Subgroups — C₂²×D₂₂₇₂ #32155568

Order 18176.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₁₃₆	C₂	1	2	yes	maximal, complemented
2	C₂×D₂₂₇₂	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₅₆₈	—	2	2	no	—
5	C₂×D₁₁₃₆	—	2	6	no	—
6	C₂×D₁₁₃₆	C₂²	1	12	yes	complemented
7	D₂₂₇₂	C₂²	1	16	yes	direct product factor
8	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₇₁	C₂²	1	6	yes	complemented
10	C₂²×D₂₈₄	—	4	2	no	—
11	C₂×D₅₆₈	—	2	12	no	—
12	C₂×D₅₆₈	—	4	6	no	—
13	D₁₁₃₆	—	2	12	no	—
14	D₁₁₃₆	C₂³	1	8	yes	complemented
15	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	D₄	1	3	yes	—
17	C₂×C₁₆×C₇₁	C₂³	1	3	yes	—
18	C₃₂×C₇₁	C₂³	1	4	yes	complemented
19	C₂³×D₇₁	—	8	2	no	—
20	C₂×D₂₈₄	—	4	12	no	—
21	C₂×D₂₈₄	—	8	6	no	—
22	D₅₆₈	—	2	8	no	—
23	D₅₆₈	—	4	12	no	—
24	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
25	C₂×C₈×C₇₁	D₈	1	3	yes	—
26	C₂×C₈×C₇₁	C₂×D₄	1	3	yes	—
27	C₁₆×C₇₁	C₂×D₄	1	3	yes	—
28	C₁₆×C₇₁	C₂⁴	1	1	yes	characteristic
29	C₂³×C₇₁	D₁₆	1	1	yes	characteristic
30	C₂²×D₇₁	—	8	12	no	—
31	C₂²×D₇₁	—	16	8	no	—
32	C₂×C₄×C₇₁	D₁₆	1	3	yes	—
33	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
34	D₂₈₄	—	4	8	no	—
35	D₂₈₄	—	8	12	no	—
36	C₈×C₇₁	C₂×D₈	1	3	yes	—
37	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
38	C₂²×C₇₁	D₃₂	1	4	yes	complemented
39	C₂²×C₇₁	C₂×D₁₆	1	3	yes	—
40	C₂×D₇₁	—	8	8	no	—
41	C₂×D₇₁	—	16	24	no	—
42	C₄×C₇₁	C₂×D₁₆	1	3	yes	—
43	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
44	C₂²×D₃₂	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	C₂×D₃₂	1	6	yes	complemented
46	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
47	D₇₁	—	16	8	no	—
48	C₂²×D₁₆	—	71	2	no	—
49	C₂×D₃₂	—	71	12	no	—
50	C₂²×C₃₂	D₇₁	1	1	yes	characteristic, complemented
51	C₇₁	C₂²×D₃₂	1	1	yes	characteristic, complemented, Sylow
52	C₂²×D₈	—	142	2	no	—
53	C₂×D₁₆	—	71	12	no	—
54	C₂×D₁₆	—	142	6	no	—
55	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
56	D₃₂	—	71	16	no	—
57	C₂×C₃₂	C₂×D₇₁	1	6	yes	complemented
58	C₂²×D₄	—	284	2	no	—
59	C₂×D₈	—	284	6	no	—
60	C₂×D₈	—	142	12	no	—
61	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
62	D₁₆	—	71	8	no	—
63	D₁₆	—	142	12	no	—
64	C₂×C₁₆	D₂₈₄	1	3	yes	—
65	C₂×C₁₆	C₂²×D₇₁	1	3	yes	—
66	C₃₂	C₂²×D₇₁	1	4	yes	complemented
67	C₂⁴	—	568	2	no	—
68	C₂×D₄	—	284	12	no	—
69	C₂×D₄	—	568	6	no	—
70	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
71	D₈	—	284	12	no	—
72	D₈	—	142	8	no	—
73	C₂×C₈	D₅₆₈	1	3	yes	—
74	C₂×C₈	C₂×D₂₈₄	1	3	yes	—
75	C₁₆	C₂×D₂₈₄	1	3	yes	—
76	C₁₆	C₂³×D₇₁	1	1	yes	characteristic
77	C₂³	—	568	12	no	—
78	C₂³	—	1136	8	no	—
79	C₂³	D₁₁₃₆	1	1	yes	characteristic
80	D₄	—	284	8	no	—
81	D₄	—	568	12	no	—
82	C₂×C₄	D₁₁₃₆	1	3	yes	—
83	C₂×C₄	C₂×D₅₆₈	1	3	yes	—
84	C₈	C₂×D₅₆₈	1	3	yes	—
85	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
86	C₂²	—	568	8	no	—
87	C₂²	—	1136	24	no	—
88	C₂²	D₂₂₇₂	1	4	yes	direct product factor
89	C₂²	C₂×D₁₁₃₆	1	3	yes	—
90	C₄	C₂×D₁₁₃₆	1	3	yes	—
91	C₄	C₂²×D₅₆₈	1	1	yes	characteristic
92	C₂	—	1136	8	no	—
93	C₂	C₂×D₂₂₇₂	1	6	yes	direct product factor
94	C₂	C₂²×D₁₁₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₁₃₆
commutator, cosocle	C₁₆×C₇₁	C₂⁴
Frattini	C₁₆	C₂³×D₇₁
Fitting	C₂²×C₃₂×C₇₁	C₂
socle	C₂³×C₇₁	D₁₆
hypercenter	C₂²×C₃₂	D₇₁
hypocenter	C₇₁	C₂²×D₃₂
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₁₃₆	C₂	semidirect
2. C₂×D₂₂₇₂	C₂	direct (N × H)
3. C₂²×C₃₂×C₇₁	C₂	semidirect
6. C₂×D₁₁₃₆	C₂²	semidirect
7. D₂₂₇₂	C₂²	direct (N × H)
9. C₂×C₃₂×C₇₁	C₂²	semidirect
14. D₁₁₃₆	C₂³	semidirect
18. C₃₂×C₇₁	C₂³	semidirect
38. C₂²×C₇₁	D₃₂	semidirect
45. C₂×C₇₁	C₂×D₃₂	semidirect
50. C₂²×C₃₂	D₇₁	semidirect
51. C₇₁	C₂²×D₃₂	semidirect
57. C₂×C₃₂	C₂×D₇₁	semidirect
66. C₃₂	C₂²×D₇₁	semidirect
88. C₂²	D₂₂₇₂	direct (N × H)
93. C₂	C₂×D₂₂₇₂	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×D₁₁₃₆	2
C₂×D₂₂₇₂	2
C₂²×C₃₂×C₇₁	2
C₂²×D₃₂	71

Subgroups — C22×D2272 #32155568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₂₂₇₂ #32155568