Subgroups — C₂×#21445506 #29677340

Order 15616.

1614 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519688	C₂	1	3	yes	maximal
2	C₂²×C₄×#92418	C₂	1	3	yes	maximal
3	#21445506	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₆₁×#376	C₂	1	1	yes	maximal, characteristic
5	C₂³×#92418	C₂²	1	1	yes	characteristic
6	C₂³×#92418	C₂²	1	3	yes	—
7	C₂×#14519688	C₄	1	24	yes	complemented
8	C₂×#14519688	C₂²	1	12	yes	—
9	C₂×C₄×#92418	C₄	1	24	yes	complemented
10	C₂×C₄×#92418	C₂²	1	12	yes	—
11	C₂²×C₄²×C₆₁	C₂²	1	3	yes	—
12	C₆₁×#376	C₄	1	4	yes	—
13	C₆₁×#376	C₂²	1	4	yes	—
14	C₂×#95×C₆₁	C₄	1	4	yes	—
15	C₂²×#92418	C₂×C₄	1	12	yes	—
16	C₂²×#92418	C₂×C₄	1	12	yes	—
17	C₂²×#92418	C₂×C₄	1	24	yes	—
18	C₂²×#92418	C₂³	1	2	yes	—
19	C₂²×#92418	C₂³	1	6	yes	—
20	#14519688	C₂×C₄	1	48	yes	complemented
21	C₄×#92418	C₂×C₄	1	48	yes	complemented
22	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
23	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
25	C₂×C₄²×C₆₁	C₂×C₄	1	12	yes	—
26	C₂×C₄²×C₆₁	C₂×C₄	1	6	yes	—
27	C₂×C₄²×C₆₁	C₂³	1	6	yes	—
28	#95×C₆₁	C₂×C₄	1	16	yes	—
29	C₂⁴×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	—	2	24	no	—
31	C₂²×C₄×C₆₁	C₄²	1	24	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	12	yes	—
33	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₆₁	C₂²×C₄	1	6	yes	—
36	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
37	C₄²×C₆₁	C₄²	1	24	yes	—
38	C₄²×C₆₁	C₂²×C₄	1	12	yes	—
39	C₄²×C₆₁	C₂²×C₄	1	12	yes	—
40	C₂×#92418	C₄²	1	16	yes	—
41	C₂×#92418	C₄²	1	48	yes	—
42	C₂×#92418	C₂²×C₄	1	12	yes	—
43	C₂×#92418	C₂²×C₄	1	12	yes	—
44	C₂×#92418	C₂²×C₄	1	24	yes	—
45	C₂³×C₆₁	C₂×C₄²	1	6	yes	—
46	C₂³×C₆₁	#131	1	8	yes	—
47	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	—	2	24	no	—
49	C₂×C₄×C₆₁	—	2	48	no	—
50	C₂×C₄×C₆₁	C₂×C₄²	1	24	yes	—
51	C₂×C₄×C₆₁	C₂×C₄²	1	6	yes	—
52	C₂×C₄×C₆₁	C₂×C₄²	1	4	yes	—
53	C₂×C₄×C₆₁	C₂×C₄²	1	12	yes	—
54	C₂×C₄×C₆₁	C₂³×C₄	1	3	yes	—
55	C₂×C₄×C₆₁	C₂³×C₄	1	3	yes	—
56	#92418	C₂×C₄²	1	8	yes	—
57	#92418	C₂×C₄²	1	24	yes	—
58	C₂×#1726	—	61	1	no	maximal, Sylow
59	C₂²×C₆₁	C₄³	1	4	yes	—
60	C₂²×C₆₁	#405	1	24	yes	—
61	C₂²×C₆₁	C₂²×C₄²	1	3	yes	—
62	C₂²×C₆₁	C₂×#131	1	4	yes	—
63	C₄×C₆₁	—	2	24	no	—
64	C₄×C₆₁	C₄³	1	4	yes	—
65	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
66	C₄×C₆₁	C₂²×C₄²	1	3	yes	—
67	C₂²×#97	—	61	3	no	—
68	C₂²×C₄×C₈	—	61	3	no	—
69	C₂×#376	D₆₁	1	1	yes	characteristic
70	#1726	—	61	8	no	—
71	C₂×C₆₁	#1726	1	8	yes	complemented
72	C₂×C₆₁	C₂×C₄³	1	1	yes	characteristic
73	C₂×C₆₁	C₂×#405	1	6	yes	—
74	C₂³×C₈	—	61	1	no	—
75	C₂³×C₈	—	61	3	no	—
76	C₂²×C₄²	C₂×D₆₁	1	3	yes	—
77	C₂×#97	—	61	24	no	—
