Subgroups — C₂×#26041118 #37536023

Order 23168.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄₄₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₈₉₆	C₂	1	1	yes	maximal, characteristic
3	#26041118	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃₂×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₇₂₄	—	2	1	no	—
6	C₂×D₁₄₄₈	—	2	1	no	—
7	Dic₁₄₄₈	—	2	1	no	—
8	Dic₁₄₄₈	C₂²	1	2	yes	complemented
9	D₂₈₉₆	—	2	1	no	—
10	D₂₈₉₆	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₈₁	C₂²	1	2	yes	complemented
13	C₂×Dic₃₆₂	—	4	1	no	—
14	C₂×D₇₂₄	—	4	1	no	—
15	Dic₇₂₄	—	2	2	no	—
16	Dic₇₂₄	—	4	1	no	—
17	D₁₄₄₈	—	2	2	no	—
18	D₁₄₄₈	—	4	1	no	—
19	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
20	C₁₆×C₁₈₁	D₄	1	1	yes	characteristic
21	C₁₆×C₁₈₁	C₂³	1	1	yes	characteristic
22	C₂²×D₁₈₁	—	8	1	no	—
23	Dic₃₆₂	—	4	2	no	—
24	Dic₃₆₂	—	8	1	no	—
25	D₇₂₄	—	4	2	no	—
26	D₇₂₄	—	8	1	no	—
27	C₂×Dic₁₈₁	—	8	1	no	—
28	C₂×C₄×C₁₈₁	D₈	1	1	yes	characteristic
29	C₈×C₁₈₁	D₈	1	1	yes	characteristic
30	C₈×C₁₈₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₈₁	D₁₆	1	1	yes	characteristic
32	C₂×D₁₈₁	—	8	2	no	—
33	C₂×D₁₈₁	—	16	2	no	—
34	C₄×C₁₈₁	D₁₆	1	1	yes	characteristic
35	C₄×C₁₈₁	C₂×D₈	1	1	yes	characteristic
36	Dic₁₈₁	—	8	2	no	—
37	C₂×C₁₈₁	SD₆₄	1	2	yes	complemented
38	C₂×C₁₈₁	C₂×D₁₆	1	1	yes	characteristic
39	D₁₈₁	—	16	2	no	—
40	C₁₈₁	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×SD₆₄	—	181	1	no	maximal, Sylow
42	C₂×Q₃₂	—	181	1	no	—
43	C₂×D₁₆	—	181	1	no	—
44	SD₆₄	—	181	4	no	—
45	C₂×C₃₂	D₁₈₁	1	1	yes	characteristic, complemented
46	C₂×Q₁₆	—	362	1	no	—
47	C₂×D₈	—	362	1	no	—
48	Q₃₂	—	362	1	no	—
49	Q₃₂	—	181	2	no	—
50	D₁₆	—	362	1	no	—
51	D₁₆	—	181	2	no	—
52	C₂×C₁₆	C₂×D₁₈₁	1	1	yes	characteristic
53	C₃₂	C₂×D₁₈₁	1	2	yes	complemented
54	C₂×Q₈	—	724	1	no	—
55	C₂×D₄	—	724	1	no	—
56	Q₁₆	—	362	2	no	—
57	Q₁₆	—	724	1	no	—
58	D₈	—	362	2	no	—
59	D₈	—	724	1	no	—
60	C₂×C₈	D₇₂₄	1	1	yes	characteristic
61	C₁₆	D₇₂₄	1	1	yes	characteristic
62	C₁₆	C₂²×D₁₈₁	1	1	yes	characteristic
63	C₂³	—	1448	1	no	—
64	Q₈	—	1448	1	no	—
65	Q₈	—	724	2	no	—
66	D₄	—	1448	1	no	—
67	D₄	—	724	2	no	—
68	C₂×C₄	—	1448	1	no	—
69	C₂×C₄	D₁₄₄₈	1	1	yes	characteristic
70	C₈	D₁₄₄₈	1	1	yes	characteristic
71	C₈	C₂×D₇₂₄	1	1	yes	characteristic
72	C₂²	—	2896	2	no	—
73	C₂²	—	1448	2	no	—
74	C₂²	D₂₈₉₆	1	1	yes	characteristic
75	C₄	—	1448	2	no	—
76	C₄	D₂₈₉₆	1	1	yes	characteristic
77	C₄	C₂×D₁₄₄₈	1	1	yes	characteristic
78	C₂	—	2896	2	no	—
79	C₂	#26041118	1	2	yes	direct product factor
80	C₂	C₂×D₂₈₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₈₉₆
commutator, cosocle	C₁₆×C₁₈₁	C₂³
Frattini	C₁₆	C₂²×D₁₈₁
Fitting	C₂×C₃₂×C₁₈₁	C₂
socle	C₂²×C₁₈₁	D₁₆
prime radical	C₂×D₂₈₉₆	C₂
hypercenter	C₂×C₃₂	D₁₈₁
hypocenter	C₁₈₁	C₂×SD₆₄
LCS	G ▷ C₁₆×C₁₈₁ ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₄₄₈	C₂	semidirect
3. #26041118	C₂	direct (N × H)
4. C₂×C₃₂×C₁₈₁	C₂	semidirect
8. Dic₁₄₄₈	C₂²	semidirect
12. C₃₂×C₁₈₁	C₂²	semidirect
37. C₂×C₁₈₁	SD₆₄	semidirect
40. C₁₈₁	C₂×SD₆₄	semidirect
45. C₂×C₃₂	D₁₈₁	semidirect
53. C₃₂	C₂×D₁₈₁	semidirect
79. C₂	#26041118	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×Dic₁₄₄₈	2
C₂×D₂₈₉₆	2
#26041118	2
C₂×C₃₂×C₁₈₁	2
C₂×SD₆₄	181

Subgroups — C2×#26041118 #37536023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26041118 #37536023