Subgroups — C₂×#27160916 #38745012

Order 25216.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#27160916	C₂	1	4	yes	maximal, direct product factor
3	C₂×#20263232	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₃₂×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₅₇₆	—	2	1	no	—
6	D₃₁₅₂	—	2	1	no	—
7	D₃₁₅₂	C₂²	1	2	yes	—
8	#20263232	C₂²	1	2	yes	complemented
9	C₂×Q₁₆×C₁₉₇	—	2	1	no	—
10	Q₃₂×C₁₉₇	—	2	1	no	—
11	Q₃₂×C₁₉₇	C₂²	1	2	yes	complemented
12	C₂×C₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×D₇₈₈	—	4	1	no	—
14	D₁₅₇₆	—	2	2	no	—
15	D₁₅₇₆	—	4	1	no	—
16	C₂×Q₈×C₁₉₇	—	4	1	no	—
17	Q₁₆×C₁₉₇	—	2	2	no	—
18	Q₁₆×C₁₉₇	—	4	1	no	—
19	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
20	C₁₆×C₁₉₇	D₄	1	1	yes	characteristic
21	C₁₆×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₂²×D₁₉₇	—	8	1	no	—
23	D₇₈₈	—	4	2	no	—
24	D₇₈₈	—	8	1	no	—
25	Q₈×C₁₉₇	—	4	2	no	—
26	Q₈×C₁₉₇	—	8	1	no	—
27	C₂×C₄×C₁₉₇	—	8	1	no	—
28	C₂×C₄×C₁₉₇	D₈	1	1	yes	characteristic
29	C₈×C₁₉₇	D₈	1	1	yes	characteristic
30	C₈×C₁₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₉₇	D₁₆	1	1	yes	characteristic
32	C₂×D₁₉₇	—	8	2	no	—
33	C₂×D₁₉₇	—	16	2	no	—
34	C₄×C₁₉₇	—	8	2	no	—
35	C₄×C₁₉₇	D₁₆	1	1	yes	characteristic
36	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
37	C₂×C₁₉₇	SD₆₄	1	2	yes	complemented
38	C₂×C₁₉₇	C₂×D₁₆	1	1	yes	characteristic
39	D₁₉₇	—	16	2	no	—
40	C₁₉₇	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×SD₆₄	—	197	1	no	maximal, Sylow
42	C₂×Q₃₂	D₁₉₇	1	1	yes	characteristic, complemented
43	C₂×D₁₆	—	197	1	no	—
44	SD₆₄	—	197	4	no	—
45	C₂×C₃₂	—	197	1	no	—
46	C₂×Q₁₆	—	2	1	no	—
47	C₂×D₈	—	394	1	no	—
48	Q₃₂	—	2	1	no	—
49	Q₃₂	C₂×D₁₉₇	1	2	yes	complemented
50	D₁₆	—	394	1	no	—
51	D₁₆	—	197	2	no	—
52	C₂×C₁₆	C₂×D₁₉₇	1	1	yes	characteristic
53	C₃₂	—	197	2	no	—
54	C₂×Q₈	—	4	1	no	—
55	C₂×D₄	—	788	1	no	—
56	Q₁₆	—	2	2	no	—
57	Q₁₆	—	4	1	no	—
58	D₈	—	788	1	no	—
59	D₈	—	394	2	no	—
60	C₂×C₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
61	C₁₆	C₂²⋊D₁₉₇	1	1	yes	characteristic
62	C₁₆	C₂²×D₁₉₇	1	1	yes	characteristic
63	C₂³	—	1576	1	no	—
64	Q₈	—	4	2	no	—
65	Q₈	—	8	1	no	—
66	D₄	—	788	2	no	—
67	D₄	—	1576	1	no	—
68	C₂×C₄	—	8	1	no	—
69	C₂×C₄	#15691343	1	1	yes	characteristic
70	C₈	#15691343	1	1	yes	characteristic
71	C₈	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
72	C₂²	—	1576	2	no	—
73	C₂²	—	3152	2	no	—
74	C₂²	#20263267	1	1	yes	characteristic
75	C₄	—	8	2	no	—
76	C₄	#20263267	1	1	yes	characteristic
77	C₄	C₂×#15691343	1	1	yes	characteristic
78	C₂	—	3152	2	no	—
79	C₂	#27160916	1	2	yes	direct product factor
80	C₂	C₂×#20263267	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263267
commutator, cosocle	C₁₆×C₁₉₇	C₂³
Frattini	C₁₆	C₂²×D₁₉₇
Fitting	C₂×Q₃₂×C₁₉₇	C₂
socle	C₂²×C₁₉₇	D₁₆
prime radical	C₂×D₃₁₅₂	C₂
hypercenter	C₂×Q₃₂	D₁₉₇
hypocenter	C₁₉₇	C₂×SD₆₄
LCS	G ▷ C₁₆×C₁₉₇ ▷ C₈×C₁₉₇ ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
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. #27160916	C₂	direct (N × H)
3. C₂×#20263232	C₂	semidirect
4. C₂×Q₃₂×C₁₉₇	C₂	semidirect
8. #20263232	C₂²	semidirect
11. Q₃₂×C₁₉₇	C₂²	semidirect
37. C₂×C₁₉₇	SD₆₄	semidirect
40. C₁₉₇	C₂×SD₆₄	semidirect
42. C₂×Q₃₂	D₁₉₇	semidirect
49. Q₃₂	C₂×D₁₉₇	semidirect
79. C₂	#27160916	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×D₃₁₅₂	2
#27160916	2
C₂×#20263232	2
C₂×Q₃₂×C₁₉₇	2
C₂×SD₆₄	197

Subgroups — C2×#27160916 #38745012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27160916 #38745012