Subgroups — C₂×C₄×C₄⋊Dic₆₁ #21448154

Order 7808.

928 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
3. C₄×C₄⋊Dic₆₁	C₂	direct (N × H)
7. C₂×C₄⋊Dic₆₁	C₄	direct (N × H)
15. C₂×C₄²×C₆₁	C₄	semidirect
32. C₄²×C₆₁	C₂×C₄	semidirect
33. C₄⋊Dic₆₁	C₂×C₄	direct (N × H)
43. C₂×C₄×C₆₁	C₄²	semidirect
44. C₂×C₄×C₆₁	C₄⋊C₄	semidirect
66. C₄×C₆₁	C₂×C₄²	semidirect
67. C₄×C₆₁	C₂×C₄⋊C₄	semidirect
70. C₂×C₆₁	C₄×C₄⋊C₄	semidirect
81. C₆₁	C₂×C₄×C₄⋊C₄	semidirect
90. C₂×C₄²	Dic₆₁	semidirect
110. C₄²	C₂×Dic₆₁	semidirect
123. C₂×C₄	C₄×Dic₆₁	semidirect
124. C₂×C₄	C₄⋊Dic₆₁	direct (N × H)
143. C₄	C₂×C₄×Dic₆₁	semidirect
144. C₄	C₂×C₄⋊Dic₆₁	direct (N × H)
145. C₂	C₄×C₄⋊Dic₆₁	direct (N × H)

Subgroups — C2×C4×C4⋊Dic61 #21448154

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×C₄⋊Dic₆₁ #21448154