Subgroups — C₂×C₄×#803 #10792511

Order 768.

1790 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂²×#803	C₂	1	4	yes	maximal
3	C₂²×C₄×C₃⋊C₈	C₂	1	2	yes	maximal
4	C₄×#803	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄×#97	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄²	C₄	1	4	yes	—
7	C₂²×C₃×C₄²	C₂²	1	2	yes	—
8	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂³×C₃⋊C₈	C₂²	1	4	yes	—
10	C₃×C₄³	C₄	1	4	yes	—
11	C₃×C₄³	C₂²	1	4	yes	—
12	C₂×#803	C₄	1	32	yes	direct product factor
13	C₂×#803	C₂²	1	16	yes	—
14	C₂×C₄×C₃⋊C₈	C₄	1	16	yes	complemented
15	C₂×C₄×C₃⋊C₈	C₂²	1	8	yes	—
16	C₂²×#97	—	3	4	no	—
17	C₂²×C₄×C₈	—	3	2	no	—
18	C₂×C₄³	D₃	1	1	yes	characteristic
19	C₄×#97	—	3	8	no	—
20	C₂³×C₃×C₄	C₂×C₄	1	4	yes	—
21	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
22	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄²	—	2	16	no	—
24	C₂×C₃×C₄²	C₂×C₄	1	8	yes	—
25	C₂×C₃×C₄²	C₂×C₄	1	16	yes	—
26	C₂×C₃×C₄²	C₂×C₄	1	16	yes	—
27	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄²	C₂³	1	2	yes	—
30	C₂×C₃×C₄²	C₂³	1	4	yes	—
31	C₂²×C₃⋊C₈	C₂×C₄	1	16	yes	—
32	C₂²×C₃⋊C₈	C₂×C₄	1	32	yes	—
33	C₂²×C₃⋊C₈	C₂³	1	8	yes	—
34	#803	C₂×C₄	1	64	yes	direct product factor
35	C₄×C₃⋊C₈	C₂×C₄	1	32	yes	complemented
36	C₂³×C₈	—	3	4	no	—
37	C₂²×C₄²	Dic₃	1	4	yes	—
38	C₂²×C₄²	C₂×D₃	1	2	yes	—
39	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂×#97	—	3	32	no	—
41	C₂×#97	—	3	16	no	—
42	C₂×C₄×C₈	—	3	16	no	—
43	C₂×C₄×C₈	—	3	8	no	—
44	C₄³	Dic₃	1	4	yes	—
45	C₄³	C₂×D₃	1	4	yes	—
46	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	—	2	16	no	—
48	C₂²×C₃×C₄	C₄²	1	16	yes	—
49	C₂²×C₃×C₄	C₄²	1	8	yes	—
50	C₂²×C₃×C₄	M₁₆	1	16	yes	—
51	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
52	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
53	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
54	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
56	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
57	C₃×C₄²	—	2	32	no	—
58	C₃×C₄²	C₄²	1	16	yes	—
59	C₃×C₄²	C₄²	1	8	yes	—
60	C₃×C₄²	C₂²×C₄	1	8	yes	—
61	C₃×C₄²	C₂²×C₄	1	16	yes	—
62	C₂×C₃⋊C₈	C₄²	1	64	yes	complemented
63	C₂×C₃⋊C₈	C₂²×C₄	1	16	yes	—
64	C₂×C₃⋊C₈	C₂²×C₄	1	32	yes	—
65	C₂³×C₄	C₂×Dic₃	1	4	yes	—
66	C₂³×C₄	C₂×Dic₃	1	2	yes	—
67	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
68	C₂²×C₈	—	3	16	no	—
69	C₂²×C₈	—	3	32	no	—
70	C₂²×C₈	—	3	8	no	—
71	C₂×C₄²	—	2	16	no	—
72	C₂×C₄²	C₄×D₃	1	8	yes	—
73	C₂×C₄²	C₄×D₃	1	16	yes	—
74	C₂×C₄²	C₂×Dic₃	1	16	yes	—
75	C₂×C₄²	C₂×Dic₃	1	4	yes	—
76	C₂×C₄²	C₂×Dic₃	1	2	yes	—
77	C₂×C₄²	C₂²×D₃	1	2	yes	—
78	C₂×C₄²	C₂²×D₃	1	4	yes	—
79	#97	—	3	64	no	—
80	C₄×C₈	—	3	32	no	—
81	C₂³×C₃	C₂×C₄²	1	2	yes	—
82	C₂³×C₃	C₂×C₄²	1	4	yes	—
83	C₂³×C₃	C₂×M₁₆	1	8	yes	—
84	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	—	2	16	no	—
86	C₂×C₃×C₄	—	2	32	no	—
87	C₂×C₃×C₄	#97	1	32	yes	complemented
88	C₂×C₃×C₄	C₂×C₄²	1	16	yes	—
89	C₂×C₃×C₄	C₂×C₄²	1	16	yes	—
90	C₂×C₃×C₄	C₂×C₄²	1	2	yes	—
91	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
92	C₂×C₃×C₄	C₂×C₄²	1	8	yes	—
93	C₂×C₃×C₄	C₂×M₁₆	1	16	yes	—
