Subgroups — C₂²×#1370 #483328

Order 512.

682 conjugacy classes of subgroups.

The subgroup lattice has 141 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂²×C₈²	C₂	1	1	yes	maximal, characteristic
3	C₂×#1370	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆	—	2	2	no	—
5	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₁₆	—	2	6	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	12	yes	—
10	C₂×C₈²	C₄	1	6	yes	—
11	C₂×C₈²	C₂²	1	6	yes	—
12	#1370	C₂²	1	16	yes	direct product factor
13	C₂³×C₈	—	2	1	no	—
14	C₂³×C₈	C₈	1	1	yes	characteristic
15	C₂³×C₈	C₈	1	1	yes	characteristic
16	C₂³×C₈	D₄	1	1	yes	characteristic
17	C₂³×C₈	Q₈	1	1	yes	characteristic
18	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	—	2	12	no	—
20	C₂²×C₁₆	—	4	8	no	—
21	C₂×C₄×C₈	—	2	3	no	—
22	C₂×C₄×C₈	C₈	1	6	yes	—
23	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
26	C₂×C₄×C₈	D₄	1	3	yes	—
27	C₂×C₄×C₈	Q₈	1	3	yes	—
28	C₂×C₄×C₈	C₂³	1	3	yes	—
29	C₄⋊C₁₆	—	2	12	no	—
30	C₄⋊C₁₆	C₂³	1	8	yes	—
31	C₈²	C₂×C₄	1	12	yes	—
32	C₈²	C₂³	1	4	yes	—
33	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
34	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂³×C₄	M₁₆	1	1	yes	characteristic
36	C₂²×C₈	—	2	8	no	—
37	C₂²×C₈	—	2	4	no	—
38	C₂²×C₈	—	2	4	no	—
39	C₂²×C₈	—	2	6	no	—
40	C₂²×C₈	C₁₆	1	8	yes	complemented
41	C₂²×C₈	C₄⋊C₄	1	3	yes	—
42	C₂²×C₈	C₄⋊C₄	1	3	yes	—
43	C₂²×C₈	C₂×C₈	1	6	yes	—
44	C₂²×C₈	C₂×C₈	1	6	yes	—
45	C₂²×C₈	D₈	1	4	yes	—
46	C₂²×C₈	Q₁₆	1	4	yes	—
47	C₂²×C₈	C₂×D₄	1	3	yes	—
48	C₂²×C₈	C₂×Q₈	1	3	yes	—
49	C₂×C₄²	C₄⋊C₄	1	3	yes	—
50	C₂×C₄²	C₂×C₈	1	3	yes	—
51	C₂×C₄²	M₁₆	1	3	yes	—
52	C₂×C₄²	C₂²×C₄	1	3	yes	—
53	C₂×C₁₆	—	2	8	no	—
54	C₂×C₁₆	—	4	24	no	—
55	C₄×C₈	—	2	6	no	—
56	C₄×C₈	C₄⋊C₄	1	6	yes	—
57	C₄×C₈	C₂×C₈	1	12	yes	—
58	C₄×C₈	C₂²×C₄	1	4	yes	—
59	C₄×C₈	C₂²×C₄	1	3	yes	—
60	C₄×C₈	C₂²×C₄	1	4	yes	—
61	C₄×C₈	C₂×D₄	1	3	yes	—
62	C₄×C₈	C₂×Q₈	1	3	yes	—
63	C₄×C₈	C₂⁴	1	1	yes	characteristic
64	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₄	—	2	4	no	—
66	C₂²×C₄	C₄⋊C₈	1	3	yes	—
67	C₂²×C₄	C₄⋊C₈	1	3	yes	—
68	C₂²×C₄	C₄⋊C₈	1	3	yes	—
69	C₂²×C₄	#107	1	4	yes	—
70	C₂²×C₄	#108	1	4	yes	—
71	C₂²×C₄	C₂×C₁₆	1	4	yes	—
72	C₂²×C₄	M₃₂	1	4	yes	—
73	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
74	C₂²×C₄	C₂²×C₈	1	3	yes	—
75	C₂²×C₄	C₂×M₁₆	1	3	yes	—
76	C₂×C₈	—	2	24	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	2	6	no	—
80	C₂×C₈	—	2	12	no	—
