Subgroups — C₃×C₇×#424 #13056935

Order 1344.

228 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	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₇×#424	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₃×C₇×M₁₆	—	2	2	no	—
8	C₂×C₃×C₇×C₈	—	2	2	no	—
9	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
11	C₃×C₇×C₄⋊C₄	C₄	1	2	yes	—
12	C₃×C₇×C₂²⋊C₄	C₄	1	2	yes	—
13	C₃×C₄²×C₇	C₂²	1	2	yes	—
14	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
16	C₇×#117	C₂×C₃	1	1	yes	characteristic
17	C₇×C₄⋊C₈	C₂×C₃	1	4	yes	complemented
18	C₃×#424	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
24	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₃×C₇×C₈	—	2	2	no	—
26	C₃×C₇×C₈	—	2	2	no	—
27	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
28	C₇×M₁₆	—	2	2	no	—
29	C₂×C₇×C₈	—	2	2	no	—
30	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
31	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
32	C₇×C₄⋊C₄	C₃×C₄	1	2	yes	—
33	C₇×C₂²⋊C₄	C₃×C₄	1	2	yes	—
34	C₄²×C₇	C₂²×C₃	1	2	yes	—
35	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic
37	C₃×#117	C₂×C₇	1	1	yes	characteristic
38	C₃×C₄⋊C₈	C₂×C₇	1	4	yes	complemented
39	C₂²×C₃×C₇	—	2	2	no	—
40	C₂²×C₃×C₇	C₄⋊C₄	1	2	yes	—
41	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₄×C₇	—	2	4	no	—
43	C₃×C₄×C₇	C₄⋊C₄	1	2	yes	—
44	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
45	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
46	#424	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	—
49	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
51	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
52	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
53	C₇×C₈	—	2	2	no	—
54	C₇×C₈	—	2	2	no	—
55	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
56	C₃×M₁₆	—	2	2	no	—
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
59	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
60	C₃×C₄⋊C₄	C₄×C₇	1	2	yes	—
61	C₃×C₂²⋊C₄	C₄×C₇	1	2	yes	—
62	C₃×C₄²	C₂²×C₇	1	2	yes	—
63	C₂×C₃×C₇	—	2	2	no	—
64	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₇	#131	1	2	yes	—
66	C₂×M₁₆	C₂×C₃×C₇	1	1	yes	characteristic
67	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic
68	#117	C₂×C₃×C₇	1	1	yes	characteristic
69	C₄⋊C₈	C₂×C₃×C₇	1	4	yes	complemented
70	C₂²×C₇	—	2	2	no	—
71	C₂²×C₇	C₃×C₄⋊C₄	1	2	yes	—
72	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₄×C₇	—	2	4	no	—
74	C₄×C₇	C₃×C₄⋊C₄	1	2	yes	—
75	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
76	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
77	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×C₄×C₇	1	4	yes	—
79	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
80	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
81	C₂×C₃×C₄	C₇×Q₈	1	2	yes	—
82	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
83	C₃×C₈	—	2	2	no	—
84	C₃×C₈	—	2	2	no	—
85	C₃×C₇	#424	1	1	yes	characteristic, direct product factor, Hall
86	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
90	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
91	C₄⋊C₄	C₃×C₄×C₇	1	2	yes	—
92	C₂²⋊C₄	C₃×C₄×C₇	1	2	yes	—
93	C₄²	C₂²×C₃×C₇	1	2	yes	—
94	C₂×C₇	—	2	2	no	—
95	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₇	C₃×#131	1	2	yes	—
97	C₂²×C₃	—	2	2	no	—
98	C₂²×C₃	C₇×C₄⋊C₄	1	2	yes	—
99	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
100	C₃×C₄	—	2	4	no	—
101	C₃×C₄	C₇×C₄⋊C₄	1	2	yes	—
102	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
103	C₃×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
104	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₃×C₄×C₇	1	4	yes	—
106	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
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	2	no	—
111	C₈	—	2	2	no	—
112	C₇	C₃×#424	1	1	yes	characteristic, direct product factor, Sylow
113	C₂×C₃	—	2	2	no	—
114	C₂×C₃	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₃	C₇×#131	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₃×C₇×C₄⋊C₄	1	2	yes	—
118	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	C₃×C₇×C₄⋊C₄	1	2	yes	—
121	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
122	C₄	C₂×C₃×C₇×Q₈	1	1	yes	characteristic
123	C₃	C₇×#424	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	C₃×C₇×#131	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
4. C₃×C₇×C₄⋊C₈	C₂	semidirect
5. C₇×#424	C₃	direct (N × H)
17. C₇×C₄⋊C₈	C₂×C₃	semidirect
18. C₃×#424	C₇	direct (N × H)
38. C₃×C₄⋊C₈	C₂×C₇	semidirect
46. #424	C₃×C₇	direct (N × H)
69. C₄⋊C₈	C₂×C₃×C₇	semidirect
85. C₃×C₇	#424	direct (N × H)
112. C₇	C₃×#424	direct (N × H)
123. C₃	C₇×#424	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
C₂²×C₃×C₇×C₈	2
C₃×C₇×#117	2
C₃×C₇×C₄⋊C₈	2
C₇×#424	3
C₃×#424	7

Subgroups — C3×C7×#424 #13056935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#424 #13056935