Subgroups — C₂²×#7463 #72234356

Order 1024.

753 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1397	C₂	1	1	yes	maximal, characteristic
2	C₂²×#1372	C₂	1	2	yes	maximal
3	C₂×#7463	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₃₂	C₄	1	1	yes	characteristic
5	C₂³×C₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×#335	—	2	2	no	—
8	C₂×#1397	C₄	1	6	yes	—
9	C₂×#1397	C₂²	1	6	yes	—
10	C₂×#1372	—	2	6	no	—
11	C₂×#1372	C₂²	1	12	yes	—
12	#7463	C₂²	1	16	yes	direct product factor
13	C₂³×C₁₆	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₁₆	Q₈	1	1	yes	characteristic
15	C₂²×C₄⋊C₈	—	4	2	no	—
16	C₂²×C₄×C₈	D₄	1	1	yes	characteristic
17	C₂²×C₃₂	—	2	4	no	—
18	C₂²×C₃₂	C₈	1	8	yes	complemented
19	C₂²×C₃₂	C₂×C₄	1	6	yes	—
20	C₂²×C₃₂	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₁₆	C₂×C₄	1	3	yes	—
22	C₂×C₄×C₁₆	D₄	1	3	yes	—
23	C₂×C₄×C₁₆	Q₈	1	3	yes	—
24	C₂×C₄×C₁₆	C₂³	1	3	yes	—
25	C₂×#335	—	2	12	no	—
26	C₂×#335	—	4	6	no	—
27	#1397	C₂×C₄	1	12	yes	—
28	#1397	C₂³	1	4	yes	—
29	#1372	—	2	12	no	—
30	#1372	C₂³	1	8	yes	—
31	C₂³×C₈	—	8	2	no	—
32	C₂³×C₈	C₄⋊C₄	1	1	yes	characteristic
33	C₂³×C₈	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄²	D₈	1	1	yes	characteristic
35	C₂²×C₁₆	—	2	4	no	—
36	C₂²×C₁₆	C₄⋊C₄	1	3	yes	—
37	C₂²×C₁₆	C₄⋊C₄	1	3	yes	—
38	C₂²×C₁₆	C₂×C₈	1	4	yes	—
39	C₂²×C₁₆	M₁₆	1	4	yes	—
40	C₂²×C₁₆	C₂²×C₄	1	3	yes	—
41	C₂²×C₁₆	C₂×Q₈	1	3	yes	—
42	C₂×C₄⋊C₈	—	4	12	no	—
43	C₂×C₄⋊C₈	—	8	6	no	—
44	C₂×C₄×C₈	C₄⋊C₄	1	3	yes	—
45	C₂×C₄×C₈	D₈	1	3	yes	—
46	C₂×C₄×C₈	Q₁₆	1	3	yes	—
47	C₂×C₄×C₈	C₂×D₄	1	3	yes	—
48	C₂×C₃₂	—	2	12	no	—
49	C₂×C₃₂	C₂×C₈	1	24	yes	complemented
50	C₂×C₃₂	C₂²×C₄	1	4	yes	—
51	C₂×C₃₂	C₂²×C₄	1	4	yes	—
52	C₄×C₁₆	C₄⋊C₄	1	6	yes	—
53	C₄×C₁₆	C₂²×C₄	1	3	yes	—
54	C₄×C₁₆	C₂×D₄	1	3	yes	—
55	C₄×C₁₆	C₂×Q₈	1	3	yes	—
56	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
57	#335	—	2	8	no	—
58	#335	—	4	12	no	—
59	C₂³×C₄	#107	1	1	yes	characteristic
60	C₂³×C₄	SD₃₂	1	2	yes	—
61	C₂²×C₈	—	2	4	no	—
62	C₂²×C₈	—	8	12	no	—
63	C₂²×C₈	—	16	8	no	—
64	C₂²×C₈	C₄⋊C₈	1	4	yes	—
65	C₂²×C₈	#107	1	3	yes	—
66	C₂²×C₈	#107	1	3	yes	—
67	C₂²×C₈	#108	1	4	yes	—
68	C₂²×C₈	C₂×C₄⋊C₄	1	3	yes	—
69	C₂²×C₈	C₂×Q₁₆	1	3	yes	—
70	C₂×C₄²	#107	1	3	yes	—
71	C₂×C₄²	SD₃₂	1	6	yes	—
72	C₂×C₄²	C₂×D₈	1	3	yes	—
73	C₂×C₁₆	—	2	6	no	—
74	C₂×C₁₆	—	2	6	no	—
75	C₂×C₁₆	C₄⋊C₈	1	12	yes	—
76	C₂×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
77	C₂×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₁₆	C₂²×C₈	1	6	yes	—
79	C₂×C₁₆	C₂×M₁₆	1	6	yes	—
