Subgroups — C₃×C₇×#423 #13056934

Order 1344.

252 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₇×#423	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₄×C₇	C₄	1	2	yes	—
6	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₃×C₇×M₁₆	—	2	4	no	—
8	C₂×C₃×C₇×C₈	C₂²	1	4	yes	—
9	C₃×C₄²×C₇	C₄	1	1	yes	characteristic
10	C₃×C₄²×C₇	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₇	C₂²	1	2	yes	—
12	C₂×C₇×M₁₆	C₂×C₃	1	2	yes	—
13	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
14	C₇×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
15	C₃×#423	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₇	—	2	4	no	—
18	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
22	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
23	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₃×C₇×C₈	—	2	4	no	—
25	C₂²×C₄×C₇	C₃×C₄	1	2	yes	—
26	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
27	C₇×M₁₆	—	2	4	no	—
28	C₂×C₇×C₈	C₂²×C₃	1	4	yes	—
29	C₄²×C₇	C₃×C₄	1	1	yes	characteristic
30	C₄²×C₇	C₃×C₄	1	1	yes	characteristic
31	C₄²×C₇	C₂²×C₃	1	2	yes	—
32	C₂×C₃×M₁₆	C₂×C₇	1	2	yes	—
33	C₂×C₃×C₄²	C₂×C₇	1	1	yes	characteristic
34	C₃×C₄⋊C₈	C₂×C₇	1	4	yes	complemented
35	C₂²×C₃×C₇	—	2	2	no	—
36	C₂²×C₃×C₇	C₄⋊C₄	1	2	yes	—
37	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₃×C₄×C₇	—	2	2	no	—
39	C₃×C₄×C₇	C₄⋊C₄	1	2	yes	—
40	C₃×C₄×C₇	M₁₆	1	4	yes	complemented
41	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
43	#423	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇	—	2	4	no	—
46	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
47	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
49	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
51	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
52	C₇×C₈	—	2	4	no	—
53	C₂²×C₃×C₄	C₄×C₇	1	2	yes	—
54	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
55	C₃×M₁₆	—	2	4	no	—
56	C₂×C₃×C₈	C₂²×C₇	1	4	yes	—
57	C₃×C₄²	C₄×C₇	1	1	yes	characteristic
58	C₃×C₄²	C₄×C₇	1	1	yes	characteristic
59	C₃×C₄²	C₂²×C₇	1	2	yes	—
60	C₂×C₃×C₇	—	2	2	no	—
61	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃×C₇	C₂×M₁₆	1	2	yes	—
63	C₂×M₁₆	C₂×C₃×C₇	1	2	yes	—
64	C₂×C₄²	C₂×C₃×C₇	1	1	yes	characteristic
65	C₄⋊C₈	C₂×C₃×C₇	1	4	yes	complemented
66	C₂²×C₇	—	2	2	no	—
67	C₂²×C₇	C₃×C₄⋊C₄	1	2	yes	—
68	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
69	C₄×C₇	—	2	2	no	—
70	C₄×C₇	C₃×C₄⋊C₄	1	2	yes	—
71	C₄×C₇	C₃×M₁₆	1	4	yes	complemented
72	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
73	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
74	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
77	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
79	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
80	C₂×C₃×C₄	C₇×Q₈	1	2	yes	—
81	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
82	C₃×C₈	—	2	4	no	—
83	C₃×C₇	#423	1	1	yes	characteristic, direct product factor, Hall
84	C₂²×C₄	C₃×C₄×C₇	1	2	yes	—
85	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	C₂×C₈	C₂²×C₃×C₇	1	4	yes	—
88	C₄²	C₃×C₄×C₇	1	1	yes	characteristic
89	C₄²	C₃×C₄×C₇	1	1	yes	characteristic
90	C₄²	C₂²×C₃×C₇	1	2	yes	—
91	C₂×C₇	—	2	2	no	—
92	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₇	C₂×C₃×M₁₆	1	2	yes	—
94	C₂²×C₃	—	2	2	no	—
95	C₂²×C₃	C₇×C₄⋊C₄	1	2	yes	—
96	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
97	C₃×C₄	—	2	2	no	—
98	C₃×C₄	C₇×C₄⋊C₄	1	2	yes	—
99	C₃×C₄	C₇×M₁₆	1	4	yes	complemented
100	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
101	C₃×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
102	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
105	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
106	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
107	C₂×C₄	C₃×C₇×D₄	1	2	yes	—
108	C₂×C₄	C₃×C₇×Q₈	1	2	yes	—
109	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₇	C₃×#423	1	1	yes	characteristic, direct product factor, Sylow
112	C₂×C₃	—	2	2	no	—
113	C₂×C₃	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃	C₂×C₇×M₁₆	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₃×C₇×C₄⋊C₄	1	2	yes	—
117	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	C₃×C₇×C₄⋊C₄	1	2	yes	—
120	C₄	C₃×C₇×M₁₆	1	4	yes	complemented
121	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
122	C₄	C₂×C₃×C₇×Q₈	1	1	yes	characteristic
123	C₃	C₇×#423	1	1	yes	characteristic, direct product factor, Sylow
124	C₂	—	2	2	no	—
125	C₂	C₂×C₃×C₇×C₄⋊C₄	1	1	yes	characteristic
126	C₂	C₂×C₃×C₇×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₇	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₇×C₄⋊C₈	C₂	semidirect
4. C₇×#423	C₃	direct (N × H)
14. C₇×C₄⋊C₈	C₂×C₃	semidirect
15. C₃×#423	C₇	direct (N × H)
34. C₃×C₄⋊C₈	C₂×C₇	semidirect
40. C₃×C₄×C₇	M₁₆	semidirect
43. #423	C₃×C₇	direct (N × H)
65. C₄⋊C₈	C₂×C₃×C₇	semidirect
71. C₄×C₇	C₃×M₁₆	semidirect
83. C₃×C₇	#423	direct (N × H)
99. C₃×C₄	C₇×M₁₆	semidirect
111. C₇	C₃×#423	direct (N × H)
120. C₄	C₃×C₇×M₁₆	semidirect
123. C₃	C₇×#423	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×C₃×C₇×M₁₆	2
C₂×C₃×C₄²×C₇	2
C₃×C₇×C₄⋊C₈	2
C₇×#423	3
C₃×#423	7

Subgroups — C3×C7×#423 #13056934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#423 #13056934