Subgroups — C₂²×#1373 #483337

Order 512.

542 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×#364	C₂	1	2	yes	maximal
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1373	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₃₂	—	2	2	no	—
5	C₂³×C₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×#364	—	2	6	no	—
9	C₂×#364	C₂²	1	12	yes	—
10	C₂×C₄×C₁₆	C₄	1	6	yes	—
11	C₂×C₄×C₁₆	C₂²	1	6	yes	—
12	#1373	C₂²	1	16	yes	direct product factor
13	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₈	D₄	1	1	yes	characteristic
15	C₂²×C₄²	Q₈	1	1	yes	characteristic
16	C₂×M₃₂	—	2	12	no	—
17	C₂×M₃₂	—	4	6	no	—
18	C₂²×C₁₆	—	2	3	no	—
19	C₂²×C₁₆	—	2	1	no	—
20	C₂²×C₁₆	—	4	2	no	—
21	C₂²×C₁₆	C₈	1	6	yes	—
22	C₂²×C₁₆	C₈	1	2	yes	—
23	C₂²×C₁₆	C₂×C₄	1	6	yes	—
24	C₂²×C₁₆	C₂×C₄	1	6	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	3	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	#364	—	2	12	no	—
30	#364	C₂³	1	8	yes	—
31	C₄×C₁₆	C₂×C₄	1	12	yes	—
32	C₄×C₁₆	C₂³	1	4	yes	—
33	C₂³×C₄	—	2	1	no	—
34	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₈	C₄⋊C₄	1	3	yes	—
36	C₂²×C₈	C₄⋊C₄	1	3	yes	—
37	C₂²×C₈	C₂×C₈	1	3	yes	—
38	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₈	M₁₆	1	3	yes	—
40	C₂²×C₈	M₁₆	1	1	yes	characteristic
41	C₂²×C₈	D₈	1	3	yes	—
42	C₂²×C₈	D₈	1	1	yes	characteristic
43	C₂²×C₈	Q₁₆	1	1	yes	characteristic
44	C₂²×C₈	Q₁₆	1	3	yes	—
45	C₂²×C₈	C₂²×C₄	1	3	yes	—
46	C₂²×C₈	C₂×D₄	1	3	yes	—
47	C₂×C₄²	—	2	3	no	—
48	C₂×C₄²	C₄⋊C₄	1	3	yes	—
49	C₂×C₄²	C₂×Q₈	1	3	yes	—
50	M₃₂	—	2	8	no	—
51	M₃₂	—	4	12	no	—
52	C₂×C₁₆	—	2	6	no	—
53	C₂×C₁₆	—	2	6	no	—
54	C₂×C₁₆	—	4	12	no	—
55	C₂×C₁₆	C₂×C₈	1	12	yes	—
56	C₂×C₁₆	C₂×C₈	1	12	yes	—
57	C₂×C₁₆	C₂²×C₄	1	4	yes	—
58	C₂×C₁₆	C₂²×C₄	1	4	yes	—
59	C₄×C₈	C₄⋊C₄	1	6	yes	—
60	C₄×C₈	C₂²×C₄	1	3	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	#108	1	1	yes	characteristic
65	C₂²×C₄	—	2	8	no	—
66	C₂²×C₄	—	2	3	no	—
67	C₂²×C₄	—	2	3	no	—
68	C₂²×C₄	C₄⋊C₈	1	3	yes	—
69	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₄	#107	1	3	yes	—
71	C₂²×C₄	#107	1	1	yes	characteristic
72	C₂²×C₄	#108	1	3	yes	—
73	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₈	C₄⋊C₈	1	6	yes	—
75	C₂×C₈	C₄⋊C₈	1	3	yes	—
76	C₂×C₈	C₄⋊C₈	1	3	yes	—
77	C₂×C₈	#107	1	3	yes	—
78	C₂×C₈	#107	1	6	yes	—
79	C₂×C₈	#107	1	3	yes	—
80	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
81	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
82	C₂×C₈	C₂²×C₈	1	3	yes	—
83	C₂×C₈	C₂²×C₈	1	3	yes	—
84	C₂×C₈	C₂×M₁₆	1	3	yes	—
85	C₂×C₈	C₂×M₁₆	1	3	yes	—
86	C₂×C₈	C₂×D₈	1	3	yes	—
87	C₂×C₈	C₂×D₈	1	3	yes	—
88	C₂×C₈	C₂×Q₁₆	1	3	yes	—
89	C₂×C₈	C₂×Q₁₆	1	3	yes	—
90	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
92	C₄²	—	2	3	no	—
93	C₄²	—	2	3	no	—
94	C₄²	C₂×C₄⋊C₄	1	3	yes	—
95	C₄²	C₂²×Q₈	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	4	8	no	—
98	C₁₆	C₂²×C₈	1	8	yes	—
99	C₂³	—	2	4	no	—
100	C₂³	#335	1	3	yes	—
101	C₂³	#335	1	1	yes	characteristic
102	C₂³	C₂×#108	1	3	yes	—
103	C₂×C₄	—	2	12	no	—
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	#335	1	6	yes	—
108	C₂×C₄	#335	1	3	yes	—
109	C₂×C₄	#335	1	3	yes	—
110	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
111	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
112	C₂×C₄	C₂×#107	1	3	yes	—
113	C₂×C₄	C₂×#107	1	3	yes	—
114	C₂×C₄	C₂×#108	1	3	yes	—
115	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₈	C₂×C₄⋊C₈	1	3	yes	—
117	C₈	C₂×C₄⋊C₈	1	3	yes	—
118	C₈	C₂×#107	1	3	yes	—
119	C₈	C₂×#107	1	3	yes	—
120	C₈	C₂³×C₈	1	1	yes	characteristic
121	C₈	C₂²×M₁₆	1	1	yes	characteristic
122	C₈	C₂²×D₈	1	1	yes	characteristic
123	C₈	C₂²×Q₁₆	1	1	yes	characteristic
124	C₂²	—	2	6	no	—
125	C₂²	—	2	6	no	—
126	C₂²	#1373	1	4	yes	direct product factor
127	C₂²	C₂×#335	1	3	yes	—
128	C₂²	C₂×#335	1	3	yes	—
129	C₂²	C₂²×#108	1	1	yes	characteristic
130	C₄	—	2	6	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂×#335	1	3	yes	—
133	C₄	C₂×#335	1	3	yes	—
134	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
135	C₄	C₂²×#107	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	3	no	—
138	C₂	C₂×#1373	1	6	yes	direct product factor
139	C₂	C₂²×#335	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₈
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	#335
prime radical	C₂⁴	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ 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×#1373	C₂	direct (N × H)
12. #1373	C₂²	direct (N × H)
126. C₂²	#1373	direct (N × H)
138. C₂	C₂×#1373	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×#364	2
C₂²×C₄×C₁₆	2
C₂×#1373	2

Subgroups — C22×#1373 #483337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1373 #483337