Subgroups — #18011563

Order 4736.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558591	C₂	1	8	yes	maximal, complemented
2	C₂×#11924610	C₂	1	6	yes	maximal, complemented
3	C₂×#11924543	C₂	1	1	yes	maximal, characteristic
4	#11924610	C₂²	1	4	yes	—
5	#11924610	C₂²	1	12	yes	complemented
6	C₂×#10596927	C₂²	1	12	yes	complemented
7	#11924543	C₄	1	4	yes	—
8	#11924543	C₂²	1	4	yes	—
9	C₂×#10596934	C₄	1	3	yes	—
10	C₂²×#64436	C₂²	1	3	yes	—
11	C₂×#41×C₃₇	C₄	1	1	yes	characteristic
12	#41×C₃₇	C₈	1	4	yes	—
13	#41×C₃₇	C₂×C₄	1	4	yes	—
14	C₂×Q₈×C₃₇	C₈	1	1	yes	characteristic
15	C₂×D₄×C₃₇	C₈	1	3	yes	—
16	C₂²×C₄×C₃₇	C₂×C₄	1	3	yes	—
17	#10596934	C₂×C₄	1	12	yes	—
18	C₂×#64436	C₂×C₄	1	3	yes	—
19	C₂×#64436	C₂×C₄	1	6	yes	—
20	C₂×#64436	C₂³	1	1	yes	characteristic
21	C₂×#64436	C₂³	1	6	yes	—
22	#10596927	C₂³	1	8	yes	—
23	C₂³×C₃₇	C₂×C₈	1	3	yes	—
24	Q₈×C₃₇	C₂×C₈	1	4	yes	—
25	D₄×C₃₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₃₇	C₂×C₈	1	3	yes	—
27	C₂×C₄×C₃₇	C₂×C₈	1	6	yes	—
28	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₃₇	C₂²×C₄	1	6	yes	—
30	#64436	C₂²×C₄	1	1	yes	characteristic
31	#64436	C₂²×C₄	1	6	yes	—
32	#64436	C₂⁴	1	1	yes	characteristic
33	C₂²×C₃₇	—	4	3	no	—
34	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
35	C₂²×C₃₇	C₂²×C₈	1	6	yes	—
36	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₃₇	C₂²×C₈	1	6	yes	—
38	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
39	#3407	—	37	1	no	maximal, Sylow
40	C₂×C₃₇	—	2	1	no	—
41	C₂×C₃₇	—	2	6	no	—
42	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
43	C₂×#131	—	37	1	no	—
44	#504	—	37	8	no	—
45	C₂×M₃₂	—	37	6	no	—
46	C₃₇	#3407	1	1	yes	characteristic, complemented, Sylow
47	C₂×#41	#3866	1	1	yes	characteristic
48	#131	—	37	4	no	—
49	#131	—	37	4	no	—
50	C₂×M₁₆	—	37	3	no	—
51	C₂²×C₈	—	37	3	no	—
52	M₃₂	—	37	4	no	—
53	M₃₂	—	37	12	no	—
54	C₂×C₁₆	—	37	12	no	—
55	#41	#64438	1	4	yes	—
56	#41	C₂×#3866	1	4	yes	—
57	C₂×Q₈	#64438	1	1	yes	characteristic
58	C₂×D₄	#64438	1	3	yes	—
59	C₂²×C₄	C₂×#3866	1	3	yes	—
60	M₁₆	—	37	12	no	—
61	C₂×C₈	—	37	3	no	—
62	C₂×C₈	—	37	6	no	—
63	C₂×C₈	—	37	6	no	—
64	C₂×C₈	—	37	1	no	—
65	C₁₆	—	37	8	no	—
66	C₂³	C₂×#64438	1	3	yes	—
67	Q₈	C₂×#64438	1	4	yes	—
68	D₄	C₂×#64438	1	12	yes	—
69	C₂×C₄	C₂×#64438	1	3	yes	—
70	C₂×C₄	C₂×#64438	1	6	yes	—
71	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
72	C₂×C₄	C₂²×#3866	1	6	yes	—
73	C₈	—	37	1	no	—
74	C₈	—	37	1	no	—
75	C₈	—	37	6	no	—
76	C₂²	—	4	3	no	—
77	C₂²	C₂²×#64438	1	1	yes	characteristic
78	C₂²	C₂²×#64438	1	6	yes	—
79	C₄	C₂²×#64438	1	1	yes	characteristic
80	C₄	C₂²×#64438	1	6	yes	—
81	C₄	C₂³×#3866	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2	6	no	—
84	C₂	C₂³×#64438	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#3866
commutator, socle	C₂×C₃₇	C₂³×C₈
Fitting, prime radical	C₂×#41×C₃₇	C₄
cosocle	#64436	C₂⁴
hypercenter	C₂×#41	#3866
hypocenter	C₃₇	#3407
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #14558591	C₂	semidirect
2. C₂×#11924610	C₂	semidirect
5. #11924610	C₂²	semidirect
6. C₂×#10596927	C₂²	semidirect
46. C₃₇	#3407	semidirect

Maximal subgroups

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

subgroup	index
#14558591	2
C₂×#11924610	2
C₂×#11924543	2
#3407	37