Subgroups — #37559756

Order 23872.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045768	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₃₇₃	C₂	1	1	yes	maximal, characteristic
3	#26045726	C₂	1	1	yes	maximal, characteristic, complemented
4	#26045725	C₂	1	1	yes	maximal, characteristic, complemented
5	#26045716	C₂	1	1	yes	maximal, characteristic, complemented
6	#26045704	C₂	1	1	yes	maximal, characteristic
7	#103×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₇₃	—	2	1	no	—
9	C₂×C₄×D₃₇₃	C₂²	1	1	yes	characteristic
10	C₂×#15688412	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
13	#20238589	—	2	1	no	—
14	#20238589	—	2	1	no	—
15	#20238587	—	2	1	no	—
16	C₂×Q₈×C₃₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₇₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₇₃	C₂²	1	1	yes	characteristic
19	C₂²×D₃₇₃	D₄	1	1	yes	characteristic
20	C₂×Dic₃₇₃	—	2	1	no	—
21	C₂×Dic₃₇₃	—	2	1	no	—
22	C₂×Dic₃₇₃	D₄	1	1	yes	characteristic
23	C₄×D₃₇₃	—	2	1	no	—
24	C₄×D₃₇₃	—	2	1	no	—
25	C₄×D₃₇₃	—	4	2	no	—
26	#15688412	—	2	1	no	—
27	Q₈×C₃₇₃	—	2	1	no	—
28	Q₈×C₃₇₃	—	2	1	no	—
29	C₂×C₄×C₃₇₃	—	2	1	no	—
30	C₂×C₄×C₃₇₃	—	2	1	no	—
31	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
32	C₈×C₃₇₃	—	2	1	no	—
33	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂×D₃₇₃	—	2	1	no	—
35	C₂×D₃₇₃	—	2	1	no	—
36	C₂×D₃₇₃	Q₁₆	1	2	yes	—
37	C₄×C₃₇₃	—	4	1	no	—
38	C₄×C₃₇₃	—	4	1	no	—
39	C₄×C₃₇₃	#41	1	1	yes	characteristic
40	C₄×C₃₇₃	#41	1	1	yes	characteristic
41	Dic₃₇₃	—	2	1	no	—
42	Dic₃₇₃	—	4	1	no	—
43	Dic₃₇₃	—	4	1	no	—
44	C₂×C₃₇₃	#123	1	1	yes	characteristic
45	C₂×C₃₇₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₃₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₃₇₃	—	2	2	no	—
48	C₃₇₃	#484	1	1	yes	characteristic, complemented, Sylow
49	#484	—	373	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	373	1	no	—
51	C₂×C₄⋊C₄	—	373	1	no	—
52	#107	—	373	1	no	—
53	#107	—	373	1	no	—
54	#103	—	373	1	no	—
55	#103	D₃₇₃	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	—	373	1	no	—
57	C₂×Q₈	C₂×D₃₇₃	1	1	yes	characteristic
58	C₂²×C₄	—	373	1	no	—
59	C₂²×C₄	—	746	1	no	—
60	C₂×C₈	—	373	1	no	—
61	C₂×C₈	C₂×D₃₇₃	1	1	yes	characteristic
62	C₄⋊C₄	—	373	1	no	—
63	C₄⋊C₄	—	373	1	no	—
64	C₄⋊C₄	—	746	1	no	—
65	C₄⋊C₄	—	746	1	no	—
66	C₄⋊C₄	C₂×D₃₇₃	1	1	yes	characteristic
67	C₂²⋊C₄	—	746	1	no	—
68	C₂³	—	373	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	373	1	no	—
74	C₂×C₄	—	1492	2	no	—
75	C₂×C₄	—	746	1	no	—
76	C₂×C₄	—	746	1	no	—
77	C₂×C₄	—	746	1	no	—
78	C₂×C₄	—	746	1	no	—
79	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	746	1	no	—
82	C₂²	—	373	2	no	—
83	C₂²	—	746	1	no	—
84	C₂²	—	746	1	no	—
85	C₂²	D₄×D₃₇₃	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	1492	1	no	—
89	C₄	—	1492	1	no	—
90	C₄	—	746	1	no	—
91	C₄	#20238609	1	1	yes	characteristic
92	C₄	#20238610	1	1	yes	characteristic
93	C₂	—	746	2	no	—
94	C₂	#26045774	1	1	yes	characteristic
95	C₂	Q₁₆×D₃₇₃	1	1	yes	characteristic
96	C₂	#26045819	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₇₃
commutator, cosocle	C₂×C₄×C₃₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₃
Fitting	#103×C₃₇₃	C₂
socle	C₂²×C₃₇₃	C₂×D₄
prime radical	C₂²×D₃₇₃	D₄
hypercenter	#103	D₃₇₃
hypocenter	C₃₇₃	#484
LCS	G ▷ C₂×C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. #26045726	C₂	semidirect
4. #26045725	C₂	semidirect
5. #26045716	C₂	semidirect
7. #103×C₃₇₃	C₂	semidirect
48. C₃₇₃	#484	semidirect
55. #103	D₃₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26045768	2
C₄⋊C₄×D₃₇₃	2
#26045726	2
#26045725	2
#26045716	2
#26045704	2
#103×C₃₇₃	2
#484	373