Subgroups — C₂×#20241523 #26136462

Order 12032.

222 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×#15688957	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688898	C₂	1	1	yes	maximal, characteristic
3	#20241523	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₄₇	C₄	1	1	yes	characteristic
7	#15688957	C₄	1	2	yes	complemented
8	#15688957	C₂²	1	2	yes	—
9	C₂×C₄₇⋊C₃₂	C₄	1	2	yes	complemented
10	#15688898	C₄	1	2	yes	—
11	#15688898	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×M₆₄	C₄	1	2	yes	complemented
14	C₄₇×M₆₄	C₂²	1	2	yes	—
15	C₂×C₃₂×C₄₇	C₄	1	2	yes	complemented
16	C₂²×C₄₇⋊C₈	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₄₇	C₈	1	1	yes	characteristic
18	C₂×C₄₇⋊C₁₆	C₈	1	2	yes	—
19	C₂×C₄₇⋊C₁₆	C₈	1	2	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₈×Dic₄₇	C₈	1	2	yes	—
22	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
23	C₄₇⋊C₃₂	C₂×C₄	1	4	yes	complemented
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
31	C₃₂×C₄₇	C₂×C₄	1	4	yes	complemented
32	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₄₇	—	2	1	no	—
41	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₄₇⋊C₈	—	2	1	no	—
44	C₂×C₄₇⋊C₈	C₂×C₈	1	2	yes	—
45	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
46	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
47	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
49	C₄₇⋊C₁₆	C₂×C₈	1	4	yes	—
50	C₂³×C₄₇	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
54	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
55	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₄₇	—	2	2	no	—
58	C₂×Dic₄₇	—	4	2	no	—
59	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
60	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
61	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
62	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
63	C₄₇⋊C₈	—	2	2	no	—
64	C₂×#1398	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	2	no	—
66	C₂²×C₄₇	#346	1	1	yes	characteristic
67	C₂²×C₄₇	#346	1	1	yes	characteristic
68	C₂²×C₄₇	C₂×M₃₂	1	1	yes	characteristic
69	C₄×C₄₇	#346	1	1	yes	characteristic
70	C₄×C₄₇	#346	1	1	yes	characteristic
71	C₄×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
72	C₄×C₄₇	C₂×M₃₂	1	1	yes	characteristic
73	Dic₄₇	—	4	2	no	—
74	C₂×M₆₄	—	47	1	no	—
75	C₂×M₆₄	D₄₇	1	1	yes	characteristic
76	C₂×#346	—	47	1	no	—
77	#1398	—	47	4	no	—
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	#1398	1	2	yes	complemented
81	C₂×C₄₇	C₂×#346	1	1	yes	characteristic
82	C₂²×C₁₆	—	47	1	no	—
83	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
84	C₂×C₄×C₈	—	47	1	no	—
85	M₆₄	—	47	2	no	—
86	M₆₄	—	47	2	no	—
87	M₆₄	Dic₄₇	1	2	yes	complemented
88	M₆₄	C₂×D₄₇	1	2	yes	—
89	C₂×C₃₂	—	47	2	no	—
90	C₂×C₃₂	Dic₄₇	1	2	yes	complemented
91	#346	—	47	2	no	—
92	#346	—	47	2	no	—
93	C₄₇	C₂×#1398	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	47	1	no	—
95	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
96	C₂×C₄²	—	47	1	no	—
97	C₂×C₁₆	—	47	2	no	—
98	C₂×C₁₆	—	47	2	no	—
99	C₂×C₁₆	—	47	2	no	—
100	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
103	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
105	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
106	C₄×C₈	—	47	2	no	—
107	C₄×C₈	—	47	2	no	—
108	C₃₂	—	47	4	no	—
109	C₃₂	C₂×Dic₄₇	1	4	yes	complemented
110	C₂²×C₄	—	94	1	no	—
111	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
112	C₂×C₈	—	47	2	no	—
113	C₂×C₈	—	94	1	no	—
114	C₂×C₈	—	94	1	no	—
115	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
116	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
117	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
118	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
119	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
121	C₄²	—	47	2	no	—
122	C₄²	—	94	1	no	—
123	C₁₆	—	47	4	no	—
124	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
125	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
126	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
127	C₁₆	C₂²×Dic₄₇	1	1	yes	characteristic
128	C₂³	#13093963	1	1	yes	characteristic
129	C₂×C₄	—	94	2	no	—
130	C₂×C₄	—	188	2	no	—
131	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
132	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
133	C₂×C₄	#13093963	1	1	yes	characteristic
134	C₂×C₄	#13093963	1	1	yes	characteristic
135	C₂×C₄	#13093963	1	1	yes	characteristic
136	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
137	C₈	—	94	2	no	—
138	C₈	C₈×Dic₄₇	1	1	yes	characteristic
139	C₈	C₈×Dic₄₇	1	1	yes	characteristic
140	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
141	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	#15688898	1	1	yes	characteristic
144	C₂²	#15688898	1	1	yes	characteristic
145	C₂²	C₂×#13093963	1	1	yes	characteristic
146	C₄	—	188	2	no	—
147	C₄	#15688898	1	1	yes	characteristic
148	C₄	#15688898	1	1	yes	characteristic
149	C₄	C₂×C₈×Dic₄₇	1	1	yes	characteristic
150	C₄	C₂×#13093963	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	2	1	no	—
153	C₂	#20241523	1	2	yes	direct product factor
154	C₂	C₂×#15688898	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₇
commutator	C₄×C₄₇	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₂×C₄₇×M₆₄	C₂
socle	C₂²×C₄₇	#346
prime radical	C₂³×C₄₇	M₃₂
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	C₂×M₆₄	D₄₇
hypocenter	C₄₇	C₂×#1398
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #20241523	C₂	direct (N × H)
7. #15688957	C₄	semidirect
9. C₂×C₄₇⋊C₃₂	C₄	semidirect
13. C₄₇×M₆₄	C₄	semidirect
15. C₂×C₃₂×C₄₇	C₄	semidirect
23. C₄₇⋊C₃₂	C₂×C₄	semidirect
31. C₃₂×C₄₇	C₂×C₄	semidirect
80. C₂×C₄₇	#1398	semidirect
87. M₆₄	Dic₄₇	semidirect
90. C₂×C₃₂	Dic₄₇	semidirect
93. C₄₇	C₂×#1398	semidirect
109. C₃₂	C₂×Dic₄₇	semidirect
153. C₂	#20241523	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×#15688957	2
C₂×#15688898	2
#20241523	2
C₂×C₄₇×M₆₄	2
C₂×#1398	47

Subgroups — C2×#20241523 #26136462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20241523 #26136462