Subgroups — C₃×C₄×C₇×Q₁₆ #13056950

Order 1344.

220 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇×Q₈	C₂	1	2	yes	maximal
3	C₃×C₇×#107	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×#103	C₂	1	2	yes	maximal
5	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×C₇×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
8	C₃×C₇×Q₁₆	C₄	1	4	yes	direct product factor
9	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₇×C₄⋊C₄	—	2	2	no	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₄²×C₇	—	2	2	no	—
14	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₂×C₇×Q₁₆	C₂×C₃	1	1	yes	characteristic
16	C₄×C₇×Q₈	C₂×C₃	1	2	yes	—
17	C₇×#107	C₂×C₃	1	1	yes	characteristic
18	C₇×#103	C₂×C₃	1	2	yes	—
19	C₄×C₇×C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄×Q₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₃×C₇×Q₈	—	2	2	no	—
22	C₃×C₇×Q₈	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄×C₇	—	2	2	no	—
24	C₂×C₃×C₄×C₇	—	2	2	no	—
25	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₃×C₇×C₈	—	2	1	no	—
29	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
31	C₇×Q₁₆	C₃×C₄	1	4	yes	direct product factor
32	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
33	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
34	C₇×C₄⋊C₄	—	2	2	no	—
35	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
36	C₄²×C₇	—	2	2	no	—
37	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₃×Q₁₆	C₂×C₇	1	1	yes	characteristic
39	C₃×C₄×Q₈	C₂×C₇	1	2	yes	—
40	C₃×#107	C₂×C₇	1	1	yes	characteristic
41	C₃×#103	C₂×C₇	1	2	yes	—
42	C₃×C₄×C₈	C₂×C₇	1	1	yes	characteristic
43	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
44	C₃×C₄×C₇	—	2	4	no	—
45	C₃×C₄×C₇	—	2	1	no	—
46	C₃×C₄×C₇	—	4	2	no	—
47	C₃×C₄×C₇	Q₁₆	1	2	yes	direct product factor
48	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₇	#41	1	1	yes	characteristic
50	C₄×Q₁₆	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
51	C₇×Q₈	—	2	2	no	—
52	C₇×Q₈	C₂×C₃×C₄	1	4	yes	—
53	C₂×C₄×C₇	—	2	2	no	—
54	C₂×C₄×C₇	—	2	2	no	—
55	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
58	C₇×C₈	—	2	1	no	—
59	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
60	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
61	C₃×Q₁₆	C₄×C₇	1	4	yes	direct product factor
62	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
66	C₃×C₄²	—	2	2	no	—
67	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
68	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
69	C₂×C₃×C₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₃×C₇	#135	1	1	yes	characteristic
71	C₂×Q₁₆	C₂×C₃×C₇	1	1	yes	characteristic
72	C₄×Q₈	C₂×C₃×C₇	1	2	yes	—
73	#107	C₂×C₃×C₇	1	1	yes	characteristic
74	#103	C₂×C₃×C₇	1	2	yes	—
75	C₄×C₈	C₂×C₃×C₇	1	1	yes	characteristic
76	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
77	C₄×C₇	—	2	4	no	—
78	C₄×C₇	—	2	1	no	—
79	C₄×C₇	—	4	2	no	—
80	C₄×C₇	C₃×Q₁₆	1	2	yes	direct product factor
81	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₄×C₇	C₃×#41	1	1	yes	characteristic
83	C₃×Q₈	—	2	2	no	—
84	C₃×Q₈	C₂×C₄×C₇	1	4	yes	—
85	C₂×C₃×C₄	—	2	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
90	C₃×C₈	—	2	1	no	—
91	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
92	C₃×C₇	C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Hall
93	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
94	Q₁₆	C₃×C₄×C₇	1	4	yes	direct product factor
95	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
96	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
99	C₄²	—	2	2	no	—
100	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
101	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
102	C₂×C₇	C₂×C₃×Q₁₆	1	1	yes	characteristic
103	C₂×C₇	C₃×#135	1	1	yes	characteristic
104	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
105	C₃×C₄	—	2	4	no	—
106	C₃×C₄	—	2	1	no	—
107	C₃×C₄	—	4	2	no	—
108	C₃×C₄	C₇×Q₁₆	1	2	yes	direct product factor
109	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
110	C₃×C₄	C₇×#41	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	Q₈	C₂×C₃×C₄×C₇	1	4	yes	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
116	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
117	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	C₂×C₃×C₄×C₇	1	2	yes	—
120	C₇	C₃×C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
122	C₂×C₃	C₂×C₇×Q₁₆	1	1	yes	characteristic
123	C₂×C₃	C₇×#135	1	1	yes	characteristic
124	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
125	C₄	—	2	4	no	—
126	C₄	—	2	1	no	—
127	C₄	—	4	2	no	—
128	C₄	C₃×C₇×Q₁₆	1	2	yes	direct product factor
129	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
130	C₄	C₃×C₇×#41	1	1	yes	characteristic
131	C₃	C₄×C₇×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
132	C₂	C₃×C₄×C₇×D₄	1	1	yes	characteristic
133	C₂	C₂×C₃×C₇×Q₁₆	1	1	yes	characteristic
134	C₂	C₃×C₇×#135	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₇	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ C₃×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄×C₇×Q₁₆	C₃	direct (N × H)
8. C₃×C₇×Q₁₆	C₄	direct (N × H)
20. C₃×C₄×Q₁₆	C₇	direct (N × H)
31. C₇×Q₁₆	C₃×C₄	direct (N × H)
47. C₃×C₄×C₇	Q₁₆	direct (N × H)
50. C₄×Q₁₆	C₃×C₇	direct (N × H)
61. C₃×Q₁₆	C₄×C₇	direct (N × H)
80. C₄×C₇	C₃×Q₁₆	direct (N × H)
92. C₃×C₇	C₄×Q₁₆	direct (N × H)
94. Q₁₆	C₃×C₄×C₇	direct (N × H)
108. C₃×C₄	C₇×Q₁₆	direct (N × H)
120. C₇	C₃×C₄×Q₁₆	direct (N × H)
128. C₄	C₃×C₇×Q₁₆	direct (N × H)
131. C₃	C₄×C₇×Q₁₆	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×Q₁₆	2
C₃×C₄×C₇×Q₈	2
C₃×C₇×#107	2
C₃×C₇×#103	2
C₃×C₄×C₇×C₈	2
C₄×C₇×Q₁₆	3
C₃×C₄×Q₁₆	7

Subgroups — C3×C4×C7×Q16 #13056950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₇×Q₁₆ #13056950