Subgroups — C₂×C₃×#1313 #10630010

Order 768.

302 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
4	C₃×#1313	C₂	1	4	yes	maximal, direct product factor
5	C₂×#1313	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×C₁₆	C₄	1	2	yes	—
7	C₂²×C₃×C₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×#346	C₄	1	2	yes	—
11	C₃×#346	C₄	1	2	yes	—
12	C₃×#346	C₂²	1	2	yes	—
13	C₃×#346	C₂²	1	2	yes	—
14	C₃×#322	C₄	1	2	yes	—
15	C₃×#322	C₂²	1	2	yes	—
16	C₂×#346	C₂×C₃	1	1	yes	characteristic
17	C₂×#346	C₂×C₃	1	1	yes	characteristic
18	C₂×#322	C₂×C₃	1	1	yes	characteristic
19	#1313	C₂×C₃	1	4	yes	direct product factor
20	C₂²×C₃×C₈	—	2	2	no	—
21	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₁₆	—	2	4	no	—
25	C₂×C₃×C₁₆	C₈	1	8	yes	complemented
26	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₁₆	C₂×C₄	1	4	yes	—
28	C₃×C₄×C₈	—	2	2	no	—
29	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
32	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₁₆	C₃×C₄	1	2	yes	—
35	C₂²×C₁₆	C₃×C₄	1	2	yes	—
36	C₂×C₄×C₈	C₃×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
38	#346	C₃×C₄	1	2	yes	—
39	#346	C₃×C₄	1	2	yes	—
40	#346	C₂²×C₃	1	2	yes	—
41	#346	C₂²×C₃	1	2	yes	—
42	#322	C₃×C₄	1	2	yes	—
43	#322	C₂²×C₃	1	2	yes	—
44	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
45	C₂²×C₃×C₄	M₁₆	1	2	yes	—
46	C₂×C₃×C₈	—	2	4	no	—
47	C₂×C₃×C₈	—	4	4	no	—
48	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
49	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂×C₈	1	4	yes	—
51	C₂×C₃×C₈	M₁₆	1	4	yes	—
52	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₄²	C₄²	1	1	yes	characteristic
55	C₃×C₄²	M₁₆	1	2	yes	—
56	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₁₆	—	2	4	no	—
58	C₃×C₁₆	C₂×C₈	1	8	yes	complemented
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
63	C₂×C₁₆	—	2	4	no	—
64	C₂×C₁₆	C₃×C₈	1	8	yes	complemented
65	C₂×C₁₆	C₂×C₃×C₄	1	4	yes	—
66	C₂×C₁₆	C₂×C₃×C₄	1	4	yes	—
67	C₄×C₈	—	2	2	no	—
68	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×C₃×C₄	1	4	yes	—
71	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
73	C₂³×C₃	#97	1	1	yes	characteristic
74	C₂×C₃×C₄	—	2	4	no	—
75	C₂×C₃×C₄	C₄×C₈	1	2	yes	—
76	C₂×C₃×C₄	#97	1	2	yes	—
77	C₂×C₃×C₄	#97	1	1	yes	characteristic
78	C₂×C₃×C₄	#97	1	2	yes	—
79	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
81	C₃×C₈	—	4	4	no	—
82	C₃×C₈	C₄×C₈	1	2	yes	—
83	C₃×C₈	#97	1	2	yes	—
84	C₃×C₈	C₂²×C₈	1	2	yes	—
85	C₃×C₈	C₂×M₁₆	1	2	yes	—
86	C₂²×C₄	C₃×C₄²	1	1	yes	characteristic
87	C₂²×C₄	C₃×M₁₆	1	2	yes	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	4	4	no	—
90	C₂×C₈	C₃×C₄²	1	1	yes	characteristic
91	C₂×C₈	C₃×C₄²	1	1	yes	characteristic
92	C₂×C₈	C₂×C₃×C₈	1	4	yes	—
93	C₂×C₈	C₃×M₁₆	1	4	yes	—
94	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
96	C₄²	C₃×C₄²	1	1	yes	characteristic
97	C₄²	C₃×M₁₆	1	2	yes	—
98	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
99	C₁₆	—	2	4	no	—
100	C₁₆	C₂×C₃×C₈	1	8	yes	complemented
101	C₂²×C₃	#322	1	2	yes	—
102	C₂²×C₃	#347	1	4	yes	—
103	C₂²×C₃	C₂×#97	1	1	yes	characteristic
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	#322	1	2	yes	—
107	C₃×C₄	C₂×C₄×C₈	1	1	yes	characteristic
108	C₃×C₄	C₂×#97	1	1	yes	characteristic
109	C₂³	C₃×#97	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	C₃×C₄×C₈	1	2	yes	—
112	C₂×C₄	C₃×#97	1	2	yes	—
113	C₂×C₄	C₃×#97	1	2	yes	—
114	C₂×C₄	C₃×#97	1	1	yes	characteristic
115	C₂×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
116	C₂×C₄	C₂×C₃×M₁₆	1	2	yes	—
117	C₈	—	4	4	no	—
118	C₈	C₃×C₄×C₈	1	2	yes	—
119	C₈	C₃×#97	1	2	yes	—
120	C₈	C₂²×C₃×C₈	1	2	yes	—
121	C₈	C₂×C₃×M₁₆	1	2	yes	—
122	C₂×C₃	#1313	1	4	yes	direct product factor
123	C₂×C₃	C₂×#322	1	1	yes	characteristic
124	C₂×C₃	C₂×#347	1	2	yes	—
125	C₂²	C₃×#322	1	2	yes	—
126	C₂²	C₃×#347	1	4	yes	—
127	C₂²	C₂×C₃×#97	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	C₃×#322	1	2	yes	—
131	C₄	C₂×C₃×C₄×C₈	1	1	yes	characteristic
132	C₄	C₂×C₃×#97	1	1	yes	characteristic
133	C₃	C₂×#1313	1	1	yes	characteristic, direct product factor, Sylow
134	C₂	C₃×#1313	1	4	yes	direct product factor
135	C₂	C₂×C₃×#322	1	1	yes	characteristic
136	C₂	C₂×C₃×#347	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₄²
commutator	C₄	C₂×C₃×C₄×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle, prime radical	C₂³×C₃	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×#1313	C₂	direct (N × H)
5. C₂×#1313	C₃	direct (N × H)
19. #1313	C₂×C₃	direct (N × H)
25. C₂×C₃×C₁₆	C₈	semidirect
58. C₃×C₁₆	C₂×C₈	semidirect
64. C₂×C₁₆	C₃×C₈	semidirect
100. C₁₆	C₂×C₃×C₈	semidirect
122. C₂×C₃	#1313	direct (N × H)
133. C₃	C₂×#1313	direct (N × H)
134. C₂	C₃×#1313	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×#346	2
C₂×C₃×#322	2
C₃×#1313	2
C₂×#1313	3

Subgroups — C2×C3×#1313 #10630010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1313 #10630010