Subgroups — #16776914

Order 3328.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#6092	C₂	1	1	yes	maximal, characteristic
2	C₈×#6092	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic
4	C₄×#6092	C₄	1	2	yes	—
5	C₄×#6092	C₄	1	2	yes	—
6	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
7	C₄×#6090	C₄	1	1	yes	characteristic
8	C₈×#1055	C₂²	1	1	yes	characteristic
9	C₂×#6092	C₈	1	4	yes	—
10	C₂×#6092	C₈	1	4	yes	—
11	C₂×C₁₃×C₁₆	C₈	1	2	yes	—
12	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×#6090	C₈	1	2	yes	—
14	C₄×#1055	C₂×C₄	1	1	yes	characteristic
15	C₄×#1055	C₂×C₄	1	1	yes	characteristic
16	#7052	—	13	1	no	maximal, Sylow
17	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₁₃	C₄²	1	1	yes	characteristic
20	C₂×#1055	C₂×C₈	1	2	yes	—
21	C₂×#1055	C₂×C₈	1	2	yes	—
22	#6092	—	2	4	no	—
23	#6092	C₁₆	1	8	yes	complemented
24	C₁₃×C₁₆	—	2	2	no	—
25	#6090	—	2	2	no	—
26	C₈×C₁₆	—	13	1	no	—
27	C₈×C₁₆	—	13	1	no	—
28	C₈×C₁₆	—	13	1	no	—
29	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
32	C₈×C₁₃	M₃₂	1	2	yes	—
33	C₈×C₁₃	M₃₂	1	2	yes	—
34	#1055	C₂×C₁₆	1	4	yes	—
35	#1055	M₃₂	1	4	yes	—
36	C₄×C₁₆	—	13	2	no	—
37	C₄×C₁₆	—	13	2	no	—
38	C₄×C₁₆	—	13	1	no	—
39	C₄×C₁₆	#261	1	1	yes	characteristic
40	C₈²	—	13	1	no	—
41	C₂²×C₁₃	C₈²	1	1	yes	characteristic
42	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
43	C₄×C₁₃	#346	1	2	yes	—
44	C₄×C₁₃	#346	1	1	yes	characteristic
45	C₄×C₁₃	#346	1	1	yes	characteristic
46	C₂×C₁₆	—	13	4	no	—
47	C₂×C₁₆	—	13	4	no	—
48	C₂×C₁₆	—	13	2	no	—
49	C₂×C₁₆	#1057	1	2	yes	—
50	C₄×C₈	—	13	1	no	—
51	C₄×C₈	—	13	1	no	—
52	C₄×C₈	C₂×#261	1	1	yes	characteristic
53	C₂×C₁₃	C₈×C₁₆	1	1	yes	characteristic
54	C₂×C₁₃	#1311	1	1	yes	characteristic
55	C₂×C₁₃	#1311	1	1	yes	characteristic
56	C₂×C₈	—	13	2	no	—
57	C₂×C₈	—	13	2	no	—
58	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
59	C₂×C₈	C₂×#1057	1	1	yes	characteristic
60	C₄²	C₄×#261	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	13	8	no	—
63	C₁₆	—	26	4	no	—
64	C₁₆	—	26	2	no	—
65	C₁₃	#7052	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄	C₈×#261	1	1	yes	characteristic
67	C₂×C₄	C₈×#261	1	1	yes	characteristic
68	C₂×C₄	C₄×#1057	1	1	yes	characteristic
69	C₈	—	13	4	no	—
70	C₈	—	13	4	no	—
71	C₈	#88098	1	2	yes	—
72	C₈	#88106	1	2	yes	—
73	C₂²	C₈×#1057	1	1	yes	characteristic
74	C₄	C₁₆×#261	1	2	yes	—
75	C₄	#11718216	1	2	yes	—
76	C₄	#11718231	1	1	yes	characteristic
77	C₄	#11718258	1	1	yes	characteristic
78	C₂	C₁₆×#1057	1	1	yes	characteristic
79	C₂	#13129747	1	1	yes	characteristic
80	C₂	#13129748	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₈×C₁₆
Fitting	C₄×C₁₃×C₁₆	C₄
socle, prime radical	C₂²×C₁₃	C₈²
cosocle	C₈×#1055	C₂²
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	#7052
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
23. #6092	C₁₆	semidirect
65. C₁₃	#7052	semidirect

Maximal subgroups

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

subgroup	index
C₈×#6092	2
C₁₆×#1055	2
#7052	13