80	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×Q₈	1	1	yes	characteristic
82	C₄⋊C₈	—	4	8	no	—
83	C₄⋊C₈	—	8	12	no	—
84	C₄×C₈	#107	1	6	yes	—
85	C₄×C₈	C₂×C₄⋊C₄	1	3	yes	—
86	C₄×C₈	C₂×D₈	1	3	yes	—
87	C₄×C₈	C₂×Q₁₆	1	3	yes	—
88	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
89	C₃₂	—	2	4	no	—
90	C₃₂	C₂²×C₈	1	8	yes	complemented
91	C₂⁴	#367	1	1	yes	characteristic
92	C₂²×C₄	—	2	8	no	—
93	C₂²×C₄	#335	1	4	yes	—
94	C₂²×C₄	#367	1	6	yes	—
95	C₂²×C₄	#367	1	3	yes	—
96	C₂²×C₄	#368	1	4	yes	—
97	C₂²×C₄	C₂×#107	1	3	yes	—
98	C₂²×C₄	C₂×SD₃₂	1	6	yes	—
99	C₂×C₈	—	2	6	no	—
100	C₂×C₈	—	2	6	no	—
101	C₂×C₈	—	8	8	no	—
102	C₂×C₈	—	16	24	no	—
103	C₂×C₈	#335	1	12	yes	—
104	C₂×C₈	C₂×C₄⋊C₈	1	6	yes	—
105	C₂×C₈	C₂×#107	1	3	yes	—
106	C₂×C₈	C₂×#107	1	3	yes	—
107	C₂×C₈	C₂×#108	1	6	yes	—
108	C₂×C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₈	C₂²×Q₁₆	1	1	yes	characteristic
110	C₄²	#367	1	6	yes	—
111	C₄²	C₂×#107	1	3	yes	—
112	C₄²	C₂×SD₃₂	1	6	yes	—
113	C₄²	C₂²×D₈	1	1	yes	characteristic
114	C₁₆	—	2	3	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	C₂×C₄⋊C₈	1	6	yes	—
117	C₁₆	C₂³×C₈	1	1	yes	characteristic
118	C₁₆	C₂²×M₁₆	1	1	yes	characteristic
119	C₂³	#1372	1	4	yes	—
120	C₂³	#1426	1	8	yes	—
121	C₂³	C₂×#367	1	3	yes	—
122	C₂×C₄	—	2	12	no	—
123	C₂×C₄	—	2	12	no	—
124	C₂×C₄	#1372	1	12	yes	—
125	C₂×C₄	C₂×#335	1	6	yes	—
126	C₂×C₄	C₂×#367	1	6	yes	—
127	C₂×C₄	C₂×#367	1	3	yes	—
128	C₂×C₄	C₂×#368	1	6	yes	—
129	C₂×C₄	C₂²×#107	1	1	yes	characteristic
130	C₂×C₄	C₂²×SD₃₂	1	2	yes	—
131	C₈	—	2	1	no	—
132	C₈	—	2	3	no	—
133	C₈	—	16	8	no	—
134	C₈	C₂×#335	1	6	yes	—
135	C₈	C₂²×C₄⋊C₈	1	1	yes	characteristic
136	C₈	C₂²×#108	1	1	yes	characteristic
137	C₂²	#7463	1	16	yes	direct product factor
138	C₂²	C₂×#1372	1	6	yes	—
139	C₂²	C₂×#1426	1	12	yes	—
140	C₂²	C₂²×#367	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	2	6	no	—
143	C₄	C₂×#1372	1	6	yes	—
144	C₄	C₂²×#335	1	1	yes	characteristic
145	C₄	C₂²×#368	1	1	yes	characteristic
146	C₂	C₂×#7463	1	12	yes	direct product factor
147	C₂	C₂²×#1372	1	1	yes	characteristic
148	C₂	C₂²×#1426	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	#367
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#7463	C₂	direct (N × H)
12. #7463	C₂²	direct (N × H)
18. C₂²×C₃₂	C₈	semidirect
49. C₂×C₃₂	C₂×C₈	semidirect
90. C₃₂	C₂²×C₈	semidirect
137. C₂²	#7463	direct (N × H)
146. C₂	C₂×#7463	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#1397	2
C₂²×#1372	2
C₂×#7463	2

Subgroups — C22×#7463 #72234356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#7463 #72234356