Subgroups — C₂×#27211709 #41000882

Order 26816.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211709	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₄₁₉	C₂	1	2	yes	maximal
3	C₂×C₄⋊Dic₄₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₁₉⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₄₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₄₁₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₄₁₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₄₁₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₁₉	C₂²	1	2	yes	—
10	C₄⋊Dic₄₁₉	C₂²	1	8	yes	—
11	C₄⋊Dic₄₁₉	C₂²	1	4	yes	—
12	Dic₄₁₉⋊C₄	C₂²	1	8	yes	—
13	C₄×Dic₄₁₉	C₂²	1	4	yes	—
14	C₂²×C₄×C₄₁₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₄₁₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₁₉	C₂²	1	4	yes	—
17	C₂×Dic₄₁₉	—	2	2	no	—
18	C₂×Dic₄₁₉	—	2	2	no	—
19	C₂×Dic₄₁₉	—	2	4	no	—
20	C₂×Dic₄₁₉	C₂³	1	2	yes	—
21	C₂×Dic₄₁₉	C₂³	1	2	yes	—
22	C₂×Dic₄₁₉	C₂³	1	4	yes	—
23	C₂³×C₄₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₁₉	—	2	4	no	—
25	C₂×C₄×C₄₁₉	Q₈	1	4	yes	—
26	C₂×C₄×C₄₁₉	C₂³	1	4	yes	—
27	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
28	C₂²×C₄₁₉	C₂×Q₈	1	2	yes	—
29	C₂²×C₄₁₉	#41	1	2	yes	—
30	C₂²×C₄₁₉	#41	1	2	yes	—
31	C₂²×C₄₁₉	C₂⁴	1	1	yes	characteristic
32	C₄×C₄₁₉	—	2	4	no	—
33	C₄×C₄₁₉	C₂×Q₈	1	4	yes	—
34	Dic₄₁₉	—	2	2	no	—
35	Dic₄₁₉	—	2	2	no	—
36	Dic₄₁₉	—	2	4	no	—
37	C₂×C₄₁₉	#125	1	4	yes	complemented
38	C₂×C₄₁₉	C₂²×Q₈	1	1	yes	characteristic
39	C₂×C₄₁₉	C₂×#41	1	1	yes	characteristic
40	C₂×C₄₁₉	C₂×#41	1	1	yes	characteristic
41	C₄₁₉	C₂×#125	1	1	yes	characteristic, complemented, Sylow
42	C₂×#125	—	419	1	no	maximal, Sylow
43	#125	—	419	8	no	—
44	C₂×C₄⋊C₄	—	419	2	no	—
45	C₂×C₄⋊C₄	—	419	2	no	—
46	C₂×C₄⋊C₄	—	419	1	no	—
47	C₂×C₄⋊C₄	D₄₁₉	1	1	yes	characteristic
48	C₂×C₄²	—	419	1	no	—
49	C₂²×C₄	—	419	1	no	—
50	C₂²×C₄	—	419	1	no	—
51	C₂²×C₄	—	419	2	no	—
52	C₂²×C₄	C₂×D₄₁₉	1	2	yes	—
53	C₂²×C₄	C₂×D₄₁₉	1	1	yes	characteristic
54	C₄⋊C₄	—	419	8	no	—
55	C₄⋊C₄	—	419	8	no	—
56	C₄⋊C₄	—	419	4	no	—
57	C₄⋊C₄	C₂×D₄₁₉	1	4	yes	—
58	C₄²	—	419	4	no	—
59	C₂³	C₂²×D₄₁₉	1	1	yes	characteristic
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	—	838	2	no	—
62	C₂×C₄	—	838	2	no	—
63	C₂×C₄	—	838	4	no	—
64	C₂×C₄	—	419	2	no	—
65	C₂×C₄	—	419	2	no	—
66	C₂×C₄	—	419	4	no	—
67	C₂×C₄	Dic₈₃₈	1	4	yes	—
68	C₂×C₄	C₂²×D₄₁₉	1	4	yes	—
69	C₂×C₄	C₂²×D₄₁₉	1	2	yes	—
70	C₂²	C₂×Dic₈₃₈	1	2	yes	—
71	C₂²	#20269410	1	2	yes	—
72	C₂²	#20269412	1	2	yes	—
73	C₂²	C₂³×D₄₁₉	1	1	yes	characteristic
74	C₄	—	2	4	no	—
75	C₄	—	838	2	no	—
76	C₄	—	838	2	no	—
77	C₄	—	838	4	no	—
78	C₄	C₂×Dic₈₃₈	1	4	yes	—
79	C₂	#27211709	1	4	yes	direct product factor
80	C₂	C₂²×Dic₈₃₈	1	1	yes	characteristic
81	C₂	C₂×#20269410	1	1	yes	characteristic
82	C₂	C₂×#20269412	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₄₁₉
commutator, cosocle	C₂²×C₄₁₉	C₂⁴
Frattini	C₂²	C₂³×D₄₁₉
Fitting	C₂×C₄⋊C₄×C₄₁₉	C₂
socle, prime radical	C₂³×C₄₁₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₄₁₉
hypocenter	C₄₁₉	C₂×#125
LCS	G ▷ C₂²×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #27211709	C₂	direct (N × H)
37. C₂×C₄₁₉	#125	semidirect
41. C₄₁₉	C₂×#125	semidirect
79. C₂	#27211709	direct (N × H)

Maximal subgroups

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

subgroup	index
#27211709	2
C₂×C₄⋊Dic₄₁₉	2
C₂×Dic₄₁₉⋊C₄	2
C₂×C₄×Dic₄₁₉	2
C₂×C₄⋊C₄×C₄₁₉	2
C₂×#125	419

Subgroups — C2×#27211709 #41000882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27211709 #41000882