Subgroups — C₂×C₇×#124 #90547

Order 448.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#124	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₇	C₂²	1	2	yes	—
7	C₂×C₇×Q₈	—	2	4	no	—
8	C₂×C₇×Q₈	C₂²	1	4	yes	complemented
9	C₂×C₇×D₄	—	2	4	no	—
10	C₂×C₇×D₄	C₂²	1	4	yes	—
11	C₂²×C₄×C₇	C₂²	1	2	yes	—
12	C₂²×C₄×C₇	C₂²	1	2	yes	—
13	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	C₂²	1	16	yes	complemented
15	C₄²×C₇	C₂²	1	4	yes	complemented
16	C₂×#124	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	—	2	4	no	—
18	C₂³×C₇	—	2	8	no	—
19	C₂³×C₇	C₂³	1	4	yes	—
20	C₂³×C₇	C₂³	1	1	yes	characteristic
21	C₇×Q₈	—	2	8	no	—
22	C₇×D₄	—	2	8	no	—
23	C₂×C₄×C₇	—	2	4	no	—
24	C₂×C₄×C₇	—	2	4	no	—
25	C₂×C₄×C₇	D₄	1	4	yes	—
26	C₂×C₄×C₇	C₂³	1	4	yes	—
27	C₂×C₄×C₇	C₂³	1	4	yes	—
28	C₂×C₄×C₇	C₂³	1	2	yes	—
29	C₂²×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
30	C₂²×D₄	C₂×C₇	1	1	yes	characteristic
31	#124	C₂×C₇	1	8	yes	direct product factor
32	C₂×C₂²⋊C₄	C₂×C₇	1	4	yes	complemented
33	C₂×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
34	C₂²×C₇	—	2	4	no	—
35	C₂²×C₇	—	2	8	no	—
36	C₂²×C₇	—	4	8	no	—
37	C₂²×C₇	C₂×D₄	1	2	yes	—
38	C₂²×C₇	#41	1	4	yes	—
39	C₂²×C₇	C₂⁴	1	1	yes	characteristic
40	C₄×C₇	—	2	4	no	—
41	C₄×C₇	—	2	4	no	—
42	C₄×C₇	C₂×D₄	1	4	yes	—
43	C₂⁴	C₂²×C₇	1	2	yes	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	C₂²×C₇	1	4	yes	complemented
46	C₂×D₄	—	2	4	no	—
47	C₂×D₄	C₂²×C₇	1	4	yes	—
48	C₂²×C₄	C₂²×C₇	1	2	yes	—
49	C₂²×C₄	C₂²×C₇	1	2	yes	—
50	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
51	C₂²⋊C₄	C₂²×C₇	1	16	yes	complemented
52	C₄²	C₂²×C₇	1	4	yes	complemented
53	C₂×C₇	—	4	4	no	—
54	C₂×C₇	#124	1	4	yes	direct product factor
55	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₇	C₂×#41	1	2	yes	—
57	C₂³	—	2	4	no	—
58	C₂³	—	2	8	no	—
59	C₂³	C₂³×C₇	1	4	yes	—
60	C₂³	C₂³×C₇	1	1	yes	characteristic
61	Q₈	—	2	8	no	—
62	D₄	—	2	8	no	—
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	C₇×D₄	1	4	yes	—
66	C₂×C₄	C₂³×C₇	1	4	yes	—
67	C₂×C₄	C₂³×C₇	1	4	yes	—
68	C₂×C₄	C₂³×C₇	1	2	yes	—
69	C₇	C₂×#124	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²	—	2	4	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	4	8	no	—
73	C₂²	C₂×C₇×D₄	1	2	yes	—
74	C₂²	C₇×#41	1	4	yes	—
75	C₂²	C₂⁴×C₇	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	4	no	—
78	C₄	C₂×C₇×D₄	1	4	yes	—
79	C₂	—	4	4	no	—
80	C₂	C₇×#124	1	4	yes	direct product factor
81	C₂	C₂²×C₇×D₄	1	1	yes	characteristic
82	C₂	C₂×C₇×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂
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. C₂²×C₇×Q₈	C₂	semidirect
3. C₇×#124	C₂	direct (N × H)
4. C₂×C₇×C₂²⋊C₄	C₂	semidirect
5. C₂×C₄²×C₇	C₂	semidirect
8. C₂×C₇×Q₈	C₂²	semidirect
14. C₇×C₂²⋊C₄	C₂²	semidirect
15. C₄²×C₇	C₂²	semidirect
16. C₂×#124	C₇	direct (N × H)
29. C₂²×Q₈	C₂×C₇	semidirect
31. #124	C₂×C₇	direct (N × H)
32. C₂×C₂²⋊C₄	C₂×C₇	semidirect
33. C₂×C₄²	C₂×C₇	semidirect
45. C₂×Q₈	C₂²×C₇	semidirect
51. C₂²⋊C₄	C₂²×C₇	semidirect
52. C₄²	C₂²×C₇	semidirect
54. C₂×C₇	#124	direct (N × H)
69. C₇	C₂×#124	direct (N × H)
80. C₂	C₇×#124	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×C₇×Q₈	2
C₂²×C₇×D₄	2
C₇×#124	2
C₂×C₇×C₂²⋊C₄	2
C₂×C₄²×C₇	2
C₂×#124	7

Subgroups — C2×C7×#124 #90547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#124 #90547