Subgroups — C₂×#20260574 #27144038

Order 12416.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₅₅₂	C₂	1	1	yes	maximal, characteristic
3	#20260574	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃₂×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Dic₃₈₈	—	2	1	no	—
6	C₂×D₇₇₆	—	2	1	no	—
7	Dic₇₇₆	—	2	1	no	—
8	Dic₇₇₆	C₂²	1	2	yes	complemented
9	D₁₅₅₂	—	2	1	no	—
10	D₁₅₅₂	C₂²	1	2	yes	—
11	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	C₃₂×C₉₇	C₂²	1	2	yes	complemented
13	C₂×Dic₁₉₄	—	4	1	no	—
14	C₂×D₃₈₈	—	4	1	no	—
15	Dic₃₈₈	—	2	2	no	—
16	Dic₃₈₈	—	4	1	no	—
17	D₇₇₆	—	2	2	no	—
18	D₇₇₆	—	4	1	no	—
19	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
20	C₁₆×C₉₇	D₄	1	1	yes	characteristic
21	C₁₆×C₉₇	C₂³	1	1	yes	characteristic
22	C₂²×D₉₇	—	8	1	no	—
23	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
24	C₂×Dic₉₇	—	8	1	no	—
25	D₃₈₈	—	4	2	no	—
26	D₃₈₈	—	8	1	no	—
27	Dic₁₉₄	—	4	2	no	—
28	Dic₁₉₄	—	8	1	no	—
29	C₈×C₉₇	D₈	1	1	yes	characteristic
30	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₉₇	D₁₆	1	1	yes	characteristic
32	C₂×D₉₇	—	8	2	no	—
33	C₂×D₉₇	—	16	2	no	—
34	C₄×C₉₇	D₁₆	1	1	yes	characteristic
35	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
36	Dic₉₇	—	8	2	no	—
37	C₂×C₉₇	SD₆₄	1	2	yes	complemented
38	C₂×C₉₇	C₂×D₁₆	1	1	yes	characteristic
39	D₉₇	—	16	2	no	—
40	C₂×SD₆₄	—	97	1	no	maximal, Sylow
41	C₉₇	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₃₂	—	97	1	no	—
43	C₂×D₁₆	—	97	1	no	—
44	SD₆₄	—	97	4	no	—
45	C₂×C₃₂	D₉₇	1	1	yes	characteristic, complemented
46	C₂×Q₁₆	—	194	1	no	—
47	C₂×D₈	—	194	1	no	—
48	Q₃₂	—	97	2	no	—
49	Q₃₂	—	194	1	no	—
50	D₁₆	—	97	2	no	—
51	D₁₆	—	194	1	no	—
52	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
53	C₃₂	C₂×D₉₇	1	2	yes	complemented
54	C₂×Q₈	—	388	1	no	—
55	C₂×D₄	—	388	1	no	—
56	Q₁₆	—	388	1	no	—
57	Q₁₆	—	194	2	no	—
58	D₈	—	388	1	no	—
59	D₈	—	194	2	no	—
60	C₂×C₈	D₃₈₈	1	1	yes	characteristic
61	C₁₆	D₃₈₈	1	1	yes	characteristic
62	C₁₆	C₂²×D₉₇	1	1	yes	characteristic
63	C₂³	—	776	1	no	—
64	Q₈	—	776	1	no	—
65	Q₈	—	388	2	no	—
66	D₄	—	776	1	no	—
67	D₄	—	388	2	no	—
68	C₂×C₄	—	776	1	no	—
69	C₂×C₄	D₇₇₆	1	1	yes	characteristic
70	C₈	D₇₇₆	1	1	yes	characteristic
71	C₈	C₂×D₃₈₈	1	1	yes	characteristic
72	C₂²	—	776	2	no	—
73	C₂²	—	1552	2	no	—
74	C₂²	D₁₅₅₂	1	1	yes	characteristic
75	C₄	—	776	2	no	—
76	C₄	D₁₅₅₂	1	1	yes	characteristic
77	C₄	C₂×D₇₇₆	1	1	yes	characteristic
78	C₂	—	1552	2	no	—
79	C₂	#20260574	1	2	yes	direct product factor
80	C₂	C₂×D₁₅₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₅₅₂
commutator, cosocle	C₁₆×C₉₇	C₂³
Frattini	C₁₆	C₂²×D₉₇
Fitting	C₂×C₃₂×C₉₇	C₂
socle	C₂²×C₉₇	D₁₆
prime radical	C₂×D₁₅₅₂	C₂
hypercenter	C₂×C₃₂	D₉₇
hypocenter	C₉₇	C₂×SD₆₄
LCS	G ▷ C₁₆×C₉₇ ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
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×Dic₇₇₆	C₂	semidirect
3. #20260574	C₂	direct (N × H)
4. C₂×C₃₂×C₉₇	C₂	semidirect
8. Dic₇₇₆	C₂²	semidirect
12. C₃₂×C₉₇	C₂²	semidirect
37. C₂×C₉₇	SD₆₄	semidirect
41. C₉₇	C₂×SD₆₄	semidirect
45. C₂×C₃₂	D₉₇	semidirect
53. C₃₂	C₂×D₉₇	semidirect
79. C₂	#20260574	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×Dic₇₇₆	2
C₂×D₁₅₅₂	2
#20260574	2
C₂×C₃₂×C₉₇	2
C₂×SD₆₄	97

Subgroups — C2×#20260574 #27144038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20260574 #27144038