Subgroups — C₂×#26013595 #36374597

Order 22336.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
2	#26013595	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19150212	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₃₄₉	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊C₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₄₉	C₄	1	8	yes	complemented
8	C₂²×Dic₃₄₉	C₂²	1	2	yes	—
9	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₃₄₉	C₂²	1	4	yes	—
12	#19150212	C₂²	1	8	yes	complemented
13	C₂²⋊Dic₃₄₉	C₂²	1	4	yes	—
14	C₄×Dic₃₄₉	C₂²	1	8	yes	complemented
15	C₂⁴×C₃₄₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₃₄₉	C₂²	1	4	yes	—
18	C₂×Dic₃₄₉	—	2	4	no	—
19	C₂×Dic₃₄₉	C₂×C₄	1	4	yes	—
20	C₂×Dic₃₄₉	C₂×C₄	1	4	yes	—
21	C₂×Dic₃₄₉	C₂×C₄	1	16	yes	complemented
22	C₂×Dic₃₄₉	C₂³	1	4	yes	—
23	C₂×Dic₃₄₉	C₂³	1	2	yes	—
24	C₂×Dic₃₄₉	C₂³	1	2	yes	—
25	C₂³×C₃₄₉	—	2	4	no	—
26	C₂³×C₃₄₉	C₂×C₄	1	4	yes	—
27	C₂³×C₃₄₉	C₂³	1	2	yes	—
28	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₃₄₉	—	2	4	no	—
30	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
31	C₂²×C₃₄₉	—	2	8	no	—
32	C₂²×C₃₄₉	—	2	4	no	—
33	C₂²×C₃₄₉	C₂²×C₄	1	4	yes	—
34	C₂²×C₃₄₉	C₂²×C₄	1	2	yes	—
35	C₂²×C₃₄₉	#41	1	4	yes	—
36	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
37	C₄×C₃₄₉	—	2	4	no	—
38	Dic₃₄₉	—	2	4	no	—
39	Dic₃₄₉	C₂²×C₄	1	4	yes	—
40	Dic₃₄₉	C₂²×C₄	1	4	yes	—
41	C₂×C₃₄₉	—	2	4	no	—
42	C₂×C₃₄₉	#117	1	4	yes	complemented
43	C₂×C₃₄₉	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₃₄₉	C₂×#41	1	2	yes	—
45	C₃₄₉	C₂×#117	1	1	yes	characteristic, complemented, Sylow
46	C₂×#117	—	349	1	no	maximal, Sylow
47	C₂³×C₄	—	349	1	no	—
48	#117	—	349	8	no	—
49	C₂×C₄⋊C₄	—	349	2	no	—
50	C₂×C₂²⋊C₄	—	349	1	no	—
51	C₂×C₂²⋊C₄	D₃₄₉	1	1	yes	characteristic
52	C₂×C₄²	—	349	2	no	—
53	C₂⁴	C₂×D₃₄₉	1	1	yes	characteristic
54	C₂²×C₄	—	349	8	no	—
55	C₂²×C₄	—	349	4	no	—
56	C₂²×C₄	—	349	1	no	—
57	C₂²×C₄	—	349	1	no	—
58	C₂²×C₄	—	349	2	no	—
59	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
60	C₄⋊C₄	—	349	8	no	—
61	C₂²⋊C₄	—	349	4	no	—
62	C₂²⋊C₄	C₂×D₃₄₉	1	4	yes	—
63	C₄²	—	349	8	no	—
64	C₂³	—	2	4	no	—
65	C₂³	C₄×D₃₄₉	1	4	yes	—
66	C₂³	C₂²×D₃₄₉	1	2	yes	—
67	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	349	4	no	—
70	C₂×C₄	—	349	4	no	—
71	C₂×C₄	—	349	16	no	—
72	C₂×C₄	—	349	2	no	—
73	C₂×C₄	—	349	2	no	—
74	C₂×C₄	—	349	4	no	—
75	C₂×C₄	—	698	4	no	—
76	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
77	C₂²	—	2	8	no	—
78	C₂²	—	2	4	no	—
79	C₂²	C₂×C₄×D₃₄₉	1	4	yes	—
80	C₂²	C₂×C₄×D₃₄₉	1	2	yes	—
81	C₂²	#19150232	1	4	yes	—
82	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	349	4	no	—
85	C₄	—	349	4	no	—
86	C₄	—	698	4	no	—
87	C₂	—	2	4	no	—
88	C₂	#26013595	1	4	yes	direct product factor
89	C₂	C₂²×C₄×D₃₄₉	1	1	yes	characteristic
90	C₂	C₂×#19150232	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator	C₂×C₃₄₉	C₂³×C₄
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₂²⋊C₄×C₃₄₉	C₂
socle	C₂³×C₃₄₉	C₂³
prime radical	C₂⁴×C₃₄₉	C₂²
cosocle	C₂²×C₃₄₉	C₂⁴
hypercenter	C₂×C₂²⋊C₄	D₃₄₉
hypocenter	C₃₄₉	C₂×#117
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #26013595	C₂	direct (N × H)
3. C₂×#19150212	C₂	semidirect
5. C₂×C₄×Dic₃₄₉	C₂	semidirect
7. C₂²×Dic₃₄₉	C₄	semidirect
12. #19150212	C₂²	semidirect
14. C₄×Dic₃₄₉	C₂²	semidirect
21. C₂×Dic₃₄₉	C₂×C₄	semidirect
42. C₂×C₃₄₉	#117	semidirect
45. C₃₄₉	C₂×#117	semidirect
88. C₂	#26013595	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
#26013595	2
C₂×#19150212	2
C₂×C₂²⋊Dic₃₄₉	2
C₂×C₄×Dic₃₄₉	2
C₂×C₂²⋊C₄×C₃₄₉	2
C₂×#117	349

Subgroups — C2×#26013595 #36374597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013595 #36374597