Subgroups — C₂×C₄×#63216 #14544980

Order 2176.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄²	#595
commutator, hypocenter	C₁₇	C₂×C₄×C₁₆
Frattini	C₂×C₄	C₂²×#595
Fitting	C₂×C₄²×C₁₇	C₄
socle, prime radical	C₂³×C₁₇	C₂×C₈
cosocle	C₂×#3622	C₂³
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#63216	C₂	direct (N × H)
7. C₂×#63216	C₄	direct (N × H)
20. #63216	C₂×C₄	direct (N × H)
22. C₂×C₄×C₁₇	C₁₆	semidirect
36. C₄×C₁₇	C₂×C₁₆	semidirect
42. C₂×C₁₇	C₄×C₁₆	semidirect
52. C₁₇	C₂×C₄×C₁₆	semidirect
64. C₂×C₄	#63216	direct (N × H)
76. C₄	C₂×#63216	direct (N × H)
80. C₂	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
C₂²×#63216	2
C₄×#63216	2
C₂×C₄×#3622	2
C₂×C₄×C₁₆	17

Subgroups — C2×C4×#63216 #14544980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#63216 #14544980