Subgroups — #16777007

Order 3328.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129858	C₂	1	1	yes	maximal, characteristic
2	#13129858	C₂	1	1	yes	maximal, characteristic
3	#13112752	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#6092	—	2	1	no	—
5	C₄⋊#6092	—	2	1	no	—
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	—	4	1	no	—
10	C₂×#6092	—	4	1	no	—
11	C₂×C₁₃×C₁₆	—	2	1	no	—
12	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×#6090	C₈	1	1	yes	characteristic
14	C₂×#6090	C₈	1	1	yes	characteristic
15	C₄×#1055	D₄	1	1	yes	characteristic
16	C₄×#1055	Q₈	1	1	yes	characteristic
17	#7099	—	13	1	no	maximal, Sylow
18	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₈×C₁₃	M₁₆	1	1	yes	characteristic
20	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×#1055	—	2	1	no	—
22	C₂×#1055	SD₁₆	1	1	yes	characteristic
23	C₂×#1055	SD₁₆	1	1	yes	characteristic
24	#6092	—	8	1	no	—
25	#6092	—	8	1	no	—
26	C₁₃×C₁₆	—	2	2	no	—
27	#6090	—	2	1	no	—
28	#6090	C₁₆	1	2	yes	complemented
29	#1367	—	13	1	no	—
30	#1367	—	13	1	no	—
31	#1311	—	13	1	no	—
32	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
34	C₂×C₄×C₁₃	#108	1	1	yes	characteristic
35	C₈×C₁₃	—	2	1	no	—
36	C₈×C₁₃	C₂×C₁₆	1	1	yes	characteristic
37	C₈×C₁₃	M₃₂	1	1	yes	characteristic
38	#1055	—	2	1	no	—
39	#1055	—	2	1	no	—
40	#1055	—	4	1	no	—
41	C₄⋊C₁₆	—	26	1	no	—
42	C₄⋊C₁₆	—	26	1	no	—
43	C₄×C₁₆	—	13	1	no	—
44	C₄×C₁₆	#261	1	1	yes	characteristic
45	C₈²	—	13	1	no	—
46	C₂²×C₁₃	#334	1	1	yes	characteristic
47	C₄×C₁₃	—	2	1	no	—
48	C₄×C₁₃	C₄⋊C₁₆	1	1	yes	characteristic
49	C₄×C₁₃	#364	1	1	yes	characteristic
50	C₄×C₁₃	#365	1	1	yes	characteristic
51	C₄×C₁₃	#365	1	1	yes	characteristic
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	13	1	no	—
54	C₂×C₁₆	—	13	1	no	—
55	C₂×C₁₆	—	52	1	no	—
56	C₂×C₁₆	—	52	1	no	—
57	C₄×C₈	—	13	1	no	—
58	C₄×C₈	—	13	1	no	—
59	C₄×C₈	C₂×#261	1	1	yes	characteristic
60	C₂×C₁₃	#1367	1	1	yes	characteristic
61	C₂×C₁₃	#1368	1	1	yes	characteristic
62	C₂×C₁₃	#1369	1	1	yes	characteristic
63	C₂×C₈	—	13	1	no	—
64	C₂×C₈	—	13	1	no	—
65	C₂×C₈	—	26	1	no	—
66	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
67	C₂×C₈	#6122	1	1	yes	characteristic
68	C₄²	C₄⋊#261	1	1	yes	characteristic
69	C₁₆	—	2	2	no	—
70	C₁₆	—	13	2	no	—
71	C₁₆	—	26	1	no	—
72	C₁₆	—	104	1	no	—
73	C₁₆	—	104	1	no	—
74	C₁₃	#7099	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄	#88101	1	1	yes	characteristic
76	C₂×C₄	#88104	1	1	yes	characteristic
77	C₂×C₄	#88112	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	26	1	no	—
80	C₈	—	26	1	no	—
81	C₈	—	52	1	no	—
82	C₈	D₁₃⋊C₁₆	1	1	yes	characteristic
83	C₈	#88098	1	1	yes	characteristic
84	C₂²	#11718255	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	#11718219	1	1	yes	characteristic
87	C₄	#11718220	1	1	yes	characteristic
88	C₄	#11718233	1	1	yes	characteristic
89	C₄	#11718260	1	1	yes	characteristic
90	C₂	#13129859	1	1	yes	characteristic
91	C₂	#13129862	1	1	yes	characteristic
92	C₂	#13129864	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88101
commutator	C₈×C₁₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#261
Fitting	C₄×C₁₃×C₁₆	C₄
socle, prime radical	C₂²×C₁₃	#334
cosocle	C₈×#1055	C₂²
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	#7099
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
28. #6090	C₁₆	semidirect
74. C₁₃	#7099	semidirect

Maximal subgroups

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

subgroup	index
#13129858	2
#13112752	2
#7099	13