Subgroups — #41000153

Order 26816.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211714	C₂	1	1	yes	maximal, characteristic
2	#27211633	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₄₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆₇₆	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₄₁₉	—	2	1	no	—
6	C₂×C₂²⋊D₄₁₉	—	2	1	no	—
7	C₂²⋊Dic₄₁₉	—	2	1	no	—
8	C₄×Dic₄₁₉	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₄₁₉	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₄₁₉	—	2	1	no	—
11	C₂²×D₄₁₉	—	2	1	no	—
12	D₁₆₇₆	—	4	1	no	—
13	C₂²⋊D₄₁₉	—	4	2	no	—
14	C₂²⋊D₄₁₉	—	4	2	no	—
15	C₂×Dic₄₁₉	—	2	1	no	—
16	C₂×Dic₄₁₉	—	4	1	no	—
17	C₂³×C₄₁₉	D₄	1	1	yes	characteristic
18	C₂³×C₄₁₉	D₄	1	1	yes	characteristic
19	D₄×C₄₁₉	—	4	1	no	—
20	C₂×C₄×C₄₁₉	—	4	1	no	—
21	C₂×C₄×C₄₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₁₉	—	2	1	no	—
23	C₂²×C₄₁₉	—	2	1	no	—
24	C₂²×C₄₁₉	—	4	1	no	—
25	C₂²×C₄₁₉	—	4	1	no	—
26	C₂²×C₄₁₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₄₁₉	—	4	1	no	—
28	C₂×D₄₁₉	—	4	1	no	—
29	C₂×D₄₁₉	—	4	1	no	—
30	C₄×C₄₁₉	—	2	1	no	—
31	C₄×C₄₁₉	—	8	1	no	—
32	Dic₄₁₉	—	2	2	no	—
33	Dic₄₁₉	—	8	1	no	—
34	C₂×C₄₁₉	—	2	1	no	—
35	C₂×C₄₁₉	—	4	1	no	—
36	C₂×C₄₁₉	—	4	1	no	—
37	C₂×C₄₁₉	#99	1	1	yes	characteristic
38	D₄₁₉	—	8	1	no	—
39	C₄₁₉	#353	1	1	yes	characteristic, complemented, Sylow
40	#353	—	419	1	no	maximal, Sylow
41	#127	—	419	1	no	—
42	#99	—	419	1	no	—
43	#99	D₄₁₉	1	1	yes	characteristic, complemented
44	C₂×D₄	—	838	1	no	—
45	C₂×D₄	—	838	1	no	—
46	C₂×D₄	—	419	1	no	—
47	C₂×D₄	C₂×D₄₁₉	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	838	1	no	—
50	C₄²	—	419	1	no	—
51	C₂³	—	838	1	no	—
52	C₂³	C₂²⋊D₄₁₉	1	1	yes	characteristic
53	C₂³	D₁₆₇₆	1	1	yes	characteristic
54	D₄	—	4	1	no	—
55	D₄	—	1676	1	no	—
56	D₄	—	1676	2	no	—
57	D₄	—	1676	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	838	1	no	—
60	C₂×C₄	—	1676	1	no	—
61	C₂×C₄	C₄×D₄₁₉	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	1676	1	no	—
67	C₂²	—	1676	1	no	—
68	C₂²	—	1676	1	no	—
69	C₂²	(C₂×D₄₁₉)⋊C₄	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	8	1	no	—
72	C₄	—	3352	1	no	—
73	C₄	—	838	2	no	—
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	3352	1	no	—
78	C₂	#27211632	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27211632
commutator	C₂×C₄×C₄₁₉	C₂×C₄
Frattini	C₂×D₄	C₂×D₄₁₉
Fitting	#99×C₄₁₉	C₂
socle	C₂×C₄₁₉	#99
prime radical	#27211714	C₂
cosocle	C₂×D₄×C₄₁₉	C₂²
hypercenter	#99	D₄₁₉
hypocenter	C₄₁₉	#353
LCS	G ▷ C₂×C₄×C₄₁₉ ▷ C₂²×C₄₁₉ ▷ C₂×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. #27211633	C₂	semidirect
3. #99×C₄₁₉	C₂	semidirect
4. C₂×D₁₆₇₆	C₄	semidirect
8. C₄×Dic₄₁₉	C₄	semidirect
39. C₄₁₉	#353	semidirect
43. #99	D₄₁₉	semidirect

Maximal subgroups

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

subgroup	index
#27211714	2
#27211633	2
#99×C₄₁₉	2
#353	419