Subgroups — C₂³×C₇×#97 #13237267

Order 1792.

4126 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×C₈	C₂	1	2	yes	maximal
2	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×#97	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₄×C₇	C₄	1	2	yes	—
5	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂³×C₇×C₈	C₄	1	32	yes	complemented
7	C₂³×C₇×C₈	C₂²	1	28	yes	—
8	C₂²×C₄²×C₇	C₄	1	14	yes	—
9	C₂²×C₄²×C₇	C₂²	1	14	yes	—
10	C₂×C₇×#97	C₂²	1	112	yes	direct product factor
11	C₂³×#97	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₇	—	2	16	no	—
14	C₂³×C₄×C₇	C₂×C₄	1	28	yes	—
15	C₂³×C₄×C₇	C₂×C₄	1	7	yes	—
16	C₂³×C₄×C₇	C₂×C₄	1	16	yes	—
17	C₂³×C₄×C₇	C₂³	1	7	yes	—
18	C₂²×C₇×C₈	C₂×C₄	1	224	yes	complemented
19	C₂²×C₇×C₈	C₂³	1	56	yes	—
20	C₂×C₄²×C₇	C₂×C₄	1	84	yes	—
21	C₂×C₄²×C₇	C₂³	1	28	yes	—
22	C₇×#97	C₂³	1	64	yes	direct product factor
23	C₂⁴×C₈	C₂×C₇	1	2	yes	—
24	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic
25	C₂²×#97	C₂×C₇	1	28	yes	direct product factor
26	C₂⁴×C₇	C₄²	1	8	yes	—
27	C₂⁴×C₇	M₁₆	1	16	yes	—
28	C₂⁴×C₇	C₂²×C₄	1	7	yes	—
29	C₂²×C₄×C₇	—	2	112	no	—
30	C₂²×C₄×C₇	C₄²	1	56	yes	—
31	C₂²×C₄×C₇	C₂²×C₄	1	56	yes	—
32	C₂²×C₄×C₇	C₂²×C₄	1	21	yes	—
33	C₂²×C₄×C₇	C₂²×C₄	1	56	yes	—
34	C₂²×C₄×C₇	C₂⁴	1	7	yes	—
35	C₂×C₇×C₈	C₂²×C₄	1	224	yes	complemented
36	C₂×C₇×C₈	C₂⁴	1	16	yes	—
37	C₄²×C₇	C₂²×C₄	1	56	yes	—
38	C₄²×C₇	C₂⁴	1	8	yes	—
39	C₂⁴×C₄	C₄×C₇	1	2	yes	—
40	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
41	C₂³×C₈	C₄×C₇	1	32	yes	complemented
42	C₂³×C₈	C₂²×C₇	1	28	yes	—
43	C₂²×C₄²	C₄×C₇	1	14	yes	—
44	C₂²×C₄²	C₂²×C₇	1	14	yes	—
45	C₂×#97	C₂²×C₇	1	112	yes	direct product factor
46	C₂³×C₇	#97	1	64	yes	direct product factor
47	C₂³×C₇	C₂×C₄²	1	28	yes	—
48	C₂³×C₇	C₂×M₁₆	1	56	yes	—
49	C₂³×C₇	C₂³×C₄	1	7	yes	—
50	C₂×C₄×C₇	—	2	112	no	—
51	C₂×C₄×C₇	C₂×C₄²	1	84	yes	—
52	C₂×C₄×C₇	C₂³×C₄	1	16	yes	—
53	C₂×C₄×C₇	C₂³×C₄	1	28	yes	—
54	C₂×C₄×C₇	C₂³×C₄	1	7	yes	—
55	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
56	C₇×C₈	C₂³×C₄	1	32	yes	complemented
57	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
58	C₂³×C₄	—	2	16	no	—
59	C₂³×C₄	C₂×C₄×C₇	1	28	yes	—
60	C₂³×C₄	C₂×C₄×C₇	1	7	yes	—
61	C₂³×C₄	C₂×C₄×C₇	1	16	yes	—
62	C₂³×C₄	C₂³×C₇	1	7	yes	—
63	C₂²×C₈	C₂×C₄×C₇	1	224	yes	complemented
64	C₂²×C₈	C₂³×C₇	1	56	yes	—
65	C₂×C₄²	C₂×C₄×C₇	1	84	yes	—
66	C₂×C₄²	C₂³×C₇	1	28	yes	—
67	#97	C₂³×C₇	1	64	yes	direct product factor
68	C₂²×C₇	C₂×#97	1	112	yes	direct product factor
69	C₂²×C₇	C₂²×C₄²	1	14	yes	—
70	C₂²×C₇	C₂²×M₁₆	1	28	yes	—
