Subgroups — #26132678

Order 12032.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241466	C₂	1	6	yes	maximal
2	C₄²×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₄₇	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
6	C₄³×C₄₇	C₄	1	1	yes	characteristic
7	C₂×C₄₇⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₄₇	C₈	1	3	yes	—
11	C₂×C₄²×C₄₇	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₄₇	—	2	6	no	—
14	C₄×C₈×C₄₇	—	2	6	no	—
15	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
16	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
17	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₄₇	—	2	3	no	—
19	C₂×C₈×C₄₇	—	2	3	no	—
20	C₂×C₈×C₄₇	M₁₆	1	6	yes	—
21	C₂×C₈×C₄₇	#41	1	6	yes	—
22	C₄²×C₄₇	—	2	6	no	—
23	C₄²×C₄₇	—	2	3	no	—
24	C₄²×C₄₇	—	2	3	no	—
25	C₄²×C₄₇	C₁₆	1	4	yes	complemented
26	C₄₇⋊C₁₆	—	4	4	no	—
27	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₄₇	—	2	3	no	—
29	C₂×C₄×C₄₇	—	2	3	no	—
30	C₂×C₄×C₄₇	—	2	3	no	—
31	C₂×C₄×C₄₇	—	2	3	no	—
32	C₂×C₄×C₄₇	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₄₇	M₃₂	1	6	yes	—
34	C₂×C₄×C₄₇	#117	1	3	yes	—
35	C₂×C₄×C₄₇	C₂×M₁₆	1	3	yes	—
36	C₈×C₄₇	—	2	6	no	—
37	C₈×C₄₇	#126	1	4	yes	—
38	#12894	—	47	1	no	maximal, Sylow
39	C₂²×C₄₇	#431	1	3	yes	—
40	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
41	C₂²×C₄₇	C₂×M₃₂	1	3	yes	—
42	C₄×C₄₇	—	2	6	no	—
43	C₄×C₄₇	—	2	6	no	—
44	C₄×C₄₇	#383	1	1	yes	characteristic
45	C₄×C₄₇	#432	1	3	yes	—
46	C₄²×C₈	D₄₇	1	1	yes	characteristic
47	#1374	—	47	6	no	—
48	C₂×C₄₇	#1839	1	1	yes	characteristic
49	C₂×C₄₇	#2162	1	3	yes	—
50	C₂×C₄₇	#2163	1	3	yes	—
51	C₂²×C₁₆	—	47	4	no	—
52	C₂×C₄×C₈	Dic₄₇	1	3	yes	—
53	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
54	C₄³	Dic₄₇	1	1	yes	characteristic
55	C₄₇	#12894	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
57	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
58	C₂×C₄²	C₄₇⋊C₈	1	3	yes	—
59	C₂×C₄²	C₄₇⋊C₈	1	1	yes	characteristic
60	C₂×C₄²	C₂×Dic₄₇	1	3	yes	—
61	C₂×C₁₆	—	94	12	no	—
62	C₄×C₈	—	2	6	no	—
63	C₄×C₈	—	2	6	no	—
64	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
65	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
66	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
67	C₂×C₈	—	2	3	no	—
68	C₂×C₈	—	2	3	no	—
69	C₂×C₈	#10625329	1	6	yes	—
70	C₂×C₈	#10625350	1	6	yes	—
71	C₄²	—	2	6	no	—
72	C₄²	—	2	3	no	—
73	C₄²	—	2	3	no	—
74	C₄²	C₄₇⋊C₁₆	1	4	yes	complemented
75	C₁₆	—	188	4	no	—
76	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×C₄₇⋊C₁₆	1	6	yes	—
82	C₂×C₄	#13093964	1	6	yes	—
83	C₂×C₄	#13093983	1	3	yes	—
84	C₂×C₄	C₂×#10625329	1	3	yes	—
85	C₈	—	2	6	no	—
86	C₈	#13094018	1	4	yes	—
87	C₂²	#15689176	1	3	yes	—
88	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
89	C₂²	C₂×#13093964	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#15688991	1	1	yes	characteristic
93	C₄	#15689180	1	3	yes	—
94	C₂	#20243531	1	1	yes	characteristic
95	C₂	#20245187	1	3	yes	—
96	C₂	#20245191	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂²×C₁₆
Fitting	C₄²×C₈×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₂²×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄²×C₈	D₄₇
hypocenter	C₄₇	#12894
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
25. C₄²×C₄₇	C₁₆	semidirect
55. C₄₇	#12894	semidirect
74. C₄²	C₄₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20241466	2
C₄²×C₈×C₄₇	2
#12894	47