Subgroups — #17997772

Order 4736.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11924533	C₂	1	1	yes	maximal, characteristic
2	#14557411	C₂	1	12	yes	maximal, complemented
3	#14557401	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#383	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₄₈	C₂²	1	6	yes	—
6	C₂²×C₄×D₃₇	C₂²	1	4	yes	—
7	#11924533	C₄	1	8	yes	complemented
8	C₂×#10596938	C₂²	1	12	yes	—
9	#11924443	C₂²	1	6	yes	—
10	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
11	#95×C₃₇	C₂²	1	6	yes	—
12	C₂³×D₃₇	C₂³	1	4	yes	—
13	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
14	C₂²×C₄×C₃₇	C₂³	1	3	yes	—
15	C₂²×Dic₃₇	C₂³	1	4	yes	—
16	C₂×D₁₄₈	—	4	6	no	—
17	C₂×D₁₄₈	—	4	6	no	—
18	C₂×D₁₄₈	C₂×C₄	1	24	yes	complemented
19	C₂×C₄×D₃₇	—	4	12	no	—
20	C₄²×C₃₇	C₂×C₄	1	4	yes	complemented
21	#10596938	—	4	12	no	—
22	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
23	C₂²×D₃₇	—	2	12	no	—
24	C₂²×D₃₇	—	2	12	no	—
25	C₂²×D₃₇	C₂²×C₄	1	8	yes	—
26	C₂×C₄×C₃₇	—	2	6	no	—
27	C₂×C₄×C₃₇	—	2	12	no	—
28	C₂×C₄×C₃₇	C₂²×C₄	1	6	yes	—
29	C₂×Dic₃₇	—	2	12	no	—
30	D₁₄₈	—	4	24	no	—
31	C₄×D₃₇	—	4	16	no	—
32	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₃₇	H(5,𝔽₂)	1	6	yes	—
34	C₂×D₃₇	—	2	24	no	—
35	C₂×D₃₇	—	4	4	no	—
36	C₂×D₃₇	—	4	12	no	—
37	C₄×C₃₇	—	2	6	no	—
38	C₄×C₃₇	—	4	4	no	—
39	Dic₃₇	—	4	4	no	—
40	#2380	—	37	1	no	maximal, Sylow
41	C₂×C₃₇	#518	1	3	yes	—
42	C₂×C₃₇	#560	1	4	yes	—
43	D₃₇	—	4	8	no	—
44	C₂×#127	—	37	1	no	—
45	#386	—	37	12	no	—
46	#383	—	37	1	no	—
47	#383	D₃₇	1	1	yes	characteristic, complemented
48	C₃₇	#2380	1	1	yes	characteristic, complemented, Sylow
49	C₂²×D₄	—	37	6	no	—
50	C₂³×C₄	—	37	4	no	—
51	#127	—	37	8	no	—
52	C₂×C₂²⋊C₄	—	37	12	no	—
53	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
54	#95	—	37	6	no	—
55	#95	C₂×D₃₇	1	6	yes	—
56	C₂⁴	—	37	4	no	—
57	C₂×D₄	—	37	24	no	—
58	C₂×D₄	—	148	6	no	—
59	C₂×D₄	—	148	6	no	—
60	C₂²×C₄	—	37	4	no	—
61	C₂²×C₄	—	148	12	no	—
62	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
63	C₂²×C₄	C₂²×D₃₇	1	3	yes	—
64	C₂²⋊C₄	—	148	12	no	—
65	C₄²	C₄×D₃₇	1	4	yes	complemented
66	C₂³	—	37	8	no	—
67	C₂³	—	74	12	no	—
68	C₂³	—	74	12	no	—
69	C₂³	C₂³×D₃₇	1	1	yes	characteristic
70	D₄	—	148	24	no	—
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	74	12	no	—
74	C₂×C₄	—	148	16	no	—
75	C₂×C₄	C₂×C₄×D₃₇	1	6	yes	—
76	C₂²	—	74	24	no	—
77	C₂²	—	148	4	no	—
78	C₂²	—	148	12	no	—
79	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
80	C₂²	#11924582	1	6	yes	—
81	C₄	—	2	6	no	—
82	C₄	—	4	4	no	—
83	C₄	—	148	4	no	—
84	C₂	—	148	8	no	—
85	C₂	#14558007	1	3	yes	—
86	C₂	#14558274	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#383	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
prime radical	C₂×#11924533	C₂
hypercenter	#383	D₃₇
hypocenter	C₃₇	#2380
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ #383

Complements

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

N	H ≅ G/N	type
2. #14557411	C₂	semidirect
3. #14557401	C₂	semidirect
4. C₃₇×#383	C₂	semidirect
7. #11924533	C₄	semidirect
18. C₂×D₁₄₈	C₂×C₄	semidirect
20. C₄²×C₃₇	C₂×C₄	semidirect
47. #383	D₃₇	semidirect
48. C₃₇	#2380	semidirect
65. C₄²	C₄×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11924533	2
#14557411	2
#14557401	2
C₃₇×#383	2
#2380	37