Subgroups — C₂×C₇×#135 #90593

Order 448.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	2	yes	maximal, complemented
2	C₇×#135	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#41	—	2	4	no	—
8	C₇×#41	C₂²	1	8	yes	complemented
9	C₂×C₇×Q₈	C₂²	1	2	yes	—
10	C₂×C₇×D₄	—	2	2	no	—
11	C₂×C₇×D₄	C₂²	1	2	yes	—
12	C₂²×C₄×C₇	—	2	2	no	—
13	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₇×Q₁₆	C₂²	1	4	yes	complemented
15	C₇×SD₁₆	C₂²	1	8	yes	complemented
16	C₇×D₈	C₂²	1	4	yes	complemented
17	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₈	C₂²	1	4	yes	complemented
20	C₂×#135	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₇	—	2	2	no	—
22	C₂³×C₇	D₄	1	1	yes	characteristic
23	C₇×Q₈	—	2	2	no	—
24	C₇×Q₈	C₂³	1	4	yes	—
25	C₇×D₄	—	2	8	no	—
26	C₇×D₄	—	2	2	no	—
27	C₇×D₄	C₂³	1	4	yes	—
28	C₂×C₄×C₇	—	2	2	no	—
29	C₂×C₄×C₇	—	2	8	no	—
30	C₂×C₄×C₇	D₄	1	2	yes	—
31	C₂×C₄×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂³	1	2	yes	—
34	C₇×C₈	C₂³	1	2	yes	—
35	C₇×C₈	C₂³	1	2	yes	—
36	C₂×#41	C₂×C₇	1	2	yes	complemented
37	#135	C₂×C₇	1	8	yes	direct product factor
38	C₂×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
39	C₂×SD₁₆	C₂×C₇	1	2	yes	complemented
40	C₂×D₈	C₂×C₇	1	1	yes	characteristic, complemented
41	C₂²×C₈	C₂×C₇	1	1	yes	characteristic, complemented
42	C₂²×C₇	—	2	2	no	—
43	C₂²×C₇	—	2	4	no	—
44	C₂²×C₇	—	4	4	no	—
45	C₂²×C₇	C₂×D₄	1	2	yes	—
46	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₇	—	2	4	no	—
48	C₄×C₇	C₂×D₄	1	2	yes	—
49	C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₄×C₇	C₂⁴	1	1	yes	characteristic
51	#41	—	2	4	no	—
52	#41	C₂²×C₇	1	8	yes	complemented
53	C₂×Q₈	C₂²×C₇	1	2	yes	—
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	C₂²×C₇	1	2	yes	—
56	C₂²×C₄	—	2	2	no	—
57	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
58	Q₁₆	C₂²×C₇	1	4	yes	complemented
59	SD₁₆	C₂²×C₇	1	8	yes	complemented
60	D₈	C₂²×C₇	1	4	yes	complemented
61	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×C₇	1	4	yes	complemented
64	C₂×C₇	—	2	2	no	—
65	C₂×C₇	—	4	4	no	—
66	C₂×C₇	#135	1	2	yes	direct product factor
67	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
68	C₂³	—	2	2	no	—
69	C₂³	C₇×D₄	1	1	yes	characteristic
70	Q₈	—	2	2	no	—
71	Q₈	C₂³×C₇	1	4	yes	—
72	D₄	—	2	8	no	—
73	D₄	—	2	2	no	—
74	D₄	C₂³×C₇	1	4	yes	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	C₇×D₄	1	2	yes	—
78	C₂×C₄	C₇×D₄	1	1	yes	characteristic
79	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
80	C₂×C₄	C₂³×C₇	1	2	yes	—
81	C₈	C₂³×C₇	1	2	yes	—
82	C₈	C₂³×C₇	1	2	yes	—
83	C₇	C₂×#135	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²	—	2	2	no	—
85	C₂²	—	2	4	no	—
86	C₂²	—	4	4	no	—
87	C₂²	C₂×C₇×D₄	1	2	yes	—
88	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
89	C₄	—	2	4	no	—
90	C₄	C₂×C₇×D₄	1	2	yes	—
91	C₄	C₂×C₇×D₄	1	1	yes	characteristic
92	C₄	C₂⁴×C₇	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	—	4	4	no	—
95	C₂	C₇×#135	1	2	yes	direct product factor
96	C₂	C₂²×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#41	C₂	semidirect
2. C₇×#135	C₂	direct (N × H)
3. C₂×C₇×Q₁₆	C₂	semidirect
4. C₂×C₇×SD₁₆	C₂	semidirect
5. C₂×C₇×D₈	C₂	semidirect
6. C₂²×C₇×C₈	C₂	semidirect
8. C₇×#41	C₂²	semidirect
14. C₇×Q₁₆	C₂²	semidirect
15. C₇×SD₁₆	C₂²	semidirect
16. C₇×D₈	C₂²	semidirect
19. C₂×C₇×C₈	C₂²	semidirect
20. C₂×#135	C₇	direct (N × H)
36. C₂×#41	C₂×C₇	semidirect
37. #135	C₂×C₇	direct (N × H)
38. C₂×Q₁₆	C₂×C₇	semidirect
39. C₂×SD₁₆	C₂×C₇	semidirect
40. C₂×D₈	C₂×C₇	semidirect
41. C₂²×C₈	C₂×C₇	semidirect
52. #41	C₂²×C₇	semidirect
58. Q₁₆	C₂²×C₇	semidirect
59. SD₁₆	C₂²×C₇	semidirect
60. D₈	C₂²×C₇	semidirect
63. C₂×C₈	C₂²×C₇	semidirect
66. C₂×C₇	#135	direct (N × H)
83. C₇	C₂×#135	direct (N × H)
95. C₂	C₇×#135	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#41	2
C₇×#135	2
C₂×C₇×Q₁₆	2
C₂×C₇×SD₁₆	2
C₂×C₇×D₈	2
C₂²×C₇×C₈	2
C₂×#135	7

Subgroups — C2×C7×#135 #90593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#135 #90593