Subgroups — #41001089

Order 26816.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211718	C₂	1	1	yes	maximal, characteristic
2	#27211716	C₂	1	2	yes	maximal
3	#27211710	C₂	1	2	yes	maximal
4	#27211708	C₂	1	4	yes	maximal
5	C₄×Dic₈₃₈	C₂	1	2	yes	maximal
6	C₄×Dic₈₃₈	C₂	1	1	yes	maximal, characteristic
7	#27211683	C₂	1	2	yes	maximal
8	C₄×Q₈×C₄₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₈₃₈	C₂²	1	2	yes	—
10	C₂×Dic₈₃₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₁₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₁₉	C₂²	1	2	yes	—
13	C₄⋊Dic₄₁₉	C₂²	1	2	yes	—
14	Dic₄₁₉⋊C₄	C₂²	1	4	yes	—
15	Dic₄₁₉⋊C₄	C₂²	1	2	yes	—
16	Dic₄₁₉⋊C₄	C₂²	1	4	yes	—
17	Dic₄₁₉⋊C₄	C₂²	1	4	yes	—
18	C₄×Dic₄₁₉	C₂²	1	4	yes	—
19	C₄×Dic₄₁₉	C₂²	1	2	yes	—
20	C₂×Q₈×C₄₁₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₄₁₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₄₁₉	C₂²	1	2	yes	—
23	C₄²×C₄₁₉	C₂²	1	2	yes	—
24	C₄²×C₄₁₉	C₂²	1	1	yes	characteristic
25	Dic₈₃₈	—	2	4	no	—
26	Dic₈₃₈	Q₈	1	4	yes	complemented
27	C₂×Dic₄₁₉	C₂³	1	2	yes	—
28	C₂×Dic₄₁₉	C₂³	1	4	yes	—
29	C₂×Dic₄₁₉	C₂³	1	2	yes	—
30	Q₈×C₄₁₉	—	2	2	no	—
31	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
32	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
34	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₄₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄₁₉	C₂⁴	1	1	yes	characteristic
37	C₄×C₄₁₉	—	2	2	no	—
38	C₄×C₄₁₉	—	2	2	no	—
39	C₄×C₄₁₉	—	2	1	no	—
40	C₄×C₄₁₉	C₂×Q₈	1	2	yes	—
41	C₄×C₄₁₉	#41	1	2	yes	—
42	Dic₄₁₉	—	2	4	no	—
43	Dic₄₁₉	—	2	2	no	—
44	Dic₄₁₉	C₂×Q₈	1	4	yes	—
45	C₂×C₄₁₉	C₂²×Q₈	1	1	yes	characteristic
46	C₂×C₄₁₉	C₂×#41	1	1	yes	characteristic
47	C₂×C₄₁₉	2₋¹⁺⁴	1	1	yes	characteristic
48	C₄₁₉	#557	1	1	yes	characteristic, complemented, Sylow
49	#557	—	419	1	no	maximal, Sylow
50	C₄⋊Q₈	—	419	2	no	—
51	C₄⋊Q₈	—	419	1	no	—
52	#125	—	419	4	no	—
53	#125	—	419	2	no	—
54	C₄×Q₈	—	419	2	no	—
55	C₄×Q₈	—	419	2	no	—
56	C₄×Q₈	—	419	1	no	—
57	C₄×Q₈	D₄₁₉	1	1	yes	characteristic
58	C₂×Q₈	—	419	2	no	—
59	C₂×Q₈	—	419	1	no	—
60	C₂×Q₈	C₂×D₄₁₉	1	1	yes	characteristic
61	C₄⋊C₄	—	419	4	no	—
62	C₄⋊C₄	—	419	2	no	—
63	C₄⋊C₄	—	419	4	no	—
64	C₄⋊C₄	—	419	4	no	—
65	C₄⋊C₄	—	419	1	no	—
66	C₄⋊C₄	—	419	2	no	—
67	C₄⋊C₄	—	419	2	no	—
68	C₄⋊C₄	C₂×D₄₁₉	1	1	yes	characteristic
69	C₄⋊C₄	C₂×D₄₁₉	1	2	yes	—
70	C₄²	—	419	4	no	—
71	C₄²	—	419	2	no	—
72	C₄²	C₂×D₄₁₉	1	2	yes	—
73	C₄²	C₂×D₄₁₉	1	1	yes	characteristic
74	Q₈	—	2	2	no	—
75	Q₈	—	838	4	no	—
76	Q₈	—	419	4	no	—
77	C₂×C₄	—	419	2	no	—
78	C₂×C₄	—	419	4	no	—
79	C₂×C₄	—	419	2	no	—
80	C₂×C₄	C₂²×D₄₁₉	1	2	yes	—
81	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₄₁₉	1	2	yes	—
83	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₄₁₉	1	1	yes	characteristic
85	C₂²	C₂³×D₄₁₉	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	2	no	—
88	C₄	—	2	1	no	—
89	C₄	—	838	4	no	—
90	C₄	—	838	2	no	—
91	C₄	—	419	4	no	—
92	C₄	Q₈×D₄₁₉	1	2	yes	—
93	C₄	#20269411	1	2	yes	—
94	C₂	C₂×Q₈×D₄₁₉	1	1	yes	characteristic
95	C₂	C₂×#20269411	1	1	yes	characteristic
96	C₂	#27211765	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁₉
commutator, socle, prime radical, cosocle	C₂²×C₄₁₉	C₂⁴
Fitting	C₄×Q₈×C₄₁₉	C₂
hypercenter	C₄×Q₈	D₄₁₉
hypocenter	C₄₁₉	#557
LCS	G ▷ C₂²×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
26. Dic₈₃₈	Q₈	semidirect
48. C₄₁₉	#557	semidirect

Maximal subgroups

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

subgroup	index
#27211718	2
#27211716	2
#27211710	2
#27211708	2
C₄×Dic₈₃₈	2
#27211683	2
C₄×Q₈×C₄₁₉	2
#557	419