Subgroups — #37560293

Order 23872.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₇₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	#26045871	C₂	1	1	yes	maximal, characteristic
3	C₄²×D₃₇₃	C₂	1	1	yes	maximal, characteristic
4	#20238620	C₄	1	2	yes	complemented
5	#20238619	—	2	1	no	—
6	D₃₇₃⋊C₈	—	2	1	no	—
7	C₄⋊#13093637	C₄	1	2	yes	complemented
8	C₂²⋊#13093637	—	2	1	no	—
9	C₄×#13093637	—	2	1	no	—
10	C₂×C₄×D₃₇₃	—	2	1	no	—
11	C₂×C₄×D₃₇₃	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₇₃	—	2	1	no	—
13	C₄×Dic₃₇₃	C₄	1	1	yes	characteristic, complemented
14	C₄²×C₃₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#13093637	—	2	2	no	—
16	#15688419	—	2	2	no	—
17	C₂²×D₃₇₃	D₄	1	1	yes	characteristic
18	C₂×Dic₃₇₃	—	2	1	no	—
19	C₂×Dic₃₇₃	D₄	1	1	yes	characteristic
20	C₄×D₃₇₃	—	2	4	no	—
21	C₄×D₃₇₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₃₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×D₃₇₃	D₄	1	1	yes	characteristic
24	C₄×D₃₇₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₃₇₃	—	2	1	no	—
26	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×D₃₇₃	—	2	2	no	—
29	C₂×D₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₃₇₃	C₄⋊C₄	1	1	yes	characteristic
31	#13093637	—	4	2	no	—
32	C₄×C₃₇₃	—	2	2	no	—
33	C₄×C₃₇₃	C₄²	1	1	yes	characteristic
34	C₄×C₃₇₃	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₇₃	—	2	2	no	—
36	Dic₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	Dic₃₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₇₃	—	2	1	no	—
39	C₂×C₃₇₃	#95	1	1	yes	characteristic
40	D₃₇₃	—	2	1	no	—
41	D₃₇₃	#104	1	2	yes	—
42	C₃₇₃	#339	1	1	yes	characteristic, complemented, Sylow
43	#339	—	373	1	no	maximal, Sylow
44	C₂×M₁₆	—	373	1	no	—
45	#117	—	373	1	no	—
46	C₂×C₄²	—	373	1	no	—
47	C₂²×C₄	—	373	1	no	—
48	C₂²×C₄	—	746	1	no	—
49	M₁₆	—	373	2	no	—
50	M₁₆	—	746	1	no	—
51	C₂×C₈	—	746	1	no	—
52	C₄⋊C₄	—	373	2	no	—
53	C₂²⋊C₄	—	746	1	no	—
54	C₄²	—	373	1	no	—
55	C₄²	—	746	1	no	—
56	C₄²	—	746	1	no	—
57	C₄²	#13093637	1	1	yes	characteristic, complemented
58	C₂³	—	373	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	373	1	no	—
61	C₂×C₄	—	373	1	no	—
62	C₂×C₄	—	373	1	no	—
63	C₂×C₄	—	373	1	no	—
64	C₂×C₄	—	373	1	no	—
65	C₂×C₄	—	746	4	no	—
66	C₂×C₄	—	746	2	no	—
67	C₂×C₄	—	746	1	no	—
68	C₂×C₄	C₂×#13093637	1	1	yes	characteristic
69	C₈	—	746	2	no	—
70	C₂²	—	373	1	no	—
71	C₂²	—	373	1	no	—
72	C₂²	—	746	2	no	—
73	C₂²	C₂²⋊#13093637	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	373	1	no	—
76	C₄	—	373	1	no	—
77	C₄	—	1492	2	no	—
78	C₄	—	746	2	no	—
79	C₄	C₄×#13093637	1	1	yes	characteristic
80	C₄	C₄⋊#13093637	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	373	2	no	—
83	C₂	—	746	1	no	—
84	C₂	(C₂×#13093637)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#13093637
commutator	C₄×C₃₇₃	C₄²
Frattini	C₂×C₄	C₂×#13093637
Fitting	C₄²×C₃₇₃	C₄
socle	C₂×C₃₇₃	#95
prime radical	C₂²×D₃₇₃	D₄
cosocle	C₂×C₄×D₃₇₃	C₂²
hypercenter	C₄²	#13093637
hypocenter	C₃₇₃	#339
LCS	G ▷ C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. #20238620	C₄	semidirect
7. C₄⋊#13093637	C₄	semidirect
13. C₄×Dic₃₇₃	C₄	semidirect
14. C₄²×C₃₇₃	C₄	semidirect
42. C₃₇₃	#339	semidirect
57. C₄²	#13093637	semidirect

Maximal subgroups

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

subgroup	index
D₃₇₃⋊M₁₆	2
#26045871	2
C₄²×D₃₇₃	2
#339	373