Subgroups — C₂×C₇×#120 #90542

Order 448.

662 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×D₄	C₂	1	3	yes	maximal, complemented
3	C₇×#120	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂⁴×C₇	—	2	12	no	—
6	C₂⁴×C₇	C₂²	1	3	yes	—
7	C₂⁴×C₇	C₂²	1	4	yes	complemented
8	C₂⁴×C₇	C₂²	1	1	yes	characteristic, complemented
9	C₂×C₇×D₄	—	2	12	no	—
10	C₂×C₇×D₄	C₂²	1	12	yes	complemented
11	C₂²×C₄×C₇	C₂²	1	3	yes	complemented
12	C₇×C₂²⋊C₄	C₂²	1	12	yes	complemented
13	C₂×#120	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₇	—	2	32	no	—
15	C₂³×C₇	—	2	12	no	—
16	C₂³×C₇	—	2	24	no	—
17	C₂³×C₇	—	2	6	no	—
18	C₂³×C₇	D₄	1	12	yes	complemented
19	C₂³×C₇	C₂³	1	6	yes	—
20	C₂³×C₇	C₂³	1	1	yes	characteristic
21	C₂³×C₇	C₂³	1	2	yes	complemented
22	C₇×D₄	—	2	24	no	—
23	C₂×C₄×C₇	—	2	6	no	—
24	C₂×C₄×C₇	C₂³	1	6	yes	complemented
25	C₂⁵	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂²×D₄	C₂×C₇	1	3	yes	complemented
27	#120	C₂×C₇	1	8	yes	direct product factor
28	C₂×C₂²⋊C₄	C₂×C₇	1	3	yes	complemented
29	C₂²×C₇	—	2	32	no	—
30	C₂²×C₇	—	2	24	no	—
31	C₂²×C₇	—	2	12	no	—
32	C₂²×C₇	—	2	6	no	—
33	C₂²×C₇	—	4	4	no	—
34	C₂²×C₇	C₂×D₄	1	6	yes	—
35	C₂²×C₇	C₂×D₄	1	12	yes	complemented
36	C₂²×C₇	C₂⁴	1	1	yes	characteristic
37	C₄×C₇	—	2	6	no	—
38	C₂⁴	—	2	12	no	—
39	C₂⁴	C₂²×C₇	1	3	yes	—
40	C₂⁴	C₂²×C₇	1	4	yes	complemented
41	C₂⁴	C₂²×C₇	1	1	yes	characteristic, complemented
42	C₂×D₄	—	2	12	no	—
43	C₂×D₄	C₂²×C₇	1	12	yes	complemented
44	C₂²×C₄	C₂²×C₇	1	3	yes	complemented
45	C₂²⋊C₄	C₂²×C₇	1	12	yes	complemented
46	C₂×C₇	—	2	12	no	—
47	C₂×C₇	—	4	2	no	—
48	C₂×C₇	#120	1	4	yes	direct product factor
49	C₂×C₇	C₂²×D₄	1	3	yes	—
50	C₂³	—	2	32	no	—
51	C₂³	—	2	12	no	—
52	C₂³	—	2	24	no	—
53	C₂³	—	2	6	no	—
54	C₂³	C₇×D₄	1	12	yes	complemented
55	C₂³	C₂³×C₇	1	6	yes	—
56	C₂³	C₂³×C₇	1	1	yes	characteristic
57	C₂³	C₂³×C₇	1	2	yes	complemented
58	D₄	—	2	24	no	—
59	C₂×C₄	—	2	6	no	—
60	C₂×C₄	C₂³×C₇	1	6	yes	complemented
61	C₇	C₂×#120	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²	—	2	32	no	—
63	C₂²	—	2	24	no	—
64	C₂²	—	2	12	no	—
65	C₂²	—	2	6	no	—
66	C₂²	—	4	4	no	—
67	C₂²	C₂×C₇×D₄	1	6	yes	—
68	C₂²	C₂×C₇×D₄	1	12	yes	complemented
69	C₂²	C₂⁴×C₇	1	1	yes	characteristic
70	C₄	—	2	6	no	—
71	C₂	—	2	12	no	—
72	C₂	—	4	2	no	—
73	C₂	C₇×#120	1	4	yes	direct product factor
74	C₂	C₂²×C₇×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₇	C₂	semidirect
2. C₂²×C₇×D₄	C₂	semidirect
3. C₇×#120	C₂	direct (N × H)
4. C₂×C₇×C₂²⋊C₄	C₂	semidirect
7. C₂⁴×C₇	C₂²	semidirect
8. C₂⁴×C₇	C₂²	semidirect
10. C₂×C₇×D₄	C₂²	semidirect
11. C₂²×C₄×C₇	C₂²	semidirect
12. C₇×C₂²⋊C₄	C₂²	semidirect
13. C₂×#120	C₇	direct (N × H)
18. C₂³×C₇	D₄	semidirect
21. C₂³×C₇	C₂³	semidirect
24. C₂×C₄×C₇	C₂³	semidirect
25. C₂⁵	C₂×C₇	semidirect
26. C₂²×D₄	C₂×C₇	semidirect
27. #120	C₂×C₇	direct (N × H)
28. C₂×C₂²⋊C₄	C₂×C₇	semidirect
35. C₂²×C₇	C₂×D₄	semidirect
40. C₂⁴	C₂²×C₇	semidirect
41. C₂⁴	C₂²×C₇	semidirect
43. C₂×D₄	C₂²×C₇	semidirect
44. C₂²×C₄	C₂²×C₇	semidirect
45. C₂²⋊C₄	C₂²×C₇	semidirect
48. C₂×C₇	#120	direct (N × H)
54. C₂³	C₇×D₄	semidirect
57. C₂³	C₂³×C₇	semidirect
60. C₂×C₄	C₂³×C₇	semidirect
61. C₇	C₂×#120	direct (N × H)
68. C₂²	C₂×C₇×D₄	semidirect
73. C₂	C₇×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₇	2
C₂²×C₇×D₄	2
C₇×#120	2
C₂×C₇×C₂²⋊C₄	2
C₂×#120	7

Subgroups — C2×C7×#120 #90542

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#120 #90542