Subgroups — #10716182

Order 768.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₉₆	C₂	1	4	yes	maximal, complemented
2	#69060	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#13725	—	3	1	no	maximal, Sylow
5	C₂×D₄₈	—	2	2	no	—
6	C₂×D₄₈	C₂²	1	4	yes	—
7	#4695	—	2	2	no	—
8	C₂×C₃×C₃₂	C₂²	1	2	yes	—
9	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	D₉₆	—	2	8	no	—
11	C₂×D₃₂	—	3	4	no	—
12	#2246	—	3	2	no	—
13	C₄×C₃₂	D₃	1	1	yes	characteristic, complemented
14	C₂×D₂₄	—	2	4	no	—
15	C₂×D₂₄	—	4	2	no	—
16	#874	—	4	2	no	—
17	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
18	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₈	D₄	1	1	yes	characteristic
20	D₄₈	—	2	8	no	—
21	D₄₈	—	4	4	no	—
22	C₃×C₃₂	D₄	1	4	yes	complemented
23	C₂×D₁₆	—	3	4	no	—
24	C₂×D₁₆	—	6	2	no	—
25	#493	—	6	2	no	—
26	D₃₂	—	6	8	no	—
27	C₂×C₃₂	C₂×D₃	1	2	yes	—
28	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
29	C₂×D₁₂	—	4	4	no	—
30	C₂×D₁₂	—	8	4	no	—
31	C₂×C₃×C₈	D₈	1	1	yes	characteristic
32	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
33	C₃×C₄²	D₈	1	1	yes	characteristic
34	D₂₄	—	4	4	no	—
35	D₂₄	—	4	4	no	—
36	D₂₄	—	8	4	no	—
37	C₃×C₁₆	D₈	1	2	yes	—
38	C₃×C₁₆	C₂×D₄	1	2	yes	—
39	C₂×D₈	—	6	4	no	—
40	C₂×D₈	—	12	2	no	—
41	#127	—	12	2	no	—
42	D₁₆	—	6	8	no	—
43	D₁₆	—	12	4	no	—
44	C₂×C₁₆	D₁₂	1	1	yes	characteristic
45	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
46	C₄×C₈	D₁₂	1	1	yes	characteristic
47	C₃₂	D₁₂	1	4	yes	complemented
48	C₂²×D₃	—	8	4	no	—
49	C₂×C₃×C₄	D₁₆	1	2	yes	—
50	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
51	D₁₂	—	8	4	no	—
52	D₁₂	—	8	4	no	—
53	D₁₂	—	16	8	no	—
54	C₃×C₈	D₁₆	1	2	yes	—
55	C₃×C₈	#127	1	1	yes	characteristic
56	C₃×C₈	C₂×D₈	1	1	yes	characteristic
57	C₂×D₄	—	12	4	no	—
58	C₂×D₄	—	24	4	no	—
59	D₈	—	12	4	no	—
60	D₈	—	12	4	no	—
61	D₈	—	24	4	no	—
62	C₂×C₈	D₂₄	1	1	yes	characteristic
63	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
64	C₄²	D₂₄	1	1	yes	characteristic
65	C₁₆	D₂₄	1	2	yes	—
66	C₁₆	C₂×D₁₂	1	2	yes	—
67	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
68	C₂×D₃	—	16	4	no	—
69	C₂×D₃	—	16	8	no	—
70	C₃×C₄	D₃₂	1	4	yes	complemented
71	C₃×C₄	#493	1	1	yes	characteristic
72	C₃×C₄	C₂×D₁₆	1	1	yes	characteristic
73	C₂³	—	24	4	no	—
74	D₄	—	24	4	no	—
75	D₄	—	24	4	no	—
76	D₄	—	48	8	no	—
77	C₂×C₄	D₄₈	1	2	yes	—
78	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
79	C₈	D₄₈	1	2	yes	—
80	C₈	#874	1	1	yes	characteristic
81	C₈	C₂×D₂₄	1	1	yes	characteristic
82	C₂×C₃	#2246	1	1	yes	characteristic
83	C₂×C₃	C₂×D₃₂	1	2	yes	—
84	D₃	—	32	4	no	—
85	C₂²	—	48	4	no	—
86	C₂²	—	48	8	no	—
87	C₂²	C₂×D₄₈	1	1	yes	characteristic
88	C₄	D₉₆	1	4	yes	complemented
89	C₄	#4695	1	1	yes	characteristic
90	C₄	C₂×D₄₈	1	1	yes	characteristic
91	C₃	#13725	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	96	4	no	—
93	C₂	#69060	1	1	yes	characteristic
94	C₂	C₂×D₉₆	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₈
commutator, cosocle	C₂×C₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₃
Fitting	C₃×C₄×C₃₂	C₂
socle	C₂²×C₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₃
hypocenter	C₃	#13725
LCS	G ▷ C₂×C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×D₉₆	C₂	semidirect
2. #69060	C₂	semidirect
3. C₃×C₄×C₃₂	C₂	semidirect
13. C₄×C₃₂	D₃	semidirect
22. C₃×C₃₂	D₄	semidirect
47. C₃₂	D₁₂	semidirect
70. C₃×C₄	D₃₂	semidirect
88. C₄	D₉₆	semidirect
91. C₃	#13725	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₉₆	2
#69060	2
C₃×C₄×C₃₂	2
#13725	3