Subgroups — #39909995

Order 26176.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27189705	C₂	1	2	yes	maximal, complemented
2	#27189702	C₂	1	4	yes	maximal, complemented
3	#27189695	C₂	1	4	yes	maximal, complemented
4	#27189692	C₂	1	4	yes	maximal, complemented
5	#125×C₄₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₁₈	C₂²	1	2	yes	—
7	C₂×D₁₆₃₆	C₂²	1	4	yes	—
8	C₂×D₁₆₃₆	C₂²	1	2	yes	—
9	C₂×C₄×D₄₀₉	C₂²	1	4	yes	—
10	#20266967	C₂²	1	4	yes	—
11	#20266965	C₂²	1	4	yes	—
12	#20266965	C₂²	1	4	yes	—
13	#20266965	C₂²	1	4	yes	—
14	C₄⋊C₄×C₄₀₉	C₂²	1	4	yes	—
15	C₄⋊C₄×C₄₀₉	C₂²	1	2	yes	—
16	C₄²×C₄₀₉	C₂²	1	1	yes	characteristic
17	C₂²×D₄₀₉	C₂³	1	4	yes	—
18	Dic₈₁₈	—	4	2	no	—
19	D₁₆₃₆	—	4	4	no	—
20	D₁₆₃₆	—	4	2	no	—
21	C₂×Dic₄₀₉	C₂³	1	4	yes	—
22	C₄×D₄₀₉	—	4	4	no	—
23	C₂×C₄×C₄₀₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₀₉	C₂³	1	2	yes	—
25	C₂×C₄×C₄₀₉	C₂³	1	4	yes	—
26	C₂²×C₄₀₉	C₂⁴	1	1	yes	characteristic
27	C₂×D₄₀₉	—	2	4	no	—
28	C₂×D₄₀₉	—	2	4	no	—
29	C₂×D₄₀₉	—	2	4	no	—
30	C₄×C₄₀₉	—	2	4	no	—
31	C₄×C₄₀₉	—	2	2	no	—
32	C₄×C₄₀₉	—	2	1	no	—
33	Dic₄₀₉	—	2	4	no	—
34	C₂×C₄₀₉	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₄₀₉	2₋¹⁺⁴	1	1	yes	characteristic
36	D₄₀₉	—	4	4	no	—
37	C₄₀₉	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	409	1	no	maximal, Sylow
39	#125	D₄₀₉	1	1	yes	characteristic, complemented
40	#124	—	409	2	no	—
41	#123	—	409	4	no	—
42	C₂²⋊Q₈	—	409	4	no	—
43	#121	—	409	4	no	—
44	C₂×Q₈	—	409	2	no	—
45	C₂×D₄	—	409	4	no	—
46	C₂×D₄	—	409	2	no	—
47	C₂²×C₄	—	409	4	no	—
48	C₄⋊C₄	—	409	4	no	—
49	C₄⋊C₄	C₂×D₄₀₉	1	4	yes	—
50	C₄⋊C₄	C₂×D₄₀₉	1	2	yes	—
51	C₂²⋊C₄	—	409	4	no	—
52	C₂²⋊C₄	—	409	4	no	—
53	C₂²⋊C₄	—	409	4	no	—
54	C₄²	C₂×D₄₀₉	1	1	yes	characteristic
55	C₂³	—	409	4	no	—
56	Q₈	—	1636	2	no	—
57	D₄	—	1636	4	no	—
58	D₄	—	1636	2	no	—
59	C₂×C₄	—	1636	4	no	—
60	C₂×C₄	—	409	4	no	—
61	C₂×C₄	C₂²×D₄₀₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₄₀₉	1	2	yes	—
63	C₂×C₄	C₂²×D₄₀₉	1	4	yes	—
64	C₂²	—	818	4	no	—
65	C₂²	—	818	4	no	—
66	C₂²	—	818	4	no	—
67	C₂²	C₂³×D₄₀₉	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	818	4	no	—
72	C₂	—	1636	4	no	—
73	C₂	#27189763	1	2	yes	—
74	C₂	#27189765	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₀₉
commutator, socle, cosocle	C₂²×C₄₀₉	C₂⁴
Fitting	#125×C₄₀₉	C₂
hypercenter	#125	D₄₀₉
hypocenter	C₄₀₉	#562
LCS	G ▷ C₂²×C₄₀₉ ▷ C₄₀₉
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #27189705	C₂	semidirect
2. #27189702	C₂	semidirect
3. #27189695	C₂	semidirect
4. #27189692	C₂	semidirect
5. #125×C₄₀₉	C₂	semidirect
37. C₄₀₉	#562	semidirect
39. #125	D₄₀₉	semidirect

Maximal subgroups

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

subgroup	index
#27189705	2
#27189702	2
#27189695	2
#27189692	2
#125×C₄₀₉	2
#562	409