Subgroups — C₂×C₆₁×#562 #21443568

Order 7808.

1152 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	C₆₁×#562	C₂	1	16	yes	maximal, direct product factor
2	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#124×C₆₁	C₂	1	2	yes	maximal, complemented
4	C₂×#123×C₆₁	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₆₁	C₂	1	4	yes	maximal, complemented
6	C₂×#121×C₆₁	C₂	1	4	yes	maximal, complemented
7	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
8	C₂²×D₄×C₆₁	C₂²	1	4	yes	—
9	C₂²×D₄×C₆₁	C₂²	1	2	yes	—
10	C₂³×C₄×C₆₁	C₂²	1	4	yes	—
11	#125×C₆₁	C₂²	1	8	yes	complemented
12	#124×C₆₁	C₂²	1	16	yes	complemented
13	#123×C₆₁	C₂²	1	32	yes	complemented
14	C₂²⋊Q₈×C₆₁	C₂²	1	32	yes	complemented
15	#121×C₆₁	C₂²	1	32	yes	complemented
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₆₁	C₂²	1	4	yes	—
22	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂⁴×C₆₁	C₂³	1	4	yes	—
24	C₂×Q₈×C₆₁	—	4	4	no	—
25	C₂×Q₈×C₆₁	C₂³	1	8	yes	—
26	C₂×D₄×C₆₁	—	4	8	no	—
27	C₂×D₄×C₆₁	—	4	4	no	—
28	C₂×D₄×C₆₁	C₂³	1	16	yes	—
29	C₂×D₄×C₆₁	C₂³	1	8	yes	—
30	C₂²×C₄×C₆₁	—	4	8	no	—
31	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₆₁	C₂³	1	16	yes	—
36	C₄⋊C₄×C₆₁	C₂³	1	16	yes	—
37	C₄⋊C₄×C₆₁	C₂³	1	16	yes	—
38	C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
39	C₂²⋊C₄×C₆₁	C₂³	1	16	yes	—
40	C₂²⋊C₄×C₆₁	C₂³	1	16	yes	—
41	C₂²⋊C₄×C₆₁	C₂³	1	16	yes	—
42	C₄²×C₆₁	C₂³	1	4	yes	—
43	C₂³×C₆₁	—	2	8	no	—
44	C₂³×C₆₁	—	2	8	no	—
45	C₂³×C₆₁	—	2	8	no	—
46	C₂³×C₆₁	C₂⁴	1	8	yes	—
47	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
48	Q₈×C₆₁	—	4	8	no	—
49	D₄×C₆₁	—	4	16	no	—
50	D₄×C₆₁	—	4	8	no	—
51	C₂×C₄×C₆₁	—	2	8	no	—
52	C₂×C₄×C₆₁	—	2	8	no	—
53	C₂×C₄×C₆₁	—	2	4	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	4	16	no	—
56	C₂×C₄×C₆₁	C₂⁴	1	2	yes	—
57	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
58	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
59	C₂×C₄×C₆₁	C₂⁴	1	8	yes	—
60	C₂²×C₆₁	—	2	8	no	—
61	C₂²×C₆₁	—	2	8	no	—
62	C₂²×C₆₁	—	2	8	no	—
63	C₂²×C₆₁	—	4	16	no	—
64	C₂²×C₆₁	H(5,𝔽₂)	1	4	yes	—
65	C₂²×C₆₁	2₋¹⁺⁴	1	2	yes	—
66	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
67	C₄×C₆₁	—	2	8	no	—
68	C₄×C₆₁	—	2	8	no	—
69	C₄×C₆₁	—	2	4	no	—
70	C₄×C₆₁	—	2	2	no	—
71	C₂×#562	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₆₁	—	4	8	no	—
73	C₂×C₆₁	#562	1	4	yes	direct product factor
74	C₂×C₆₁	C₂×H(5,𝔽₂)	1	2	yes	—
75	C₂×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
76	#562	C₂×C₆₁	1	16	yes	direct product factor
77	C₂×#125	C₂×C₆₁	1	1	yes	characteristic, complemented
78	C₂×#124	C₂×C₆₁	1	2	yes	complemented
79	C₂×#123	C₂×C₆₁	1	4	yes	complemented
80	C₂×C₂²⋊Q₈	C₂×C₆₁	1	4	yes	complemented
81	C₂×#121	C₂×C₆₁	1	4	yes	complemented
82	C₆₁	C₂×#562	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×Q₈	C₂²×C₆₁	1	2	yes	—
84	C₂²×D₄	C₂²×C₆₁	1	4	yes	—
85	C₂²×D₄	C₂²×C₆₁	1	2	yes	—
86	C₂³×C₄	C₂²×C₆₁	1	4	yes	—
87	#125	C₂²×C₆₁	1	8	yes	complemented
88	#124	C₂²×C₆₁	1	16	yes	complemented
89	#123	C₂²×C₆₁	1	32	yes	complemented
90	C₂²⋊Q₈	C₂²×C₆₁	1	32	yes	complemented
