Subgroups — C₂×#20300917 #28411501

Order 14656.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14270080	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₂₂₉	C₂	1	1	yes	maximal, characteristic
3	#20300917	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₂₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₉	C₄	1	1	yes	characteristic
6	C₂²×Dic₂₂₉	C₄	1	1	yes	characteristic
7	C₂×#14270080	—	2	2	no	—
8	C₂×#14270080	C₂²	1	2	yes	—
9	C₄×Dic₂₂₉	C₄	1	2	yes	—
10	C₄×Dic₂₂₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₂₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₉	—	2	2	no	—
13	C₂×C₈×C₂₂₉	C₂²	1	2	yes	—
14	C₂×Dic₂₂₉	—	2	2	no	—
15	C₂×Dic₂₂₉	C₈	1	4	yes	complemented
16	C₂×Dic₂₂₉	C₂×C₄	1	2	yes	—
17	C₂×Dic₂₂₉	C₂×C₄	1	2	yes	—
18	#14270080	—	2	2	no	—
19	C₂³×C₂₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₉	D₄	1	2	yes	—
22	C₂×C₄×C₂₂₉	Q₈	1	2	yes	—
23	C₂×C₄×C₂₂₉	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₉	—	2	2	no	—
25	C₂²×C₂₂₉	C₄⋊C₄	1	2	yes	—
26	C₂²×C₂₂₉	C₂×C₈	1	2	yes	—
27	C₂²×C₂₂₉	M₁₆	1	2	yes	—
28	C₂²×C₂₂₉	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₂₉	C₄⋊C₄	1	2	yes	—
30	C₄×C₂₂₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₂₉	C₂×Q₈	1	1	yes	characteristic
32	Dic₂₂₉	—	2	2	no	—
33	Dic₂₂₉	C₂×C₈	1	4	yes	complemented
34	C₂×C₂₂₉	C₄⋊C₈	1	4	yes	complemented
35	C₂×C₂₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₂₂₉	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₂₂₉	C₂×M₁₆	1	1	yes	characteristic
38	C₂₂₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₄⋊C₈	—	229	1	no	maximal, Sylow
40	C₂²×C₈	—	229	1	no	—
41	C₂²×C₈	D₂₂₉	1	1	yes	characteristic
42	C₂×C₄²	—	229	1	no	—
43	C₄⋊C₈	—	229	4	no	—
44	C₂²×C₄	—	229	1	no	—
45	C₂²×C₄	—	229	1	no	—
46	C₂²×C₄	C₂×D₂₂₉	1	1	yes	characteristic
47	C₂×C₈	—	2	2	no	—
48	C₂×C₈	—	458	2	no	—
49	C₂×C₈	—	229	2	no	—
50	C₂×C₈	C₂×D₂₂₉	1	2	yes	—
51	C₄²	—	229	2	no	—
52	C₄²	—	229	2	no	—
53	C₂³	C₄×D₂₂₉	1	1	yes	characteristic
54	C₂×C₄	—	458	2	no	—
55	C₂×C₄	—	229	2	no	—
56	C₂×C₄	—	229	4	no	—
57	C₂×C₄	—	229	2	no	—
58	C₂×C₄	C₄×D₂₂₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₂₉	1	2	yes	—
60	C₂×C₄	Dic₄₅₈	1	2	yes	—
61	C₂×C₄	C₂²×D₂₂₉	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₈	—	458	2	no	—
64	C₂²	#16814645	1	2	yes	—
65	C₂²	C₈×D₂₂₉	1	2	yes	—
66	C₂²	#16814649	1	2	yes	—
67	C₂²	C₂×C₄×D₂₂₉	1	1	yes	characteristic
68	C₄	—	458	2	no	—
69	C₄	—	229	4	no	—
70	C₄	#16814645	1	2	yes	—
71	C₄	C₂×C₂²⋊D₂₂₉	1	1	yes	characteristic
72	C₄	C₂×Dic₄₅₈	1	1	yes	characteristic
73	C₂	#20300917	1	4	yes	direct product factor
74	C₂	C₂×#16814645	1	1	yes	characteristic
75	C₂	C₂×C₈×D₂₂₉	1	1	yes	characteristic
76	C₂	C₂×#16814649	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₂₉
commutator	C₂×C₂₂₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₂₂₉
Fitting	C₂²×C₈×C₂₂₉	C₂
socle, prime radical	C₂³×C₂₂₉	C₂×C₄
cosocle	C₂×C₄×C₂₂₉	C₂³
hypercenter	C₂²×C₈	D₂₂₉
hypocenter	C₂₂₉	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #20300917	C₂	direct (N × H)
15. C₂×Dic₂₂₉	C₈	semidirect
33. Dic₂₂₉	C₂×C₈	semidirect
34. C₂×C₂₂₉	C₄⋊C₈	semidirect
38. C₂₂₉	C₂×C₄⋊C₈	semidirect
73. C₂	#20300917	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×#14270080	2
C₂×C₄×Dic₂₂₉	2
#20300917	2
C₂²×C₈×C₂₂₉	2
C₂×C₄⋊C₈	229

Subgroups — C2×#20300917 #28411501

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20300917 #28411501