Subgroups — C₂×C₃×#2253 #10652331

Order 768.

494 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#499	C₂	1	2	yes	maximal
2	C₃×#2253	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×#367	C₂	1	2	yes	maximal
4	C₂×C₃×#366	C₂	1	2	yes	maximal
5	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×#2253	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#125	—	2	2	no	—
8	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#499	—	2	4	no	—
11	C₃×#499	C₂²	1	8	yes	—
12	C₂×C₃×#107	C₂²	1	4	yes	—
13	C₂×C₃×#106	—	2	2	no	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×#367	C₂²	1	8	yes	—
16	C₃×#366	C₂²	1	8	yes	—
17	C₃×C₄×C₁₆	C₂²	1	4	yes	—
18	C₂×#499	C₂×C₃	1	2	yes	—
19	#2253	C₂×C₃	1	8	yes	direct product factor
20	C₂×#367	C₂×C₃	1	2	yes	—
21	C₂×#366	C₂×C₃	1	2	yes	—
22	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×#125	—	2	8	no	—
26	C₃×#125	—	4	4	no	—
27	C₂×C₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₃×C₄⋊C₄	—	4	4	no	—
29	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₃×C₁₆	Q₈	1	4	yes	—
31	C₂×C₃×C₁₆	Q₈	1	4	yes	—
32	C₂×C₃×C₁₆	C₂³	1	2	yes	—
33	C₂×C₃×C₁₆	C₂³	1	2	yes	—
34	C₃×#107	—	2	4	no	—
35	C₃×#107	C₂³	1	8	yes	—
36	C₃×#106	—	2	8	no	—
37	C₃×C₄×C₈	D₄	1	2	yes	—
38	C₃×C₄×C₈	C₂³	1	2	yes	—
39	C₂×#125	—	2	2	no	—
40	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
42	#499	—	2	4	no	—
43	#499	C₂²×C₃	1	8	yes	—
44	C₂×#107	C₂²×C₃	1	4	yes	—
45	C₂×#106	—	2	2	no	—
46	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
47	#367	C₂²×C₃	1	8	yes	—
48	#366	C₂²×C₃	1	8	yes	—
49	C₄×C₁₆	C₂²×C₃	1	4	yes	—
50	C₂²×C₃×C₄	—	4	4	no	—
51	C₂²×C₃×C₄	D₈	1	2	yes	—
52	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₈	Q₁₆	1	4	yes	—
54	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
56	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
58	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
59	C₃×C₄⋊C₄	—	2	8	no	—
60	C₃×C₄⋊C₄	—	4	4	no	—
61	C₃×C₄⋊C₄	—	4	16	no	—
62	C₃×C₄²	D₈	1	2	yes	—
63	C₃×C₄²	C₂×D₄	1	2	yes	—
64	C₃×C₁₆	C₂×Q₈	1	4	yes	—
65	C₃×C₁₆	C₂×Q₈	1	4	yes	—
66	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
67	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
68	#125	—	2	8	no	—
69	#125	—	4	4	no	—
70	C₂×C₄⋊C₄	—	2	4	no	—
71	C₂×C₄⋊C₄	—	4	4	no	—
72	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
73	C₂×C₁₆	C₃×Q₈	1	4	yes	—
74	C₂×C₁₆	C₃×Q₈	1	4	yes	—
75	C₂×C₁₆	C₂³×C₃	1	2	yes	—
76	C₂×C₁₆	C₂³×C₃	1	2	yes	—
77	#107	—	2	4	no	—
78	#107	C₂³×C₃	1	8	yes	—
79	#106	—	2	8	no	—
80	C₄×C₈	C₃×D₄	1	2	yes	—
81	C₄×C₈	C₂³×C₃	1	2	yes	—
82	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	4	8	no	—
85	C₂×C₃×C₄	—	8	8	no	—
86	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
87	C₂×C₃×C₄	C₂×D₈	1	4	yes	—
88	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
90	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
91	C₃×C₈	C₄⋊Q₈	1	2	yes	—
92	C₃×C₈	C₂×Q₁₆	1	4	yes	—
93	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
94	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
95	C₂²×C₄	—	4	4	no	—
96	C₂²×C₄	C₃×D₈	1	2	yes	—
97	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
98	C₂×C₈	C₃×Q₁₆	1	4	yes	—
99	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
101	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
102	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
103	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
104	C₄⋊C₄	—	2	8	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	4	16	no	—
107	C₄²	C₃×D₈	1	2	yes	—
108	C₄²	C₂×C₃×D₄	1	2	yes	—
109	C₁₆	C₂×C₃×Q₈	1	4	yes	—
110	C₁₆	C₂×C₃×Q₈	1	4	yes	—
111	C₂²×C₃	#500	1	2	yes	—
112	C₂²×C₃	#508	1	4	yes	—
113	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
114	C₃×C₄	—	2	4	no	—
115	C₃×C₄	—	8	8	no	—
116	C₃×C₄	#500	1	2	yes	—
117	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
119	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	8	8	no	—
123	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
124	C₂×C₄	C₂×C₃×D₈	1	4	yes	—
125	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
126	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
127	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
128	C₈	C₃×C₄⋊Q₈	1	2	yes	—
129	C₈	C₂×C₃×Q₁₆	1	4	yes	—
130	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
131	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
132	C₂×C₃	#2253	1	4	yes	direct product factor
133	C₂×C₃	C₂×#500	1	1	yes	characteristic
134	C₂×C₃	C₂×#508	1	2	yes	—
135	C₂²	C₃×#500	1	2	yes	—
136	C₂²	C₃×#508	1	4	yes	—
137	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	8	8	no	—
140	C₄	C₃×#500	1	2	yes	—
141	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
142	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
143	C₃	C₂×#2253	1	1	yes	characteristic, direct product factor, Sylow
144	C₂	C₃×#2253	1	4	yes	direct product factor
145	C₂	C₂×C₃×#500	1	1	yes	characteristic
146	C₂	C₂×C₃×#508	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2253	C₂	direct (N × H)
6. C₂×#2253	C₃	direct (N × H)
19. #2253	C₂×C₃	direct (N × H)
132. C₂×C₃	#2253	direct (N × H)
143. C₃	C₂×#2253	direct (N × H)
144. C₂	C₃×#2253	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×#499	2
C₃×#2253	2
C₂×C₃×#367	2
C₂×C₃×#366	2
C₂×C₃×C₄×C₁₆	2
C₂×#2253	3

Subgroups — C2×C3×#2253 #10652331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2253 #10652331