91	#121	C₂²×C₆₁	1	32	yes	complemented
92	C₂×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
93	C₂×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
94	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
95	C₂×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
96	C₂×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
97	C₂×C₂²⋊C₄	C₂²×C₆₁	1	4	yes	—
98	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
99	C₂⁴	C₂³×C₆₁	1	4	yes	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	C₂³×C₆₁	1	8	yes	—
102	C₂×D₄	—	4	8	no	—
103	C₂×D₄	—	4	4	no	—
104	C₂×D₄	C₂³×C₆₁	1	16	yes	—
105	C₂×D₄	C₂³×C₆₁	1	8	yes	—
106	C₂²×C₄	—	4	8	no	—
107	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
108	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
109	C₂²×C₄	C₂³×C₆₁	1	4	yes	—
110	C₂²×C₄	C₂³×C₆₁	1	4	yes	—
111	C₂²×C₄	C₂³×C₆₁	1	16	yes	—
112	C₄⋊C₄	C₂³×C₆₁	1	16	yes	—
113	C₄⋊C₄	C₂³×C₆₁	1	16	yes	—
114	C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
115	C₂²⋊C₄	C₂³×C₆₁	1	16	yes	—
116	C₂²⋊C₄	C₂³×C₆₁	1	16	yes	—
117	C₂²⋊C₄	C₂³×C₆₁	1	16	yes	—
118	C₄²	C₂³×C₆₁	1	4	yes	—
119	C₂³	—	2	8	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	8	no	—
122	C₂³	C₂⁴×C₆₁	1	8	yes	—
123	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
124	Q₈	—	4	8	no	—
125	D₄	—	4	16	no	—
126	D₄	—	4	8	no	—
127	C₂×C₄	—	2	8	no	—
128	C₂×C₄	—	2	8	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	16	no	—
132	C₂×C₄	C₂⁴×C₆₁	1	2	yes	—
133	C₂×C₄	C₂⁴×C₆₁	1	4	yes	—
134	C₂×C₄	C₂⁴×C₆₁	1	8	yes	—
135	C₂×C₄	C₂⁴×C₆₁	1	8	yes	—
136	C₂²	—	2	8	no	—
137	C₂²	—	2	8	no	—
138	C₂²	—	2	8	no	—
139	C₂²	—	4	16	no	—
140	C₂²	H(5,𝔽₂)×C₆₁	1	4	yes	—
141	C₂²	2₋¹⁺⁴×C₆₁	1	2	yes	—
142	C₂²	C₂⁵×C₆₁	1	1	yes	characteristic
143	C₄	—	2	8	no	—
144	C₄	—	2	8	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₂	—	4	8	no	—
148	C₂	C₆₁×#562	1	4	yes	direct product factor
149	C₂	C₂×H(5,𝔽₂)×C₆₁	1	2	yes	—
150	C₂	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₆₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#562	C₂	direct (N × H)
2. C₂×#125×C₆₁	C₂	semidirect
3. C₂×#124×C₆₁	C₂	semidirect
4. C₂×#123×C₆₁	C₂	semidirect
5. C₂×C₂²⋊Q₈×C₆₁	C₂	semidirect
6. C₂×#121×C₆₁	C₂	semidirect
11. #125×C₆₁	C₂²	semidirect
12. #124×C₆₁	C₂²	semidirect
13. #123×C₆₁	C₂²	semidirect
14. C₂²⋊Q₈×C₆₁	C₂²	semidirect
15. #121×C₆₁	C₂²	semidirect
71. C₂×#562	C₆₁	direct (N × H)
73. C₂×C₆₁	#562	direct (N × H)
76. #562	C₂×C₆₁	direct (N × H)
77. C₂×#125	C₂×C₆₁	semidirect
78. C₂×#124	C₂×C₆₁	semidirect
79. C₂×#123	C₂×C₆₁	semidirect
80. C₂×C₂²⋊Q₈	C₂×C₆₁	semidirect
81. C₂×#121	C₂×C₆₁	semidirect
82. C₆₁	C₂×#562	direct (N × H)
87. #125	C₂²×C₆₁	semidirect
88. #124	C₂²×C₆₁	semidirect
89. #123	C₂²×C₆₁	semidirect
90. C₂²⋊Q₈	C₂²×C₆₁	semidirect
91. #121	C₂²×C₆₁	semidirect
148. C₂	C₆₁×#562	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#562	2
C₂×#125×C₆₁	2
C₂×#124×C₆₁	2
C₂×#123×C₆₁	2
C₂×C₂²⋊Q₈×C₆₁	2
C₂×#121×C₆₁	2
C₂×#562	61

Subgroups — C2×C61×#562 #21443568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₁×#562 #21443568