Subgroups — C₂²×#10587538 #14545745

Order 2176.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587538	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#63216	C₂	1	2	yes	maximal, complemented
3	C₂³×#3622	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₁₇	C₄	1	1	yes	characteristic
5	C₂²×#3622	C₄	1	6	yes	—
6	C₂²×#3622	C₄	1	1	yes	characteristic
7	C₂²×#3622	C₂²	1	1	yes	characteristic
8	C₂²×#3622	C₂²	1	6	yes	—
9	#10587538	C₂²	1	16	yes	direct product factor
10	C₂×#63216	C₂²	1	12	yes	complemented
11	C₂⁴×C₁₇	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₁₇	C₈	1	6	yes	—
13	C₂²×C₄×C₁₇	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₇	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#3622	C₂×C₄	1	3	yes	—
17	C₂×#3622	C₂×C₄	1	12	yes	—
18	C₂×#3622	C₂×C₄	1	6	yes	—
19	C₂×#3622	C₂³	1	3	yes	—
20	C₂×#3622	C₂³	1	4	yes	—
21	#63216	C₂³	1	8	yes	complemented
22	C₂³×C₁₇	—	2	4	no	—
23	C₂³×C₁₇	C₂×C₈	1	6	yes	—
24	C₂³×C₁₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₇	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₁₇	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₁₇	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
30	#3622	C₂²×C₄	1	3	yes	—
31	#3622	C₂²×C₄	1	4	yes	—
32	#3622	C₂⁴	1	1	yes	characteristic
33	C₂²×M₃₂	—	17	1	no	maximal, Sylow
34	C₂²×C₁₇	—	2	12	no	—
35	C₂²×C₁₇	M₃₂	1	4	yes	complemented
36	C₂²×C₁₇	C₂²×C₈	1	4	yes	—
37	C₂²×C₁₇	C₂²×C₈	1	3	yes	—
38	C₄×C₁₇	C₂²×C₈	1	4	yes	—
39	C₄×C₁₇	C₂²×C₈	1	3	yes	—
40	C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
41	C₂³×C₈	—	17	1	no	—
42	C₂×M₃₂	—	17	12	no	—
43	C₂²×C₁₆	—	17	2	no	—
44	C₂×C₁₇	—	2	4	no	—
45	C₂×C₁₇	C₂×M₃₂	1	6	yes	complemented
46	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
47	C₂³×C₄	#595	1	1	yes	characteristic
48	C₂²×C₈	—	17	6	no	—
49	C₂²×C₈	—	17	6	no	—
50	C₂²×C₈	—	17	1	no	—
51	C₂²×C₈	—	17	1	no	—
52	M₃₂	—	17	16	no	—
53	C₂×C₁₆	—	17	12	no	—
54	C₁₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
55	C₂⁴	#3624	1	1	yes	characteristic
56	C₂²×C₄	#3624	1	1	yes	characteristic
57	C₂²×C₄	#3624	1	6	yes	—
58	C₂²×C₄	C₂×#595	1	6	yes	—
59	C₂²×C₄	C₂×#595	1	1	yes	characteristic
60	C₂×C₈	—	17	3	no	—
61	C₂×C₈	—	17	4	no	—
62	C₂×C₈	—	17	12	no	—
63	C₂×C₈	—	17	3	no	—
64	C₂×C₈	—	17	6	no	—
65	C₁₆	—	17	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×#3624	1	6	yes	—
68	C₂³	C₂×#3624	1	1	yes	characteristic
69	C₂×C₄	C₂×#3624	1	12	yes	—
70	C₂×C₄	C₂×#3624	1	6	yes	—
71	C₂×C₄	C₂×#3624	1	3	yes	—
72	C₂×C₄	C₂²×#595	1	4	yes	—
73	C₂×C₄	C₂²×#595	1	3	yes	—
74	C₈	—	17	1	no	—
75	C₈	—	17	3	no	—
76	C₈	—	17	4	no	—
77	C₂²	—	2	12	no	—
78	C₂²	#10587538	1	4	yes	direct product factor
79	C₂²	C₂²×#3624	1	4	yes	—
80	C₂²	C₂²×#3624	1	3	yes	—
81	C₄	C₂²×#3624	1	4	yes	—
82	C₄	C₂²×#3624	1	3	yes	—
83	C₄	C₂³×#595	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#10587538	1	6	yes	direct product factor
86	C₂	C₂³×#3624	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#595
commutator	C₂×C₁₇	C₂³×C₈
Frattini	C₄	C₂³×#595
Fitting	C₂³×C₄×C₁₇	C₄
socle	C₂³×C₁₇	C₂×C₈
prime radical	C₂⁴×C₁₇	C₈
cosocle	#3622	C₂⁴
hypercenter	C₂³×C₄	#595
hypocenter	C₁₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#10587538	C₂	direct (N × H)
2. C₂²×#63216	C₂	semidirect
9. #10587538	C₂²	direct (N × H)
10. C₂×#63216	C₂²	semidirect
21. #63216	C₂³	semidirect
35. C₂²×C₁₇	M₃₂	semidirect
45. C₂×C₁₇	C₂×M₃₂	semidirect
54. C₁₇	C₂²×M₃₂	semidirect
78. C₂²	#10587538	direct (N × H)
85. C₂	C₂×#10587538	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×#10587538	2
C₂²×#63216	2
C₂³×#3622	2
C₂²×M₃₂	17

Subgroups — C22×#10587538 #14545745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10587538 #14545745