Subgroups — #37559403

Order 23872.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20238609	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₃₇₃	C₂	1	2	yes	maximal, complemented
3	#26045710	C₂	1	2	yes	maximal, complemented
4	#26045709	C₂	1	1	yes	maximal, characteristic, complemented
5	#100×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#20238609	—	2	2	no	—
7	#20238609	—	2	2	no	—
8	C₂×Dic₇₄₆	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₃₇₃	C₄	1	2	yes	—
10	C₂²×Dic₃₇₃	C₄	1	2	yes	—
11	C₂×C₄×D₃₇₃	C₂²	1	1	yes	characteristic
12	#20238594	C₂²	1	2	yes	—
13	#20238593	—	2	2	no	—
14	C₈×D₃₇₃	—	2	2	no	—
15	C₂×D₄×C₃₇₃	C₂²	1	1	yes	characteristic
16	M₁₆×C₃₇₃	C₂²	1	2	yes	—
17	C₂²×D₃₇₃	C₂×C₄	1	1	yes	characteristic
18	Dic₇₄₆	—	2	2	no	—
19	C₂²⋊D₃₇₃	—	2	4	no	—
20	C₂×Dic₃₇₃	—	2	2	no	—
21	C₂×Dic₃₇₃	—	2	2	no	—
22	C₂×Dic₃₇₃	C₂×C₄	1	2	yes	—
23	C₂×Dic₃₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×D₃₇₃	D₄	1	2	yes	—
25	C₄×D₃₇₃	D₄	1	2	yes	—
26	#15688412	—	2	2	no	—
27	C₂³×C₃₇₃	C₂×C₄	1	2	yes	—
28	D₄×C₃₇₃	—	2	2	no	—
29	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
30	C₈×C₃₇₃	—	2	2	no	—
31	C₂²×C₃₇₃	—	2	2	no	—
32	C₂²×C₃₇₃	—	4	2	no	—
33	C₂²×C₃₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×D₃₇₃	—	4	1	no	—
35	C₂×D₃₇₃	C₂²⋊C₄	1	2	yes	—
36	C₄×C₃₇₃	C₂×D₄	1	2	yes	—
37	Dic₃₇₃	—	2	2	no	—
38	Dic₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₃₇₃	—	2	1	no	—
41	C₂×C₃₇₃	—	4	2	no	—
42	C₂×C₃₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	D₃₇₃	—	2	2	no	—
44	C₃₇₃	#413	1	1	yes	characteristic, complemented, Sylow
45	#413	—	373	1	no	maximal, Sylow
46	C₂×#41	—	373	1	no	—
47	C₂×M₁₆	—	373	2	no	—
48	#101	—	373	2	no	—
49	#100	—	373	1	no	—
50	#100	D₃₇₃	1	1	yes	characteristic, complemented
51	#41	—	746	2	no	—
52	#41	—	746	2	no	—
53	C₂×Q₈	—	373	1	no	—
54	C₂×D₄	—	373	2	no	—
55	C₂×D₄	C₂×D₃₇₃	1	1	yes	characteristic
56	C₂²×C₄	—	373	1	no	—
57	C₂²×C₄	—	373	2	no	—
58	M₁₆	—	373	2	no	—
59	M₁₆	—	746	2	no	—
60	M₁₆	C₂×D₃₇₃	1	2	yes	—
61	C₂×C₈	—	746	2	no	—
62	C₂³	—	373	1	no	—
63	C₂³	C₄×D₃₇₃	1	2	yes	—
64	Q₈	—	746	2	no	—
65	D₄	—	2	2	no	—
66	D₄	—	746	4	no	—
67	C₂×C₄	—	373	1	no	—
68	C₂×C₄	—	373	2	no	—
69	C₂×C₄	—	373	2	no	—
70	C₂×C₄	—	373	2	no	—
71	C₂×C₄	—	746	2	no	—
72	C₂×C₄	—	746	2	no	—
73	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	746	2	no	—
76	C₂²	—	2	2	no	—
77	C₂²	—	4	2	no	—
78	C₂²	—	373	2	no	—
79	C₂²	—	1492	1	no	—
80	C₂²	C₂×C₄×D₃₇₃	1	1	yes	characteristic
81	C₄	—	373	1	no	—
82	C₄	—	373	1	no	—
83	C₄	—	746	2	no	—
84	C₄	D₄×D₃₇₃	1	2	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	4	2	no	—
87	C₂	—	746	2	no	—
88	C₂	C₂²⋊C₄×D₃₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₃₇₃
commutator	C₂²×C₃₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₇₃
Fitting	#100×C₃₇₃	C₂
socle	C₂×C₃₇₃	C₂×C₂²⋊C₄
prime radical	C₂×#20238609	C₂
cosocle	C₂×C₄×C₃₇₃	C₂³
hypercenter	#100	D₃₇₃
hypocenter	C₃₇₃	#413
LCS	G ▷ C₂²×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. M₁₆×D₃₇₃	C₂	semidirect
3. #26045710	C₂	semidirect
4. #26045709	C₂	semidirect
5. #100×C₃₇₃	C₂	semidirect
44. C₃₇₃	#413	semidirect
50. #100	D₃₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20238609	2
M₁₆×D₃₇₃	2
#26045710	2
#26045709	2
#100×C₃₇₃	2
#413	373