Subgroups — C₂×C₇×#2867 #13213858

Order 1792.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#127	C₂	1	14	yes	maximal, complemented
2	C₇×#2867	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₇×D₄	C₂²	1	28	yes	—
5	C₂×C₇×#127	—	2	112	no	—
6	C₂×C₇×#127	C₂²	1	112	yes	complemented
7	C₂²×C₄²×C₇	C₂²	1	7	yes	—
8	C₄³×C₇	C₂²	1	8	yes	complemented
9	C₂×#2867	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂⁵×C₇	C₂³	1	8	yes	—
11	C₂²×C₇×D₄	—	2	336	no	—
12	C₂²×C₇×D₄	C₂³	1	112	yes	—
13	C₂³×C₄×C₇	C₂³	1	7	yes	—
14	C₇×#127	—	2	448	no	—
15	C₂×C₄²×C₇	D₄	1	56	yes	complemented
16	C₂×C₄²×C₇	C₂³	1	28	yes	—
17	C₂²×#127	C₂×C₇	1	14	yes	complemented
18	#2867	C₂×C₇	1	16	yes	direct product factor
19	C₂×C₄³	C₂×C₇	1	1	yes	characteristic, complemented
20	C₂⁴×C₇	—	2	112	no	—
21	C₂⁴×C₇	C₂⁴	1	16	yes	—
22	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
23	C₂×C₇×D₄	—	2	672	no	—
24	C₂×C₇×D₄	—	4	448	no	—
25	C₂²×C₄×C₇	C₂×D₄	1	84	yes	—
26	C₂²×C₄×C₇	C₂⁴	1	14	yes	—
27	C₄²×C₇	C₂×D₄	1	112	yes	complemented
28	C₂³×D₄	C₂²×C₇	1	28	yes	—
29	C₂×#127	—	2	112	no	—
30	C₂×#127	C₂²×C₇	1	112	yes	complemented
31	C₂²×C₄²	C₂²×C₇	1	7	yes	—
32	C₄³	C₂²×C₇	1	8	yes	complemented
33	C₂³×C₇	—	2	112	no	—
34	C₂³×C₇	—	4	224	no	—
35	C₂³×C₇	C₂²×D₄	1	14	yes	—
36	C₂³×C₇	C₂⁵	1	1	yes	characteristic
37	C₇×D₄	—	4	448	no	—
38	C₂×C₄×C₇	#127	1	112	yes	complemented
39	C₂×C₄×C₇	C₂²×D₄	1	84	yes	—
40	C₂⁵	C₂³×C₇	1	8	yes	—
41	C₂²×D₄	—	2	336	no	—
42	C₂²×D₄	C₂³×C₇	1	112	yes	—
43	C₂³×C₄	C₂³×C₇	1	7	yes	—
44	#127	—	2	448	no	—
45	C₂×C₄²	C₇×D₄	1	56	yes	complemented
46	C₂×C₄²	C₂³×C₇	1	28	yes	—
47	C₂²×C₇	—	4	112	no	—
48	C₂²×C₇	—	8	64	no	—
49	C₂²×C₇	C₂×#127	1	28	yes	—
50	C₂²×C₇	C₂³×D₄	1	7	yes	—
51	C₄×C₇	C₂×#127	1	56	yes	complemented
52	C₂⁴	—	2	112	no	—
53	C₂⁴	C₂⁴×C₇	1	16	yes	—
54	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
55	C₂×D₄	—	2	672	no	—
56	C₂×D₄	—	4	448	no	—
57	C₂²×C₄	C₂×C₇×D₄	1	84	yes	—
58	C₂²×C₄	C₂⁴×C₇	1	14	yes	—
59	C₄²	C₂×C₇×D₄	1	112	yes	complemented
60	C₂×C₇	—	8	16	no	—
61	C₂×C₇	#2867	1	8	yes	direct product factor
62	C₂×C₇	C₂²×#127	1	7	yes	—
63	C₂³	—	2	112	no	—
64	C₂³	—	4	224	no	—
65	C₂³	C₂²×C₇×D₄	1	14	yes	—
66	C₂³	C₂⁵×C₇	1	1	yes	characteristic
67	D₄	—	4	448	no	—
68	C₂×C₄	C₇×#127	1	112	yes	complemented
69	C₂×C₄	C₂²×C₇×D₄	1	84	yes	—
70	C₇	C₂×#2867	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²	—	4	112	no	—
72	C₂²	—	8	64	no	—
73	C₂²	C₂×C₇×#127	1	28	yes	—
74	C₂²	C₂³×C₇×D₄	1	7	yes	—
75	C₄	C₂×C₇×#127	1	56	yes	complemented
76	C₂	—	8	16	no	—
77	C₂	C₇×#2867	1	8	yes	direct product factor
78	C₂	C₂²×C₇×#127	1	7	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×#127	C₂	semidirect
2. C₇×#2867	C₂	direct (N × H)
3. C₂×C₄³×C₇	C₂	semidirect
6. C₂×C₇×#127	C₂²	semidirect
8. C₄³×C₇	C₂²	semidirect
9. C₂×#2867	C₇	direct (N × H)
15. C₂×C₄²×C₇	D₄	semidirect
17. C₂²×#127	C₂×C₇	semidirect
18. #2867	C₂×C₇	direct (N × H)
19. C₂×C₄³	C₂×C₇	semidirect
27. C₄²×C₇	C₂×D₄	semidirect
30. C₂×#127	C₂²×C₇	semidirect
32. C₄³	C₂²×C₇	semidirect
38. C₂×C₄×C₇	#127	semidirect
45. C₂×C₄²	C₇×D₄	semidirect
51. C₄×C₇	C₂×#127	semidirect
59. C₄²	C₂×C₇×D₄	semidirect
61. C₂×C₇	#2867	direct (N × H)
68. C₂×C₄	C₇×#127	semidirect
70. C₇	C₂×#2867	direct (N × H)
75. C₄	C₂×C₇×#127	semidirect
77. C₂	C₇×#2867	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×#127	2
C₇×#2867	2
C₂×C₄³×C₇	2
C₂×#2867	7

Subgroups — C2×C7×#2867 #13213858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2867 #13213858