71	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
72	C₄×C₇	—	2	16	no	—
73	C₄×C₇	C₂²×C₄²	1	14	yes	—
74	C₄×C₇	C₂⁴×C₄	1	2	yes	—
75	C₂⁴	C₄²×C₇	1	8	yes	—
76	C₂⁴	C₇×M₁₆	1	16	yes	—
77	C₂⁴	C₂²×C₄×C₇	1	7	yes	—
78	C₂²×C₄	—	2	112	no	—
79	C₂²×C₄	C₄²×C₇	1	56	yes	—
80	C₂²×C₄	C₂²×C₄×C₇	1	56	yes	—
81	C₂²×C₄	C₂²×C₄×C₇	1	21	yes	—
82	C₂²×C₄	C₂²×C₄×C₇	1	56	yes	—
83	C₂²×C₄	C₂⁴×C₇	1	7	yes	—
84	C₂×C₈	C₂²×C₄×C₇	1	224	yes	complemented
85	C₂×C₈	C₂⁴×C₇	1	16	yes	—
86	C₄²	C₂²×C₄×C₇	1	56	yes	—
87	C₄²	C₂⁴×C₇	1	8	yes	—
88	C₂×C₇	C₂²×#97	1	28	yes	direct product factor
89	C₂×C₇	C₂³×C₄²	1	1	yes	characteristic
90	C₂×C₇	C₂³×M₁₆	1	2	yes	—
91	C₂³	C₇×#97	1	64	yes	direct product factor
92	C₂³	C₂×C₄²×C₇	1	28	yes	—
93	C₂³	C₂×C₇×M₁₆	1	56	yes	—
94	C₂³	C₂³×C₄×C₇	1	7	yes	—
95	C₂×C₄	—	2	112	no	—
96	C₂×C₄	C₂×C₄²×C₇	1	84	yes	—
97	C₂×C₄	C₂³×C₄×C₇	1	16	yes	—
98	C₂×C₄	C₂³×C₄×C₇	1	7	yes	—
99	C₂×C₄	C₂³×C₄×C₇	1	28	yes	—
100	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
101	C₈	C₂³×C₄×C₇	1	32	yes	complemented
102	C₇	C₂³×#97	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²	C₂×C₇×#97	1	112	yes	direct product factor
104	C₂²	C₂²×C₄²×C₇	1	14	yes	—
105	C₂²	C₂²×C₇×M₁₆	1	28	yes	—
106	C₂²	C₂⁴×C₄×C₇	1	1	yes	characteristic
107	C₄	—	2	16	no	—
108	C₄	C₂²×C₄²×C₇	1	14	yes	—
109	C₄	C₂⁴×C₄×C₇	1	2	yes	—
110	C₂	C₂²×C₇×#97	1	28	yes	direct product factor
111	C₂	C₂³×C₄²×C₇	1	1	yes	characteristic
112	C₂	C₂³×C₇×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄×C₇	C₂²
commutator	C₂	C₂³×C₄²×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
socle, prime radical	C₂⁵×C₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₇×#97	C₂	direct (N × H)
6. C₂³×C₇×C₈	C₄	semidirect
10. C₂×C₇×#97	C₂²	direct (N × H)
11. C₂³×#97	C₇	direct (N × H)
18. C₂²×C₇×C₈	C₂×C₄	semidirect
22. C₇×#97	C₂³	direct (N × H)
25. C₂²×#97	C₂×C₇	direct (N × H)
35. C₂×C₇×C₈	C₂²×C₄	semidirect
41. C₂³×C₈	C₄×C₇	semidirect
45. C₂×#97	C₂²×C₇	direct (N × H)
46. C₂³×C₇	#97	direct (N × H)
56. C₇×C₈	C₂³×C₄	semidirect
63. C₂²×C₈	C₂×C₄×C₇	semidirect
67. #97	C₂³×C₇	direct (N × H)
68. C₂²×C₇	C₂×#97	direct (N × H)
84. C₂×C₈	C₂²×C₄×C₇	semidirect
88. C₂×C₇	C₂²×#97	direct (N × H)
91. C₂³	C₇×#97	direct (N × H)
101. C₈	C₂³×C₄×C₇	semidirect
102. C₇	C₂³×#97	direct (N × H)
103. C₂²	C₂×C₇×#97	direct (N × H)
110. C₂	C₂²×C₇×#97	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×C₈	2
C₂³×C₄²×C₇	2
C₂²×C₇×#97	2
C₂³×#97	7

Subgroups — C23×C7×#97 #13237267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×#97 #13237267