Subgroups — C₂²×Dic₁₁₃₆ #32155575

Order 18176.

462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₆₈	C₂	1	2	yes	maximal
2	C₂×Dic₁₁₃₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₈₄	—	2	2	no	—
5	C₂×Dic₅₆₈	—	2	6	no	—
6	C₂×Dic₅₆₈	C₂²	1	12	yes	—
7	Dic₁₁₃₆	C₂²	1	16	yes	direct product factor
8	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₇₁	C₂²	1	6	yes	—
10	C₂²×Dic₁₄₂	—	4	2	no	—
11	C₂×Dic₂₈₄	—	2	12	no	—
12	C₂×Dic₂₈₄	—	4	6	no	—
13	Dic₅₆₈	—	2	12	no	—
14	Dic₅₆₈	C₂³	1	8	yes	—
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	—
19	C₂×Dic₁₄₂	—	4	12	no	—
20	C₂×Dic₁₄₂	—	8	6	no	—
21	C₂²×Dic₇₁	—	8	2	no	—
22	Dic₂₈₄	—	2	8	no	—
23	Dic₂₈₄	—	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	C₂×C₄×C₇₁	D₁₆	1	3	yes	—
31	C₂×C₄×C₇₁	C₂×D₈	1	3	yes	—
32	C₂×Dic₇₁	—	8	12	no	—
33	Dic₁₄₂	—	4	8	no	—
34	Dic₁₄₂	—	8	12	no	—
35	C₈×C₇₁	C₂×D₈	1	3	yes	—
36	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
37	C₂²×C₇₁	Q₆₄	1	4	yes	complemented
38	C₂²×C₇₁	C₂×D₁₆	1	3	yes	—
39	C₄×C₇₁	C₂×D₁₆	1	3	yes	—
40	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
41	Dic₇₁	—	8	8	no	—
42	C₂²×Q₆₄	—	71	1	no	maximal, Sylow
43	C₂×C₇₁	C₂×Q₆₄	1	6	yes	complemented
44	C₂×C₇₁	C₂²×D₁₆	1	1	yes	characteristic
45	C₂²×Q₃₂	—	71	2	no	—
46	C₂×Q₆₄	—	71	12	no	—
47	C₂²×C₃₂	D₇₁	1	1	yes	characteristic
48	C₇₁	C₂²×Q₆₄	1	1	yes	characteristic, complemented, Sylow
49	C₂²×Q₁₆	—	142	2	no	—
50	C₂×Q₃₂	—	71	12	no	—
51	C₂×Q₃₂	—	142	6	no	—
52	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
53	Q₆₄	—	71	16	no	—
54	C₂×C₃₂	C₂×D₇₁	1	6	yes	—
55	C₂²×Q₈	—	284	2	no	—
56	C₂×Q₁₆	—	284	6	no	—
57	C₂×Q₁₆	—	142	12	no	—
58	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
59	Q₃₂	—	71	8	no	—
60	Q₃₂	—	142	12	no	—
61	C₂×C₁₆	D₂₈₄	1	3	yes	—
62	C₂×C₁₆	C₂²×D₇₁	1	3	yes	—
63	C₃₂	C₂²×D₇₁	1	4	yes	—
64	C₂×Q₈	—	284	12	no	—
65	C₂×Q₈	—	568	6	no	—
66	C₂²×C₄	—	568	2	no	—
67	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
68	Q₁₆	—	284	12	no	—
69	Q₁₆	—	142	8	no	—
70	C₂×C₈	D₅₆₈	1	3	yes	—
71	C₂×C₈	C₂×D₂₈₄	1	3	yes	—
72	C₁₆	C₂×D₂₈₄	1	3	yes	—
73	C₁₆	C₂³×D₇₁	1	1	yes	characteristic
74	C₂³	D₁₁₃₆	1	1	yes	characteristic
75	Q₈	—	284	8	no	—
76	Q₈	—	568	12	no	—
77	C₂×C₄	—	568	12	no	—
78	C₂×C₄	D₁₁₃₆	1	3	yes	—
79	C₂×C₄	C₂×D₅₆₈	1	3	yes	—
80	C₈	C₂×D₅₆₈	1	3	yes	—
81	C₈	C₂²×D₂₈₄	1	1	yes	characteristic
82	C₂²	Dic₁₁₃₆	1	4	yes	direct product factor
83	C₂²	C₂×D₁₁₃₆	1	3	yes	—
84	C₄	—	568	8	no	—
85	C₄	C₂×D₁₁₃₆	1	3	yes	—
86	C₄	C₂²×D₅₆₈	1	1	yes	characteristic
87	C₂	C₂×Dic₁₁₃₆	1	6	yes	direct product factor
88	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, prime radical	C₂³×C₇₁	D₁₆
hypercenter	C₂²×C₃₂	D₇₁
hypocenter	C₇₁	C₂²×Q₆₄
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
2. C₂×Dic₁₁₃₆	C₂	direct (N × H)
7. Dic₁₁₃₆	C₂²	direct (N × H)
37. C₂²×C₇₁	Q₆₄	semidirect
43. C₂×C₇₁	C₂×Q₆₄	semidirect
48. C₇₁	C₂²×Q₆₄	semidirect
82. C₂²	Dic₁₁₃₆	direct (N × H)
87. C₂	C₂×Dic₁₁₃₆	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×Dic₅₆₈	2
C₂×Dic₁₁₃₆	2
C₂²×C₃₂×C₇₁	2
C₂²×Q₆₄	71

Subgroups — C22×Dic1136 #32155575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₁₁₃₆ #32155575