆₁	C₂×C₄³	1	1	yes	characteristic
84	C₂×C₆₁	C₂×C₄×C₄⋊C₄	1	3	yes	—
85	C₂×C₆₁	C₄×#117	1	3	yes	—
86	C₂×C₆₁	C₄²×D₄	1	4	yes	—
87	C₂×C₆₁	C₄²×Q₈	1	4	yes	—
88	C₂²×C₄⋊C₄	—	61	4	no	—
89	C₂²×C₄²	—	61	12	no	—
90	C₂²×C₄²	C₂×D₆₁	1	3	yes	—
91	C₂²×C₄²	C₂×D₆₁	1	3	yes	—
92	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
93	C₄×C₄⋊C₄	—	61	48	no	—
94	C₂×#95	—	61	12	no	—
95	C₄³	—	122	8	no	—
96	C₄³	Dic₆₁	1	8	yes	complemented
97	C₆₁	C₄²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	—	61	8	no	—
99	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
100	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
101	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
102	C₂×C₄⋊C₄	—	61	48	no	—
103	C₂×C₄²	—	2	12	no	—
104	C₂×C₄²	—	61	48	no	—
105	C₂×C₄²	—	122	24	no	—
106	C₂×C₄²	—	122	24	no	—
107	C₂×C₄²	Dic₁₂₂	1	4	yes	—
108	C₂×C₄²	C₄×D₆₁	1	24	yes	—
109	C₂×C₄²	D₂₄₄	1	4	yes	—
110	C₂×C₄²	C₂×Dic₆₁	1	12	yes	—
111	C₂×C₄²	C₂×Dic₆₁	1	12	yes	—
112	C₂×C₄²	C₂×Dic₆₁	1	4	yes	—
113	#95	—	61	48	no	—
114	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
115	C₂²×C₄	—	2	12	no	—
116	C₂²×C₄	—	61	48	no	—
117	C₂²×C₄	—	122	8	no	—
118	C₂²×C₄	—	122	24	no	—
119	C₂²×C₄	C₂×Dic₁₂₂	1	6	yes	—
120	C₂²×C₄	C₂×C₄×D₆₁	1	12	yes	—
121	C₂²×C₄	C₂×C₄×D₆₁	1	12	yes	—
122	C₂²×C₄	C₂×C₄×D₆₁	1	12	yes	—
123	C₂²×C₄	C₂×D₂₄₄	1	6	yes	—
124	C₂²×C₄	#11758222	1	12	yes	—
125	C₂²×C₄	C₂²×Dic₆₁	1	6	yes	—
126	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
127	C₂²×C₄	C₂²×Dic₆₁	1	6	yes	—
128	C₄⋊C₄	—	61	64	no	—
129	C₄²	—	2	24	no	—
130	C₄²	—	122	96	no	—
131	C₄²	C₄×Dic₆₁	1	48	yes	complemented
132	C₄²	C₄⋊Dic₆₁	1	16	yes	direct product factor
133	C₂³	C₂²×C₄×D₆₁	1	6	yes	—
134	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
135	C₂³	C₂²×D₂₄₄	1	1	yes	characteristic
136	C₂³	C₂²×Dic₁₂₂	1	1	yes	characteristic
137	C₂³	C₂×#11758222	1	6	yes	—
138	C₂×C₄	—	2	12	no	—
139	C₂×C₄	—	2	24	no	—
140	C₂×C₄	—	61	32	no	—
141	C₂×C₄	—	122	48	no	—
142	C₂×C₄	—	122	48	no	—
143	C₂×C₄	C₄²×D₆₁	1	16	yes	—
144	C₂×C₄	C₂×C₄×Dic₆₁	1	12	yes	—
145	C₂×C₄	C₂×C₄×Dic₆₁	1	12	yes	—
146	C₂×C₄	C₂×C₄×Dic₆₁	1	12	yes	—
147	C₂×C₄	C₂×C₄⋊Dic₆₁	1	12	yes	—
148	C₂×C₄	#14519736	1	12	yes	—
149	C₂×C₄	C₄×D₂₄₄	1	24	yes	—
150	C₂×C₄	C₄×Dic₁₂₂	1	24	yes	—
151	C₂²	C₂×C₄²×D₆₁	1	4	yes	—
152	C₂²	C₂²×C₄×Dic₆₁	1	3	yes	—
153	C₂²	C₂²×C₄⋊Dic₆₁	1	1	yes	characteristic
154	C₂²	C₂×#14519736	1	3	yes	—
155	C₂²	C₂×C₄×D₂₄₄	1	6	yes	—
156	C₂²	C₂×C₄×Dic₁₂₂	1	6	yes	—
157	C₂²	C₄×#11758222	1	12	yes	—
158	C₄	—	2	12	no	—
159	C₄	—	122	32	no	—
160	C₄	C₄²×Dic₆₁	1	8	yes	complemented
161	C₄	C₄×C₄⋊Dic₆₁	1	24	yes	direct product factor
162	C₂	C₂×C₄²×Dic₆₁	1	1	yes	characteristic
163	C₂	C₂×C₄×C₄⋊Dic₆₁	1	3	yes	—
164	C₂	C₄×#14519736	1	3	yes	—
165	C₂	C₄²×D₂₄₄	1	4	yes	—
166	C₂	C₄²×Dic₁₂₂	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₆₁
commutator	C₂×C₆₁	C₂×C₄³
Frattini	C₂⁴	C₂³×D₆₁
Fitting	C₂×C₄³×C₆₁	C₂
socle, prime radical, cosocle	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂×C₄³	D₆₁
hypocenter	C₆₁	C₄²×C₄⋊C₄
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
7. C₄×C₄⋊Dic₆₁	C₄	direct (N × H)
13. C₄³×C₆₁	C₄	semidirect
45. C₄²×C₆₁	C₄²	semidirect
46. C₄²×C₆₁	C₄⋊C₄	semidirect
47. C₄⋊Dic₆₁	C₄²	direct (N × H)
76. C₄×C₆₁	C₄³	semidirect
77. C₄×C₆₁	C₄×C₄⋊C₄	semidirect
96. C₄³	Dic₆₁	semidirect
97. C₆₁	C₄²×C₄⋊C₄	semidirect
131. C₄²	C₄×Dic₆₁	semidirect
132. C₄²	C₄⋊Dic₆₁	direct (N × H)
160. C₄	C₄²×Dic₆₁	semidirect
161. C₄	C₄×C₄⋊Dic₆₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊Dic₆₁	2
C₂×C₄²×Dic₆₁	2
C₄×#14519684	2
C₂×C₄³×C₆₁	2
C₄²×C₄⋊C₄	61

Subgroups — C42×C4⋊Dic61 #29601092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊Dic₆₁ #29601092