78	C₂×#97	—	61	12	no	—
79	C₂×C₄×C₈	—	61	24	no	—
80	C₂×C₄×C₈	—	61	12	no	—
81	#376	Dic₆₁	1	4	yes	—
82	#376	C₂×D₆₁	1	4	yes	—
83	C₂×#95	Dic₆₁	1	4	yes	—
84	C₆₁	C₂×#1726	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₄	C₂×Dic₆₁	1	6	yes	—
86	C₂³×C₄	C₂²×D₆₁	1	1	yes	characteristic
87	C₂²×C₈	—	61	12	no	—
88	C₂²×C₈	—	61	12	no	—
89	C₂²×C₈	—	61	24	no	—
90	C₂²×C₈	—	61	2	no	—
91	C₂²×C₈	—	61	6	no	—
92	C₂×C₄²	C₄×D₆₁	1	12	yes	—
93	C₂×C₄²	C₄×D₆₁	1	12	yes	—
94	C₂×C₄²	C₂×Dic₆₁	1	6	yes	—
95	C₂×C₄²	C₂²×D₆₁	1	6	yes	—
96	#97	—	61	48	no	—
97	C₄×C₈	—	61	48	no	—
98	#95	C₂×Dic₆₁	1	16	yes	—
99	C₂⁴	C₂²×Dic₆₁	1	1	yes	characteristic
100	C₂²×C₄	—	2	24	no	—
101	C₂²×C₄	C₄×Dic₆₁	1	24	yes	—
102	C₂²×C₄	C₂×C₄×D₆₁	1	6	yes	—
103	C₂²×C₄	C₂×C₄×D₆₁	1	6	yes	—
104	C₂²×C₄	C₂²×Dic₆₁	1	12	yes	—
105	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
106	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
107	C₂×C₈	—	61	12	no	—
108	C₂×C₈	—	61	12	no	—
109	C₂×C₈	—	61	24	no	—
110	C₂×C₈	—	61	16	no	—
111	C₂×C₈	—	61	48	no	—
112	C₄²	C₄×Dic₆₁	1	24	yes	—
113	C₄²	C₂×C₄×D₆₁	1	12	yes	—
114	C₄²	C₂×C₄×D₆₁	1	12	yes	—
115	C₂³	C₂×C₄×Dic₆₁	1	6	yes	—
116	C₂³	#14519784	1	8	yes	—
117	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
118	C₂×C₄	—	2	48	no	—
119	C₂×C₄	—	2	24	no	—
120	C₂×C₄	C₄²×D₆₁	1	4	yes	—
121	C₂×C₄	C₄²×D₆₁	1	12	yes	—
122	C₂×C₄	C₂×C₄×Dic₆₁	1	24	yes	—
123	C₂×C₄	C₂×C₄×Dic₆₁	1	6	yes	—
124	C₂×C₄	C₂²×C₄×D₆₁	1	3	yes	—
125	C₂×C₄	C₂²×C₄×D₆₁	1	3	yes	—
126	C₈	—	61	8	no	—
127	C₈	—	61	24	no	—
128	C₂²	C₄²×Dic₆₁	1	4	yes	—
129	C₂²	#16839718	1	24	yes	—
130	C₂²	C₂²×C₄×Dic₆₁	1	3	yes	—
131	C₂²	C₂×#14519784	1	4	yes	—
132	C₄	—	2	24	no	—
133	C₄	C₄²×Dic₆₁	1	4	yes	—
134	C₄	C₂×C₄²×D₆₁	1	1	yes	characteristic
135	C₄	C₂×C₄²×D₆₁	1	3	yes	—
136	C₂	#21445506	1	8	yes	direct product factor
137	C₂	C₂×C₄²×Dic₆₁	1	1	yes	characteristic
138	C₂	C₂×#16839718	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₁
commutator	C₂×C₆₁	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₂×C₆₁×#376	C₂
socle, prime radical	C₂⁴×C₆₁	C₂²×C₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	C₂×#376	D₆₁
hypocenter	C₆₁	C₂×#1726
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
3. #21445506	C₂	direct (N × H)
7. C₂×#14519688	C₄	semidirect
9. C₂×C₄×#92418	C₄	semidirect
20. #14519688	C₂×C₄	semidirect
21. C₄×#92418	C₂×C₄	semidirect
71. C₂×C₆₁	#1726	semidirect
84. C₆₁	C₂×#1726	semidirect
136. C₂	#21445506	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519688	2
C₂²×C₄×#92418	2
#21445506	2
C₂×C₆₁×#376	2
C₂×#1726	61

Subgroups — C2×#21445506 #29677340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445506 #29677340