Subgroups — #34033116

Order 18688.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14556681	C₂	1	2	yes	maximal
2	C₈×#11924031	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556681	—	2	2	no	—
4	C₄×#11924031	C₄	1	1	yes	characteristic
5	C₄×#11924031	C₂²	1	1	yes	characteristic
6	C₈×#10596831	C₄	1	1	yes	characteristic
7	#14556681	—	4	2	no	—
8	C₂×#11924031	—	2	1	no	—
9	C₂×#11924031	D₄	1	1	yes	characteristic
10	C₂×#11924031	Q₈	1	1	yes	characteristic
11	C₄×#10596831	C₈	1	1	yes	characteristic
12	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₇₃	C₈	1	1	yes	characteristic
14	#11924031	—	2	2	no	—
15	#11924031	—	2	1	no	—
16	#11924031	D₈	1	1	yes	characteristic
17	#11924031	Q₁₆	1	1	yes	characteristic
18	C₂×#10596831	C₁₆	1	1	yes	characteristic
19	C₂×#10596831	C₁₆	1	1	yes	characteristic
20	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
21	C₂×#10596831	M₁₆	1	1	yes	characteristic
22	C₂×C₈×C₇₃	—	2	1	no	—
23	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
27	C₈×C₇₃	—	2	2	no	—
28	#10596831	—	2	1	no	—
29	#10596831	—	2	1	no	—
30	#10596831	C₃₂	1	2	yes	complemented
31	#10596831	#107	1	1	yes	characteristic
32	#10596831	#108	1	1	yes	characteristic
33	C₂²×C₇₃	C₄⋊C₁₆	1	1	yes	characteristic
34	C₄×C₇₃	—	2	1	no	—
35	C₄×C₇₃	#335	1	1	yes	characteristic
36	C₄×C₇₃	#364	1	1	yes	characteristic
37	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
38	C₄×C₇₃	M₆₄	1	1	yes	characteristic
39	#7458	—	73	1	no	maximal, Sylow
40	C₂×C₇₃	#1370	1	1	yes	characteristic
41	C₂×C₇₃	C₄⋊C₃₂	1	1	yes	characteristic
42	C₂×C₇₃	#1422	1	1	yes	characteristic
43	C₄⋊C₃₂	—	73	2	no	—
44	C₈×C₁₆	—	73	1	no	—
45	C₇₃	#7458	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃₂	—	146	2	no	—
47	C₄×C₁₆	—	73	1	no	—
48	C₄×C₁₆	—	73	1	no	—
49	C₈²	—	73	1	no	—
50	C₂×C₁₆	—	73	1	no	—
51	C₂×C₁₆	—	73	1	no	—
52	C₂×C₁₆	—	146	1	no	—
53	C₄×C₈	—	73	1	no	—
54	C₄×C₈	—	73	1	no	—
55	C₄×C₈	#10596842	1	1	yes	characteristic
56	C₃₂	—	292	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	73	1	no	—
59	C₂×C₈	—	73	1	no	—
60	C₂×C₈	—	73	1	no	—
61	C₂×C₈	—	73	1	no	—
62	C₄²	C₂×#10596842	1	1	yes	characteristic
63	C₁₆	—	73	1	no	—
64	C₁₆	—	73	1	no	—
65	C₁₆	—	146	1	no	—
66	C₁₆	—	146	2	no	—
67	C₂×C₄	C₄⋊#10596842	1	1	yes	characteristic
68	C₂×C₄	#14556686	1	1	yes	characteristic
69	C₂×C₄	#14556687	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	73	1	no	—
72	C₈	—	73	1	no	—
73	C₈	—	73	2	no	—
74	C₈	—	146	1	no	—
75	C₈	—	146	1	no	—
76	C₂²	#17989690	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#17989677	1	1	yes	characteristic
79	C₄	#17989691	1	1	yes	characteristic
80	C₄	#17989694	1	1	yes	characteristic
81	C₄	#17989696	1	1	yes	characteristic
82	C₂	#22626996	1	1	yes	characteristic
83	C₂	#22627030	1	1	yes	characteristic
84	C₂	#22627032	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#10596842
commutator	C₄×C₇₃	C₂×C₃₂
Frattini	C₄²	C₂×#10596842
Fitting	C₄×C₈×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	C₄⋊C₁₆
cosocle	C₄×#11924031	C₂²
hypercenter	C₄×C₈	#10596842
hypocenter	C₇₃	#7458
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
30. #10596831	C₃₂	semidirect
45. C₇₃	#7458	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14556681	2
C₈×#11924031	2
#7458	73