Subgroups — C₄×C₆₁×#366 #29516105

Order 15616.

366 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#366	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₆₁	C₂	1	2	yes	maximal
4	C₆₁×#1380	C₂	1	2	yes	maximal
5	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×#107×C₆₁	C₂²	1	4	yes	—
8	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
10	C₆₁×#366	C₄	1	8	yes	direct product factor
11	C₄×C₁₆×C₆₁	C₄	1	4	yes	complemented
12	C₆₁×#340	—	2	2	no	—
13	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
16	C₂×C₄²×C₆₁	—	4	2	no	—
17	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₆₁	—	2	2	no	—
19	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
22	#107×C₆₁	—	2	4	no	—
23	#107×C₆₁	C₂×C₄	1	8	yes	—
24	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	Q₈	1	2	yes	—
26	#95×C₆₁	—	4	2	no	—
27	C₂²×C₄×C₆₁	—	4	4	no	—
28	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₆₁	#41	1	2	yes	—
37	C₄⋊C₄×C₆₁	—	2	8	no	—
38	C₄⋊C₄×C₆₁	—	4	4	no	—
39	C₄²×C₆₁	—	8	4	no	—
40	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
41	C₄²×C₆₁	Q₁₆	1	2	yes	—
42	C₁₆×C₆₁	—	2	2	no	—
43	C₁₆×C₆₁	C₄²	1	4	yes	complemented
44	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	—	4	8	no	—
47	C₂×C₄×C₆₁	—	8	4	no	—
48	C₂×C₄×C₆₁	—	8	4	no	—
49	C₂×C₄×C₆₁	#107	1	2	yes	—
50	C₂×C₄×C₆₁	#107	1	2	yes	—
51	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
52	C₂×C₄×C₆₁	Q₃₂	1	2	yes	—
53	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₆₁	#135	1	2	yes	—
57	C₈×C₆₁	—	2	2	no	—
58	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₆₁	#117	1	1	yes	characteristic
60	C₈×C₆₁	C₄×Q₈	1	2	yes	—
61	C₄×#366	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
63	C₂²×C₆₁	C₄×D₈	1	2	yes	—
64	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
65	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
66	C₂²×C₆₁	#508	1	2	yes	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	8	8	no	—
69	C₄×C₆₁	#366	1	4	yes	direct product factor
70	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
71	C₄×C₆₁	#427	1	1	yes	characteristic
72	C₄×C₆₁	C₄×Q₁₆	1	2	yes	—
73	C₂×#366	C₂×C₆₁	1	2	yes	—
74	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic
75	C₄×#107	C₂×C₆₁	1	2	yes	—
76	#1380	C₂×C₆₁	1	2	yes	—
77	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
78	C₂×C₆₁	C₂×#366	1	1	yes	characteristic
79	C₂×C₆₁	#2158	1	1	yes	characteristic
80	C₂×C₆₁	C₄×D₁₆	1	2	yes	—
81	C₂×C₆₁	C₄×Q₃₂	1	2	yes	—
82	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
83	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
84	C₂×#107	C₂²×C₆₁	1	4	yes	—
85	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	2	2	no	—
87	#366	C₄×C₆₁	1	8	yes	direct product factor
88	C₄×C₁₆	C₄×C₆₁	1	4	yes	complemented
89	#340	—	2	2	no	—
90	C₆₁	C₄×#366	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
92	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄²	—	4	2	no	—
95	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
98	C₂×C₁₆	C₂×C₄×C₆₁	1	2	yes	—
99	C₂×C₁₆	C₂×C₄×C₆₁	1	4	yes	—
100	#107	—	2	4	no	—
101	#107	C₂×C₄×C₆₁	1	8	yes	—
102	C₄×C₈	C₂×C₄×C₆₁	1	2	yes	—
103	C₄×C₈	Q₈×C₆₁	1	2	yes	—
104	#95	—	4	2	no	—
105	C₂²×C₄	—	4	4	no	—
106	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
107	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	C₄⋊C₄×C₆₁	1	2	yes	—
111	C₂×C₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
112	C₂×C₈	C₂²×C₄×C₆₁	1	2	yes	—
113	C₂×C₈	C₂×Q₈×C₆₁	1	1	yes	characteristic
114	C₂×C₈	#41×C₆₁	1	2	yes	—
115	C₄⋊C₄	—	2	8	no	—
116	C₄⋊C₄	—	4	4	no	—
117	C₄²	—	8	4	no	—
118	C₄²	C₄⋊C₄×C₆₁	1	2	yes	—
119	C₄²	Q₁₆×C₆₁	1	2	yes	—
120	C₁₆	—	2	2	no	—
121	C₁₆	C₄²×C₆₁	1	4	yes	complemented
122	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	8	no	—
125	C₂×C₄	—	8	4	no	—
126	C₂×C₄	—	8	4	no	—
127	C₂×C₄	#107×C₆₁	1	2	yes	—
128	C₂×C₄	#107×C₆₁	1	2	yes	—
129	C₂×C₄	D₁₆×C₆₁	1	2	yes	—
130	C₂×C₄	Q₃₂×C₆₁	1	2	yes	—
131	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
132	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
133	C₂×C₄	C₂×Q₁₆×C₆₁	1	1	yes	characteristic
134	C₂×C₄	#135×C₆₁	1	2	yes	—
135	C₈	—	2	2	no	—
136	C₈	C₂×C₄²×C₆₁	1	1	yes	characteristic
137	C₈	#117×C₆₁	1	1	yes	characteristic
138	C₈	C₄×Q₈×C₆₁	1	2	yes	—
139	C₂²	C₂×#107×C₆₁	1	1	yes	characteristic
140	C₂²	C₄×D₈×C₆₁	1	2	yes	—
141	C₂²	C₂×D₁₆×C₆₁	1	1	yes	characteristic
142	C₂²	C₂×Q₃₂×C₆₁	1	1	yes	characteristic
143	C₂²	C₆₁×#508	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	8	8	no	—
146	C₄	C₆₁×#366	1	4	yes	direct product factor
147	C₄	C₄×C₄⋊C₄×C₆₁	1	1	yes	characteristic
148	C₄	C₆₁×#427	1	1	yes	characteristic
149	C₄	C₄×Q₁₆×C₆₁	1	2	yes	—
150	C₂	C₄×#107×C₆₁	1	1	yes	characteristic
151	C₂	C₂×C₆₁×#366	1	1	yes	characteristic
152	C₂	C₆₁×#2158	1	1	yes	characteristic
153	C₂	C₄×D₁₆×C₆₁	1	2	yes	—
154	C₂	C₄×Q₃₂×C₆₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₁	D₈
commutator	C₈	C₂×C₄²×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle, prime radical	C₂³×C₆₁	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₆₁×#366	C₄	direct (N × H)
11. C₄×C₁₆×C₆₁	C₄	semidirect
43. C₁₆×C₆₁	C₄²	semidirect
61. C₄×#366	C₆₁	direct (N × H)
69. C₄×C₆₁	#366	direct (N × H)
87. #366	C₄×C₆₁	direct (N × H)
88. C₄×C₁₆	C₄×C₆₁	semidirect
90. C₆₁	C₄×#366	direct (N × H)
121. C₁₆	C₄²×C₆₁	semidirect
146. C₄	C₆₁×#366	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×C₆₁×#366	2
C₂×C₄×C₁₆×C₆₁	2
C₄×#107×C₆₁	2
C₆₁×#1380	2
C₄×#366	61

Subgroups — C4×C61×#366 #29516105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₆₁×#366 #29516105