Subgroups — #17996468

Order 4736.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₉₆	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₄₈	C₂	1	2	yes	maximal
3	#14557361	C₂	1	1	yes	maximal, characteristic
4	#14557338	C₂	1	2	yes	maximal
5	C₃₇×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₄₈	C₂²	1	2	yes	—
7	C₄×Dic₇₄	—	2	2	no	—
8	Dic₂₉₆	C₄	1	4	yes	complemented
9	#11924470	C₂²	1	2	yes	—
10	#11924461	—	2	2	no	—
11	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₄	—	2	2	no	—
15	C₂×C₈×C₃₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
17	C₄²×C₃₇	D₄	1	1	yes	characteristic
18	C₄⋊Dic₃₇	—	2	2	no	—
19	#10596936	—	4	2	no	—
20	C₄×Dic₃₇	—	4	2	no	—
21	Dic₁₄₈	—	2	2	no	—
22	Dic₁₄₈	C₂×C₄	1	4	yes	—
23	C₁₆×C₃₇	—	2	1	no	—
24	C₁₆×C₃₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₇	D₈	1	2	yes	—
26	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₃₇	—	4	2	no	—
28	C₂×Dic₃₇	—	4	2	no	—
29	Dic₇₄	—	2	4	no	—
30	Dic₇₄	—	4	2	no	—
31	C₈×C₃₇	—	2	1	no	—
32	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₃₇	#41	1	1	yes	characteristic
34	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₃₇	—	2	2	no	—
36	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₃₇	#135	1	1	yes	characteristic
38	Dic₃₇	—	4	4	no	—
39	Dic₃₇	—	8	2	no	—
40	#2176	—	37	1	no	maximal, Sylow
41	C₂×C₃₇	C₄×D₈	1	1	yes	characteristic
42	C₂×C₃₇	#510	1	2	yes	—
43	C₂×Q₃₂	—	37	1	no	—
44	C₄×Q₁₆	—	37	2	no	—
45	#367	—	37	1	no	—
46	#358	—	37	2	no	—
47	#346	D₃₇	1	1	yes	characteristic
48	C₃₇	#2176	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₁₆	—	37	2	no	—
50	C₄×Q₈	—	74	2	no	—
51	Q₃₂	—	37	4	no	—
52	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₃₇	1	1	yes	characteristic
54	#107	—	37	2	no	—
55	#103	—	74	2	no	—
56	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
57	C₂×Q₈	—	74	2	no	—
58	Q₁₆	—	37	4	no	—
59	Q₁₆	—	74	2	no	—
60	C₂×C₈	D₁₄₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
62	C₄⋊C₄	—	74	2	no	—
63	C₄⋊C₄	—	148	2	no	—
64	C₄²	—	148	2	no	—
65	C₄²	D₁₄₈	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₃₇	1	2	yes	—
68	Q₈	—	74	4	no	—
69	Q₈	—	148	2	no	—
70	C₂×C₄	—	148	2	no	—
71	C₂×C₄	—	148	2	no	—
72	C₂×C₄	D₂₉₆	1	2	yes	—
73	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
76	C₈	#10596962	1	1	yes	characteristic
77	C₂²	C₂×D₂₉₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	296	2	no	—
80	C₄	—	148	4	no	—
81	C₄	C₄×D₁₄₈	1	1	yes	characteristic
82	C₄	#11924558	1	1	yes	characteristic
83	C₂	C₄×D₂₉₆	1	1	yes	characteristic
84	C₂	#14557976	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₉₆
commutator	C₈×C₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃₇
Fitting	C₃₇×#346	C₂
socle, prime radical	C₂²×C₃₇	C₂×D₈
cosocle	C₂×C₈×C₃₇	C₂³
hypercenter	#346	D₃₇
hypocenter	C₃₇	#2176
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₂₉₆	C₄	semidirect
48. C₃₇	#2176	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₉₆	2
C₄×Dic₁₄₈	2
#14557361	2
#14557338	2
C₃₇×#346	2
#2176	37