Subgroups — C₂×#23757110 #34102436

Order 20096.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₁₂	C₂	1	1	yes	maximal, characteristic
2	#23757110	C₂	1	4	yes	maximal, direct product factor
3	C₂×#19097677	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₃₂×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₂₅₆	—	2	1	no	—
6	D₂₅₁₂	—	2	1	no	—
7	D₂₅₁₂	C₂²	1	2	yes	—
8	#19097677	C₂²	1	2	yes	complemented
9	C₂×Q₁₆×C₁₅₇	—	2	1	no	—
10	Q₃₂×C₁₅₇	—	2	1	no	—
11	Q₃₂×C₁₅₇	C₂²	1	2	yes	complemented
12	C₂×C₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×D₆₂₈	—	4	1	no	—
14	D₁₂₅₆	—	2	2	no	—
15	D₁₂₅₆	—	4	1	no	—
16	C₂×Q₈×C₁₅₇	—	4	1	no	—
17	Q₁₆×C₁₅₇	—	2	2	no	—
18	Q₁₆×C₁₅₇	—	4	1	no	—
19	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
20	C₁₆×C₁₅₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₅₇	C₂³	1	1	yes	characteristic
22	C₂²×D₁₅₇	—	8	1	no	—
23	D₆₂₈	—	4	2	no	—
24	D₆₂₈	—	8	1	no	—
25	Q₈×C₁₅₇	—	4	2	no	—
26	Q₈×C₁₅₇	—	8	1	no	—
27	C₂×C₄×C₁₅₇	—	8	1	no	—
28	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
29	C₈×C₁₅₇	D₈	1	1	yes	characteristic
30	C₈×C₁₅₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₅₇	D₁₆	1	1	yes	characteristic
32	C₂×D₁₅₇	—	8	2	no	—
33	C₂×D₁₅₇	—	16	2	no	—
34	C₄×C₁₅₇	—	8	2	no	—
35	C₄×C₁₅₇	D₁₆	1	1	yes	characteristic
36	C₄×C₁₅₇	C₂×D₈	1	1	yes	characteristic
37	C₂×C₁₅₇	SD₆₄	1	2	yes	complemented
38	C₂×C₁₅₇	C₂×D₁₆	1	1	yes	characteristic
39	D₁₅₇	—	16	2	no	—
40	C₁₅₇	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×SD₆₄	—	157	1	no	maximal, Sylow
42	C₂×Q₃₂	D₁₅₇	1	1	yes	characteristic, complemented
43	C₂×D₁₆	—	157	1	no	—
44	SD₆₄	—	157	4	no	—
45	C₂×C₃₂	—	157	1	no	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×D₈	—	314	1	no	—
48	Q₃₂	—	2	1	no	—
49	Q₃₂	C₂×D₁₅₇	1	2	yes	complemented
50	D₁₆	—	314	1	no	—
51	D₁₆	—	157	2	no	—
52	C₂×C₁₆	C₂×D₁₅₇	1	1	yes	characteristic
53	C₃₂	—	157	2	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×D₄	—	628	1	no	—
56	Q₁₆	—	2	2	no	—
57	Q₁₆	—	4	1	no	—
58	D₈	—	314	2	no	—
59	D₈	—	628	1	no	—
60	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
61	C₁₆	C₂²⋊D₁₅₇	1	1	yes	characteristic
62	C₁₆	C₂²×D₁₅₇	1	1	yes	characteristic
63	C₂³	—	1256	1	no	—
64	Q₈	—	4	2	no	—
65	Q₈	—	8	1	no	—
66	D₄	—	628	2	no	—
67	D₄	—	1256	1	no	—
68	C₂×C₄	—	8	1	no	—
69	C₂×C₄	#14578939	1	1	yes	characteristic
70	C₈	#14578939	1	1	yes	characteristic
71	C₈	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
72	C₂²	—	2512	2	no	—
73	C₂²	—	1256	2	no	—
74	C₂²	#19097712	1	1	yes	characteristic
75	C₄	—	8	2	no	—
76	C₄	#19097712	1	1	yes	characteristic
77	C₄	C₂×#14578939	1	1	yes	characteristic
78	C₂	—	2512	2	no	—
79	C₂	#23757110	1	2	yes	direct product factor
80	C₂	C₂×#19097712	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19097712
commutator, cosocle	C₁₆×C₁₅₇	C₂³
Frattini	C₁₆	C₂²×D₁₅₇
Fitting	C₂×Q₃₂×C₁₅₇	C₂
socle	C₂²×C₁₅₇	D₁₆
prime radical	C₂×D₂₅₁₂	C₂
hypercenter	C₂×Q₃₂	D₁₅₇
hypocenter	C₁₅₇	C₂×SD₆₄
LCS	G ▷ C₁₆×C₁₅₇ ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #23757110	C₂	direct (N × H)
3. C₂×#19097677	C₂	semidirect
4. C₂×Q₃₂×C₁₅₇	C₂	semidirect
8. #19097677	C₂²	semidirect
11. Q₃₂×C₁₅₇	C₂²	semidirect
37. C₂×C₁₅₇	SD₆₄	semidirect
40. C₁₅₇	C₂×SD₆₄	semidirect
42. C₂×Q₃₂	D₁₅₇	semidirect
49. Q₃₂	C₂×D₁₅₇	semidirect
79. C₂	#23757110	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×D₂₅₁₂	2
#23757110	2
C₂×#19097677	2
C₂×Q₃₂×C₁₅₇	2
C₂×SD₆₄	157

Subgroups — C2×#23757110 #34102436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757110 #34102436