Subgroups — C₂×#26013627 #36374661

Order 22336.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₃₉₆	C₂	1	1	yes	maximal, characteristic
3	#26013627	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19150210	C₂	1	4	yes	maximal, complemented
5	C₂×C₄²×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₄₉	C₂²	1	2	yes	—
7	C₂×Dic₆₉₈	—	2	4	no	—
8	C₂×Dic₆₉₈	C₂²	1	4	yes	complemented
9	C₂×D₁₃₉₆	—	2	4	no	—
10	C₂×D₁₃₉₆	C₂²	1	4	yes	—
11	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
12	#19150210	C₂²	1	16	yes	complemented
13	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
15	C₄²×C₃₄₉	C₂²	1	4	yes	complemented
16	C₂²×D₃₄₉	—	2	4	no	—
17	C₂²×D₃₄₉	—	2	8	no	—
18	C₂²×D₃₄₉	C₂³	1	4	yes	—
19	Dic₆₉₈	—	2	8	no	—
20	D₁₃₉₆	—	2	8	no	—
21	C₂×Dic₃₄₉	—	2	4	no	—
22	C₂×Dic₃₄₉	C₂³	1	4	yes	—
23	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	—	2	4	no	—
25	C₂×C₄×C₃₄₉	D₄	1	4	yes	—
26	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
27	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
28	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
29	C₂²×C₃₄₉	#41	1	4	yes	—
30	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₄₉	—	2	4	no	—
32	C₂×D₃₄₉	—	2	8	no	—
33	C₂×D₃₄₉	—	4	8	no	—
34	C₄×C₃₄₉	—	2	4	no	—
35	C₄×C₃₄₉	C₂×D₄	1	4	yes	—
36	Dic₃₄₉	—	2	4	no	—
37	C₂×C₃₄₉	#124	1	4	yes	complemented
38	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
39	C₂×C₃₄₉	C₂×#41	1	2	yes	—
40	D₃₄₉	—	4	4	no	—
41	C₃₄₉	C₂×#124	1	1	yes	characteristic, complemented, Sylow
42	C₂×#124	—	349	1	no	maximal, Sylow
43	C₂²×Q₈	—	349	1	no	—
44	C₂²×D₄	—	349	1	no	—
45	#124	—	349	8	no	—
46	C₂×C₂²⋊C₄	—	349	4	no	—
47	C₂×C₄²	D₃₄₉	1	1	yes	characteristic, complemented
48	C₂⁴	—	349	2	no	—
49	C₂×Q₈	—	349	4	no	—
50	C₂×Q₈	—	698	4	no	—
51	C₂×D₄	—	349	4	no	—
52	C₂×D₄	—	698	4	no	—
53	C₂²×C₄	—	349	2	no	—
54	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
55	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
56	C₂²⋊C₄	—	349	16	no	—
57	C₄²	C₂×D₃₄₉	1	4	yes	complemented
58	C₂³	—	349	4	no	—
59	C₂³	—	698	4	no	—
60	C₂³	—	698	8	no	—
61	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
62	Q₈	—	698	8	no	—
63	D₄	—	698	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	349	4	no	—
66	C₂×C₄	—	698	4	no	—
67	C₂×C₄	D₁₃₉₆	1	4	yes	—
68	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
69	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
70	C₂²	—	1396	8	no	—
71	C₂²	—	698	4	no	—
72	C₂²	—	698	8	no	—
73	C₂²	C₂×D₁₃₉₆	1	2	yes	—
74	C₂²	#19150233	1	4	yes	—
75	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	698	4	no	—
78	C₄	C₂×D₁₃₉₆	1	4	yes	—
79	C₂	—	1396	4	no	—
80	C₂	#26013627	1	4	yes	direct product factor
81	C₂	C₂²×D₁₃₉₆	1	1	yes	characteristic
82	C₂	C₂×#19150233	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator, cosocle	C₂²×C₃₄₉	C₂⁴
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₄²×C₃₄₉	C₂
socle	C₂³×C₃₄₉	C₂³
prime radical	C₂²×D₁₃₉₆	C₂
hypercenter	C₂×C₄²	D₃₄₉
hypocenter	C₃₄₉	C₂×#124
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₆₉₈	C₂	semidirect
3. #26013627	C₂	direct (N × H)
4. C₂×#19150210	C₂	semidirect
5. C₂×C₄²×C₃₄₉	C₂	semidirect
8. C₂×Dic₆₉₈	C₂²	semidirect
12. #19150210	C₂²	semidirect
15. C₄²×C₃₄₉	C₂²	semidirect
37. C₂×C₃₄₉	#124	semidirect
41. C₃₄₉	C₂×#124	semidirect
47. C₂×C₄²	D₃₄₉	semidirect
57. C₄²	C₂×D₃₄₉	semidirect
80. C₂	#26013627	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₉₈	2
C₂²×D₁₃₉₆	2
#26013627	2
C₂×#19150210	2
C₂×C₄²×C₃₄₉	2
C₂×#124	349

Subgroups — C2×#26013627 #36374661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013627 #36374661