Subgroups — #17994727

Order 4736.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14557399	C₂	1	2	yes	maximal
2	#14557296	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	—
5	#11924443	—	2	4	no	—
6	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₃₇	C₄	1	4	yes	complemented
9	C₂²×C₄×C₃₇	D₄	1	2	yes	—
10	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
11	C₂²×Dic₃₇	—	2	4	no	—
12	C₂×C₈×C₃₇	—	2	4	no	—
13	C₂×C₈×C₃₇	C₂×C₄	1	4	yes	—
14	C₄²×C₃₇	C₂×C₄	1	2	yes	—
15	C₄²×C₃₇	Q₈	1	2	yes	—
16	C₄⋊Dic₃₇	—	2	8	no	—
17	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₇	—	2	4	no	—
19	C₂×C₄×C₃₇	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₃₇	#41	1	4	yes	—
23	C₂×Dic₃₇	—	4	4	no	—
24	C₂×Dic₃₇	—	4	8	no	—
25	C₈×C₃₇	—	2	4	no	—
26	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₃₇	#124	1	2	yes	—
28	C₂²×C₃₇	#135	1	4	yes	—
29	C₄×C₃₇	—	2	4	no	—
30	C₄×C₃₇	#117	1	2	yes	—
31	C₄×C₃₇	#125	1	2	yes	—
32	Dic₃₇	—	8	4	no	—
33	#1835	—	37	1	no	maximal, Sylow
34	C₂×C₃₇	#382	1	1	yes	characteristic
35	C₂×C₃₇	#427	1	2	yes	—
36	C₂×C₃₇	#488	1	4	yes	—
37	C₂×C₄×C₈	D₃₇	1	1	yes	characteristic
38	#382	—	37	2	no	—
39	#340	—	37	4	no	—
40	C₃₇	#1835	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	C₂×D₃₇	1	2	yes	—
42	C₂×C₄⋊C₄	—	37	4	no	—
43	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
44	C₄×C₈	Dic₃₇	1	4	yes	complemented
45	#95	—	74	4	no	—
46	C₂²×C₄	—	74	4	no	—
47	C₂²×C₄	D₁₄₈	1	2	yes	—
48	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	C₂×Dic₃₇	1	4	yes	—
51	C₄⋊C₄	—	74	8	no	—
52	C₄²	Dic₇₄	1	2	yes	—
53	C₄²	C₂×Dic₃₇	1	2	yes	—
54	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	148	4	no	—
57	C₂×C₄	—	148	8	no	—
58	C₂×C₄	C₄⋊Dic₃₇	1	4	yes	—
59	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
60	C₂×C₄	#10596962	1	4	yes	—
61	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
62	C₈	—	2	4	no	—
63	C₂²	C₂×C₄⋊Dic₃₇	1	1	yes	characteristic
64	C₂²	#11924524	1	2	yes	—
65	C₂²	#11924558	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	296	4	no	—
68	C₄	#11924495	1	2	yes	—
69	C₄	#11924528	1	2	yes	—
70	C₂	#14557399	1	1	yes	characteristic
71	C₂	#14557571	1	2	yes	—
72	C₂	#14557888	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×C₄×C₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₃₇
hypocenter	C₃₇	#1835
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×C₈×C₃₇	C₄	semidirect
40. C₃₇	#1835	semidirect
44. C₄×C₈	Dic₃₇	semidirect

Maximal subgroups

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

subgroup	index
#14557399	2
#14557296	2
C₂×C₄×C₈×C₃₇	2
#1835	37