Subgroups — C₂×#26013546 #36373988

Order 22336.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₄₉	C₂	1	2	yes	maximal
2	#26013546	C₂	1	4	yes	maximal, direct product factor
3	C₂³×C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₄₉	—	2	4	no	—
5	C₂²×Dic₃₄₉	—	2	4	no	—
6	C₂²×Dic₃₄₉	C₄	1	8	yes	complemented
7	C₂²×Dic₃₄₉	C₂²	1	4	yes	—
8	C₂⁴×C₃₄₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₃₄₉	—	2	4	no	—
10	C₂²×C₄×C₃₄₉	C₄	1	4	yes	complemented
11	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
12	C₂×Dic₃₄₉	—	2	16	no	—
13	C₂×Dic₃₄₉	—	2	4	no	—
14	C₂×Dic₃₄₉	—	2	4	no	—
15	C₂×Dic₃₄₉	C₂×C₄	1	8	yes	complemented
16	C₂³×C₃₄₉	C₂×C₄	1	4	yes	—
17	C₂³×C₃₄₉	C₂×C₄	1	2	yes	—
18	C₂³×C₃₄₉	D₄	1	4	yes	—
19	C₂³×C₃₄₉	D₄	1	2	yes	—
20	C₂³×C₃₄₉	Q₈	1	2	yes	—
21	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₃₄₉	—	2	8	no	—
23	C₂×C₄×C₃₄₉	—	2	4	no	—
24	C₂×C₄×C₃₄₉	C₂×C₄	1	4	yes	complemented
25	C₂²×C₃₄₉	C₄²	1	4	yes	—
26	C₂²×C₃₄₉	C₂²⋊C₄	1	8	yes	—
27	C₂²×C₃₄₉	C₂²⋊C₄	1	4	yes	—
28	C₂²×C₃₄₉	C₄⋊C₄	1	8	yes	—
29	C₂²×C₃₄₉	C₄⋊C₄	1	4	yes	—
30	C₂²×C₃₄₉	C₂²×C₄	1	2	yes	—
31	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
33	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₃₄₉	—	2	4	no	—
36	Dic₃₄₉	—	2	8	no	—
37	C₂×C₃₄₉	#95	1	8	yes	complemented
38	C₂×C₃₄₉	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	2	yes	—
40	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₃₄₉	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₃₄₉	C₂×#95	1	1	yes	characteristic, complemented, Sylow
44	C₂×#95	—	349	1	no	maximal, Sylow
45	C₂³×C₄	—	349	2	no	—
46	C₂³×C₄	D₃₄₉	1	1	yes	characteristic
47	#95	—	349	4	no	—
48	C₂⁴	C₂×D₃₄₉	1	1	yes	characteristic
49	C₂²×C₄	—	2	4	no	—
50	C₂²×C₄	—	349	8	no	—
51	C₂²×C₄	—	349	4	no	—
52	C₂²×C₄	—	698	4	no	—
53	C₂²×C₄	—	698	4	no	—
54	C₂²×C₄	Dic₃₄₉	1	4	yes	complemented
55	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
56	C₂³	C₄×D₃₄₉	1	4	yes	—
57	C₂³	C₂×Dic₃₄₉	1	2	yes	—
58	C₂³	C₂²⋊D₃₄₉	1	4	yes	—
59	C₂³	D₁₃₉₆	1	2	yes	—
60	C₂³	Dic₆₉₈	1	2	yes	—
61	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
62	C₂×C₄	—	2	8	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	349	8	no	—
65	C₂×C₄	—	698	16	no	—
66	C₂×C₄	—	698	4	no	—
67	C₂×C₄	—	698	4	no	—
68	C₂×C₄	C₂×Dic₃₄₉	1	4	yes	complemented
69	C₂²	C₄×Dic₃₄₉	1	4	yes	—
70	C₂²	#19150210	1	8	yes	—
71	C₂²	C₂²⋊Dic₃₄₉	1	4	yes	—
72	C₂²	#19150212	1	8	yes	—
73	C₂²	C₄⋊Dic₃₄₉	1	4	yes	—
74	C₂²	C₂×C₄×D₃₄₉	1	2	yes	—
75	C₂²	C₂²×Dic₃₄₉	1	1	yes	characteristic
76	C₂²	C₂×C₂²⋊D₃₄₉	1	2	yes	—
77	C₂²	C₂×D₁₃₉₆	1	1	yes	characteristic
78	C₂²	C₂×Dic₆₉₈	1	1	yes	characteristic
79	C₄	—	2	4	no	—
80	C₄	—	698	8	no	—
81	C₂	#26013546	1	8	yes	direct product factor
82	C₂	C₂×C₄×Dic₃₄₉	1	1	yes	characteristic
83	C₂	C₂×#19150210	1	2	yes	—
84	C₂	C₂×C₂²⋊Dic₃₄₉	1	1	yes	characteristic
85	C₂	C₂×#19150212	1	2	yes	—
86	C₂	C₂×C₄⋊Dic₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₄₉
commutator	C₂×C₃₄₉	C₂×C₄²
Frattini	C₂³	C₂²×D₃₄₉
Fitting	C₂³×C₄×C₃₄₉	C₂
socle, prime radical	C₂⁴×C₃₄₉	C₂²
cosocle	C₂³×C₃₄₉	C₂³
hypercenter	C₂³×C₄	D₃₄₉
hypocenter	C₃₄₉	C₂×#95
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. #26013546	C₂	direct (N × H)
6. C₂²×Dic₃₄₉	C₄	semidirect
10. C₂²×C₄×C₃₄₉	C₄	semidirect
15. C₂×Dic₃₄₉	C₂×C₄	semidirect
24. C₂×C₄×C₃₄₉	C₂×C₄	semidirect
37. C₂×C₃₄₉	#95	semidirect
43. C₃₄₉	C₂×#95	semidirect
54. C₂²×C₄	Dic₃₄₉	semidirect
68. C₂×C₄	C₂×Dic₃₄₉	semidirect
81. C₂	#26013546	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₄₉	2
#26013546	2
C₂³×C₄×C₃₄₉	2
C₂×#95	349

Subgroups — C2×#26013546 #36373988

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013546 #36373988