Subgroups — C₂×#11730763 #13559890

Order 1792.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6437	C₂	1	14	yes	maximal, complemented
2	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#11730763	C₂	1	16	yes	maximal, direct product factor
4	C₂³×D₂₈	C₂²	1	28	yes	—
5	C₂²×C₄²×C₇	C₂²	1	7	yes	—
6	C₂×#6437	—	2	112	no	—
7	C₂×#6437	C₂²	1	112	yes	complemented
8	C₄³×C₇	C₂²	1	8	yes	complemented
9	C₂×#2867	—	7	1	no	maximal, Sylow
10	C₂⁴×D₇	C₂³	1	8	yes	—
11	C₂³×C₄×C₇	C₂³	1	7	yes	—
12	C₂²×D₂₈	—	2	336	no	—
13	C₂²×D₂₈	C₂³	1	112	yes	—
14	C₂×C₄²×C₇	D₄	1	56	yes	complemented
15	C₂×C₄²×C₇	C₂³	1	28	yes	—
16	#6437	—	2	448	no	—
17	C₂²×#127	—	7	14	no	—
18	#2867	—	7	16	no	—
19	C₂×C₄³	D₇	1	1	yes	characteristic, complemented
20	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
21	C₂³×D₇	—	2	112	no	—
22	C₂³×D₇	C₂⁴	1	16	yes	—
23	C₂²×C₄×C₇	C₂×D₄	1	84	yes	—
24	C₂²×C₄×C₇	C₂⁴	1	14	yes	—
25	C₂×D₂₈	—	2	672	no	—
26	C₂×D₂₈	—	4	448	no	—
27	C₄²×C₇	C₂×D₄	1	112	yes	complemented
28	C₂³×D₄	—	7	28	no	—
29	C₂×#127	—	7	112	no	—
30	C₂×#127	—	14	112	no	—
31	C₂²×C₄²	C₂×D₇	1	7	yes	—
32	C₄³	C₂×D₇	1	8	yes	complemented
33	C₂³×C₇	C₂²×D₄	1	14	yes	—
34	C₂³×C₇	C₂⁵	1	1	yes	characteristic
35	C₂²×D₇	—	2	112	no	—
36	C₂²×D₇	—	4	224	no	—
37	C₂×C₄×C₇	#127	1	112	yes	complemented
38	C₂×C₄×C₇	C₂²×D₄	1	84	yes	—
39	D₂₈	—	4	448	no	—
40	C₂⁵	—	7	8	no	—
41	C₂²×D₄	—	7	112	no	—
42	C₂²×D₄	—	14	336	no	—
43	C₂³×C₄	C₂²×D₇	1	7	yes	—
44	#127	—	14	448	no	—
45	C₂×C₄²	D₂₈	1	56	yes	complemented
46	C₂×C₄²	C₂²×D₇	1	28	yes	—
47	C₂²×C₇	C₂×#127	1	28	yes	—
48	C₂²×C₇	C₂³×D₄	1	7	yes	—
49	C₂×D₇	—	4	112	no	—
50	C₂×D₇	—	8	64	no	—
51	C₄×C₇	C₂×#127	1	56	yes	complemented
52	C₂⁴	—	7	16	no	—
53	C₂⁴	—	14	112	no	—
54	C₂⁴	C₂³×D₇	1	1	yes	characteristic
55	C₂×D₄	—	14	672	no	—
56	C₂×D₄	—	28	448	no	—
57	C₂²×C₄	C₂×D₂₈	1	84	yes	—
58	C₂²×C₄	C₂³×D₇	1	14	yes	—
59	C₄²	C₂×D₂₈	1	112	yes	complemented
60	C₂×C₇	#2867	1	8	yes	complemented
61	C₂×C₇	C₂²×#127	1	7	yes	—
62	D₇	—	8	16	no	—
63	C₂³	—	14	112	no	—
64	C₂³	—	28	224	no	—
65	C₂³	C₂²×D₂₈	1	14	yes	—
66	C₂³	C₂⁴×D₇	1	1	yes	characteristic
67	D₄	—	28	448	no	—
68	C₂×C₄	#6437	1	112	yes	complemented
69	C₂×C₄	C₂²×D₂₈	1	84	yes	—
70	C₇	C₂×#2867	1	1	yes	characteristic, complemented, Sylow
71	C₂²	—	28	112	no	—
72	C₂²	—	56	64	no	—
73	C₂²	C₂×#6437	1	28	yes	—
74	C₂²	C₂³×D₂₈	1	7	yes	—
75	C₄	C₂×#6437	1	56	yes	complemented
76	C₂	—	56	16	no	—
77	C₂	#11730763	1	8	yes	direct product factor
78	C₂	C₂²×#6437	1	7	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇
commutator, cosocle	C₂³×C₇	C₂⁵
Frattini	C₂³	C₂⁴×D₇
Fitting	C₂×C₄³×C₇	C₂
socle	C₂⁴×C₇	C₂⁴
hypercenter	C₂×C₄³	D₇
hypocenter	C₇	C₂×#2867
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
1. C₂²×#6437	C₂	semidirect
2. C₂×C₄³×C₇	C₂	semidirect
3. #11730763	C₂	direct (N × H)
7. C₂×#6437	C₂²	semidirect
8. C₄³×C₇	C₂²	semidirect
14. C₂×C₄²×C₇	D₄	semidirect
19. C₂×C₄³	D₇	semidirect
27. C₄²×C₇	C₂×D₄	semidirect
32. C₄³	C₂×D₇	semidirect
37. C₂×C₄×C₇	#127	semidirect
45. C₂×C₄²	D₂₈	semidirect
51. C₄×C₇	C₂×#127	semidirect
59. C₄²	C₂×D₂₈	semidirect
60. C₂×C₇	#2867	semidirect
68. C₂×C₄	#6437	semidirect
70. C₇	C₂×#2867	semidirect
75. C₄	C₂×#6437	semidirect
77. C₂	#11730763	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×#6437	2
C₂×C₄³×C₇	2
#11730763	2
C₂×#2867	7

Subgroups — C2×#11730763 #13559890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11730763 #13559890