Subgroups — #41000751

Order 26816.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269398	C₂	1	2	yes	maximal, complemented
2	C₂×#20269396	C₂	1	2	yes	maximal, complemented
3	#27211715	C₂	1	1	yes	maximal, characteristic, complemented
4	#27211629	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₇₆	—	2	2	no	—
7	C₂×D₁₆₇₆	C₂²	1	2	yes	—
8	#20269398	—	2	4	no	—
9	#20269396	—	2	4	no	—
10	C₂×C₄₁₉⋊C₈	C₂²	1	2	yes	—
11	C₂×Q₈×C₄₁₉	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₄₁₉	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₄₁₉	—	2	2	no	—
14	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄₁₉	—	2	2	no	—
16	D₁₆₇₆	—	2	2	no	—
17	D₁₆₇₆	—	2	2	no	—
18	D₁₆₇₆	—	4	4	no	—
19	C₄₁₉⋊C₈	D₄	1	4	yes	complemented
20	C₂³×C₄₁₉	—	2	1	no	—
21	Q₈×C₄₁₉	—	2	2	no	—
22	D₄×C₄₁₉	—	2	2	no	—
23	C₂×C₄×C₄₁₉	—	2	1	no	—
24	C₂×C₄×C₄₁₉	D₄	1	2	yes	—
25	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄₁₉	—	4	1	no	—
27	C₂²×C₄₁₉	—	4	2	no	—
28	C₂²×C₄₁₉	C₂×D₄	1	1	yes	characteristic
29	C₂×D₄₁₉	—	4	2	no	—
30	C₂×D₄₁₉	—	4	4	no	—
31	C₄×C₄₁₉	—	2	2	no	—
32	C₄×C₄₁₉	—	4	1	no	—
33	C₄×C₄₁₉	C₂×D₄	1	2	yes	—
34	C₂×C₄₁₉	—	8	1	no	—
35	C₂×C₄₁₉	#127	1	1	yes	characteristic
36	C₂×C₄₁₉	AGL(1,ℤ/8)	1	2	yes	—
37	D₄₁₉	—	8	2	no	—
38	C₄₁₉	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	419	1	no	maximal, Sylow
40	C₂×SD₁₆	—	419	2	no	—
41	C₂×D₈	—	419	2	no	—
42	#127	—	419	1	no	—
43	#124	D₄₁₉	1	1	yes	characteristic, complemented
44	#97	—	419	1	no	—
45	C₂×Q₈	C₂×D₄₁₉	1	1	yes	characteristic
46	C₂×D₄	—	838	2	no	—
47	C₂×D₄	—	419	2	no	—
48	C₂×D₄	C₂×D₄₁₉	1	1	yes	characteristic
49	SD₁₆	—	838	4	no	—
50	D₈	—	838	4	no	—
51	C₂×C₈	—	419	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	838	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	838	2	no	—
59	D₄	—	838	2	no	—
60	D₄	—	1676	4	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₄₁₉	1	2	yes	—
63	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
64	C₈	—	419	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	1676	2	no	—
68	C₂²	—	1676	4	no	—
69	C₂²	C₂×C₂²⋊D₄₁₉	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₄₁₉	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	3352	2	no	—
75	C₂	#27211714	1	1	yes	characteristic
76	C₂	#27211743	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁₉
commutator, cosocle	C₂×C₄×C₄₁₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₁₉
Fitting	#124×C₄₁₉	C₂
socle	C₂²×C₄₁₉	C₂×D₄
hypercenter	#124	D₄₁₉
hypocenter	C₄₁₉	#496
LCS	G ▷ C₂×C₄×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#20269398	C₂	semidirect
2. C₂×#20269396	C₂	semidirect
3. #27211715	C₂	semidirect
4. #27211629	C₂	semidirect
5. #124×C₄₁₉	C₂	semidirect
19. C₄₁₉⋊C₈	D₄	semidirect
38. C₄₁₉	#496	semidirect
43. #124	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269398	2
C₂×#20269396	2
#27211715	2
#27211629	2
#124×C₄₁₉	2
#496	419