Subgroups — C₂×C₆₇×#2867 #30715893

Order 17152.

8094 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#127×C₆₇	C₂	1	14	yes	maximal, complemented
2	C₆₇×#2867	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄×C₆₇	C₂²	1	28	yes	—
5	C₂×#127×C₆₇	—	2	112	no	—
6	C₂×#127×C₆₇	C₂²	1	112	yes	complemented
7	C₂²×C₄²×C₆₇	C₂²	1	7	yes	—
8	C₄³×C₆₇	C₂²	1	8	yes	complemented
9	C₂⁵×C₆₇	C₂³	1	8	yes	—
10	C₂²×D₄×C₆₇	—	2	336	no	—
11	C₂²×D₄×C₆₇	C₂³	1	112	yes	—
12	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
13	#127×C₆₇	—	2	448	no	—
14	C₂×C₄²×C₆₇	D₄	1	56	yes	complemented
15	C₂×C₄²×C₆₇	C₂³	1	28	yes	—
16	C₂⁴×C₆₇	—	2	112	no	—
17	C₂⁴×C₆₇	C₂⁴	1	16	yes	—
18	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
19	C₂×D₄×C₆₇	—	2	672	no	—
20	C₂×D₄×C₆₇	—	4	448	no	—
21	C₂²×C₄×C₆₇	C₂×D₄	1	84	yes	—
22	C₂²×C₄×C₆₇	C₂⁴	1	14	yes	—
23	C₄²×C₆₇	C₂×D₄	1	112	yes	complemented
24	C₂³×C₆₇	—	2	112	no	—
25	C₂³×C₆₇	—	4	224	no	—
26	C₂³×C₆₇	C₂²×D₄	1	14	yes	—
27	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
28	D₄×C₆₇	—	4	448	no	—
29	C₂×C₄×C₆₇	#127	1	112	yes	complemented
30	C₂×C₄×C₆₇	C₂²×D₄	1	84	yes	—
31	C₂²×C₆₇	—	4	112	no	—
32	C₂²×C₆₇	—	8	64	no	—
33	C₂²×C₆₇	C₂×#127	1	28	yes	—
34	C₂²×C₆₇	C₂³×D₄	1	7	yes	—
35	C₄×C₆₇	C₂×#127	1	56	yes	complemented
36	C₂×#2867	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₆₇	—	8	16	no	—
38	C₂×C₆₇	#2867	1	8	yes	direct product factor
39	C₂×C₆₇	C₂²×#127	1	7	yes	—
40	C₂²×#127	C₂×C₆₇	1	14	yes	complemented
41	#2867	C₂×C₆₇	1	16	yes	direct product factor
42	C₂×C₄³	C₂×C₆₇	1	1	yes	characteristic, complemented
43	C₆₇	C₂×#2867	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×D₄	C₂²×C₆₇	1	28	yes	—
45	C₂×#127	—	2	112	no	—
46	C₂×#127	C₂²×C₆₇	1	112	yes	complemented
47	C₂²×C₄²	C₂²×C₆₇	1	7	yes	—
48	C₄³	C₂²×C₆₇	1	8	yes	complemented
49	C₂⁵	C₂³×C₆₇	1	8	yes	—
50	C₂²×D₄	—	2	336	no	—
51	C₂²×D₄	C₂³×C₆₇	1	112	yes	—
52	C₂³×C₄	C₂³×C₆₇	1	7	yes	—
53	#127	—	2	448	no	—
54	C₂×C₄²	D₄×C₆₇	1	56	yes	complemented
55	C₂×C₄²	C₂³×C₆₇	1	28	yes	—
56	C₂⁴	—	2	112	no	—
57	C₂⁴	C₂⁴×C₆₇	1	16	yes	—
58	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
59	C₂×D₄	—	2	672	no	—
60	C₂×D₄	—	4	448	no	—
61	C₂²×C₄	C₂×D₄×C₆₇	1	84	yes	—
62	C₂²×C₄	C₂⁴×C₆₇	1	14	yes	—
63	C₄²	C₂×D₄×C₆₇	1	112	yes	complemented
64	C₂³	—	2	112	no	—
65	C₂³	—	4	224	no	—
66	C₂³	C₂²×D₄×C₆₇	1	14	yes	—
67	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
68	D₄	—	4	448	no	—
69	C₂×C₄	#127×C₆₇	1	112	yes	complemented
70	C₂×C₄	C₂²×D₄×C₆₇	1	84	yes	—
71	C₂²	—	4	112	no	—
72	C₂²	—	8	64	no	—
73	C₂²	C₂×#127×C₆₇	1	28	yes	—
74	C₂²	C₂³×D₄×C₆₇	1	7	yes	—
75	C₄	C₂×#127×C₆₇	1	56	yes	complemented
76	C₂	—	8	16	no	—
77	C₂	C₆₇×#2867	1	8	yes	direct product factor
78	C₂	C₂²×#127×C₆₇	1	7	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#127×C₆₇	C₂	semidirect
2. C₆₇×#2867	C₂	direct (N × H)
3. C₂×C₄³×C₆₇	C₂	semidirect
6. C₂×#127×C₆₇	C₂²	semidirect
8. C₄³×C₆₇	C₂²	semidirect
14. C₂×C₄²×C₆₇	D₄	semidirect
23. C₄²×C₆₇	C₂×D₄	semidirect
29. C₂×C₄×C₆₇	#127	semidirect
35. C₄×C₆₇	C₂×#127	semidirect
36. C₂×#2867	C₆₇	direct (N × H)
38. C₂×C₆₇	#2867	direct (N × H)
40. C₂²×#127	C₂×C₆₇	semidirect
41. #2867	C₂×C₆₇	direct (N × H)
42. C₂×C₄³	C₂×C₆₇	semidirect
43. C₆₇	C₂×#2867	direct (N × H)
46. C₂×#127	C₂²×C₆₇	semidirect
48. C₄³	C₂²×C₆₇	semidirect
54. C₂×C₄²	D₄×C₆₇	semidirect
63. C₄²	C₂×D₄×C₆₇	semidirect
69. C₂×C₄	#127×C₆₇	semidirect
75. C₄	C₂×#127×C₆₇	semidirect
77. C₂	C₆₇×#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×#127×C₆₇	2
C₆₇×#2867	2
C₂×C₄³×C₆₇	2
C₂×#2867	67

Subgroups — C2×C67×#2867 #30715893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2867 #30715893