Subgroups — C₆₁×#2330 #21442371

Order 7808.

444 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	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#389	C₂	1	4	yes	maximal
3	C₆₁×#389	C₂	1	2	yes	maximal
4	C₆₁×#387	C₂	1	4	yes	maximal
5	C₆₁×#384	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₆₁	C₂	1	2	yes	maximal
7	C₆₁×#376	C₂	1	1	yes	maximal, characteristic
8	#125×C₆₁	C₄	1	8	yes	complemented
9	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
11	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₆₁	C₂²	1	4	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	#95×C₆₁	C₂²	1	2	yes	—
19	#95×C₆₁	C₂²	1	2	yes	—
20	#95×C₆₁	C₂²	1	4	yes	—
21	#95×C₆₁	C₂²	1	4	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
26	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₆₁	—	2	4	no	—
30	C₄⋊C₄×C₆₁	—	4	4	no	—
31	C₄⋊C₄×C₆₁	—	4	4	no	—
32	C₄⋊C₄×C₆₁	C₂×C₄	1	16	yes	—
33	C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
34	C₄²×C₆₁	—	2	2	no	—
35	C₄²×C₆₁	—	4	4	no	—
36	C₄²×C₆₁	C₂×C₄	1	4	yes	—
37	C₄²×C₆₁	Q₈	1	4	yes	—
38	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	—	2	4	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	—	2	4	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	—	2	1	no	—
47	C₂×C₄×C₆₁	—	2	4	no	—
48	C₂×C₄×C₆₁	—	2	4	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	4	no	—
51	C₂×C₄×C₆₁	—	2	2	no	—
52	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₆₁	C₂²×C₄	1	8	yes	—
54	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
56	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₆₁	#41	1	2	yes	—
58	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×C₆₁	C₂×#41	1	1	yes	characteristic
61	C₂²×C₆₁	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₆₁	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₆₁	—	2	8	no	—
64	C₄×C₆₁	—	2	4	no	—
65	C₄×C₆₁	—	2	4	no	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	—	4	4	no	—
68	C₄×C₆₁	—	4	1	no	—
69	C₄×C₆₁	C₄×Q₈	1	4	yes	—
70	#2330	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
72	C₂×C₆₁	#520	1	2	yes	—
73	C₂×C₆₁	#540	1	1	yes	characteristic
74	C₂×C₆₁	#541	1	1	yes	characteristic
75	C₂×C₆₁	#544	1	2	yes	—
76	C₂×#125	C₂×C₆₁	1	1	yes	characteristic
77	#389	C₂×C₆₁	1	4	yes	—
78	#389	C₂×C₆₁	1	2	yes	—
79	#387	C₂×C₆₁	1	4	yes	—
80	#384	C₂×C₆₁	1	1	yes	characteristic
81	C₄×C₄⋊C₄	C₂×C₆₁	1	2	yes	—
82	#376	C₂×C₆₁	1	1	yes	characteristic
83	C₆₁	#2330	1	1	yes	characteristic, direct product factor, Sylow
84	#125	C₄×C₆₁	1	8	yes	complemented
85	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
87	C₂×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
88	C₂×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₆₁	1	4	yes	—
90	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₆₁	1	4	yes	—
92	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
93	C₂×C₄²	C₂²×C₆₁	1	1	yes	characteristic
94	#95	C₂²×C₆₁	1	2	yes	—
95	#95	C₂²×C₆₁	1	2	yes	—
96	#95	C₂²×C₆₁	1	4	yes	—
97	#95	C₂²×C₆₁	1	4	yes	—
98	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
99	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
100	C₂²×C₄	C₂³×C₆₁	1	4	yes	—
101	C₂²×C₄	C₂³×C₆₁	1	4	yes	—
102	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
103	C₂²×C₄	C₂³×C₆₁	1	2	yes	—
104	C₂²×C₄	C₂³×C₆₁	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	C₂×C₄×C₆₁	1	16	yes	—
109	C₄⋊C₄	C₂×C₄×C₆₁	1	8	yes	—
110	C₄²	—	2	2	no	—
111	C₄²	—	4	4	no	—
112	C₄²	C₂×C₄×C₆₁	1	4	yes	—
113	C₄²	Q₈×C₆₁	1	4	yes	—
114	C₂³	C₂⁴×C₆₁	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₂²×C₄×C₆₁	1	4	yes	—
129	C₂×C₄	C₂²×C₄×C₆₁	1	8	yes	—
130	C₂×C₄	C₂²×C₄×C₆₁	1	2	yes	—
131	C₂×C₄	C₂×Q₈×C₆₁	1	4	yes	—
132	C₂×C₄	C₂×Q₈×C₆₁	1	2	yes	—
133	C₂×C₄	#41×C₆₁	1	2	yes	—
134	C₂²	C₂³×C₄×C₆₁	1	1	yes	characteristic
135	C₂²	C₂²×Q₈×C₆₁	1	1	yes	characteristic
136	C₂²	C₂×#41×C₆₁	1	1	yes	characteristic
137	C₂²	H(5,𝔽₂)×C₆₁	1	2	yes	—
138	C₂²	2₋¹⁺⁴×C₆₁	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	2	8	no	—
141	C₄	—	2	4	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	4	no	—
144	C₄	—	4	1	no	—
145	C₄	C₄×Q₈×C₆₁	1	4	yes	—
146	C₂	C₂×C₄×Q₈×C₆₁	1	1	yes	characteristic
147	C₂	C₆₁×#520	1	2	yes	—
148	C₂	C₆₁×#540	1	1	yes	characteristic
149	C₂	C₆₁×#541	1	1	yes	characteristic
150	C₂	C₆₁×#544	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₆₁
Frattini, cosocle	C₂³	C₂⁴×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
8. #125×C₆₁	C₄	semidirect
70. #2330	C₆₁	direct (N × H)
83. C₆₁	#2330	direct (N × H)
84. #125	C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₆₁	2
C₆₁×#389	2
C₆₁×#387	2
C₆₁×#384	2
C₄×C₄⋊C₄×C₆₁	2
C₆₁×#376	2
#2330	61

Subgroups — C61×#2330 #21442371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#2330 #21442371