Subgroups — #419×C₄₀₉ #39908798

Order 26176.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₄₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₄₀₉	C₂	1	1	yes	maximal, characteristic
3	#117×C₄₀₉	C₂	1	1	yes	maximal, characteristic
4	#103×C₄₀₉	C₂	1	4	yes	maximal, complemented
5	C₂×Q₈×C₄₀₉	—	2	2	no	—
6	C₂×Q₈×C₄₀₉	—	2	1	no	—
7	C₂×Q₈×C₄₀₉	C₄	1	4	yes	complemented
8	C₂×Q₈×C₄₀₉	C₂²	1	2	yes	—
9	C₂²×C₄×C₄₀₉	—	2	1	no	—
10	C₂²×C₄×C₄₀₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₄₀₉	—	2	2	no	—
12	C₂×C₈×C₄₀₉	C₂²	1	2	yes	—
13	C₄⋊C₄×C₄₀₉	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₄₀₉	—	2	1	no	—
15	C₄²×C₄₀₉	—	2	1	no	—
16	C₂³×C₄₀₉	D₄	1	1	yes	characteristic
17	Q₈×C₄₀₉	—	2	4	no	—
18	Q₈×C₄₀₉	—	2	2	no	—
19	Q₈×C₄₀₉	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₄₀₉	—	2	4	no	—
21	C₂×C₄×C₄₀₉	—	2	2	no	—
22	C₂×C₄×C₄₀₉	—	2	2	no	—
23	C₂×C₄×C₄₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₄₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₀₉	D₄	1	2	yes	—
26	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
27	C₈×C₄₀₉	—	2	2	no	—
28	C₂²×C₄₀₉	—	2	2	no	—
29	C₂²×C₄₀₉	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄₀₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₀₉	—	2	4	no	—
32	C₄×C₄₀₉	—	4	2	no	—
33	C₄×C₄₀₉	C₂²⋊C₄	1	2	yes	—
34	C₄×C₄₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₄₀₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄₀₉	—	2	2	no	—
37	C₂×C₄₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄₀₉	#137	1	2	yes	—
39	C₄₀₉	#419	1	1	yes	characteristic, direct product factor, Sylow
40	#419	C₄₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂²×Q₈	C₂×C₄₀₉	1	1	yes	characteristic
42	C₂×M₁₆	C₂×C₄₀₉	1	1	yes	characteristic
43	#117	C₂×C₄₀₉	1	1	yes	characteristic
44	#103	C₂×C₄₀₉	1	4	yes	complemented
45	C₂×Q₈	—	2	2	no	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	C₄×C₄₀₉	1	4	yes	complemented
48	C₂×Q₈	C₂²×C₄₀₉	1	2	yes	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₂²×C₄₀₉	1	1	yes	characteristic
51	M₁₆	—	2	2	no	—
52	C₂×C₈	C₂²×C₄₀₉	1	2	yes	—
53	C₄⋊C₄	C₂²×C₄₀₉	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₂³	D₄×C₄₀₉	1	1	yes	characteristic
57	Q₈	—	2	4	no	—
58	Q₈	—	2	2	no	—
59	Q₈	C₂×C₄×C₄₀₉	1	4	yes	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	C₂×C₄×C₄₀₉	1	2	yes	—
64	C₂×C₄	D₄×C₄₀₉	1	1	yes	characteristic
65	C₂×C₄	D₄×C₄₀₉	1	2	yes	—
66	C₂×C₄	C₂³×C₄₀₉	1	1	yes	characteristic
67	C₈	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²⋊C₄×C₄₀₉	1	2	yes	—
70	C₂²	C₂×D₄×C₄₀₉	1	1	yes	characteristic
71	C₄	—	2	4	no	—
72	C₄	—	4	2	no	—
73	C₄	C₂²⋊C₄×C₄₀₉	1	2	yes	—
74	C₄	C₂²×C₄×C₄₀₉	1	1	yes	characteristic
75	C₄	C₂×D₄×C₄₀₉	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×C₂²⋊C₄×C₄₀₉	1	1	yes	characteristic
78	C₂	#137×C₄₀₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₀₉	C₂×D₄
commutator	C₄	C₂²×C₄×C₄₀₉
Frattini, cosocle	C₂×C₄	C₂³×C₄₀₉
prime radical	C₂³×C₄₀₉	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₀₉ ◁ C₂²×C₄×C₄₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. #103×C₄₀₉	C₂	semidirect
7. C₂×Q₈×C₄₀₉	C₄	semidirect
39. C₄₀₉	#419	direct (N × H)
40. #419	C₄₀₉	direct (N × H)
44. #103	C₂×C₄₀₉	semidirect
47. C₂×Q₈	C₄×C₄₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₈×C₄₀₉	2
C₂×M₁₆×C₄₀₉	2
#117×C₄₀₉	2
#103×C₄₀₉	2
#419	409

Subgroups — #419×C409 #39908798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #419×C₄₀₉ #39908798