81	C₂×C₈	#107	1	12	yes	—
82	C₂×C₈	C₂×C₁₆	1	24	yes	complemented
83	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
84	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
85	C₂×C₈	C₂²×C₈	1	4	yes	—
86	C₂×C₈	C₂²×C₈	1	4	yes	—
87	C₂×C₈	C₂×D₈	1	6	yes	—
88	C₂×C₈	C₂×Q₁₆	1	6	yes	—
89	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
91	C₄²	C₄⋊C₈	1	6	yes	—
92	C₄²	C₂×C₄⋊C₄	1	3	yes	—
93	C₄²	C₂²×C₈	1	3	yes	—
94	C₄²	C₂×M₁₆	1	3	yes	—
95	C₄²	C₂³×C₄	1	1	yes	characteristic
96	C₁₆	—	4	8	no	—
97	C₂³	#335	1	4	yes	—
98	C₂³	C₄⋊C₁₆	1	4	yes	—
99	C₂³	#364	1	4	yes	—
100	C₂³	C₂×C₄⋊C₈	1	3	yes	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	#335	1	12	yes	—
104	C₂×C₄	C₄⋊C₁₆	1	12	yes	—
105	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
106	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
107	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
108	C₂×C₄	C₂×#107	1	6	yes	—
109	C₂×C₄	C₂×#108	1	6	yes	—
110	C₂×C₄	C₂²×C₁₆	1	6	yes	—
111	C₂×C₄	C₂×M₃₂	1	6	yes	—
112	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
114	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
115	C₈	—	2	8	no	—
116	C₈	—	2	4	no	—
117	C₈	—	2	3	no	—
118	C₈	—	2	1	no	—
119	C₈	C₂×#107	1	6	yes	—
120	C₈	C₂²×C₁₆	1	8	yes	complemented
121	C₈	C₂²×D₈	1	1	yes	characteristic
122	C₈	C₂²×Q₁₆	1	1	yes	characteristic
123	C₂²	#1370	1	16	yes	direct product factor
124	C₂²	C₂×#335	1	6	yes	—
125	C₂²	C₂×C₄⋊C₁₆	1	6	yes	—
126	C₂²	C₂×#364	1	6	yes	—
127	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	3	no	—
130	C₄	C₂×#335	1	6	yes	—
131	C₄	C₂×C₄⋊C₁₆	1	6	yes	—
132	C₄	C₂²×#107	1	1	yes	characteristic
133	C₄	C₂²×#108	1	1	yes	characteristic
134	C₄	C₂³×C₁₆	1	1	yes	characteristic
135	C₄	C₂²×M₃₂	1	1	yes	characteristic
136	C₂	C₂×#1370	1	12	yes	direct product factor
137	C₂	C₂²×#335	1	1	yes	characteristic
138	C₂	C₂²×C₄⋊C₁₆	1	1	yes	characteristic
139	C₂	C₂²×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	D₄
commutator	C₄	C₂³×C₁₆
Frattini, cosocle	C₄×C₈	C₂⁴
socle, prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#1370	C₂	direct (N × H)
12. #1370	C₂²	direct (N × H)
40. C₂²×C₈	C₁₆	semidirect
82. C₂×C₈	C₂×C₁₆	semidirect
120. C₈	C₂²×C₁₆	semidirect
123. C₂²	#1370	direct (N × H)
136. C₂	C₂×#1370	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×C₄⋊C₁₆	2
C₂²×C₈²	2
C₂×#1370	2

Subgroups — C22×#1370 #483328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1370 #483328