94	C₂×C₃×C₄	C₂³×C₄	1	2	yes	—
95	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
96	C₃⋊C₈	C₂×C₄²	1	32	yes	complemented
97	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
98	C₂²×C₄	—	2	16	no	—
99	C₂²×C₄	#208	1	16	yes	—
100	C₂²×C₄	C₄×Dic₃	1	16	yes	—
101	C₂²×C₄	C₄×Dic₃	1	8	yes	—
102	C₂²×C₄	C₂×C₄×D₃	1	4	yes	—
103	C₂²×C₄	C₂×C₄×D₃	1	8	yes	—
104	C₂²×C₄	C₂²×Dic₃	1	8	yes	—
105	C₂²×C₄	C₂²×Dic₃	1	4	yes	—
106	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
107	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
108	C₂×C₈	—	3	64	no	—
109	C₂×C₈	—	3	16	no	—
110	C₂×C₈	—	3	32	no	—
111	C₄²	—	2	32	no	—
112	C₄²	C₄×Dic₃	1	16	yes	—
113	C₄²	C₄×Dic₃	1	8	yes	—
114	C₄²	C₂×C₄×D₃	1	8	yes	—
115	C₄²	C₂×C₄×D₃	1	16	yes	—
116	C₂²×C₃	C₄³	1	4	yes	—
117	C₂²×C₃	C₂×#97	1	8	yes	—
118	C₂²×C₃	C₄×M₁₆	1	16	yes	—
119	C₂²×C₃	C₂²×C₄²	1	1	yes	characteristic
120	C₂²×C₃	C₂²×C₄²	1	2	yes	—
121	C₂²×C₃	C₂²×M₁₆	1	4	yes	—
122	C₃×C₄	—	2	16	no	—
123	C₃×C₄	C₄³	1	4	yes	—
124	C₃×C₄	C₂×#97	1	16	yes	complemented
125	C₃×C₄	C₂²×C₄²	1	4	yes	—
126	C₂³	C₂×#208	1	8	yes	—
127	C₂³	C₂×C₄×Dic₃	1	2	yes	—
128	C₂³	C₂×C₄×Dic₃	1	4	yes	—
129	C₂³	C₂³×Dic₃	1	1	yes	characteristic
130	C₂×C₄	—	2	32	no	—
131	C₂×C₄	—	2	16	no	—
132	C₂×C₄	#803	1	32	yes	direct product factor
133	C₂×C₄	C₄²×D₃	1	16	yes	—
134	C₂×C₄	C₂×#208	1	16	yes	—
135	C₂×C₄	C₂×C₄×Dic₃	1	16	yes	—
136	C₂×C₄	C₂×C₄×Dic₃	1	8	yes	—
137	C₂×C₄	C₂×C₄×Dic₃	1	2	yes	—
138	C₂×C₄	C₂×C₄×Dic₃	1	4	yes	—
139	C₂×C₄	C₂²×C₄×D₃	1	2	yes	—
140	C₂×C₄	C₂²×C₄×D₃	1	4	yes	—
141	C₈	—	3	32	no	—
142	C₂×C₃	C₄×#97	1	8	yes	complemented
143	C₂×C₃	C₂×C₄³	1	1	yes	characteristic
144	C₂×C₃	C₂²×#97	1	2	yes	—
145	C₂×C₃	C₂×C₄×M₁₆	1	4	yes	—
146	C₂²	C₂×#803	1	8	yes	—
147	C₂²	C₄×#208	1	16	yes	—
148	C₂²	C₄²×Dic₃	1	4	yes	—
149	C₂²	C₂²×#208	1	4	yes	—
150	C₂²	C₂²×C₄×Dic₃	1	1	yes	characteristic
151	C₂²	C₂²×C₄×Dic₃	1	2	yes	—
152	C₄	—	2	16	no	—
153	C₄	C₂×#803	1	16	yes	direct product factor
154	C₄	C₄²×Dic₃	1	4	yes	—
155	C₄	C₂×C₄²×D₃	1	4	yes	—
156	C₃	C₂×C₄×#97	1	1	yes	characteristic, complemented, Sylow
157	C₂	C₄×#803	1	8	yes	direct product factor
158	C₂	C₂²×#803	1	2	yes	—
159	C₂	C₂×C₄×#208	1	4	yes	—
160	C₂	C₂×C₄²×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄²	C₂×D₃
commutator	C₂×C₃	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄³	C₂
socle, prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂×C₄³	D₃
hypocenter	C₃	C₂×C₄×#97
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
4. C₄×#803	C₂	direct (N × H)
12. C₂×#803	C₄	direct (N × H)
14. C₂×C₄×C₃⋊C₈	C₄	semidirect
34. #803	C₂×C₄	direct (N × H)
35. C₄×C₃⋊C₈	C₂×C₄	semidirect
62. C₂×C₃⋊C₈	C₄²	semidirect
87. C₂×C₃×C₄	#97	semidirect
96. C₃⋊C₈	C₂×C₄²	semidirect
124. C₃×C₄	C₂×#97	semidirect
132. C₂×C₄	#803	direct (N × H)
142. C₂×C₃	C₄×#97	semidirect
153. C₄	C₂×#803	direct (N × H)
156. C₃	C₂×C₄×#97	semidirect
157. C₂	C₄×#803	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×C₄³	2
C₂²×#803	2
C₂²×C₄×C₃⋊C₈	2
C₄×#803	2
C₂×C₄×#97	3

Subgroups — C2×C4×#803 #10792511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#803 #10792511