Subgroups — C₄²×#63216 #17960415

Order 4352.

558 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#63216	C₂	1	6	yes	maximal
2	C₄²×#3622	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	C₂²	1	4	yes	—
4	C₄×#63216	C₄	1	24	yes	direct product factor
5	C₂×C₄×#3622	C₄	1	3	yes	—
6	C₂×C₄×#3622	C₂²	1	3	yes	—
7	C₄³×C₁₇	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₈	1	4	yes	—
9	C₂×C₄²×C₁₇	C₂×C₄	1	3	yes	—
10	C₂²×#3622	C₂×C₄	1	3	yes	—
11	C₂²×#3622	C₂³	1	1	yes	characteristic
12	C₂×#63216	C₂×C₄	1	24	yes	—
13	C₄×#3622	C₈	1	12	yes	—
14	C₄×#3622	C₂×C₄	1	12	yes	—
15	C₂²×C₄×C₁₇	C₂×C₈	1	6	yes	—
16	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
17	C₄²×C₁₇	C₁₆	1	16	yes	complemented
18	C₄²×C₁₇	C₄²	1	6	yes	—
19	C₄²×C₁₇	C₂×C₈	1	6	yes	—
20	C₂×#3622	C₄²	1	6	yes	—
21	C₂×#3622	C₂×C₈	1	12	yes	—
22	C₂×#3622	C₂²×C₄	1	6	yes	—
23	#63216	C₄²	1	16	yes	direct product factor
24	C₄²×C₁₆	—	17	1	no	maximal, Sylow
25	C₂³×C₁₇	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	C₄×C₈	1	12	yes	—
27	C₂×C₄×C₁₇	C₂×C₁₆	1	24	yes	—
28	C₂×C₄×C₁₇	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₁₇	C₂²×C₈	1	3	yes	—
30	#3622	C₄×C₈	1	12	yes	—
31	#3622	C₂×C₄²	1	4	yes	—
32	C₂×C₄×C₁₆	—	17	6	no	—
33	C₄²×C₈	—	17	1	no	—
34	C₂²×C₁₇	C₂×C₄×C₈	1	3	yes	—
35	C₂²×C₁₇	C₂²×C₁₆	1	4	yes	—
36	C₄×C₁₇	C₄×C₁₆	1	24	yes	complemented
37	C₄×C₁₇	C₄³	1	1	yes	characteristic
38	C₄×C₁₇	C₂×C₄×C₈	1	3	yes	—
39	C₂²×C₁₆	—	17	4	no	—
40	C₂×C₄×C₈	—	17	3	no	—
41	C₂×C₄×C₈	—	17	3	no	—
42	C₄³	#595	1	1	yes	characteristic
43	C₄×C₁₆	—	17	24	no	—
44	C₂×C₁₇	C₄²×C₈	1	1	yes	characteristic
45	C₂×C₁₇	C₂×C₄×C₁₆	1	6	yes	—
46	C₂²×C₈	—	17	3	no	—
47	C₂²×C₈	—	17	1	no	—
48	C₂×C₄²	#3624	1	4	yes	—
49	C₂×C₄²	C₂×#595	1	3	yes	—
50	C₂×C₁₆	—	17	24	no	—
51	C₄×C₈	—	17	12	no	—
52	C₄×C₈	—	17	12	no	—
53	C₁₇	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₄	C₂×#3624	1	6	yes	—
55	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
56	C₂×C₈	—	17	12	no	—
57	C₂×C₈	—	17	6	no	—
58	C₂×C₈	—	17	6	no	—
59	C₄²	#63216	1	16	yes	direct product factor
60	C₄²	#63242	1	6	yes	—
61	C₄²	C₄×#595	1	6	yes	—
62	C₁₆	—	17	16	no	—
63	C₂³	C₂²×#3624	1	1	yes	characteristic
64	C₂×C₄	C₂×#63216	1	24	yes	—
65	C₂×C₄	C₄×#3624	1	12	yes	—
66	C₂×C₄	C₂×#63242	1	3	yes	—
67	C₂×C₄	C₂×C₄×#595	1	3	yes	—
68	C₈	—	17	4	no	—
69	C₈	—	17	12	no	—
70	C₂²	C₂×C₄×#3624	1	3	yes	—
71	C₂²	C₂²×#63216	1	4	yes	—
72	C₄	C₄×#63216	1	24	yes	direct product factor
73	C₄	C₄²×#595	1	1	yes	characteristic
74	C₄	C₄×#63242	1	3	yes	—
75	C₂	C₄²×#3624	1	1	yes	characteristic
76	C₂	C₂×C₄×#63216	1	6	yes	—

	subgroup	G/H
center, hypercenter	C₄³	#595
commutator, hypocenter	C₁₇	C₄²×C₁₆
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₄³×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂²×C₈
cosocle	C₂²×#3622	C₂³
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₄³

Complements

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

N	H ≅ G/N	type
4. C₄×#63216	C₄	direct (N × H)
17. C₄²×C₁₇	C₁₆	semidirect
23. #63216	C₄²	direct (N × H)
36. C₄×C₁₇	C₄×C₁₆	semidirect
53. C₁₇	C₄²×C₁₆	semidirect
59. C₄²	#63216	direct (N × H)
72. C₄	C₄×#63216	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×C₄×#63216	2
C₄²×#3622	2
C₄²×C₁₆	17

Subgroups — C42×#63216 #17960415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#63216 #17960415