Subgroups — #27173444

Order 12928.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265014	C₂	1	2	yes	maximal, complemented
3	#20265012	C₂	1	2	yes	maximal, complemented
4	#20265011	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₀₈	—	2	1	no	—
7	D₁₆₁₆	—	2	1	no	—
8	D₁₆₁₆	C₂²	1	2	yes	—
9	#15692161	C₂²	1	2	yes	complemented
10	#135×C₁₀₁	—	2	1	no	—
11	Q₃₂×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
12	SD₃₂×C₁₀₁	—	2	1	no	—
13	D₁₆×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×D₄₀₄	—	4	1	no	—
16	D₈₀₈	—	2	2	no	—
17	D₈₀₈	—	4	1	no	—
18	#41×C₁₀₁	—	4	1	no	—
19	Q₁₆×C₁₀₁	—	2	1	no	—
20	SD₁₆×C₁₀₁	—	4	1	no	—
21	D₈×C₁₀₁	—	2	1	no	—
22	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
23	C₁₆×C₁₀₁	D₄	1	1	yes	characteristic
24	C₁₆×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₂²×D₁₀₁	—	8	1	no	—
26	Q₈×C₁₀₁	—	4	1	no	—
27	D₄×C₁₀₁	—	4	1	no	—
28	D₄×C₁₀₁	—	8	1	no	—
29	C₂×C₄×C₁₀₁	—	8	1	no	—
30	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
31	D₄₀₄	—	4	2	no	—
32	D₄₀₄	—	8	1	no	—
33	C₈×C₁₀₁	D₈	1	1	yes	characteristic
34	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₀₁	—	8	1	no	—
36	C₂²×C₁₀₁	D₁₆	1	1	yes	characteristic
37	C₂×D₁₀₁	—	8	2	no	—
38	C₂×D₁₀₁	—	16	2	no	—
39	C₄×C₁₀₁	—	8	1	no	—
40	C₄×C₁₀₁	D₁₆	1	1	yes	characteristic
41	C₄×C₁₀₁	C₂×D₈	1	1	yes	characteristic
42	C₂×C₁₀₁	—	2	1	no	—
43	C₂×C₁₀₁	—	16	1	no	—
44	C₂×C₁₀₁	C₂×D₁₆	1	1	yes	characteristic
45	D₁₀₁	—	16	2	no	—
46	#2263	—	101	1	no	maximal, Sylow
47	C₁₀₁	#2263	1	1	yes	characteristic, complemented, Sylow
48	#508	D₁₀₁	1	1	yes	characteristic, complemented
49	C₂×D₁₆	—	101	1	no	—
50	SD₆₄	—	101	2	no	—
51	D₃₂	—	101	2	no	—
52	M₆₄	—	101	1	no	—
53	#135	—	2	1	no	—
54	C₂×D₈	—	202	1	no	—
55	Q₃₂	C₂×D₁₀₁	1	1	yes	characteristic, complemented
56	SD₃₂	—	2	1	no	—
57	D₁₆	—	101	2	no	—
58	D₁₆	—	202	1	no	—
59	D₁₆	C₂×D₁₀₁	1	1	yes	characteristic, complemented
60	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
61	C₃₂	—	101	2	no	—
62	#41	—	4	1	no	—
63	C₂×D₄	—	404	1	no	—
64	Q₁₆	—	2	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	D₈	—	404	1	no	—
68	D₈	—	202	2	no	—
69	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
70	C₁₆	C₂²⋊D₁₀₁	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₀₁	1	1	yes	characteristic
72	C₂³	—	808	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	8	1	no	—
76	D₄	—	808	1	no	—
77	D₄	—	404	2	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	#13108059	1	1	yes	characteristic
80	C₈	#13108059	1	1	yes	characteristic
81	C₈	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
82	C₂²	—	8	1	no	—
83	C₂²	—	808	2	no	—
84	C₂²	—	1616	2	no	—
85	C₂²	#15692196	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	#15692196	1	1	yes	characteristic
88	C₄	C₂×#13108059	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	1	no	—
91	C₂	—	1616	2	no	—
92	C₂	C₂×#15692196	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#15692196
commutator, cosocle	C₁₆×C₁₀₁	C₂³
Frattini	C₁₆	C₂²×D₁₀₁
Fitting	#508×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂×D₁₆
hypercenter	#508	D₁₀₁
hypocenter	C₁₀₁	#2263
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₆₁₆	C₂	semidirect
2. #20265014	C₂	semidirect
3. #20265012	C₂	semidirect
4. #20265011	C₂	semidirect
5. #508×C₁₀₁	C₂	semidirect
9. #15692161	C₂²	semidirect
11. Q₃₂×C₁₀₁	C₂²	semidirect
13. D₁₆×C₁₀₁	C₂²	semidirect
47. C₁₀₁	#2263	semidirect
48. #508	D₁₀₁	semidirect
55. Q₃₂	C₂×D₁₀₁	semidirect
59. D₁₆	C₂×D₁₀₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₁₆₁₆	2
#20265014	2
#20265012	2
#20265011	2
#508×C₁₀₁	2
#2263	101